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+Implementation of hyperbolic complex numbers in Julia language
+Anna V. Korolkova,1, \ast Migran N. Gevorkyan,1, \dagger and Dmitry S. Kulyabov1, 2, ‡
+1Peoples’ Friendship University of Russia (RUDN University),
+6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
+2Joint Institute for Nuclear Research
+6 Joliot-Curie, Dubna, Moscow region, 141980, Russian Federation
+Background: Hyperbolic complex numbers are used in the description of hyperbolic spaces. One
+of the well-known examples of such spaces is the Minkowski space, which plays a leading role in
+the problems of the special theory of relativity and electrodynamics. However, such numbers are
+not very common in different programming languages. Purpose: Of interest is the implementation
+of hyperbolic complex in scientific programming languages, in particular, in the Julia language.
+Methods: The Julia language is based on the concept of multiple dispatch. This concept is an
+extension of the concept of polymorphism for object-oriented programming languages. To implement
+hyperbolic complex numbers, the multiple dispatching approach of the Julia language was used.
+Results: The result is a library that implements hyperbolic numbers. Conclusions: Based on
+the results of the study, we can conclude that the concept of multiple dispatching in scientific
+programming languages is convenient and natural.
+Keywords: Julia programming language, multiple dispatch, abstract data types, type conversion, parametric
+structures, hyperbolic complex numbers
+I.
+INTRODUCTION
+The Julia programming language [1, 2] is a promising language for scientific computing. At the moment, the Julia
+language has reached a stable state. By design, Julia solves the problem of two languages. This problem lies in the fact
+that for rapid prototyping, data processing and visualization, an interpreted dynamic language or a mathematical
+package (Python, Matlab, etc.) is used, and for intensive numerical calculations, the program has to be rewritten in a
+compiled language with static typing (C/ C++, Fortran).
+An illustration of this problem can be seen in Python, which has gained wide popularity as an interface language-glue.
+Numerous wrapper libraries were written on it, which used Python code to call C/C++ and Fortran functions from
+precompiled libraries. For example, the well-known library NumPy [3] consists of 51% C code and only 47% Python
+code (the remaining percentages are divided between C++, Fortran, JavaScript and Unix shell).
+The Julia language combines the flexibility of dynamically typed interpreted languages with the performance of
+statically typed compiled languages.
+The basic part of the Julia language is very similar to other scientific programming languages, so it does not cause
+difficulties in mastering. However, Julia’s core is built around the concept of multiple dispatch [4], which is rare in other
+languages. It is in this mechanism that the essential difference of Julia from other languages lies, and its understanding
+is essential for the full use of all the advantages of Julia.
+A.
+Paper structure
+In the article, the authors paid great attention to illustrating the mechanism of multiple dispatch and other
+mechanisms that are closely related to it.
+In the first part of the article, we give the necessary definitions and illustrate the concept of multiple dispatch
+with simple examples that allow you to understand the syntax associated with this part of the language and capture
+the essence of this approach. In the second part, we give an example of the implementation of hyperbolic complex
+numbers in the Julia language. This example allows you to touch not only multiple dispatch, but also the type casting
+mechanism, the abstract type hierarchy, overloading arithmetic operators, and specifying user-defined data types.
+\ast korolkova-av@rudn.ru
+\dagger gevorkyan-mn@rudn.ru
+‡ kulyabov-ds@rudn.ru
+arXiv:2301.01707v1  [cs.MS]  4 Jan 2023
+
+2
+II.
+MULTIPLE DISPATCH
+A.
+Common definitions
+Dynamic dispatch is a mechanism that allows you to choose which of the many implementations of a polymorphic
+function (or operator) should be called in a given case [5]. In this case, the choice of one or another implementation
+is carried out at the stage of program execution. Multiple dispatch is based on dynamic dispatch. In this case, the
+choice of implementation of a polymorphic function is made based on the type, number, and order of the function’s
+arguments. This is how runtime polymorphic dispatch is implemented [6, 7]. Note also that in addition to the term
+multiple dispatch, the term multimethod is also used.
+The mechanism of multiple dispatch is similar to the mechanism of overloading functions and operators, implemented,
+for example, in the C++ language. Function overloading, however, is done exclusively at compile time, while multiple
+dispatch should work at runtime as well (runtime polymorphism).
+B.
+Multiple dispatch in Julia
+To illustrate the mechanism of multiple dispatch, we will give the following code example in the Julia language.
+function f(x, y)
+println("Generic implementation")
+return x + y
+end
+function f(x)
+println("For single argument")
+return x
+end
+function f(x::Integer, y::Integer)
+println("Implementation for integers")
+return x + y
+end
+function f(x::String, y::String)
+println("Implementation for strings")
+return x * " " * y
+end
+function f(x::Tuple{Int, Int}, y::Tuple{Int, Int})
+println("Implementation for tuples of two integer elements")
+return (x[1], x[2], y[1], y[2])
+end
+In this example, we have created five implementations of the f function, which differ from each other in different
+signatures. In terms of the Julia language, this means that one function f now has four different methods. In the first
+two methods, we did not use type annotations, so the type of the arguments will be determined either at compile
+time or at run time (as in interpreted languages). It is also worth noting that Julia uses dynamic JIT compilation
+(just-in-time), so the compilation stage is not explicitly separated from the execution stage for the user.
+The arguments of the following three methods are annotated with types, so they will only be called if the types
+match the annotations. In the f for strings, the * concatenation operator is used. The choice of the multiplication sign
+* instead of the more traditional addition sign + is justified by the creators of the language by the fact that string
+concatenation is not a commuting operation, so it is more logical to use the multiplication sign for it, rather than the
+addition sign, which is often used to denote commuting operations.
+The following code snippet illustrates how multiple dispatch works at compile time. The @show macro is used to
+print out the name of the function and the arguments passed to it.
+@show f(2.0, 1)
+@show f(2, 2)
+
+3
+@show f(0x2, 0x1) # numbers in hexadecimal system
+@show f("Text", "line")
+@show f(3)
+@show f([1, 2], [3, 4])
+@show f((1, 2), (3, 4))
+• In the first line, we passed real (floating-point) type arguments to the function, so a generic implementation
+call was made. Since the operator + is defined for floating point numbers, the function succeeded and gave the
+correct result.
+• Methods for integers were called in the second and third lines. Note that the Integer type is an abstract type
+and includes signed and unsigned integers from 1 to 16 bytes in size, defined in the language core. Numbers
+written in hexadecimal are interpreted by default as unsigned integers.
+• The method for strings was called on the fourth line. In the fifth line, the method for one argument.
+• The sixth line passed two arrays as arguments. The + operation is defined for arrays, so the function ran without
+error and returned an element-wise sum.
+• In the seventh line, the function arguments are tuples consisting of two integers. Since we defined a method for
+such a combination of arguments, the function worked correctly.
+Generic implementation
+f(2.0, 1) = 3.0
+Implementation for integers
+f(2, 2) = 4
+Implementation for integers
+f(0x02, 0x01) = 0x03
+Implementation for strings
+f("Text", "line") = "Text line"
+For single argument
+f(3) = 3
+Generic implementation
+f([1, 2], [3, 4]) = [4, 6]
+Implementation for tuples of two integer elements
+f((1, 2), (3, 4)) = (1, 2, 3, 4)
+The above examples will work correctly in languages that support function overloading and do not demonstrate the
+specifics of dynamic dispatching, since the types of arguments are known at the compilation stage and are available to
+the translator.
+To test the work of dynamic method calls, consider the following code:
+print("Enter an integer:")
+# Read a string and convert to an integer type
+@show n = parse(Int32, readline())
+if n > 0
+x = 1.2; y = 0.1
+else
+x = 1; y = 2
+end
+f(x, y)
+Here, the types of variable values x and y are not known at compile time, as they depend on what number the user
+enters during program execution. However, for the case of integer x and y the corresponding method is called.
+III.
+HYPERBOLIC NUMBERS
+We will use hyperbolic numbers to illustrate the multiple dispatch capabilities of the Julia language, so we will limit
+ourselves to the definition and basic arithmetic operations.
+
+4
+Hyperbolic numbers [8–11], along with elliptic and parabolic numbers, are a generalization of complex numbers.
+Hyperbolic numbers can be defined as follows:
+z = x + jy, j2 = 1, j \not = \pm 1.
+The quantity j will be called the hyperbolic imaginary unit, and the quantities x and y will be called the real and
+imaginary parts, respectively.
+For two hyperbolic numbers z1 = x1 + jy1 and z2 = x2 + jy2 the following arithmetic operations are performed.
+Addition: z1 + z2 = (x1 + x2) + j(y1 + y2).
+Multiplication: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1).
+Conjugation: z\ast = x  - jy.
+Inverse number: z - 1 =
+x
+x2 + y2  - j
+y
+x2  - y2 .
+Division: z1
+z2
+= x1x2  - y1y2
+x2
+2  - y2
+2
++ jx1y1  - x1y2
+x2
+2  - y2
+2
+.
+The implementation of hyperbolic numbers is in many respects similar to the implementation of complex ones.
+Operators +, -, * must be overloaded, and /, root extraction, exponentiation, elementary math functions, etc. At
+the same time, for the purposes of illustrating the mechanism of operation of multiple dispatching, it is arithmetic
+operations that are of primary interest. This is due to the fact that elementary functions take only one argument,
+and it is enough to define only one method for them. In the case of arithmetic operators, it is necessary to provide
+combinations of arguments of different numeric types. So, for example, it should be possible to add a hyperbolic
+number to an integer, rational, irrational number, which automatically affects not only multiple dispatch, but also
+type casting mechanisms, an abstract type hierarchy, and default constructor overloading.
+Therefore, we will confine ourselves to examples of the implementation of precisely arithmetic operations and that’s
+all, without touching on the more mathematically complex calculations of various elementary functions of a hyperbolic
+number.
+Note that in addition to the term hyperbolic numbers, there are also terms in the literature: double numbers, split
+complex numbers, perplex numbers, hyperbolic numbers [8, 12–15].
+IV.
+IMPLEMENTATION OF HYPERBOLIC NUMBERS IN JULIA
+A.
+Declaring a Data Structure
+The implementation of hyperbolic numbers in Julia was based on the code for complex numbers available in
+the official Julia repository. We also used the developments obtained in the implementation of parabolic complex
+numbers [16]. New type Hyperbolic defined with an immutable structure:
+struct Hyperbolic{T<:Real} <: Number
+"Real part"
+re::T
+"Imaginary part"
+jm::T
+end
+The structure is simple and contains only two fields of parametric type T. This requires that the type T was a
+subtype of the abstract type Real (syntax T<:Real). The type Hyperbolic is a subtype of the abstract type Number
+(see Fig. 1). Thus, hyperbolic numbers are built into an already existing hierarchy of numeric types.
+After the structure is defined, a new object of type Hyperbolic can be created by calling the default constructor.
+So, for example, the number h = 1 + j3 is given as follows:
+h = Hyperbolic{Float64}(1, 3)
+After creation, you can access the fields of the structure as h.re and h.jm, but an attempt changing the value of a
+field of an already existing object will result in an error, since structs are immutable entities.
+h = Hyperbolic(1, 3)
+
+5
+Number
+Hyperbolic
+Complex
+Real
+Integer
+Signed
+Int8
+Int16
+Int32
+Int64
+Int128
+Bool
+Unsigned
+UInt8
+UInt16
+UInt32
+UInt64
+UInt128
+Rational
+AbstractFloat
+Float16
+Float32
+Float64
+Legend:
+Abstract type
+Primitive type
+Structure
+Figure 1. Location of Hyperbolic Numbers in Julia’s Type Hierarchy
+However, if the argument types are different, then the default constructor will not be able to implicitly cast and
+create a new object. In this case, you must explicitly specify the parametric type
+# Float64 и Int64
+h = Hyperbolic(1.0, 3) # Error
+h = Hyperbolic{Float64}(1.0, 3) # Correct
+B.
+Additional constructors
+The default constructor is a normal function whose name is the same as the type name. By creating additional
+methods for this function, you can create additional constructors to handle various special cases.
+So, for example, in order not to specify a parametric type every time, you should add a new constructor of the
+following form:
+"""Constructor №2"""
+function Hyperbolic(x::Real, y::Real)
+return Hyperbolic(promote(x, y)...)
+end
+The promote function casts the arguments passed to it to a common type and returns the result as a tuple. Postfix
+operator ... unpacks the tuple and passes its elements as arguments to the constructor function. The language core
+defines casting rules for all subtypes of the Real abstract type, so now the constructor will work correctly for any
+combination of arguments, as long as the T<:Real rule is fulfilled. For example, the following code will work correctly:
+# Rational и Float64
+h = Hyperbolic(1//3, pi)
+>> Hyperbolic{Float64}(0.5, 3.141592653589793)
+We passed a rational number (type Rational) and a built-in global constant (number \pi ) of type Float64 to the
+constructor. After that, the type casting rule worked and both arguments were cast to the type Float64 as more
+general.
+Declaring two more additional constructors will allow you to specify hyperbolic numbers with zero imaginary part:
+"""Constructor №3"""
+function Hyperbolic{T}(x::Real) where {T<:Real}
+return Hyperbolic{T}(x, 0)
+end
+"""Constructor №4"""
+function Hyperbolic(x::Real)
+return Hyperbolic(promote(x, 0)...)
+end
+Constructor number 3 is a parametric function that is declared using the where construct. The T is a subtype of the
+abstract type Real. Constructor number 4 works similarly to constructor number 2.
+Two more constructors will allow you to pass other hyperbolic numbers as an argument to the constructor.
+
+6
+"""Constructor №5"""
+function Hyperbolic{T}(h::Hyperbolic) where {T<:Real}
+Hyperbolic{T}(h.re, h.jm)
+end
+"""Constructor №6"""
+function Hyperbolic(h::Hyperbolic)
+return Hyperbolic(promote(h.re, h.jm)...)
+end
+For more convenience, you can also create a separate constant for the imaginary cost j:
+const jm = Hyperbolic(0, 1)
+C.
+Data printing
+To be able to print hyperbolic type values in a compact and readable form, you should add the appropriate methods
+to the show function from the Base module.
+function Base.show(io::IO, h::Hyperbolic)
+print(io, h.re, "+", h.jm, "j")
+end
+Function show is used when printing data to the console, in particular, it is called by the println and macro @show.
+The code and output listings below will assume that the show method has been added for hyperbolic numbers.
+D.
+Type casting
+Before proceeding to the implementation of methods for arithmetic operations with hyperbolic numbers, it is
+necessary to define the rules for type casting. To do this, create a new method for the function promote_rule from
+the Base module.
+function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real}
+return Hyperbolic{promote_type(T, S)}
+end
+function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real,
+S<:Real}
+\lhook →
+return Hyperbolic{promote_type(T, S)}
+end
+As arguments in promote_rule parametric types are specified, which should be cast to one enclosing type. In our
+case, this is possible if one of the types is a subtype of Real, then the enclosing type is Hyperbolic.
+After adding methods for promote_rule, it becomes possible to use functions promote, promote_type and convert.
+>>h = Hyperbolic(1 // 2)
+>>promote(h, 1)
+(1//2+0//1j, 1//1+0//1j)
+>>promote_type(Hyperbolic{Int64}, Float32)
+Hyperbolic{Float32}
+The first function is already familiar to us. The second allows you to infer the enclosing type not of specific variable
+values, but of the types themselves. A type in Julia is an object of the first kind (type DataType) and can be assigned
+to other variables, passed as function arguments, and so on.
+Function convert allows you to convert the type specific value, for example:
+>>convert(Hyperbolic, 1)
+1+0j
+After adding methods for type casting, you can start adding methods for arithmetic operations. A feature of Julia is
+the implementation of arithmetic operations not in the form of operators, but in the form of functions. For example,
+the following calls are correct:
+
+7
+>>+(1,2)
+3
+>>+(1,2,3,4)
+10
+>>+((i for i in 1:10)...) # числа от 1 до 10
+55
+In this regard, adding methods for arithmetic operations is no different from the corresponding process for other
+functions.
+Adding methods for unary operations + and - is carried out as follows:
+Base.:+(h::Hyperbolic) = Hyperbolic(+h.re, +h.jm)
+Base.:-(h::Hyperbolic) = Hyperbolic(-h.re, -h.jm)
+This is an abbreviated function declaration.
+Similarly, methods are added for binary addition, subtraction, multiplication, and division. Here is the code for
+addition and multiplication.
+# Binary + and *
+function Base.:+(x::Hyperbolic, y::Hyperbolic)
+xx = x.re + y.re
+yy = x.jm + y.jm
+Hyperbolic(xx, yy)
+end
+function Base.:*(x::Hyperbolic, y::Hyperbolic)
+xx = x.re * y.re + x.jm * y.jm
+yy = x.re * y.jm + x.je * y.re
+return Hyperbolic(xx, yy)
+end
+V.
+CONCLUSION
+We examined the mechanism of multiple dispatch underlying the Julia language, using the example of the implemen-
+tation of hyperbolic numbers. This example allowed us to touch upon such concepts of the language as the hierarchy of
+data types, composite data types, type casting mechanisms, function overloading (creating new methods for functions
+in terms of the Julia language), etc.
+ACKNOWLEDGMENTS
+This paper has been supported by the RUDN University Strategic Academic Leadership Program.
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+
+Реализация гиперболических комплексных чисел на языке Julia
+А. В. Королькова,1, \ast М. Н. Геворкян,1, \dagger and Д. С. Кулябов1, 2, ‡
+1Российский университет дружбы народов,
+117198, Москва, ул. Миклухо-Маклая, д. 6
+2Объединённый институт ядерных исследований,
+ул. Жолио-Кюри 6, Дубна, Московская область, Россия, 141980
+Предпосылки. Гиперболические комплексные числа применяются при описании гиперболи-
+ческих пространств. Одним из известных примером таких пространств является пространство
+Минковского, играющее ведущее значение в задачах частной теории относительности, электро-
+динамики. Однако такие числа не очень распространены в разных языках программирования.
+Цель. Представляет интерес реализация гиперболических комплексных в языках научного
+программирования, в частности, в языке Julia. Методы. В основе языка Julia лежит концепция
+множественной диспетчеризации (multiple dispatch). Эта концепция является расширением
+концепции полиморфизма для объектно-ориентированных языков программирования. Для
+реализации гиперболических комплексных чисел использован подход множественной дис-
+петчеризацию языка Julia. Результаты. В результате получена библиотека, реализующая
+гиперболические числа. Выводы. По результатам исследования можно сделать вывод об
+удобстве и естественности концепции множественной диспетчеризации в языках научного
+программирования.
+Keywords: язык программирования Julia, множественная диспетчеризация, абстрактные типы данных,
+конвертация типов, параметрические структуры, гиперболические комплексные числа
+I.
+ВВЕДЕНИЕ
+Язык программирования Julia [1, 2] — это перспективный язык, предназначенный для научных вычислений. В
+настоящий момент язык Julia достиг стабильного состояния. По замыслу разработчиков Julia решае�� проблему
+двух языков. Данная проблема заключается в том, что для быстрого прототипирования, обработки данных и
+визуализации используется интерпретируемый динамический язык или математический пакет (Python, Matlab и
+т.д.), а для интенсивных численных расчётов программу приходится переписывать на компилируемом языке со
+статической типизацией (C/C++, Fortran).
+Иллюстрацию данной проблемы можно увидеть на примере языка Python, который приобрел широкую попу-
+лярность в качестве интерфейсного «языка-клея». На нем было написано большое количество библиотек-обёрток,
+которые использовали Python-код для вызова C/C++ и Fortran функций из предварительно скомпилированных
+библиотек. Так, например, известная библиотека NumPy [3] на 51% состоит из кода на языке Си и лишь на 47%
+из кода на языке Python (оставшиеся проценты делят между собой C++, Fortran, JavaScript и Unix shell).
+Язык Julia совмещает в себе гибкости интерпретируемых языков с динамической типизацией и производитель-
+ность компилируемых языков со статической типизацией.
+Базовая часть языка Julia крайне схожа с другими языками научного программирования поэтому не вызывает
+трудности при освоении. Однако ядро Julia построено вокруг концепцию множественной диспетчеризации
+(multiple dispatch) [4], которая редко встречается в других языках. Именно в этом механизме лежит существенное
+отличие Julia от других языков и его понимание существенно для полноценного использования всех преимуществ
+Julia.
+A.
+Структура статьи
+В статье авторы уделили большое внимание иллюстрации механизма множественной диспетчеризации и
+других механизмов, которые близко с ней связаны.
+В первой части статьи мы даем необходимые определения и иллюстрируем концепцию множественной
+диспетчеризации на простых примерах, позволяющих понять синтаксис, связанный с этой частью языка и
+\ast korolkova-av@rudn.ru
+\dagger gevorkyan-mn@rudn.ru
+‡ kulyabov-ds@rudn.ru
+arXiv:2301.01707v1  [cs.MS]  4 Jan 2023
+
+2
+уловить суть данного подхода. Во второй части мы приводим пример реализации гиперболических комплексных
+чисел на языке Julia. Данный пример позволяет затронуть не только множественную диспетчеризацию, но и
+механизм приведения типов, иерархию абстрактных типов, перегрузку арифметических операторов и задание
+пользовательских типов данных.
+II.
+МНОЖЕСТВЕННАЯ ДИСПЕТЧЕРИЗАЦИЯ
+A.
+Общие определения
+Динамическая диспетчеризация (dynamic dispatch) — это механизм, который позволяет выбрать какую из
+множества реализаций полиморфной функции (или оператора) следует вызвать в данном конкретном случае [5].
+При этом выбор той или иной реализации осуществляется на стадии выполнения программы. Множественная
+диспетчеризация основывается на динамической диспетчеризации. В этом случае выбор реализации полиморф-
+ной функции делается исходя из типа, количества и порядка следования аргументов функции. Таким образом
+реализуется полиморфизм времени выполнения (runtime polymorphic dispatch) [6, 7]. Заметим также, что кроме
+термина «множественная диспетчеризация», также употребляется термин мультиметод.
+Механизм множественной диспетчеризации похож на механизм перегрузки функций и операторов, реали-
+зованный, например, в языке C++. Перегрузка функций, однако, осуществляется исключительно на стадии
+компиляции, тогда как множественная диспетчеризация должна работать также и на стадии выполнения
+программы (полиморфизм времени выполнения).
+B.
+Множественная диспетчеризация в Julia
+Для иллюстрации механизма множественной диспетчеризации приведём следующий пример кода на языке
+Julia.
+function f(x, y)
+println("Общая реализация")
+return x + y
+end
+function f(x)
+println("Для одного аргумента")
+return x
+end
+function f(x::Integer, y::Integer)
+println("Реализация для целых чисел")
+return x + y
+end
+function f(x::String, y::String)
+println("Реализация для строк")
+return x * " " * y
+end
+function f(x::Tuple{Int, Int}, y::Tuple{Int, Int})
+println("Реализация для кортежей из двух целочисленных элементов")
+return (x[1], x[2], y[1], y[2])
+end
+В данном примере мы создали пять реализаций функции f, которые отличаются друг от друга разными
+сигнатурами. В терминах языка Julia это означает, что у одной функции f теперь существует четыре разных
+метода. В первых двух методах мы не использовали аннотаций типов, поэтому тип аргументов будет определен
+или на стадии компиляции или на стадии выполнения программы (как в интерпретируемых языках). Стоит
+также отметит, что Julia использует динамическую JIT-компиляцию (just-in-time), поэтому стадия компиляции
+от стадии выполнения отделена для пользователя не явным образом.
+
+3
+Аргументы трех следующих методов аннотированы типами, поэтому будут вызываться только в случае
+совпадения типов с аннотациями. В методе f для строк используется оператор конкатенации *. Выбор знака
+умножения * вместо более традиционного знака сложения + обосновывается создателями языка тем, что
+конкатенация строк операция не коммутирующая, поэтому более логично использовать для нее знак умножения,
+а не сложения, которым чаще все принято обозначать коммутирующие операции.
+Следующий фрагмент кода иллюстрирует работу множественной диспетчеризации на стадии компиляции.
+Макрос @show служит для распечатки имени функции и переданных ей аргументов.
+@show f(2.0, 1)
+@show f(2, 2)
+@show f(0x2, 0x1) # числа в шестнадцатеричной системе
+@show f("Строка", "текста")
+@show f(3)
+@show f([1, 2], [3, 4])
+@show f((1, 2), (3, 4))
+• В первой строке мы передали функции аргументы вещественного типа (с плавающей точкой), поэтому
+был осуществлен вызов общей реализации. Так как для чисел с плавающей точкой определен оператор +,
+то функция выполнилась успешно и дала правильный результат.
+• Во второй и третей строках были вызваны методы для целых чисел. Заметим, что тип Integer является
+абстрактным типом и включает в себя знаковые и беззнаковые целые числа размером от 1 до 16 байт,
+определённые в ядре языка. Числа, записанные в шестнадцатерич��ой системе счисления интерпретируются
+по умолчанию как беззнаковые целые.
+• В четвертой строке был вызван метод для строк. В пятой строке метод для одного аргумента.
+• В шестой строке в качестве аргументов переданы два массива. Операция + определена для массивов,
+поэтому функция выполнилась без ошибок и вернула поэлементную сумму.
+• В седьмой строке аргументами функции являются кортежи, состоящие из двух целых чисел. Так как нами
+был определен метод для такой комбинации аргументов – функция отработала корректно.
+Общая реализация
+f(2.0, 1) = 3.0
+Реализация для целых чисел
+f(2, 2) = 4
+Реализация для целых чисел
+f(0x02, 0x01) = 0x03
+Реализация для строк
+f("Строка", "текста") = "Строка текста"
+Для одного аргумента
+f(3) = 3
+Общая реализация
+f([1, 2], [3, 4]) = [4, 6]
+Реализация для кортежей из двух целочисленных элементов
+f((1, 2), (3, 4)) = (1, 2, 3, 4)
+Приведённые примеры корректно сработают и в языках, поддерживающих перегрузку функций и не демон-
+стрируют специфику динамической диспетчеризации, так как типы аргументов известны на стадии компиляции
+и доступны транслятору.
+Для проверки работы именно динамического вызова методов рассмотрим следующий код:
+print("Введите целое число:")
+# Считываем строку и конвертируем в целый тип
+@show n = parse(Int32, readline())
+if n > 0
+x = 1.2; y = 0.1
+else
+x = 1; y = 2
+end
+f(x, y)
+
+4
+Здесь типы значений переменных x и y не известны на стадии компиляции, так как зависят от того, какое
+число введёт пользователь во время выполнения программы. Тем не менее, для случая целочисленных x и y
+вызывается соответствующий метод.
+III.
+ГИПЕРБОЛИЧЕСКИЕ ЧИСЛА
+Мы будем использовать гиперболические числа для иллюстрации возможностей множественной диспетчериза-
+ции языка Julia, поэтому ограничимся лишь определением и основными арифметическими операциями.
+Гиперболические числа [8–11], наряду с эллиптическими и параболическими числами, являются обобщением
+комплексных чисел. Гиперболические числа можно определить следующим образом:
+z = x + jy, j2 = 1, j \not = \pm 1.
+Величину j будем называть гиперболической мнимой единицей, а величины x и y действительной и мнимой
+частями соответственно.
+Для двух гиперболических чисел z1 = x1 + jy1 и z2 = x2 + jy2 выполняются следующие арифметические
+операции.
+Сложение: z1 + z2 = (x1 + x2) + j(y1 + y2).
+Умножение: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1).
+Сопряжение: z\ast = x  - jy.
+Обратное число: z - 1 =
+x
+x2 + y2  - j
+y
+x2  - y2 .
+Деление: z1
+z2
+= x1x2  - y1y2
+x2
+2  - y2
+2
++ jx1y1  - x1y2
+x2
+2  - y2
+2
+.
+Реализация гиперболических чисел во многом аналогична реализации комплексных. Необходимо перегрузить
+операторы +, -, * и /, функции извлечения корня, возведения в степень, элементарные математические функции
+и т.д. При этом для целей иллюстрации механизма работы множественной диспетчеризации основной интерес
+представляют именно арифметические операции. Это обусловлено тем, что элементарные функции принимают
+только один аргумент и для них достаточно определить только один метод. В случае же арифметических
+операторов необходимо предусмотреть комбинации аргументов разных числовых типов. Так, например, должна
+иметься возможность сложения гиперболического числа с целым, рациональны, иррациональным числом, что
+автоматически затрагивает не только множественную диспетчеризацию, но и механизмы приведения типов,
+иерархию абстрактных типов и перегрузку конструктора по умолчанию.
+Поэтому мы ограничимся примерами реализации именно арифметических операций и все, не затронув более
+сложные в математическом плане вычисления разнообразных элементарных функций от гиперболического
+числа.
+Отметим, что кроме термина гиперболические числа, в литературе встречаются также термины: двойные
+числа, расщепленные комплексные числа, комплексные числа гиперболического типа (double numbers, split
+complex numbers, perplex numbers, hyperbolic numbers) [8, 12–15].
+IV.
+РЕАЛИЗАЦИЯ ГИПЕРБОЛИЧЕСКИХ ЧИСЕЛ В JULIA
+A.
+Объявление структуры данных
+При реализации гиперболических чисел в Julia за основу был взят код для комплексных чисел, доступный в
+официальном репозитории Julia. Также использовались наработки, полученные при реализации параболических
+комплексных чисел [16]. Новый тип Hyperbolic определяется с помощью неизменяемой структуры:
+struct Hyperbolic{T<:Real} <: Number
+"Real part"
+re::T
+"Imaginary part"
+jm::T
+end
+
+5
+Number
+Hyperbolic Complex
+Real
+Integer
+Signed
+Int8
+Int16
+Int32
+Int64
+Int128
+Bool
+Unsigned
+UInt8
+UInt16
+UInt32
+UInt64 UInt128
+Rational
+AbstractFloat
+Float16
+Float32
+Float64
+Легенда:
+Абстрактный тип
+Примитивный тип
+Структура
+Рис. 1. Местоположение гиперболических чисел в иерархии типов Julia
+Структура проста и содержит всего два поля параметрического типа T. При этом требуется, чтобы тип T был
+подтипом абстрактного типа Real (синтаксис T<:Real). Сам тип Hyperbolic является подтипом абстрактного
+типа Number (см рис. 1). Таким образом гиперболические числа встраиваются в уже существующую иерархию
+числовых типов.
+После определения структуры новый объект типа Hyperbolic можно создать путем вызова конструктора по
+умолчанию. Так, например, число h = 1 + j3 задается следующим образом:
+h = Hyperbolic{Float64}(1, 3)
+После создания можно обращаться к полям структуры как h.re и h.jm, но попытка изменения значения поля
+уже существующего объекта приведёт к ошибке, так как структуры являются неизменяемыми сущностями.
+Если оба аргумента конструктора имеют один и тот же тип T, то его можно явно не указывать в фигурных
+скобках, так как он будет выведен автоматически из типа передаваемых аргументов.
+h = Hyperbolic(1, 3)
+Однако, если типы аргументов отличаются, то конструктор по умолчанию не сможет осуществить неявное
+приведение типов и создать новый объект. В этом случае необходимо явно указывать параметрический тип
+# Float64 и Int64
+h = Hyperbolic(1.0, 3) # Error
+h = Hyperbolic{Float64}(1.0, 3) # Correct
+B.
+Дополнительные конструкторы
+Конструктор по умолчанию представляет собой обычную функцию, имя которой совпадает с именем типа. Со-
+здавая дополнительные методы для этой функции можно создать дополнительные конструкторы для обработки
+различных частных случаев.
+Так, например, чтобы не указывать всякий раз параметрический тип, следует добавить новый конструктор
+следующего вида:
+"""Constructor №2"""
+function Hyperbolic(x::Real, y::Real)
+return Hyperbolic(promote(x, y)...)
+end
+Функция promote осуществляет приведение типов переданных ей аргументов к общему типу и возвращает
+результат в виде кортежа. Постфиксный оператор ... распаковывает картеж и передает его элементы в виде
+аргументов в функцию-конструктор. В ядре языка определены правила приведения для всех подтипов абстракт-
+ного типа Real, поэтому теперь конструктор будет корректно работать для любой комбинации аргументов,
+главное чтобы выполнялось правило T<:Real. Например, следующий код сработает корректно:
+# Rational и Float64
+h = Hyperbolic(1//3, pi)
+>> Hyperbolic{Float64}(0.5, 3.141592653589793)
+Мы передали в конструктор рациональное число (тип Rational) и встроенную глобальную константу (число
+\pi ) типа Float64. После чего сработало правило приведения типов и оба аргументы были приведены к типу
+Float64 как к более общему.
+
+6
+Объявление еще двух дополнительных конструкторов позволит задавать гиперболические числа с нулевой
+мнимой частью:
+"""Constructor №3"""
+function Hyperbolic{T}(x::Real) where {T<:Real}
+return Hyperbolic{T}(x, 0)
+end
+"""Constructor №4"""
+function Hyperbolic(x::Real)
+return Hyperbolic(promote(x, 0)...)
+end
+Конструктор №3 является параметрической функцией, которая объявляется с использованием конструк-
+ции where. Параметр T является подтипом абстрактного типа Real. Конструктор №4 работает аналогично
+конструктору №2.
+Ещё два конструктора позволят передавать в качестве аргумента конструктора другие гиперболические числа.
+"""Constructor №5"""
+function Hyperbolic{T}(h::Hyperbolic) where {T<:Real}
+Hyperbolic{T}(h.re, h.jm)
+end
+"""Constructor №6"""
+function Hyperbolic(h::Hyperbolic)
+return Hyperbolic(promote(h.re, h.jm)...)
+end
+Для большего удобства также можно создать отдельную константу для мнимой единицы j:
+const jm = Hyperbolic(0, 1)
+C.
+Вывод данных
+Для возможности распечатывать значения гиперболического типа в компактном и читаемом виде, следует
+добавить соответствующие методы для функции show из модуля Base.
+function Base.show(io::IO, h::Hyperbolic)
+print(io, h.re, "+", h.jm, "j")
+end
+Функция show используется при распечатке данных в консоль, в частности ее вызывают функция println
+и макрос @show. В приведенных далее листингах кода и результатов его работы будет предполагаться, что
+добавлен метод show для гиперболических чисел.
+D.
+Приведение типов
+Прежде чем переходить к реализации методов для арифметических операций с гиперболическими числами,
+необходимо определить правила приведения типов. Для этого следует создать новый метод для функции
+promote_rule из модуля Base.
+function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real}
+return Hyperbolic{promote_type(T, S)}
+end
+function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real,
+S<:Real}
+\lhook →
+return Hyperbolic{promote_type(T, S)}
+end
+В качестве аргументов в promote_rule указываются параметрические типы, которые следует привести к
+одному объемлющему типу. В нашем случае это возможно, если один из типов является подтипом Real, тогда
+объемлющим типом будет тип Hyperbolic.
+
+7
+После добавления методов для promote_rule становится возможным использовать функции promote,
+promote_type и convert.
+>>h = Hyperbolic(1 // 2)
+>>promote(h, 1)
+(1//2+0//1j, 1//1+0//1j)
+>>promote_type(Hyperbolic{Int64}, Float32)
+Hyperbolic{Float32}
+Первая функция нам уже знакома. Вторая же позволяет выводить объемлющий тип не конкретных значений
+переменных, а самих типов. Тип в Julia является объектом первого рода (тип DataType) и его можно присваивать
+другим переменным, передавать в качестве аргументов функции и т.д.
+Функция convert позволяет преобразовать тип конкретного значения, например:
+>>convert(Hyperbolic, 1)
+1+0j
+E.
+Арифметические операции над гиперболическими числами
+После добавления методов для приведения типов, можно приступить к добавлению методов для арифметиче-
+ских операций. Особенностью Julia является реализация арифметических операций не в виде операторов, а в
+виде функций. Так, например, корректны следующие вызовы:
+>>+(1,2)
+3
+>>+(1,2,3,4)
+10
+>>+((i for i in 1:10)...) # числа от 1 до 10
+55
+В связи с этим, добавление методов для арифметических операций ничем не отличается от соответствующего
+процесса для других функций.
+Добавление методов для унарных операций + и - осуществляется следующим образом:
+Base.:+(h::Hyperbolic) = Hyperbolic(+h.re, +h.jm)
+Base.:-(h::Hyperbolic) = Hyperbolic(-h.re, -h.jm)
+Здесь используется сокращенная запись объявления функции.
+Аналогично добавляются методы для бинарного сложения, вычитания, умножения и деления. Приведем здесь
+код для сложения и умножения.
+# Binary + and *
+function Base.:+(x::Hyperbolic, y::Hyperbolic)
+xx = x.re + y.re
+yy = x.jm + y.jm
+Hyperbolic(xx, yy)
+end
+function Base.:*(x::Hyperbolic, y::Hyperbolic)
+xx = x.re * y.re + x.jm * y.jm
+yy = x.re * y.jm + x.je * y.re
+return Hyperbolic(xx, yy)
+end
+V.
+ЗАКЛЮЧЕНИЕ
+Мы рассмотрели механизм множественной диспетчеризации, лежащий в основе языка Julia, на примере
+реализации гиперболических чисел. Данный пример позволил затронуть такие понятия языка как иерархия
+типов данных, составные типы данных, механизмы приведения типов, перегрузка функций (создание новых
+методов для функций в терминах языка Julia) и т.д.
+
+8
+БЛАГОДАРНОСТИ
+Публикация выполнена при поддержке Программы стратегического академического лидерства РУДН.
+[1] Bezanson J., Edelman A., Karpinski S., Shah V. B. Julia: A fresh approach to numerical computing // SIAM Review. ----
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+6th International Conference "Distributed Computing and Grid-technologies in Science and Education". ---- 2014. ---- P. 27.
+[3] Oliphant T. E. Guide to NumPy. ---- 2nd edition. ---- CreateSpace Independent Publishing Platform, 2015. ---- ISBN: 978-
+1517300074.
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+Proceedings of the ACM on Programming Languages. ---- 2018. ---- oct. ---- Vol. 2, no. OOPSLA. ---- P. 1--27.
+[5] Driesen K., H\"olzle U., Vitek J. Message Dispatch on Pipelined Processors // ECOOP'95 --- Object-Oriented Programming,
+9th European Conference, \r Aarhus, Denmark, August 7--11, 1995 / Ed. by M. Tokoro, R. Pareschi. ---- Lecture Notes in
+Computer Science. Springer Berlin Heidelberg, 1995. ---- 253--282 p. ---- ISBN: 9783540601609.
+[6] Muschevici R., Potanin A., Tempero E., Noble J. Multiple dispatch in practice // OOPSLA'08: Proceedings of the 23rd
+ACM SIGPLAN conference on Object-oriented programming systems languages and applications. ---- ACM Press, 2008. ----
+10. ---- P. 563--582.
+[7] Gowda S., Ma Y., Cheli A., Gw\'o\'zzd\'z M., Shah V. B., Edelman A., Rackauckas C. High-Performance Symbolic-Numerics
+via Multiple Dispatch // ACM Communications in Computer Algebra. ---- 2022. ---- jan. ---- Vol. 55, no. 3. ---- P. 92--96.
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+история. — 1961. — Т. 6 из Математическое просвещение, сер. 2. — С. 61–106. — Режим доступа: http://mi.mathnet.
+ru/mp680.
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+Т. 19, № 5 (119). — С. 51–113.
+[10] Kulyabov D. S., Korolkova A. V., Sevastianov L. A. Complex Numbers for Relativistic Operations. ---- 2021. ---- Dec.
+[11] Kulyabov D. S., Korolkova A. V., Gevorkyan M. N. Hyperbolic numbers as Einstein numbers // Journal of Physics:
+Conference Series. ---- 2020. ---- may. ---- Vol. 1557. ---- P. 012027.
+[12] Fjelstad P. Extending special relativity via the perplex numbers // American Journal of Physics. ---- 1986. ---- may. ---- Vol. 54,
+no. 5. ---- P. 416--422.
+[13] Band W. Comments on Extending relativity via the perplex numbers // American Journal of Physics. ---- 1988. ---- may. ----
+Vol. 56, no. 5. ---- P. 469--469.
+[14] Rooney J. On the Three Types of Complex Number and Planar Transformations // Environment and Planning B: Planning
+and Design. ---- 1978. ---- Vol. 5, no. 1. ---- P. 89--99.
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+Machine Science. ---- P. 55--62.
+[16] Gevorkyan M. N., Korolkova A. V., Kulyabov D. S. Approaches to the implementation of generalized complex numbers in
+the Julia language // Workshop on information technology and scientific computing in the framework of the X International
+Conference Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems (ITTMM-
+2020) / Ed. by D. S. Kulyabov, K. E. Samouylov, L. A. Sevastianov. ---- Vol. 2639 of CEUR Workshop Proceedings. ---- Aachen,
+2020. ---- apr. ---- P. 141--157. ---- Access mode: http://ceur-ws.org/Vol-2639/paper-13.pdf.
+

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+Learning Transformations To Reduce the Geometric Shift in Object Detection
+Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2
+CVLab, EPFL1, ClearSpace SA2
+firstname.lastname@epfl.ch
+Abstract
+The performance of modern object detectors drops when
+the test distribution differs from the training one. Most of
+the methods that address this focus on object appearance
+changes caused by, e.g., different illumination conditions,
+or gaps between synthetic and real images. Here, by con-
+trast, we tackle geometric shifts emerging from variations in
+the image capture process, or due to the constraints of the
+environment causing differences in the apparent geometry
+of the content itself. We introduce a self-training approach
+that learns a set of geometric transformations to minimize
+these shifts without leveraging any labeled data in the new
+domain, nor any information about the cameras. We evalu-
+ate our method on two different shifts, i.e., a camera’s field
+of view (FoV) change and a viewpoint change. Our results
+evidence that learning geometric transformations helps de-
+tectors to perform better in the target domains.
+1. Introduction
+While modern object detectors [1, 2, 17, 23, 24] achieve
+impressive results, their performance decreases when the
+test data depart from the training distribution. This prob-
+lem arises in the presence of appearance variations due to,
+for example, differing illumination or weather conditions.
+Considering the difficulty and cost of acquiring annotated
+data in the test (i.e., target) domain, Unsupervised Domain
+Adaptation (UDA) has emerged as the standard strategy to
+address such scenarios [3,4,9,26,38].
+In this context, much effort has been made to learn do-
+main invariant features, such that the source and target dis-
+tributions in this feature space are similar. This has led to
+great progress in situations where the appearance of the ob-
+jects changes drastically from one domain to the other, as
+in case of real-to-sketch adaptation (e.g., Pascal VOC [10]
+to Comics [15]), or weather adaptation (e.g., Cityscapes [6]
+to Foggy Cityscapes [27]). Nevertheless, such object ap-
+pearance changes are not the only sources of domain shifts.
+They can also have geometric origins.
+For example, as
+shown in Fig. 1, they can be due to a change in camera view-
+Figure 1.
+Geometric shifts.
+(Left) Due to a different FoV,
+the cars highlighted in green, undergo different distortions even
+though they appear in similar image regions. (Right) Different
+camera viewpoints (front facing vs downward facing) yield dif-
+ferent distortions and occlusion patterns for pedestrian detection.
+(Bottom) The distributions of pedestrian bounding box sizes in
+Cityscapes [6] and MOT [8] differ significantly as the pedestrians
+are usually far away or in the periphery in Cityscapes. The top im-
+ages are taken from Cityscapes [6], and the bottom-left and right
+ones from KITTI [12] and MOT [8], respectively.
+point or field-of-view (FoV), or a change of object scale due
+to different scene setups. In practice, such geometric shifts
+typically arise from a combination of various factors, in-
+cluding but not limited to the ones mentioned above.
+In this paper, we introduce a domain adaptation approach
+tackling such geometric shifts. To the best of our knowl-
+edge, the recent work of [13] constitutes the only attempt at
+considering such geometric distortions. However, it intro-
+duces a method solely dedicated to FoV variations, assum-
+ing that the target FoV is fixed and known. Here, we de-
+1
+arXiv:2301.05496v1  [cs.CV]  13 Jan 2023
+
+rkplatz/F
+Cityscapes
+Apparent Bbox Distribution
+MOT20velop a more general framework able to cope with a much
+broader family of geometric shifts.
+To this end, we model geometric transformations as a
+combination of multiple homographies. We show both the-
+oretically and empirically that this representation is suffi-
+cient to encompass a broad variety of complex geometric
+transformations. We then design an aggregator block that
+can be incorporated to the detector to provide it with the
+capacity to tackle geometric shifts. We use this modified
+detector to generate pseudo labels for the target domain,
+which let us optimize the homographies so as to reduce the
+geometric shift.
+Our contributions can be summarized as follows. (i) We
+tackle the problem of general geometric shifts for object
+detection. (ii) We learn a set of homographies using unla-
+beled target data, which alleviates the geometric bias arising
+in source-only training. (iii) Our method does not require
+prior information about the target geometric distortions and
+generalizes to a broad class of geometric shifts. Our ex-
+periments demonstrate the benefits of our approach in sev-
+eral scenarios. In the presence of FoV shifts, our approach
+yields similar performance to the FoV-dedicated framework
+of [13] but without requiring any camera information. As
+such, it generalizes better to other FoVs. Furthermore, we
+show the generality of our method by using it to adapt to
+a new camera viewpoint in the context of pedestrian detec-
+tion.
+2. Related Work
+Unsupervised Domain Adaptation (UDA).
+UDA for
+image recognition [11, 21, 22, 30, 32, 35, 36] and object de-
+tection [3,4,9,20,26,38] has made a great progress in the
+past few years. The common trend in both tasks consists
+of learning domain invariant features. For object detection,
+this entails aligning the global (e.g., illumination, weather)
+and local (foreground objects) features in the two domains.
+In this context, [3,5,26,28] align image- and instance-level
+features in the two domains via adversarial learning [11];
+[33] learns category-specific attention maps to better align
+specific image regions; [38] clusters the proposed object re-
+gions using k-means clustering and uses the centroids for
+instance-level alignment. While this successfully tackles
+domain shifts caused by object appearance variations, it
+fails to account for the presence of shifts due to the image
+capture process itself, such as changes in camera intrinsics
+or viewpoint. The only initial step at considering a geo-
+metric shift is the work of [13], which shows the existence
+of an FoV gap in driving datasets [6, 12] and proposes a
+Position Invariant Transform (PIT) that corrects the distor-
+tions caused specifically by an FoV change. In essence, PIT
+undistorts the images by assuming knowledge of the target
+FoV. By contrast, here, we introduce an approach that gen-
+eralizes to a broad family of geometric shifts by learning
+transformations without requiring any camera information.
+Self-training.
+Self-training, generally employed in the
+semi-supervised setting, offers an alternative to learning
+domain-invariant features and utilize unlabeled data to im-
+prove a detector’s performance. In this context, [29] uses
+a student-teacher architecture where the teacher model is
+trained with supervised data and generates pseudo-labels
+on unannotated data. These pseudo-labels are then used
+to train a student model.
+While effective in the stan-
+dard semi-supervised learning scenario, the quality of the
+pseudo-labels obtained with this approach tends to deteri-
+orate when the labeled and unlabeled data present a distri-
+bution shift. [9, 20] have therefore extended this approach
+to domain adaptation by using the Mean Teacher strategy
+of [31] to generate reliable pseudo-labels in the target do-
+main. Other approach include the use of CycleGAN [37]
+generated images to train an unbiased teacher model [9],
+and that of different augmentation strategies to generate ro-
+bust pseudo-labels [20]. Our approach also follows a self-
+training strategy but, while these works focus on object ap-
+pearance shifts, we incorporate learnable blocks to address
+geometric shifts. As shown in our experiment, this lets us
+outperform the state-of-the-art AdaptTeacher [20].
+Learning
+Geometric
+Transformations.
+End-to-end
+learning of geometric transformations has been used to
+boost the performance of deep networks.
+For example,
+Spatial Transformer Networks (STNs) [16] reduce the
+classification error by learning to correct for affine trans-
+formations; deformable convolutions [7] model geometric
+transformations by applying the convolution kernels to
+non-local neighborhoods. These methods work well when
+annotations are available for supervision, and make the
+network invariant to the specific geometric transformations
+seen during training. Here, by contrast, we seek to learn
+transformations in an unsupervised manner and allow the
+network to generalize to unknown target transformations.
+3. Modeling Geometric Transformations
+In the context of UDA, multiple geometric differences
+can be responsible for the gap between the domains. Some
+can be characterized by the camera parameters, such as a
+change in FoV (intrinsic) or viewpoint (extrinsic), whereas
+others are content specific, such as a difference in road
+width between different countries. Ultimately, the geomet-
+ric shift is typically a combination of different geometric
+operations. Since the parameters of these operations are un-
+known, we propose to bridge the domain gap by learning a
+geometric transform. Specifically, we aggregate the results
+of multiple perspective transforms, i.e., homographies, to
+obtain a differentiable operation that can emulate a wide
+variety of geometric transforms.
+2
+
+3.1. Theoretical Model
+Let us first show that, given sufficiently many homogra-
+phies, one can perfectly reproduce any mapping between
+R2 \ (0, 0) and R2.
+Single homography for a single point.
+First, we show
+that a single homography with 4 degrees of freedom can
+map a point p ∈ R2 \(0, 0) to any other point in R2. To this
+end, let
+H =
+�
+�
+sx
+0
+0
+0
+sy
+0
+lx
+ly
+1
+�
+�
+(1)
+be a homography, with (sx, sy) the scaling factors on the x-
+and y-axis, respectively, and (lx, ly) the perspective factors
+in x and y, respectively. For any destination point d ∈ R2,
+there exists a set of parameters (sx, sy, lx, ly) such that d =
+H × p. One such set is ( dx
+px , dy
+py , 0, 0).
+Emulating any geometric transformation
+Now that we
+have shown that a single homography can move a point to
+any other point in R2, we describe a simple protocol to emu-
+late any geometric transform. Given an unknown geometric
+transform T : R2\(0, 0) → R2, we aim to emulate T with a
+set of homographies. In general, for an image I ∈ R3×h×w,
+we can restrict the domain of T to only image coordinates.
+To this end, we can define a set of homographies Hi ∈ H
+for i in {1, 2, 3, ..., h × w}, where the parameters of Hi are
+chosen to mimic the transform T for location i of the image.
+In this protocol, the aggregation mechanism is trivial since
+each homography is in charge of remapping a single pixel
+coordinate of the original space.
+While this works in theory, this is of course not viable
+in practice since it would require too many homographies.
+With a smaller number of homographies, each transform
+needs to remap multiple points, and a more sophisticated
+aggregation mechanism is required. Specifically, the ag-
+gregation mechanism needs to select which transform is in
+charge of remapping which point. In the next section, we
+empirically show that this strategy lets us closely approxi-
+mate the spherical projection mapping used in PIT [13].
+3.2. Approximating PIT with Homographies
+To demonstrate the possibility offered by aggregating
+multiple homographies, we design an approximation of PIT
+using only homographies. PIT proposes to correct for an
+FoV gap by remapping images to a spherical surface. Dur-
+ing this transformation, regions further from the center of
+a scene are compressed with a higher ratio. This variable
+compression of the space cannot be reproduced by a single
+homography transformation. To overcome this limitation,
+we combine the results of multiple homographies that all
+have different compression rates (scaling parameters). For
+the aggregation mechanism, we use the optimal strategy by
+selecting for each pixel the homography that approximates
+best the PIT mapping. As shown in Fig. 2, this combination
+closely approximates the PIT results with only 5 homogra-
+phies. Further analysis of these experiments is available in
+the supplementary material in Fig. A.3.
+Figure 2. Approximating PIT with homographies. We show the
+original image (top), the PIT [13] correction (middle), and our ap-
+proximation of PIT using 5 homographies. Note that 5 homogra-
+phies are sufficient to closely match the PIT spherical correction.
+3.3. Homographies in a Learning Setup
+In the two previous sections, we have demonstrated both
+theoretically and empirically the flexibility of aggregating
+homographies. This makes this representation an ideal can-
+didate for domain adaptation since the geometric shift be-
+tween the domains is unknown and can be a combination
+of different transforms, such as FoV change, viewpoint
+change, camera distortion, or appearance distortion. As will
+be discussed in the next section, by learning jointly the set
+of perspective transforms and the aggregation mechanism
+on real data, our model can reduce the geometric shift be-
+tween the two domains without prior knowledge about this
+domain gap.
+4. Method
+Let us now introduce our approach to reducing the ge-
+ometric shift in object detection. Following the standard
+UDA setting, let Ds = {(Is, Bs, Cs)} be a labeled source
+dataset containing images Is = {Ii
+s}Ns
+1
+with correspond-
+ing object bounding boxes Bs = {bi
+s}Ns
+1
+and object classes
+Cs = {ci
+s}Ns
+1 . Furthermore, let Dt = {It} denote an un-
+labeled target dataset for which only images It = {Ii
+t}Nt
+1
+3
+
+Original
+PIT
+Approximation 5 H.Figure 3. Architecture: The input image is first transformed by a set of trainable homographies. The feature maps extracted from
+the transformed images are then unwarped by the inverse homographies to achieve spatial consistency. We then combine the unwarped
+feature maps using a trainable aggregator, whose output is passed to a detection head. The blocks shown in green correspond to standard
+FasterRCNN operations. The � symbol represents the concatenation operation.
+are available, without annotations. Here, we tackle the case
+where the two domains differ by geometric shifts but as-
+sume no knowledge about the nature of these shifts. Below,
+we first introduce the architecture we developed to handle
+this and then our strategy to train this model.
+4.1. Model Architecture
+The overall architecture of our approach is depicted in
+Fig. 3. In essence, and as discussed in Sec. 3, we charac-
+terize the geometric changes between the source and target
+data by a set of transformations T = {Hi}N
+1 . Each Hi in
+T is a homography of the same form as in Eq. (1). For our
+method to remain general, we assume the transformations to
+be unknown, and our goal, therefore, is to learn T to bridge
+the gap between the domains. This requires differentiabil-
+ity w.r.t. the transformation parameters, which we achieve
+using the sampling strategy proposed in [16].
+As shown in Fig. 3, the input image is transformed by the
+individual homographies in T , and the transformed images
+are fed to a modified FasterRCNN [24] detector. Specif-
+ically, we extract a feature map FHi ∈ RH×W ×C for
+each transformed image via a feature extractor shared by
+all transformations. To enforce spatial correspondence be-
+tween the different FHis, we unwarp them with H−1
+i
+.
+We then introduce an aggregator Aθg, parameterized by
+θg, whose goal is to learn a common representation given
+a fixed number of unwarped feature maps F′
+Hi. To achieve
+this, the aggregator takes as input
+G = F′
+H1 ⊕ F′
+H2 ⊕ ... ⊕ F′
+HN ∈ RH×W ×C×N ,
+(2)
+where ⊕ represents concatenation in the channel dimension.
+The aggregator outputs a feature map Aθg(G) ∈ RH×W ×C,
+whose dimension is independent of the number of transfor-
+mations. This output is then passed to a detection head to
+obtain the objects’ bounding boxes and class labels.
+4.2. Model Training
+Our training procedure relies on three steps: (i) Fol-
+lowing common practice in UDA, we first train the Faster-
+RCNN detector with source-only data; (ii) We then intro-
+duce the aggregator and train it so that it learns to com-
+bine different homographies using the labeled source data;
+(iii) Finally, we learn the optimal transformations for adap-
+tation using both the source and target data via a Mean
+Teacher [31] strategy.
+Aggregator Training.
+To train the aggregator, we ran-
+domly sample a set of homographies T ∈ RN×4 in each
+training iteration.1 This gives the aggregator the ability to
+robustly combine diverse input transformations but requires
+strong supervision to avoid training instabilities. We, there-
+fore, perform this step using the source data.
+The loss function for a set of transformed images T (Is)
+is then defined as in standard FasterRCNN training with
+a combination of classification and regression terms [24].
+That is, we train the aggregator by solving
+min
+θg Lcls(T (Is)) + Lreg(T (Is)) ,
+(3)
+1As our homographies involve only 4 parameters, with a slight abuse
+of notation, we say that Hi ∈ R4.
+4
+
+Homography
+Feature
+Homography
+Detection
+Aggregator
+Projections
+Extractor
+Projections
+Headwhere
+Lcls(T (Is)) = Lrpn
+cls + Lroi
+cls ,
+(4)
+Lreg(T (Is)) = Lrpn
+reg + Lroi
+reg .
+(5)
+Lrpn
+·
+and Lroi
+·
+correspond to the Region Proposal Network
+(RPN) loss terms and the Region of Interest (RoI) ones, re-
+spectively. During this process, we freeze the parameters
+θb of the base network, i.e, feature extractor and detection
+head, which were first trained on the source data without ag-
+gregator. Ultimately, the aggregator provides the network
+with the capacity to encode different transformations that
+are not seen in the source domain. The third training step
+then aims to learn the best transformation for successful ob-
+ject detection in the target domain.
+Learning the Transformations.
+As we have no annota-
+tions in the target domain, we exploit a Mean Teacher (MT)
+strategy to learn the optimal transformations. To this end,
+our starting point is the detector with a trained aggregator
+and a set of random transformations T . The MT strategy is
+illustrated in Fig. 4. In essence, MT training [31] involves
+two copies of the model: A student model, with parameters
+θst = {T st, θst
+b , θst
+g }, that will be used during inference,
+and a teacher model, with parameters θte = {T te, θte
+b , θte
+g },
+that is updated as an Exponentially Moving Average (EMA)
+of the student model.
+That is, the student’s parameters
+are computed with standard backpropagation, whereas the
+teacher’s ones are updated as
+θte ← αθte + (1 − α)θst .
+(6)
+The student model is trained using both source and tar-
+get detection losses. Since the target domain does not have
+annotations, the teacher model is used to generate pseudo-
+labels. These labels might be noisy, and hence we only keep
+the predictions with a confidence score above a threshold τ.
+Furthermore, non-maxima suppression (NMS) is used to re-
+move the highly-overlapping bounding box predictions.
+Formally, given a source image Is and a target image It,
+the student model is trained by solving
+min
+T st,θst
+g ,θst
+b
+Ldet(T (Is)) + λLdet(T (It)) ,
+(7)
+where λ controls the target domain contribution and
+Ldet(T (Is)) = Lcls(T (Is)) + Lreg(T (Is)) ,
+(8)
+Ldet(T (It)) = Lcls(T (It)) .
+(9)
+Similarly to [18,20], we update the student model with only
+the classification loss in the target domain to help stabilize
+training.
+Figure 4. Mean Teacher formalism. The student model is trained
+with ground-truth labels in the source domain and pseudo labels in
+the target one. These pseudo labels are produced by the teacher
+model, which corresponds to an exponentially moving average
+(EMA) of the student network.
+5. Experiments
+We demonstrate the effectiveness and generality of our
+method on different geometric shifts. First, to compare to
+the only other work that modeled a geometric shift [13],
+we tackle the problem of a change in FoV between the
+source and target domain. Note that, in contrast to [13], we
+do not assume knowledge of the target FoV. Furthermore,
+while [13] was dedicated to FoV adaptation, our approach
+generalizes to other geometric shifts. We demonstrate this
+on the task of pedestrian detection under a viewpoint shift.
+We compare our method with the state-of-the-art Adapt-
+Teacher [20], which also uses a Mean Teacher, but focuses
+on appearance shifts. In the remainder of this section, we
+describe our experimental setup and discuss our results.
+5.1. Datasets
+Cityscapes [6]
+contains 2975 training and 500 test im-
+ages with annotations provided for 8 categories (person,
+car, train, rider, truck, motorcycle, bicycle and bus). The
+average horizontal (FoVx) and vertical (FoVy) FoVs of the
+capturing cameras are 50°and 26°, respectively. We use this
+dataset as the source domain for both FoV adaptation and
+viewpoint adaptation.
+KITTI [12]
+is also a street-view dataset containing 6684
+images annotated with the car category.
+The horizontal
+(FoVx) and vertical (FoVy) FoVs of the camera are 90°and
+34°, respectively.
+We use this dataset as target domain
+for FoV adaptation, as the viewpoint is similar to that of
+Cityscapes. Following [13], we use 5684 images for unsu-
+pervised training and 1000 images for evaluation.
+MOT [8]
+is a multi-object tracking dataset. We use the in-
+door mall sequence, MOT20-02, consisting of 2782 frames
+5
+
+Feature
+H1
+Hi-1
+Extractor
+Feature
+Detection
+H2
+Aggregator
+Extractor
+Head
+..
+...
+Feature
+-1
+Extractor
+Mean Teacher
+Ldet
+Feature
+H1
+H-1
+Extractor
+Feature
+Detection
+H2
+H2-1
+Ldet
+Aggregator
+Extractor
+Head
+Feature
+Hn
+-1
+Extractor
+Studentannotated with the person category. We employ this dataset
+as target domain for viewpoint adaptation. We use the first
+2000 frame for unsupervised training and last 782 for eval-
+uation.
+5.2. Adaptation Tasks and Metric
+FoV adaptation.
+As in [13], we consider the case of
+an increasing FoV using Cityscapes as source domain and
+KITTI as target domain. The horizontal and vertical FoVs
+increase from (50°, 26°) in Cityscapes to (90°, 34°) in
+KITTI. Therefore, as can be seen in Fig. 1, the KITTI
+images have a higher distortion in the corners than the
+Cityscapes ones. Similarly to PIT [13], we use the car cat-
+egory in our experiments.
+FoV generalization.
+Following PIT [13], we study the
+generalization of our approach to new FoVs by cropping
+the KITTI images to mimic different FoV changes in the
+horizontal direction (FoVx). Specifically, we treat FoVx =
+50° as the source domain and the cropped images with FoVx
+= {70°, 80°, 90°} as different target domains. We evaluate
+our approach on car on these different pairs of domains.
+Viewpoint adaptation.
+This task entails detecting objects
+seen from a different viewpoint in the source and target do-
+mains. We use the front-facing Cityscapes images as source
+domain and the downward-facing MOT ones as target one.
+As the MOT data depicts pedestrians, we use the bounding
+boxes corresponding to the person category in Cityscapes.2
+Metric.
+In all of our experiments, we use the Average Pre-
+cision (AP) as our metric. Specifically, following [13], we
+report the AP@0.5, which considers the predictions as true
+positives if they match the ground-truth label and have an
+intersection over union (IOU) score of more than 0.5 with
+the ground-truth bounding boxes.
+5.3. Implementation Details
+We use the Detectron2 [34] implementation of Faster-
+RCNN [24] with a ResNet50 [14] backbone as our base
+architecture. In all of our experiments, the images are re-
+sized so that the shorter side has 800 pixels while maintain-
+ing the aspect ratio. The base network is first trained on
+source-only images with random cropping and random flip-
+ping augmentation for 24k iterations with batch size 8. We
+use the Stochastic Gradient Descent (SGD) optimizer with
+a learning rate of 0.01, scaled down by a 0.1 factor after
+18k iterations. We use ImageNet [25] pretrained weights to
+initialize the ResNet50 backbone.
+2In Cityscapes, a person may be labeled as either person or rider. Since
+the rider label is used for people riding a vehicle, we omit these cases.
+Figure 5. FoV Adaptation: Qualitative Results. We visualize
+a car detection result in the Cityscapes-to-KITTI FoV adaptation
+scenario. The top left image corresponds to the ground truth, the
+bottom left to the Mean Teacher result, which confuses the orange
+container with a car, the bottom right to the Mean Teacher adapta-
+tion + PIT FoV adaptation result, which also mistakes the orange
+container for a car and further detects the speed limit on the road.
+Our approach, on the top right, correctly matches the ground truth.
+We then incorporate the aggregator in the trained base
+architecture.
+The aggregator architecture contains three
+convolutional layers with a kernel size of 3 × 3, and one
+1 × 1 convolutional layer.
+We first train the aggregator
+on the source data with the base frozen and using ran-
+dom transformations T .
+The transformations are gener-
+ated by randomly sampling each Hi parameters as sx, sy ∼
+U[0.5,2.0], U[0.5,2.0] and lx, ly ∼ U[−0.5,0.5], U[−0.5,0.5]. We
+train the aggregator for 30k iterations using a batch size of
+8 and the SGD optimizer with a learning rate of 1e−4.
+The student and teacher models are then initialized with
+this detector and the random T = {Hi}N
+i=1. We optimize T
+using Adam [19], while the base and aggregator networks
+are optimized by SGD. The learning rate is set to 1e−3 and
+scaled down by a factor 0.1 after 10k iterations for the SGD
+optimizer. For the first 10k iterations in FoV adaptation and
+for 2k iterations for viewpoint adaptation, we only train T
+keeping base and aggregator frozen. The α coefficient for
+the EMA update is set to 0.99; the confidence threshold
+τ = 0.6; λ = {0.01, 0.1} for FoV and viewpoint adapta-
+tion, respectively. The Mean Teacher framework is trained
+using both the source and target data. We set N = 5, unless
+otherwise specified, and use a batch size of 4, containing 2
+source and 2 target images. We apply random color jitter-
+ing on both the source and target data as in [20, 31]. All
+of our models are trained on a single NVIDIA V100 GPU.
+A detailed hyper-parameter study is provided in the supple-
+mentary material.
+5.4. Comparison with the State of the Art
+We compare our approach with the following baselines.
+FR: FasterRCNN trained only on the source data with
+random crop augmentation; AT: AdaptTeacher [20]; MT:
+Mean Teacher initialized with FR and trained with ran-
+dom color jittering on both the source and target data (i.e.,
+this corresponds to our mean teacher setup in Sec. 4.2
+but without the aggregator and without transformations T );
+6
+
+GT
+Ours
+M
+MT+PITMethod
+Car AP@0.5
+FR [24]
+76.1
+AT [20]
+77.2
+FR+PIT
+77.6
+MT
+78.3
+MT+PIT [13]
+79.7
+Ours
+80.4 ± 0.15
+Table 1. FoV Adaptation.
+Car AP@0.5 for FoVx
+Method
+50°
+70°
+80°
+90°
+FR [24]
+94.3
+90.2
+86.8
+80.6
+FR+PIT [13]
+93.6
+91.4
+89.2
+85.9
+Ours-h
+94.1± 0.16
+93.1 ± 0.33
+91.8 ± 0.40
+88.8 ± 0.21
+Table 2. FoV Generalization.
+FR+PIT: Same setup as FR but with the images corrected
+with PIT [13]; MT+PIT: Same setup as MT but with the
+images corrected with PIT. We refer to our complete ap-
+proach (Sec. 4.2) as Ours. For the task of FoV generaliza-
+tion, we report our results as Ours-h to indicate that we only
+optimize the homographies (5×4 parameters) in T to adapt
+to the new FoVs while keeping the base and aggregator net-
+works frozen. This matches the setup of PIT [13], which
+also corrects the images according to the new FoVs. As
+Ours and Ours-h are trained with randomly initialized T ,
+we report the average results and standard deviations over
+three independent runs.
+FoV adaptation.
+The results of Cityscapes → KITTI FoV
+adaptation are provided in Tab. 1. Both MT+PIT and Ours
+both bridge the FoV gap, outperforming the MT baseline.
+Note, however, that we achieve this by learning the trans-
+formations, without requiring any camera-specific informa-
+tion, which is needed by PIT. Note also that MT outper-
+forms FR by learning a better representation in the target do-
+main, even though FR is trained with strong augmentation,
+such as random cropping. AT underperforms because its
+strong augmentation strategy fails to generalize for datasets
+having prominent geometric shifts. Our improvement over
+MT evidences that learning transformations helps to over-
+come geometric shifts. We optimize with N = 9, homo-
+graphies in this setup. Fig. 5 shows a qualitative example.
+Different homographies look into different image regions
+and the aggregator learns how to combine the activations
+corresponding to objects as depicted in Fig. 7.
+FoV generalization.
+Tab. 2 summarizes the results ob-
+tained by using different FoVs as target domains while fix-
+Method
+Pedestrian AP@0.5
+FR [24]
+43.7
+AT [20]
+63.5
+MT
+64.7
+Ours
+65.3± 0.37
+Table 3. Viewpoint Adaptation.
+Figure 6. Varying the number of homographies. We evaluate
+the effect of N on the FoV adaptation task.
+ing the source FoV to 50°. Since both the source and tar-
+get images are taken from KITTI, the domain gap is only
+caused by a FoV change.
+Note that the performance of
+FR drops quickly as the FoV gap increases. Ours-h out-
+performs FR+PIT by a growing margin as the FoV gap in-
+creases. This shows that learning transformations helps to
+generalize better to different amounts of geometric shifts.
+Viewpoint adaptation.
+As shown in Fig. 1, a change in
+the camera viewpoint yields differences in the observed dis-
+tortions and type of occlusions. The results in Tab. 3 show
+the benefits of our method over MT in this case. Note that
+PIT, which was designed for FoV changes, cannot be ap-
+plied to correct for a viewpoint change. Other baselines out-
+perform FR, as they use pseudo labels to fix the difference
+in bounding box distribution, as shown in Fig. 1. These
+results illustrate the generality of our method to different
+kinds of geometric shifts. Qualitative results for this task
+can be found in Fig. A.10.
+5.5. Additional Analyses
+Variable number of homographies.
+Let us now study
+the influence of the number of homographies in T .
+To
+this end, we vary this number between 1 and 9. In Fig. 6,
+we plot the resulting APs for the Cityscapes-to-KITTI FoV
+adaptation task. Increasing the number of transformations
+results in a steady increase in performance, which nonethe-
+less tends to plateau starting at 4 homographies. Due to lim-
+ited compute resources, we couldn’t run experiments with
+7
+
+80.5
+80.0
+AP@0.5
+79.5
+79.0
+78.5
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Number of HomographyFigure 7. Feature Maps: Top row: predictions of our network and
+feature map after aggregator. Left column: Image I, transformed
+by learned homographies; Right Column: Feature maps F warped
+by corresponding H−1 which are input to the aggregator. Each
+transform distorts the image regions differently. Most of the cars
+are on the left side and of small size in the image. H1 distorts
+the left side leading to no activation(H−1
+1 F1) for the object. H3
+which causes the zoom-in effect has the strongest activation as the
+smaller objects are visible better here. These maps are generated
+by taking maximum over channel dimension.
+more than 9 homographies. This confirms the intuition that
+a higher number of perspective transformations can better
+capture the geometric shift between two domains. There-
+fore, we conducted all experiments with the maximum num-
+ber of homographies allowed by our compute resources.
+Only optimizing T .
+We also run the Ours-h baseline in
+the FoV and viewpoint adaptation scenarios. The result-
+ing APs are 78.2 and 49.8, respectively. By learning only
+the 20 (5 × 4) homography parameters, our approach out-
+performs FR (in Tab. 1 and Tab. 3, respectively) by a large
+margin in both cases. This confirms that our training strat-
+egy is able to efficiently optimize T to bridge the geometric
+gap between different domains. We visualize in Fig. A.9 in
+the supplementary material some transformations learned
+for FoV adaptation by Ours-h. Note that they converge to
+diverse homographies that mimic a different FoV, correctly
+reflecting the adaptation task.
+Diversity in T .
+To show that our approach can learn
+a diverse set of transformations that help in the adapta-
+tion task, we initialize all the homographies with iden-
+tity. Fig. 8 depicts the diversity of the learned homogra-
+phies on the FoV adaptation task.
+Even though we do
+not enforce diversity, our approach learns a diverse set of
+transformations.
+With these learned homorgraphies, our
+model achieves 79.5 AP@0.5 score for the FoV adaptation
+task. We show additional results in the supplementary ma-
+terial Sec. 4 and Sec. 5.
+Figure 8. Diversity in T : We train 5 homographies initialized as
+Hi = I. We plot the evolution of sx for different homograhies
+as training proceeds. Each homography is shown in a different
+color. Note that the values for the different homographies become
+diverse. The best score is achieved at iteration = 22k, indicated
+with the vertical line.
+Limitations.
+Our approach assumes that the geometric
+gap between two domains can be bridged by a set of per-
+spective transformations. We have shown that with enough
+transformations this is true. However, using a large num-
+ber of homographies comes at a computational cost. The
+computational overhead leads to an increment in the infer-
+ence time from 0.062s to 0.096s for N = 5 on an A100
+Nvidia GPU with image dimension 402 × 1333. Neverthe-
+less, our simple implementation shows promising results,
+and we will work on reducing this overhead in future work.
+Moreover since the optimization of the homography set is
+done at the dataset level, only certain transformations are
+beneficial to a given image. In the future, we therefore in-
+tend to condition the homography on the input image, which
+would reduce the total number of homographies needed.
+6. Conclusion
+We have introduced an approach to bridge the gap be-
+tween two domains caused by geometric shifts by learning
+a set of homographies. We have shown the effectiveness our
+method on two different kinds of shifts, without relying on
+any annotations in the target domain, including information
+about the nature of the geometric shifts. Our analyses have
+evidenced that optimizing the transformations alone brings
+in improvement over the base detector and increasing the
+number of learnt homographies helps further. In the future,
+we plan to learn transformations that are conditioned on the
+input image to model image-dependent geometric shifts.
+8
+
+H1
+1.4
+H2
+H3
+1.3
+H4
+H5
+1.2
+1.1
+1.0
+0.9
+0.8
+0.7
+0.6
+20
+5620
+11220
+16820
+22420
+28020
+IterationsReferences
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+
+Supplementary Material: Learning Transformations To Reduce the Geometric
+Shift in Object Detection
+Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2
+CVLab, EPFL1, ClearSpace SA2
+firstname.lastname@epfl.ch
+1. Transformations through Homography
+We use homography to introduce varied perspective
+transformations so that they can distort the same image re-
+gions differently as seen in Fig. A.1. This helps the detector
+to learn robust object features and simultaneously optimize
+an aggregator with a different set of homographies which
+can bridge the gap between two domains.
+2. Feature Maps Activation
+We show in Fig. A.2 how different homographies gen-
+erate activation in the feature maps. Not all homographies
+look at the same image region, therefore the task of the ag-
+gregator is to bring in the activations from different trans-
+formations together.
+3. Other Aggregator Architecture
+We implement aggregator using standard functions to
+combine {FHi}N
+i=1. Tab. A.1 illustrates this study for FoV
+adaptation, where the training is done under mean teacher
+formalism to learn |T | = N = 5. We see that these non-
+learnable aggregators are able to outperform MT baseline
+(Sec. 5.4, in the main paper) suggesting that including trans-
+formations helps to bridge the geometric shifts.
+Function
+Car AP@0.5
+sum
+78.1± 0.14
+mean
+78.7± 0.05
+max
+78.7± 0.12
+min+max
+78.9± 0.43
+MT
+78.3
+Ours
+79.9± 0.14
+Table A.1. Aggregator Architecture without learnable parameters
+4. Diversity in T
+In order to show that diverse transformations are learned,
+we set Hi = I and train our mean teacher formulation.
+Fig. A.4 shows diverse set of transformations learned in
+FoV adaptation task. Even though we do not enforce di-
+versity among homographies, it is learned through our ap-
+proach.
+5. Evolution of T
+We provide qualitative results for T learned in FoV and
+Viewpoint adaptation, Fig. A.5 and Fig. A.8, respectively.
+The qualitative results for the same adaptation task can be
+seen in Fig. A.6 and Fig. A.7, respectively.
+6. Hyperparameter details
+Augmentations.
+We use Detectron2 [2]s implementation
+for random crop and torchvision 1 for color jittering.
+Kind
+Details
+Random Crop
+Relative Range: [0.3, 1]
+Color Jitter
+Brightness=.5, Hue=.3
+Table A.2. Augmentations
+FasterRCNN [1] training.
+We train our base network
+with random crop strategy on with only source data, which
+is Cityscapes for both the adaptation tasks.
+The trained
+model achieves 74.7 and 58.4 AP@0.5 score on the source
+domain validation set for car and person detection, respec-
+tively.
+Mean Teacher Training
+For our mean teacher setup
+(Sec. 4.2, in the main paper), we choose τ = 0.6 as the con-
+fidence threshold for the pseudo-labels and evaluate con-
+tribution of target domain loss for different λ. Fig. A.11
+summarizes this study. We see that method performs worse
+when we have equal contribution from both source and tar-
+get domain loss λ = 1, as the false positives in the target
+1https://pytorch.org/vision/stable/transforms.
+html
+1
+arXiv:2301.05496v1  [cs.CV]  13 Jan 2023
+
+Figure A.1. Transformations: Here we demonstrate how the two objects in the original image undergo different perspective transforma-
+tions. Our task is to learn robust object features under such transformations and use them to bring the two domains closer while being
+agnostic to the camera parameters. We train with a multiple set of transformations to change the same image region differently. With our
+trainable aggregator, we can then combine features from different regions to help in improving the detector’s performance.
+domain quickly deteriorate the training. Fig. A.12, evalua-
+tion for different values of τ.
+7. Architecture details
+Our aggregator architecture consists of three convolution
+layers along with BatchNorm and Relu layers after each
+convolution. Tab. A.3 shows the details of different layers.
+Here, C = 1024 corresponds to the output of the feature
+extractor.
+Table A.3. Aggregator Architecture for |T | = N
+# Channels
+Layer
+Input
+Output
+Conv2d 3 × 3
+N × C
+N × C/2
+BatchNorm + Relu
+N × C/2
+N × C/2
+Conv2d 3 × 3
+N × C/2
+C
+BatchNorm + Relu
+C
+C
+Conv2d 1 × 1
+C
+C
+BatchNorm + Relu
+C
+C
+References
+[1] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.
+Faster r-cnn: towards real-time object detection with region
+proposal networks. IEEE transactions on pattern analysis and
+machine intelligence, 39(6):1137–1149, 2016. 1
+[2] Yuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen
+Lo, and Ross Girshick. Detectron2. https://github.
+com/facebookresearch/detectron2, 2019. 1
+2
+
+Figure A.2. Feature Maps: Top row: predictions of our network and feature map after aggregator. Left column: Image I, transformed
+by 5 learnt homographies; Right Column: Feature maps F warped by corresponding H−1 which are input to aggregator. Each transform
+distorts the image regions differently. Most of the cars are on the left side and of small size in the image. H1 distorts the left side leading to
+no activation(H−1
+1 F1) for the object. H3 which causes zoom-in effect has the strongest activation as the smaller objects are visible better
+here. Overall aggregator feature map contains activation from the region where the objects exist. The aggregator has learnt how to combine
+regions with activations under different homographies. The feature maps are generated by taking maximum over channel dimension.
+3
+
+Figure A.3. Approximating PIT with homographies. Left column: Visualization of each homography use to approximate PIT with 5
+transforms; the top one is the identitity, and the following ones are in order of increasing compression. Center column: Contribution of
+each homography to the final remapping. Right column: The top figure shows the per pixel coordinate error when compared to the PIT
+remapping as a function of the number of homographies used in the approximation; the three bottom figures depict the coordinate error
+maps for 1, 5, and 25 homographies used to approximate PIT (note the scale change in pixel coordinate error).
+4
+
+sx
+sy
+lx
+ly
+Figure A.4. Diversity in T : We train |T | = 5 initialized with Hi = I. Homographies parameterized by sx, sy, lx, ly evolve as the training
+proceeds and tend to become diverse. Each homography is shown in different color. Even though we do not enforce any diversity, our
+approach learns diverse set of transformations. With these learned homorgraphies, we achieve 79.5 AP@0.5 score for FoV adaptation task.
+The best score is achieved at iteration = 22k shown with the vertical line.
+5
+
+H1
+1.4
+H2
+H3
+1.3
+H4
+H5
+1.2
+1.1
+1.0
+0.9
+0.8
+0.7
+0.6
+20
+5620
+11220
+16820
+22420
+28020
+Iterations1.7
+H1
+H2
+1.6
+H3
+H4
+H5
+1.5
+1.4
+1.3
+1.2
+1.1
+1.0
+0.9
+20
+5620
+11220
+16820
+22420
+28020
+Iterations0.3
+H1
+H2
+H3
+0.2
+H4
+H5
+0.1
+0.0
+-0.1
+-0.2
+-0.3
+20
+5620
+11220
+16820
+22420
+28020
+Iterations0.20
+H1
+H2
+0.15
+H3
+H4
+H5
+0.10
+0.05
+0.00
+-0.05
+-0.10
+-0.15
+-0.20
+20
+5620
+11220
+16820
+22420
+28020
+Iterationssx
+sy
+lx
+ly
+Figure A.5. Quantitative results for the corresponding results in Figure A.6. The randomly initialized transforms, parameterized by
+sx, sy, lx, ly, evolve to achieve the best score at 28k iterations (shown by the vertical bar). The colors represent different homographies.
+Some set of parameters converges to similar value but overall each homography is unique.
+6
+
+0.4
+0.2
+H1
+H2
+0.0
+H3
+H4
+H5
+-0.2
+-0.4
+20
+5620
+11220
+16820
+22420
+28020
+Iterations0.4
+0.3
+0.2
+0.1
+0.0
+-0.1
+H1
+H2
+H3
+-0.2
+H4
+H5
+20
+5620
+11220
+16820
+22420
+28020
+IterationsH1
+1.8
+H2
+H3
+H4
+1.6
+H5
+1.4
+1.2
+1.0
+0.8
+0.6
+20
+5620
+11220
+16820
+22420
+28020
+Iterations1.8
+1.6
+1.4
+1.2
+1.0
+0.8
+H1
+H2
+H3
+0.6
+H4
+H5
+0.4
+20
+5620
+11220
+16820
+22420
+28020
+IterationsFigure A.6. FoV adaptation: The randomly initialized homographies evolve as the training progresses to improve the overall AP score.
+We train with 5 homographies and show how they transform an image for the corresponding FoV adaptation task.
+7
+
+PredFigure A.7. Viewpoint adaptation: The randomly initialized homographies evolve as the training progresses to improve the overall AP
+score. We train with 5 homographies and show how they transform an image for the corresponding viewpoint adaptation task.
+8
+
+Predsx
+sy
+lx
+ly
+Figure A.8. Quantitative results for the corresponding results in Figure A.7. The randomly initialized transforms, parameterized by
+sx, sy, lx, ly, evolve to achieve the best score at 8k iterations (shown by the vertical bar). The colors represent different homographies.
+Some sy parameters start at a similar value but eventually diverge.
+9
+
+1.8
+1.6
+1.4
+H1
+H2
+H3
+1.2
+H4
+H5
+1.0
+0.8
+0.6
+20
+1820
+3620
+5420
+7220
+9020
+Iterations1.4
+1.2
+1.0
+0.8
+H1
+H2
+H3
+0.6
+H4
+H5
+20
+1820
+3620
+5420
+7220
+9020
+IterationsH1
+0.3
+H2
+H3
+H4
+0.2
+H5
+0.1
+0.0
+-0.1
+-0.2
+-0.3
+-0.4
+-0.5
+20
+1820
+3620
+5420
+7220
+9020
+IterationsH1
+H2
+0.3
+H3
+H4
+H5
+0.2
+0.1
+0.0
+-0.1
+-0.2
+20
+1820
+3620
+5420
+7220
+9020
+IterationsFigure A.9. Evolution of T . We showcase how two homographies, H1 and H5, evolve across the training iterations and influence
+the prediction scores. Starting from random homographies at iteration 0, the transformations converge to homographies suited for FoV
+adaptation. The detection scores consequently increase throughout the training process. Moreover, this increase in detection score is
+reflected in the overall AP@0.5 score, which jumps from 74.1 to 78.2.
+Figure A.10. Viewpoint Adaptation: Qualitative Results. We visualize results for viewpoint adaptation between Cityscapes and MOT20-
+02. The left image depicts the ground truth, the middle one the results of Mean Teacher adaptation, and the right one those of our approach.
+Our approach recovers more detections (e.g., the woman near the stroller in the center-left) while having fewer false positives (overlapping
+box in bottom-left corner of the MT results).
+10
+
+88%
+94%
+93%GT
+MT
+OursFigure A.11. Study on λ for τ = 0.6, |T | = 5
+Figure A.12. Study on τ for FoV and Viewpoint adaptation using
+λ = 0.01, 0.1, respectively. Here ,|T | = 5 is used for the study.
+11
+
+81
+79
+77
+75
+73
+71
+5
+69
+AP@O.
+67
+65
+63
+61
+59
+Fov adapt.
+57
+Viewpoint adapt
+55
+1
+0.1
+0.01
+0.001
+入81
+79
+77
+75
+73
+71
+5
+AP@O.!
+69
+67
+65
+63
+61
+59
+FoV adapt.
+57
+Viewpoint adapt.
+55
+0.5
+0.6
+0.7
+0.8
+T

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+Fair Credit Scorer through Bayesian Approach
+Zhuo Zhao
+Department of Applied Mathematics
+Johns Hopkins University
+zzhao62@jhu.edu
+Abstract
+Machine learning currently plays an increasingly important role in people’s lives
+in areas such as credit scoring, auto-driving, disease diagnosing, and insurance
+quoting. However, in many of these areas, machine learning models have performed
+unfair behaviors against some sub-populations, such as some particular groups of
+race, sex, and age. These unfair behaviors can be on account of the pre-existing
+bias in the training dataset due to historical and social factors. In this paper, we
+focus on a real-world application of credit scoring and construct a fair prediction
+model by introducing latent variables to remove the correlation between protected
+attributes, such as sex and age, with the observable feature inputs, including house
+and job. For detailed implementation, we apply Bayesian approaches, including
+the Markov Chain Monte Carlo simulation, to estimate our proposed fair model.
+1
+Introduction
+Nowadays, Machine Learning methods are used to automate decisions in a variety of areas, including
+determining credit scores Nanni and Lumini [2009], classifying tumor components from MRI images
+Lundervold and Lundervold [2019], detecting pedestrians on the road Dollar et al. [2011], and
+understanding natural languages Goldberg and Levy [2014], etc. However, machine learning methods
+are heavily dependent on data Mitchell and Mitchell [1997] and this data-dependent nature makes the
+learned models sensitive to the latent bias existing in the training datasets Mehrabi et al. [2021]. Thus,
+the final decisions made by the learned models are unfairly biased against certain sub-populations,
+differentiated by some sensitive/protected attributes, such as race, sex, or age, etc. For example,
+cameras sometimes fail to recognize whether Asian blink their eyes Sharp [2009] and the beauty
+pageant judged by AI would prefer light skin Guardian [2016]. However, we would expect AI to give
+the same decision independent from the protected attributes and thus we concern about the fairness
+of machine learning methods Mehrabi et al. [2021].
+In this paper, we focus on constructing fair machine learning models to predict the credit score, with
+using the German Credit Risk dataset Hoffman [2016] (Sec. 3). The goal is to predict the credit
+score based on some observable variables, including housing and job information. However, this
+personal financial information, such as income, housing, and saving, are usually highly correlated
+to gender and age due to historical and social reasons Rennison and Planty [2003]. Therefore, it
+is necessary to learn an effective model to filter the prediction bias against sex and age, caused by
+the latent correlation between these observable variables and the protected attributes. In detail, we
+analyze and compare from the fairness perspective across the full model Montgomery et al. [2021],
+unaware model Dwork et al. [2012], and fair model based on causals and counterfactuals Kusner et al.
+[2017] (Sec. 4). Then, we apply the Markov Chain Monte Carlo (MCMC) simulation Mooney [1997]
+and the Gibbs’ sampling Gelfand [2000] to solve the corresponding parameters in these models and
+evaluate the performances (Sec. 5 and Sec. 6).
+arXiv:2301.08412v1  [cs.LG]  20 Jan 2023
+
+2
+Related work
+Fairness. Many recent works (Calders and Verwer [2010], Bolukbasi et al. [2016], Dwork et al.
+[2012], Hardt et al. [2016], Joseph et al. [2016], Kusner et al. [2017]) have been focusing on fairness
+in machine learning algorithms. Bolukbasi et al. [2016] pointed out that there is a risk of amplifying
+the bias introduced from the dataset, if using machine learning algorithms without taking effects to
+handle the pre-existing bias. For example, in the word embedding, learned over Google News with
+pre-existing gender stereotypes, the gender-neutral words widely spread along a latent embedding
+direction capturing gender difference, such as "receptionist" falling far along the direction related
+to "female" Bolukbasi et al. [2016]. Calders and Verwer [2010] modifies the Naive Bayes classifier
+by adding independence restriction toward sensitive attributes. Dwork et al. [2012] proposes a
+task-specific metric to evaluate the similarity between individuals relative to the classification task
+and optimizes over the proposed metric with the goal that similar individuals are treated similarly
+in the classification task. Kusner et al. [2017] focuses on causal inference and counterfactual, with
+introducing the latent confounding variables, which are related to the observable variables but
+independent from the protected attributes. Our work is based on the Kusner et al. [2017] idea to
+construct a fair prediction model over the German Credit Risk dataset Hoffman [2016].
+3
+Dataset
+We consider the Kaggle German Credit Risk dataset Hoffman [2016] to analyze and compare different
+types of unfair models and our method for constructing a fair model using Bayesian approaches. In
+this dataset, each entry represents a person who takes credit from a bank. The objective is to predict
+the credit amount of a person based on his/her attributes. "Sex" and "age" are the sensitive/protected
+attributes related to the bias during training and prediction in the unfairness problem. Feature "job" is
+a binary variable representing whether a person has a job or not. Feature "house" is a binary variable
+that indicates whether or not a person owns a house. The "credit amount" is our prediction target.
+The dataset is composed of 1000 records. We randomly pick 800 records for training and 200 records
+for testing. Figure 1 shows the detailed distributions of all these features in the whole dataset. In
+Figure 2, we illustrate the covariance between all the input features and the prediction target. We can
+observe a high correlation from the sensitive / protected attributes, i.e. "sex" and "age", to the "job"
+and "house". Thus, it is necessary to consider the issue of fairness when constructing a prediction
+model over "job" and "house".
+Figure 1: Distribution of features in the German Credit Risk dataset Hoffman [2016]. "Age" and
+"sex" are the sensitive / protected attributes. "Job" and "house" are the observable variables. "Credit
+amount" is the prediction target.
+2
+
+700
+140
+160
+600
+120
+500
+100
+140
+60
+300
+120
+40 
+200
+100
+100
+20
+Count
+70
+0.2
+0.4
+0.6
+0.8
+1.0
+age
+sex
+80
+800
+700
+700
+600
+60
+600
+500
+40
+8 400
+300
+300
+20
+200
+200
+100
+100
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.4
+0
+5000
+10000
+15000
+job
+0.2
+0.6
+house
+credit amtFigure 2: Illustration of the covariance matrix between all the input features and the prediction target.
+Here, we observe a high correlation from "age" and "sex" (the sensitive/protected attributes) to "job"
+and "house" (the observable variables).
+4
+Methods
+Full Model: The full model Montgomery et al. [2021] completely ignores fairness issues and
+includes sensitive variables like sex and age in the learning process. It is easy to understand that the
+full model is unfair because the predictions depend on sex and age. Figure 3 presents the directed
+acyclic graph (DAG) of the full model. In the full model, all the features are assumed to be connected.
+Unaware Model: The unaware model Dwork et al. [2012] does not use sensitive variables
+in the learning and prediction process, but it is still unfair. Even though the sensitive variables do not
+influence the target directly in the learning and prediction processes, it still has an indirect impact on
+the target through the non-sensitive variables. In our example, to predict a person’s credit amount,
+sex may influence whether a person can get a job. The job attribute still preserves the information of
+sex. Simply ignoring the sex attribute will not fully eliminate its impact on the predictions. Figure 3
+presents the DAG of an unaware model. The attributes under the grey circles are unobserved. In the
+unaware model, sex and age are not directly connected with the credit amount, but they are connected
+with job and house. It is still unfair because the change of sex and age will change the status of job
+and house, and thus influence the credit amount predictions.
+Figure 3: Two types of unfair models. Left: full model, which builds regression over all possible
+attributes without the consideration of fairness. Right: unaware model, which excludes sensi-
+tive/protected attributes, i.e. sex and age in our case.
+Fair Model: In order to build a fair model, we need to find a proxy variable that is independent
+of sensitive variables but still preserves the information in the credit amount prediction Kusner
+et al. [2017]. We can introduce the concept of latent confounding variable to resolve this issue.
+3
+
+1.0
+age
+0.8
+ sex
+0.6
+job
+0.4
+credit_amt house
+0.2
+0.0
+age
+sex
+job
+house
+credit_amtSex
+Job
+Sex
+Job
+Credit
+Credit
+House
+Age
+House
+Age
+Full Model
+Unaware ModelThe confounding variable is a variable that influences both the independent variable and dependent
+variables. In our fair model, we assume that there is an unobserved confounder C that reflects how
+reliable a person is in paying back the loan. The confounder should be independent of the sensitive
+variables to make the model fair. Figure 4 shows the DAG of the fair model structure. In the inference
+stage, we assume that job, house, and credit amount are confounded by the unobserved reliability
+level C and C is independent of sex and age. The reason is that sex and age can neither determine
+nor be related to how reliable a person is in paying back loans. Meanwhile, reliability is co-related to
+a person’s job performance, housing situation, and also credit amount. Then, in the prediction stage,
+we only use the inferred C as our feature to predict the credit amount. In this way, the predicting
+process does not contain any information about sex or age, and thus this procedure is an effective,
+fair learning algorithm in our scenario.
+Figure 4: DAG of the fair model. Here, we introduce the latent confounding variable "unobserved
+reliability level", which is independent to "sex" and "age" (the sensitive/protected attributes) but
+related to "job", "house", and "credit amount". Left: during the inference stage, we estimate this
+latent "reliability" feature with Bayesian approaches. Right: during the prediction stage, we only use
+this inferred "reliability" feature to predict the "credit amount".
+5
+Experiments
+We can represent the DAG of the fair model in a probabilistic way. We sample the two binary
+variables, job and house from two Bernoulli distributions and sample the confounder from the normal
+distribution. In the meantime, we choose the Poisson distribution as a prior for the credit amount.
+Our choices of priors correspond to the nature of the data. The job and house features are binary. And
+the credit amount is a positive attribute with a shape alike the Poisson distribution. The probabilistic
+model can be written as:
+Job ∼ Bernoulli(logit(bj + Sex × βj,s + Age × βj,a + C × βj,c))
+(1)
+House ∼ Bernoulli(logit(bh + Sex × βh,s + Age × βh,a + C × βh,c))
+(2)
+Credit ∼ Poisson(Exp(Sex × βc,s + Age × βc,a + C × βc,c))
+(3)
+C ∼ Normal(0, 1)
+where
+C ⊥ Sex,
+C ⊥ Age
+(4)
+The parameters we need to find are in the set Θ = {βm,n, bm} where m = j, h and n = s, a, c. We
+assume that these parameters are sampled from the normal distributions:
+βm,n ∼ N(0, 1)
+(5)
+bm ∼ N(0, 1)
+(6)
+We implement the Metropolis–Hastings algorithm to infer the probabilistic model. M-H algorithm
+Hastings [1970] is a Markov Chain Monte Carlo (MCMC) method for obtaining a sequence of
+4
+
+Job
+Sex
+Unobserved
+Unobserved
+House
+Credit
+Reliability Level
+Reliability Level
+Age
+Credit
+Fair model - Inference
+Fair model - PredictionAlgorithm 1 Infer C by Metropolis–Hastings
+for i = 1 to N do
+Choose J(C∗
+i |C(s)
+i
+) = uniform(C(s)
+i
+− δ, C(s)
+i
++ δ);
+Set an initial state C0
+i ;
+for s = 1 to 5000 do
+Sample C∗
+i ∼ J(C∗
+i |C(s)
+i
+);
+Compute the acceptance ratio r =
+p(C∗
+i |y)
+p(C(s)
+i
+|y) =
+p(y|C∗
+i )p(C∗
+i )
+p(y|C(s)
+i
+)p(C(s)
+i
+);
+sample u ∼ uniform(0, 1);
+if u < r then;
+C(s+1)
+i
+= C∗
+i ;
+else
+C(s+1)
+i
+= C(s)
+i
+;
+end if
+end for
+end for
+random samples from a probability distribution from which direct sampling is difficult. Algorithm 1
+explains how to infer the reliability level C.
+Once we obtain the posteriors of the inferred reliability level C, we can fit a new model using kernel
+g(.) based on the C in the prediction stage. In our experiment, since there is a nonlinear relationship
+between credit amount and "Reliability Level" in our inference stage setup (Poisson), we decide to
+use random-forest as the kernel function g(.) in our second stage prediction.
+Credit ∼ g(C)
+(7)
+6
+Results
+In this section, we provide experimental results and a discussion of the MCMC process performance.
+Specifically, in Sec. 6.1, we firstly present the MCMC estimation result and the convergence analysis
+on the fair model’s latent confounding variable C and parameters. Then, we compare the prediction
+and fairness performance across the three types of models in Sec. 6.2.
+6.1
+Fair model’s MCMC performance:
+Figure 5: Auto-correlation plots of parameters in Eq. 1 and Eq. 2 throughout the MCMC process.
+"alpha" refers to the constant offset term b in the equations.
+In Figure 5, we illustrate the auto-correlation plot of the model’s parameters in Eq. 1 and Eq. 2. We
+observe a clear decrease in auto-correlation throughout the MCMC process. Thus, this is an efficient
+MCMC process that leads to convergence. Further, in Figure 6, we provide the posterior estimation
+and the trace plot of the fair model parameters throughout the MCMC process. Though we still
+5
+
+qoreaq
+alpha job
+1.00
+1.00
+0.75
+0.75
+0.50
+0.50
+0.25
+0.25
+0.00
+0.005
+0.25
+0.25
+0.50
+0.50
+0.75
+0.75
+-1.000
+-1.000
+20
+40
+60
+80
+100 0
+20
+100 0
+60
+80
+100
+20
+60
+80
+100
+alpha_house
+ouse
+beta
+ouse
+0,2
+1.00
+1.00
+0.75 
+0.75 
+0.50
+0.50
+0.25
+0.25
+0.00
+0.00 ,
+-0.25
+-0.25
+-0.50
+-0.50
+-0.75
+0.75
+-1.000
+100 0
+100
+20
+40
+60
+80
+100Figure 6: Posterior estimation (left column) and trace plot (right column) of parameters in Eq. 1 and
+Eq. 2 throughout the MCMC process. "alpha" refers to the constant offset term b in the equations.
+observe some fluctuations till the end of the process, however, this is reasonable and acceptable. The
+reason is that we are applying over a real-world dataset, rather than a simulated dataset. Therefore,
+it is impossible to make our assumed distributions perfectly capture the behavior of the real-world
+dataset. Then, in Table 1, we provide the confidence interval over the posterior estimation of the fair
+model’s parameters.
+6.2
+Performance comparison across models:
+In this section, we compare how three distinct models perform while making predictions. In Table
+2, we present the R2 of three models in both training and testing environments. The full model
+outperforms the unaware model in both fitting and predicting by including sensitive information. It is
+surprising to see that the fair model outperforms the other two unfair models with R2 = 0.801 in the
+training set and R2 = 0.768 in the testing set. It turns out that our fair model does not only resolve
+the fairness issue but distills the information on the reliability level. The fair model is robust enough
+to be used to make fair and accurate predictions.
+7
+Conclusion
+In this paper, we have presented a fair model focusing on predicting the German credit score with
+considering the job and housing features. Specifically, we introduce the latent confounding variable
+"reliability level", which is independent of the protected attributes, i.e., "sex" and "age", but related
+to other observable variables and the prediction goal. For implementation, we apply the MCMC
+6
+
+betajob
+beta job
+4
+0
+2
+1000
+2000
+3000
+4000
+alpha house
+alpha_house
+4
+3
+2
+1
+0
+1
+2
+m
+4
+0
+1000
+2000
+3000
+4000
+beta house
+beta house
+-2
+0
+0
+1000
+2000
+3000
+4000
+beta_credit
+beta credit
+0.0
+2.5
+5.0
+-7.5
+-8
+-6
+-4
+0
+0
+1000
+2000
+3000
+4000
+C
+c
+4
+-2
+0
+2
+0
+1000
+2000
+3000
+4000std
+5%
+median
+95%
+ess_bulk
+ess_tail
+bj
+1.02
+-1.66
+0.03
+1.71
+4643.63
+3709.55
+βj,s
+0.98
+-1.32
+0.27
+1.88
+7128.36
+3907.04
+βj,a
+0.65
+-2.64
+-1.57
+-0.50
+1502.60
+2245.58
+βj,c
+0.47
+2.82
+3.46
+4.36
+2058.64
+2494.02
+bh
+1.01
+-1.61
+0.03
+1.67
+5113.35
+3167.94
+βh,s
+0.99
+-1.73
+-0.11
+1.55
+5506.02
+3900.82
+βh,a
+0.67
+-0.04
+1.05
+2.17
+1625.86
+2583.99
+βh,c
+0.46
+3.00
+3.65
+4.50
+1896.31
+2939.44
+βc,s
+0.54
+-7.78
+-6.85
+-5.98
+3255.68
+3206.54
+βc,a
+0.52
+-3.17
+-2.26
+-1.46
+4326.49
+3800.31
+βc,c
+0.23
+-0.37
+0.01
+0.38
+4455.72
+3211.78
+Table 1: The confidence intervals of the parameters estimated in Eq. 1 and Eq. 2 through the MCMC
+process.
+R2
+Full Model
+Unaware Model
+Fair Model Random Forest Kernel
+Training
+0.597
+0.466
+0.801
+Testing
+0.521
+0.424
+0.768
+Table 2: The R2 of three types of models defined in Sec. 4.
+approach to solve for the latent confounding variable and the parameters of the model. Compared
+with tradition models, our model effectively eliminates the bias related to sex and age and thus
+achieves a fair prediction of the credit amount. For the future work, we recommend trying different
+types of assumptions on the distribution for the variables over the German Credit Risk dataset and
+checking the effects on the choice of distributions over the convergence of the MCMC process and
+the final prediction.
+References
+Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai. Man is
+to computer programmer as woman is to homemaker? debiasing word embeddings. Advances in
+neural information processing systems, 29, 2016.
+Toon Calders and Sicco Verwer. Three naive bayes approaches for discrimination-free classification.
+Data mining and knowledge discovery, 21(2):277–292, 2010.
+Piotr Dollar, Christian Wojek, Bernt Schiele, and Pietro Perona. Pedestrian detection: An evaluation
+of the state of the art. IEEE transactions on pattern analysis and machine intelligence, 34(4):
+743–761, 2011.
+Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairness through
+awareness. In Proceedings of the 3rd innovations in theoretical computer science conference,
+pages 214–226, 2012.
+Alan E Gelfand. Gibbs sampling. Journal of the American statistical Association, 95(452):1300–1304,
+2000.
+Yoav Goldberg and Omer Levy. word2vec explained: deriving mikolov et al.’s negative-sampling
+word-embedding method. arXiv preprint arXiv:1402.3722, 2014.
+The Guardian.
+A beauty contest was judged by ai and the robots didn’t like dark
+skin,
+Sep 2016.
+URL https://www.theguardian.com/technology/2016/sep/08/
+artificial-intelligence-beauty-contest-doesnt-like-black-people.
+Moritz Hardt, Eric Price, and Nati Srebro. Equality of opportunity in supervised learning. Advances
+in neural information processing systems, 29, 2016.
+W Keith Hastings. Monte carlo sampling methods using markov chains and their applications. 1970.
+7
+
+Donald Hoffman. German credit risk, Dec 2016. URL https://www.kaggle.com/datasets/
+uciml/german-credit.
+Matthew Joseph, Michael Kearns, Jamie Morgenstern, Seth Neel, and Aaron Roth. Rawlsian fairness
+for machine learning. arXiv preprint arXiv:1610.09559, 1(2):19, 2016.
+Matt J Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. Advances in
+neural information processing systems, 30, 2017.
+Alexander Selvikvåg Lundervold and Arvid Lundervold. An overview of deep learning in medical
+imaging focusing on mri. Zeitschrift für Medizinische Physik, 29(2):102–127, 2019.
+Ninareh Mehrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan. A survey
+on bias and fairness in machine learning. ACM Computing Surveys (CSUR), 54(6):1–35, 2021.
+Tom M Mitchell and Tom M Mitchell. Machine learning, volume 1. McGraw-hill New York, 1997.
+Douglas C Montgomery, Elizabeth A Peck, and G Geoffrey Vining. Introduction to linear regression
+analysis. John Wiley & Sons, 2021.
+Christopher Z Mooney. Monte carlo simulation. Number 116. Sage, 1997.
+Loris Nanni and Alessandra Lumini. An experimental comparison of ensemble of classifiers for
+bankruptcy prediction and credit scoring. Expert systems with applications, 36(2):3028–3033,
+2009.
+Callie Rennison and Mike Planty. Nonlethal intimate partner violence: Examining race, gender, and
+income patterns. Violence and victims, 18(4):433–443, 2003.
+Gwen Sharp.
+Nikon camera says asians:
+People are always blinking - sociolog-
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+URL https://thesocietypages.org/socimages/2009/05/29/
+nikon-camera-says-asians-are-always-blinking/.
+8
+

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+page_content='Fair Credit Scorer through Bayesian Approach  Zhuo Zhao  Department of Applied Mathematics  Johns Hopkins University  zzhao62@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='edu  Abstract  Machine learning currently plays an increasingly important role in people’s lives  in areas such as credit scoring, auto-driving, disease diagnosing, and insurance  quoting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' However, in many of these areas, machine learning models have performed  unfair behaviors against some sub-populations, such as some particular groups of  race, sex, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' These unfair behaviors can be on account of the pre-existing  bias in the training dataset due to historical and social factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In this paper, we  focus on a real-world application of credit scoring and construct a fair prediction  model by introducing latent variables to remove the correlation between protected  attributes, such as sex and age, with the observable feature inputs, including house  and job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' For detailed implementation, we apply Bayesian approaches, including  the Markov Chain Monte Carlo simulation, to estimate our proposed fair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  1  Introduction  Nowadays, Machine Learning methods are used to automate decisions in a variety of areas, including  determining credit scores Nanni and Lumini [2009], classifying tumor components from MRI images  Lundervold and Lundervold [2019], detecting pedestrians on the road Dollar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2011], and  understanding natural languages Goldberg and Levy [2014], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' However, machine learning methods  are heavily dependent on data Mitchell and Mitchell [1997] and this data-dependent nature makes the  learned models sensitive to the latent bias existing in the training datasets Mehrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Thus,  the final decisions made by the learned models are unfairly biased against certain sub-populations,  differentiated by some sensitive/protected attributes, such as race, sex, or age, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' For example,  cameras sometimes fail to recognize whether Asian blink their eyes Sharp [2009] and the beauty  pageant judged by AI would prefer light skin Guardian [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' However, we would expect AI to give  the same decision independent from the protected attributes and thus we concern about the fairness  of machine learning methods Mehrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  In this paper, we focus on constructing fair machine learning models to predict the credit score, with  using the German Credit Risk dataset Hoffman [2016] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The goal is to predict the credit  score based on some observable variables, including housing and job information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' However, this  personal financial information, such as income, housing, and saving, are usually highly correlated  to gender and age due to historical and social reasons Rennison and Planty [2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Therefore, it  is necessary to learn an effective model to filter the prediction bias against sex and age, caused by  the latent correlation between these observable variables and the protected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In detail, we  analyze and compare from the fairness perspective across the full model Montgomery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2021],  unaware model Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2012], and fair model based on causals and counterfactuals Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  [2017] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Then, we apply the Markov Chain Monte Carlo (MCMC) simulation Mooney [1997]  and the Gibbs’ sampling Gelfand [2000] to solve the corresponding parameters in these models and  evaluate the performances (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 5 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='08412v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='LG]  20 Jan 2023  2  Related work  Fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Many recent works (Calders and Verwer [2010], Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2016], Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  [2012], Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2016], Joseph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2016], Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2017]) have been focusing on fairness  in machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2016] pointed out that there is a risk of amplifying  the bias introduced from the dataset, if using machine learning algorithms without taking effects to  handle the pre-existing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' For example, in the word embedding, learned over Google News with  pre-existing gender stereotypes, the gender-neutral words widely spread along a latent embedding  direction capturing gender difference, such as "receptionist" falling far along the direction related  to "female" Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Calders and Verwer [2010] modifies the Naive Bayes classifier  by adding independence restriction toward sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2012] proposes a  task-specific metric to evaluate the similarity between individuals relative to the classification task  and optimizes over the proposed metric with the goal that similar individuals are treated similarly  in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2017] focuses on causal inference and counterfactual, with  introducing the latent confounding variables, which are related to the observable variables but  independent from the protected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Our work is based on the Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2017] idea to  construct a fair prediction model over the German Credit Risk dataset Hoffman [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  3  Dataset  We consider the Kaggle German Credit Risk dataset Hoffman [2016] to analyze and compare different  types of unfair models and our method for constructing a fair model using Bayesian approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In  this dataset, each entry represents a person who takes credit from a bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The objective is to predict  the credit amount of a person based on his/her attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' "Sex" and "age" are the sensitive/protected  attributes related to the bias during training and prediction in the unfairness problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Feature "job" is  a binary variable representing whether a person has a job or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Feature "house" is a binary variable  that indicates whether or not a person owns a house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The "credit amount" is our prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  The dataset is composed of 1000 records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' We randomly pick 800 records for training and 200 records  for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Figure 1 shows the detailed distributions of all these features in the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In  Figure 2, we illustrate the covariance between all the input features and the prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' We can  observe a high correlation from the sensitive / protected attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' "sex" and "age", to the "job"  and "house".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Thus, it is necessary to consider the issue of fairness when constructing a prediction  model over "job" and "house".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Figure 1: Distribution of features in the German Credit Risk dataset Hoffman [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' "Age" and  "sex" are the sensitive / protected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' "Job" and "house" are the observable variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' "Credit  amount" is the prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  2  700  140  160  600  120  500  100  140  60  300  120  40  200  100  100  20  Count  70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='0  age  sex  80  800  700  700  600  60  600  500  40  8 400  300  300  20  200  200  100  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='4  0  5000  10000  15000  job  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='6  house  credit amtFigure 2: Illustration of the covariance matrix between all the input features and the prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Here, we observe a high correlation from "age" and "sex" (the sensitive/protected attributes) to "job"  and "house" (the observable variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  4  Methods  Full Model: The full model Montgomery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2021] completely ignores fairness issues and  includes sensitive variables like sex and age in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' It is easy to understand that the  full model is unfair because the predictions depend on sex and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Figure 3 presents the directed  acyclic graph (DAG) of the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In the full model, all the features are assumed to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Unaware Model: The unaware model Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2012] does not use sensitive variables  in the learning and prediction process, but it is still unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Even though the sensitive variables do not  influence the target directly in the learning and prediction processes, it still has an indirect impact on  the target through the non-sensitive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In our example, to predict a person’s credit amount,  sex may influence whether a person can get a job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The job attribute still preserves the information of  sex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Simply ignoring the sex attribute will not fully eliminate its impact on the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Figure 3  presents the DAG of an unaware model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The attributes under the grey circles are unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In the  unaware model, sex and age are not directly connected with the credit amount, but they are connected  with job and house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' It is still unfair because the change of sex and age will change the status of job  and house, and thus influence the credit amount predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Figure 3: Two types of unfair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Left: full model, which builds regression over all possible  attributes without the consideration of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Right: unaware model, which excludes sensi-  tive/protected attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' sex and age in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Fair Model: In order to build a fair model, we need to find a proxy variable that is independent  of sensitive variables but still preserves the information in the credit amount prediction Kusner  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' We can introduce the concept of latent confounding variable to resolve this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='0  age  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='8  sex  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='6  job  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='4  credit_amt house  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='0  age  sex  job  house  credit_amtSex  Job  Sex  Job  Credit  Credit  House  Age  House  Age  Full Model  Unaware ModelThe confounding variable is a variable that influences both the independent variable and dependent  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In our fair model, we assume that there is an unobserved confounder C that reflects how  reliable a person is in paying back the loan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The confounder should be independent of the sensitive  variables to make the model fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Figure 4 shows the DAG of the fair model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In the inference  stage, we assume that job, house, and credit amount are confounded by the unobserved reliability  level C and C is independent of sex and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The reason is that sex and age can neither determine  nor be related to how reliable a person is in paying back loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Meanwhile, reliability is co-related to  a person’s job performance, housing situation, and also credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Then, in the prediction stage,  we only use the inferred C as our feature to predict the credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In this way, the predicting  process does not contain any information about sex or age, and thus this procedure is an effective,  fair learning algorithm in our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Figure 4: DAG of the fair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Here, we introduce the latent confounding variable "unobserved  reliability level", which is independent to "sex" and "age" (the sensitive/protected attributes) but  related to "job", "house", and "credit amount".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Left: during the inference stage, we estimate this  latent "reliability" feature with Bayesian approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Right: during the prediction stage, we only use  this inferred "reliability" feature to predict the "credit amount".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  5  Experiments  We can represent the DAG of the fair model in a probabilistic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' We sample the two binary  variables, job and house from two Bernoulli distributions and sample the confounder from the normal  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In the meantime, we choose the Poisson distribution as a prior for the credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Our choices of priors correspond to the nature of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The job and house features are binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' And  the credit amount is a positive attribute with a shape alike the Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The probabilistic  model can be written as:  Job ∼ Bernoulli(logit(bj + Sex × βj,s + Age × βj,a + C × βj,c))  (1)  House ∼ Bernoulli(logit(bh + Sex × βh,s + Age × βh,a + C × βh,c))  (2)  Credit ∼ Poisson(Exp(Sex × βc,s + Age × βc,a + C × βc,c))  (3)  C ∼ Normal(0, 1)  where  C ⊥ Sex,  C ⊥ Age  (4)  The parameters we need to find are in the set Θ = {βm,n, bm} where m = j, h and n = s, a, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' We  assume that these parameters are sampled from the normal distributions:  βm,n ∼ N(0, 1)  (5)  bm ∼ N(0, 1)  (6)  We implement the Metropolis–Hastings algorithm to infer the probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' M-H algorithm  Hastings [1970] is a Markov Chain Monte Carlo (MCMC) method for obtaining a sequence of  4  Job  Sex  Unobserved  Unobserved  House  Credit  Reliability Level  Reliability Level  Age  Credit  Fair model - Inference  Fair model - PredictionAlgorithm 1 Infer C by Metropolis–Hastings  for i = 1 to N do  Choose J(C∗  i |C(s)  i  ) = uniform(C(s)  i  − δ, C(s)  i  + δ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Set an initial state C0  i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  for s = 1 to 5000 do  Sample C∗  i ∼ J(C∗  i |C(s)  i  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Compute the acceptance ratio r =  p(C∗  i |y)  p(C(s)  i  |y) =  p(y|C∗  i )p(C∗  i )  p(y|C(s)  i  )p(C(s)  i  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  sample u ∼ uniform(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  if u < r then;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  C(s+1)  i  = C∗  i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  else  C(s+1)  i  = C(s)  i  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  end if  end for  end for  random samples from a probability distribution from which direct sampling is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Algorithm 1  explains how to infer the reliability level C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Once we obtain the posteriors of the inferred reliability level C, we can fit a new model using kernel  g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=') based on the C in the prediction stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In our experiment, since there is a nonlinear relationship  between credit amount and "Reliability Level" in our inference stage setup (Poisson), we decide to  use random-forest as the kernel function g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=') in our second stage prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Credit ∼ g(C)  (7)  6  Results  In this section, we provide experimental results and a discussion of the MCMC process performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Specifically, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='1, we firstly present the MCMC estimation result and the convergence analysis  on the fair model’s latent confounding variable C and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Then, we compare the prediction  and fairness performance across the three types of models in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='1  Fair model’s MCMC performance:  Figure 5: Auto-correlation plots of parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 2 throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  "alpha" refers to the constant offset term b in the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  In Figure 5, we illustrate the auto-correlation plot of the model’s parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' We  observe a clear decrease in auto-correlation throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Thus, this is an efficient  MCMC process that leads to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Further, in Figure 6, we provide the posterior estimation  and the trace plot of the fair model parameters throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Though we still  5  qoreaq  alpha job  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
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+page_content='000  20  40  60  80  100 0  20  100 0  60  80  100  20  60  80  100  alpha_house  ouse  beta  ouse  0,2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
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+page_content='000  100 0  100  20  40  60  80  100Figure 6: Posterior estimation (left column) and trace plot (right column) of parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 1 and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 2 throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' "alpha" refers to the constant offset term b in the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  observe some fluctuations till the end of the process, however, this is reasonable and acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The  reason is that we are applying over a real-world dataset, rather than a simulated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Therefore,  it is impossible to make our assumed distributions perfectly capture the behavior of the real-world  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Then, in Table 1, we provide the confidence interval over the posterior estimation of the fair  model’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='2  Performance comparison across models:  In this section, we compare how three distinct models perform while making predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In Table  2, we present the R2 of three models in both training and testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The full model  outperforms the unaware model in both fitting and predicting by including sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' It is  surprising to see that the fair model outperforms the other two unfair models with R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='801 in the  training set and R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='768 in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' It turns out that our fair model does not only resolve  the fairness issue but distills the information on the reliability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' The fair model is robust enough  to be used to make fair and accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  7  Conclusion  In this paper, we have presented a fair model focusing on predicting the German credit score with  considering the job and housing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Specifically, we introduce the latent confounding variable  "reliability level", which is independent of the protected attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=', "sex" and "age", but related  to other observable variables and the prediction goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' For implementation, we apply the MCMC  6  betajob  beta job  4  0  2  1000  2000  3000  4000  alpha house  alpha_house  4  3  2  1  0  1  2  m  4  0  1000  2000  3000  4000  beta house  beta house  2  0  0  1000  2000  3000  4000  beta_credit  beta credit  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
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+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  approach to solve for the latent confounding variable and the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Compared  with tradition models, our model effectively eliminates the bias related to sex and age and thus  achieves a fair prediction of the credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' For the future work, we recommend trying different  types of assumptions on the distribution for the variables over the German Credit Risk dataset and  checking the effects on the choice of distributions over the convergence of the MCMC process and  the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  References  Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Man is  to computer programmer as woman is to homemaker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' debiasing word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Advances in  neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Toon Calders and Sicco Verwer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Three naive bayes approaches for discrimination-free classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Data mining and knowledge discovery, 21(2):277–292, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Piotr Dollar, Christian Wojek, Bernt Schiele, and Pietro Perona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Pedestrian detection: An evaluation  of the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' IEEE transactions on pattern analysis and machine intelligence, 34(4):  743–761, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Fairness through  awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' In Proceedings of the 3rd innovations in theoretical computer science conference,  pages 214–226, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Alan E Gelfand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Gibbs sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Journal of the American statistical Association, 95(452):1300–1304,  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Yoav Goldberg and Omer Levy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' word2vec explained: deriving mikolov et al.’s negative-sampling  word-embedding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' arXiv preprint arXiv:1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='3722, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  The Guardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  A beauty contest was judged by ai and the robots didn’t like dark  skin,  Sep 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='theguardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='com/technology/2016/sep/08/  artificial-intelligence-beauty-contest-doesnt-like-black-people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Moritz Hardt, Eric Price, and Nati Srebro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Equality of opportunity in supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Advances  in neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  W Keith Hastings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Monte carlo sampling methods using markov chains and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  7  Donald Hoffman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' German credit risk, Dec 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='com/datasets/  uciml/german-credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Matthew Joseph, Michael Kearns, Jamie Morgenstern, Seth Neel, and Aaron Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Rawlsian fairness  for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' arXiv preprint arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='09559, 1(2):19, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Matt J Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Counterfactual fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Advances in  neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Alexander Selvikvåg Lundervold and Arvid Lundervold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' An overview of deep learning in medical  imaging focusing on mri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Zeitschrift für Medizinische Physik, 29(2):102–127, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Ninareh Mehrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' A survey  on bias and fairness in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' ACM Computing Surveys (CSUR), 54(6):1–35, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Tom M Mitchell and Tom M Mitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Machine learning, volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' McGraw-hill New York, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Douglas C Montgomery, Elizabeth A Peck, and G Geoffrey Vining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Introduction to linear regression  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' John Wiley & Sons, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Christopher Z Mooney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Monte carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Number 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
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+page_content='  Loris Nanni and Alessandra Lumini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' An experimental comparison of ensemble of classifiers for  bankruptcy prediction and credit scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Expert systems with applications, 36(2):3028–3033,  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Callie Rennison and Mike Planty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Nonlethal intimate partner violence: Examining race, gender, and  income patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content=' Violence and victims, 18(4):433–443, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Gwen Sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  Nikon camera says asians:  People are always blinking - sociolog-  ical images,  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  URL https://thesocietypages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='org/socimages/2009/05/29/  nikon-camera-says-asians-are-always-blinking/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'}

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+CONVERSATIONAL AUTOMATED PROGRAM REPAIR
+Chunqiu Steven Xia, Lingming Zhang
+University of Illinois at Urbana-Champaign
+{chunqiu2, lingming}@illinois.edu
+ABSTRACT
+Automated Program Repair (APR) can help developers automatically generate
+patches for bugs. Due to the impressive performance obtained using Large Pre-
+Trained Language Models (LLMs) on many code related tasks, researchers have
+started to directly use LLMs for APR. However, prior approaches simply repeat-
+edly sample the LLM given the same constructed input/prompt created from the
+original buggy code, which not only leads to generating the same incorrect patches
+repeatedly but also miss the critical information in testcases. To address these lim-
+itations, we propose conversational APR, a new paradigm for program repair that
+alternates between patch generation and validation in a conversational manner.
+In conversational APR, we iteratively build the input to the model by combining
+previously generated patches with validation feedback. As such, we leverage the
+long-term context window of LLMs to not only avoid generating previously incor-
+rect patches but also incorporate validation feedback to help the model understand
+the semantic meaning of the program under test. We evaluate 10 different LLM
+including the newly developed ChatGPT model to demonstrate the improvement
+of conversational APR over the prior LLM for APR approach.
+1
+INTRODUCTION
+Bugs in software can cause significant financial losses Matteson (2018) and create dangerous health
+and safety problems Hanbury (2019). Due to the high manual cost of fixing bugs O’Dell (2017),
+Automated Program Repair (APR) Gazzola et al. (2019) is a promising solution to reduce developer
+work by automatically generating patches given the buggy code and failing testcases.
+Traditionally, APR approaches commonly use the paradigm of Generate and Validate (G&V), where
+APR tools will first generate a list of candidate patches given the original buggy code and then
+validate each one sequentially until a plausible patch that passes all the testcases is found. Plausible
+patch is then passed on to a human developer where they have to determine if this is a correct
+patch that correctly fixes the underlying bug. Traditional APR approaches such as template-based
+tools Ghanbari et al. (2019); Liu et al. (2019); Lou et al. (2020) have been proven useful in fixing
+bugs with pre-defined templates to match buggy and corresponding fix code patterns. Recently,
+researchers have designed learning-based APR tools Ye et al. (2022); Zhu et al. (2021); Jiang et al.
+(2021) which build a Neural Machine Translation (NMT) model by training on pairs of buggy and
+patch code. However, these learning-based APR tools suffer from lack of patch variety as it can
+only repair the types of bugs that are a part of the buggy/patch training data. Furthermore, these bug
+fixing datasets can be difficult to construct as it require scraping open-source bug fix commits which
+may contain many false positives, adding noise to the dataset.
+Recognizing the limitation of prior learning-based APR tools, researchers have started to look
+into directly leveraging Large Pre-Trained Language Models (LLMs) for APR without fine-tuning.
+LLMs have proven their ability in various code generation tasks Austin et al. (2021). Xia & Zhang
+(2022) first introduced cloze-style APR where a LLM directly fill-in the correct code given its sur-
+rounding context. Other studies Prenner et al. (2022); Kolak et al. (2022); Xia et al. (2022) have also
+investigated directly applying different types of LLMs for APR by smartly applying prompts or giv-
+ing original buggy code as context. Typically, directly applying LLMs for APR involves creating a
+common prompt/prefix which can be just the buggy context (zero-shot) or combining buggy context
+with a few examples of bug fixes (few-shot) as input to the model. Following the G&V paradigm,
+1
+arXiv:2301.13246v1  [cs.SE]  30 Jan 2023
+
+prior approach will sample the LLMs multiple times to obtain candidate patches. However, this
+pipeline has the following limitations:
+First, sampling from the same prefix/prompt multiple times can lead to many repeated patches due
+to the probabilistic nature of sampling. This means the LLMs may waste a lot of compute and
+time generating the same patches which have already been validated as incorrect by the testsuite.
+Second, prompts provided to the LLMs for APR are created only from the original buggy code and
+does not include any of the testcase information. Such information like the expected input and output
+examples that can help LLMs understand the functionality of the buggy program are not provided.
+Third, prior approaches also fail to consider the outputs produced by the generated incorrect patches.
+Previously incorrect patches may fail on a particular corner case, which can be exposed by looking
+at the test output and providing it to the LLM to address it in future patches.
+Our Work. We propose conversational APR – a new paradigm of using LLMs for APR that di-
+rectly leverages the testcase validation information to provide feedback to LLMs in a conversational
+manner. In conversational APR, we interleave patch generation with validation where LLM first
+generates a patch, we then validate it against testsuite to provide feedback and prompt LLM with
+the new feedback information to generate a new patch. While in this paper we consider simple test-
+case input/output/error validation feedback, one can apply conversational APR with a wild range of
+possible feedback information such as human evaluation of the patch. We refer to the process of
+generating a patch followed by validation as a turn where a conversation chain is made up of mul-
+tiple turns in sequence. In the start of the conversation chain, we begin with an initial prompt and
+sample the LLM to obtain a candidate patch. As we continue the conversation, the input given to the
+LLM in each turn is a concatenation of all previously incorrect patches along with their associated
+testcase feedback within the same conversation chain. A conversational chain is terminated once a
+patch that passes all the testcases are found or the maximum chain length is reached (i.e., maximum
+number of turns). In the latter case, we start a new conversation chain with the initial prompt again.
+Compared with prior LLM for APR tools which only use the buggy code snippet as inputs, conver-
+sational APR incorporates patch validation in the form of validation feedback to help the model un-
+derstand the reason why previously generated patches are incorrect. Such feedback can contain the
+incorrect and expected test outputs or indicate if the generated patch contains compilation/runtime
+errors. Furthermore, while prior LLM for APR tools continuously sample from the same input, our
+approach iteratively builds the input by including previously incorrect patches. As such, the LLM,
+through its long context window, can recognize previous generations and avoid repeatedly generat-
+ing an already validated incorrect patch. We evaluated our conversational APR by using 10 popular
+LLMs, where we found that our approach not only improves the number of bugs fixed but also
+can arrive at the correct patch faster compared with sampling-based baseline. Furthermore, we also
+evaluate the recently developed ChatGPT Schulman et al. (2022)1, a dialogue focused LLM trained
+using reinforcement learning and highlight the performance of conversational APR when using a
+LLM designed for conversation/dialogue.
+2
+BACKGROUND & RELATED WORK
+2.1
+LLMS FOR APR
+To combat the reliance on training using bug-fixing datasets to build learning-based APR tools based
+on NMT models, researchers directly applied LLMs for APR without any fine-tuning. Xia & Zhang
+(2022) proposed AlphaRepair, the first cloze-style APR to directly leverage LLMs for APR in a
+zero-shot setting by removing the buggy line and replacing it with masked tokens. AlphaRepair
+then queries the CodeBERT Feng et al. (2020) model to fill-in the masked tokens with the correct
+tokens to generate patches. Prenner et al. (2022) investigated the ability for Codex Chen et al. (2021)
+to repair bugs using a simple prompting method to generate a complete patched function given the
+original buggy function. Kolak et al. (2022) evaluated the scaling effect of LLMs for APR by using
+4 LLMs of different model sizes to generate a single line fix given only the original buggy prefix
+(i.e., removing all lines after and including the buggy line of the buggy function). Recently, Xia et al.
+(2022) conducted an extensive study on directly applying LLMs for APR. In the study, they adopt
+1While we perform repair using ChatGPT, no part of this paper is written by ChatGPT. :)
+2
+
+several repair settings, including few-shot generation using a few examples of bug fixes, cloze-style
+APR and also single line generation.
+The findings across these prior work is consistent in showing that directly using LLMs for APR
+achieves comparable if not better performance compared to prior APR tools. However, these pro-
+posed LLMs for APR techniques almost exclusively use sampling where patches are generated by
+sampling from the same input over and over again, leading to many repeated patches. Furthermore,
+the inputs to the LLMs are only constructed from the original buggy function, missing the rich infor-
+mation in the form of testcases. In this work, our conversational APR approach aims to bridge these
+limitations in LLMs for APR by constructing new inputs based on prior incorrect patches to avoid
+sampling repeated patches and providing the validation feedback to add another dimension of input
+apart from original buggy code to help the model understand the semantic meaning of the program.
+2.2
+MULTI-STEP PROGRAM REASONING AND SYNTHESIS USING LLMS
+A related research direction is in applying multi-step reasoning for code understanding and synthe-
+sis. Nye et al. (2021) trains a LLM designed for program understanding by introducing the idea
+of a “scratchpad” in which the LLM predicts the intermediate states of a program along with the
+final execution results. Chen et al. (2022) extends the chain-of-thoughts Wei et al. (2022) prompting
+style in NLP to propose program-of-thoughts where the prompt contains an explicit command to
+construct the program step-by-step. However, these work still generates a complete result (i.e., final
+program execution or code), albeit with intermediate results, in one shot, whereas our conversational
+APR samples multiple times LLMs with different inputs to obtain one output plausible patch.
+Different from one-shot methods, Austin et al. (2021) investigated the ability for LLMs to use hu-
+man feedback in a conversational manner for program synthesis. The approach works by keeping a
+conversation of previously generated code and correcting any mistake using natural language feed-
+back provided by human developers. Nijkamp et al. (2022) manually created a multi-step synthesis
+dataset where each target program is broken down into multiple smaller steps where only a few lines
+of code needs to be generated. They then sample the model multiple times to iteratively complete
+each smaller step and concatenate them together to form the final program. While these described
+techniques involve iteratively sampling from the model with new feedback similar to a conversa-
+tional manner, our work can automatically create this feedback through testcase execution without
+any human-in-the-loop.
+3
+CONVERSATIONAL APR
+We propose a conversational APR approach to prompt LLM patch generation by combining previ-
+ously generated patches and validation feedback in a conversational manner. Contrasting with the
+classic Generate and Validate (G&V) APR approach that first generates a large number of candidate
+patches and then validate each one to find a list of plausible patches, conversational APR interleaves
+generation and validation to provide immediate feedback for the new candidate patch. Different
+from previous APR tools which make use of LLMs through sampling given the same prefix/context
+for each bug, conversational APR approach aims to incorporate feedback information after each
+generation (if the candidate patch failed to pass all tests) as new context for subsequent generations.
+Specifically, the feedback information includes both the incorrect generated patch and its associated
+failed testcase information.
+Conversational APR involves iteratively obtaining new candidate patches from the LLM by using
+previously generated patches/validation results as feedback. We refer to this process as a turn, where
+each turn includes three different steps: 1) construct new a prompt based on prior feedback 2) sam-
+ple the model to produce a sample output function 3) validate the sample output function against
+testcases to obtain validation feedback. Multiple turns in sequence is defined as a chain. The ter-
+minating conditions are that the sample output patch is able to pass all testcases (i.e., a plausible
+patch is obtained) or the maximum number of turns (length of the chain) is reached. Note that each
+turn (all three steps) are done automatically without needing any human-in-the-loop, this allows
+conversational APR to be an automatic approach for program repair.
+3
+
+Turn 1:
+Turn 2:
+The following code is buggy.
+def sieve(max):
+    primes = []
+    for n in range(2, max):
+        if any(n%p for p in primes):
+            primes.append(n)
+    return primes
+Please provide a fixed version.
+def sieve(max):
+    primes = []
+    for n in range(2, max+1):
+        if not any(n%p for p in primes):
+            primes.append(n)
+    return primes
+The fixed version is still not correct.
+def sieve(max):
+    primes = []
+    for n in range(2, max):
+        if all(n%p for p in primes):
+            primes.append(n)
+    return primes
+def sieve(max):
+    primes = []
+    for n in range(2, max+1):
+        if all(n%p for p in primes):
+            primes.append(n)
+    return primes
+sieve(4) returns [2, 4] but it should return [2, 3]
+Please provide a fixed version.
+The fixed version is still not correct.
+sieve(2) returns [] but it should return [2]
+Please provide a fixed version.
+S
+I
+F1
+I
+S1
+F1
+concatenate
+S2
+F2
+I
+S1
+F1
+S2
+F2
+concatenate
+S3
+Turn 3:
+F3
+Initial 
+Prompt
+sample
+output
+validation 
+feedback
+sample model
+sample model
+sample model
+run 
+testcase
+run 
+testcase
+run 
+testcase
+sample
+output
+validation 
+feedback
+sample
+output
+The fixed version is correct!
+validation 
+feedback
+def sieve(max):
+    primes = []
+    for n in range(2, max):
+        if any(n%p for p in primes):
+            primes.append(n)
+    return primes
+def sieve(max):
+    primes = []
+    for n in range(2, max):
+        if all(n % p for p in primes):
+            primes.append(n)
+    return primes
+original buggy function
+plausible patch
+S1
+I
+S2
+S3
+F3
+F2
+F1
+I
+S1
+I
+S1
+F1
+F1
+S2
+F2
+F3
+S3
+I
+S1
+F1
+S2
+F2
+Figure 1: Overview of conversational APR with an illustrative example in fixing the buggy
+sieve function
+3.1
+PIPELINE & EXAMPLE
+Figure 1 shows an illustrative example of a conversation chain (multiple turns) and an overview
+of the pipeline of the conversational APR approach. We first take in as input the original buggy
+function and a set of testcases which contains some failing tests that expose the underlying bug.
+In the example, the buggy function (sieve) attempts to use to sieve algorithm to calculate the list
+of prime numbers below the integer input (max). The location of the bug occurs on line 4 where
+the buggy function incorrectly uses any instead of all. This bug is exposed by the testcase of
+sieve(2) = [2] where the buggy function incorrectly returns an empty array [].
+• Turn 1: We first create an initial prompt
+I using the original buggy function which contains
+natural language description to indicate that the function is buggy (The following code is
+buggy) and the task we want the LLM to solve (Please provide a fixed version). We
+then sample the model using the initial prompt
+I to obtain the first sample output function S1 .
+The change is made to line 4 where the function in S1 negated the original if condition. We then
+validate S1 against the list of tests and found that while the new patch is able to successfully pass
+the previous failing test of sieve(2) = [2], it returns [2, 4] for sieve(4) when the correct
+output should be [2, 3]. This validation information F1 is collected as feedback to use during
+the next conversation turn.
+• Turn 2: Different from turn 1, where the input to the LLM is just the initial prompt
+I , now we
+provide the model also with the previously generated patch and its failing testcase. In short, we
+construct the validation feedback F1 by using the failing testcase and indicate to the model that the
+previous sample S1 is still not correct (The fixed version is still not correct) and
+the new task (Please provide another fixed version). We then concatenate the initial
+prompt, first sample output function and the validation feedback { I , S1 , F1 } together as the input
+to the LLM. As such, the model is able to not only use the original buggy function but also use the
+previously generated sample and its testcase feedback to generate a new patched function. Similar
+to turn 1, we obtain S2 and F2 where the correct line 4 is obtained (switching any to all) but the
+candidate patch function incorrectly reduced the upper range of the for loop by 1.
+4
+
+• Turn 3: Similar to turn 2, we first construct the new validation feedback F2 from the previous
+failing test case. We then concatenate all previously sampled output along with its validation
+feedback in sequence to produce { I , S1 , F1 , S2 , F2 }. Using this input, we then sample the LLM
+again to produce the next candidate patch S3 . We observe that this candidate patch correctly fixes
+the underlying bug and this is indicated by its validation F3 where it is able to pass all the testcases.
+The program repair process is then terminated as we have obtained our plausible patch S3 .
+Compared to prior approach in APR based on LLMs which simply samples from a pre-defined
+prompt/context, conversational APR leverages the previously missing key feedback information in
+the form of testcase results to prompt future patch generations. The testcase feedback not only tells
+the LLM that the previous patches are incorrect (i.e. leading to more unique patches) but also pro-
+vides input and output examples which helps the model to understand the underlying functionality
+of the function (i.e. leading to more correct patches).
+3.2
+DESIGN DECISIONS
+In the above example illustrated in Figure 1, we show the overall pipeline of conversational APR.
+However, there are different design decisions which can impact the performance of the approach:
+Prompt engineering. Prompting has been shown to be an effective way of leveraging LLMs on
+various downstream tasks without needing any explicit fine-tuning. In conversational APR approach,
+we follow the style of prior work Xia et al. (2022) in providing a short and concise prompt with
+respect to the description of the input and the task we want to model to solve. Additionally, we
+follow prior guidelines and kept the prompt to be open-ended rather than to restrict the generation
+with a close-ended prompt. One particular important prompt constructing is validation feedback
+in providing the failing testcase to the LLM. In the Figure 1 example, we provide a functional
+prompt that directly invokes the function and highlight the discrepancy between output and expected
+testcase output. We refer to this as functional prompt since it directly calls the function with input
+parameters similar to what one would do in code. In Section 6.2, we compare this style of validation
+prompting with other methods including without any testcase information to demonstrate the benefit
+of including validation feedback to the model.
+Maximum chain length. Recall that a conversation chain refers to the continuous sequence of turns
+to fix a bug. A chain is demonstrated in Figure 1 with a chain length of 3. Along with finding a
+plausible patch, a preset value for the maximum chain length is also a terminating condition since
+the LLM used will have a maximum context window and cannot take in arbitrary length inputs.
+Once this maximum chain length is reached, conversational APR will restart from the beginning
+(i.e., by crafting initial prompt again) with a new chain conversation. The maximum chain length
+is a parameter which controls how much history the LLM may receive. A maximum chain length
+of 1 refers to the base case of sampling from the initial prompt over and over again, meaning the
+model does not know any of the previously generated incorrect patches. A higher maximum chain
+length means the model can see multiple previously failed patches, however this also may not be
+beneficial as it can cause the LLM to repeat some of the earlier patches or get stuck on a particular
+implementation of the function. In Section 6.2, we evaluate the effect of the chain length has on
+repair performance.
+4
+DATASETS
+In this section, we describe the LLMs used in our evaluation and also the repair benchmark used to
+evaluate our proposed technique.
+4.1
+LLMS
+In our work, we evaluate 10 different LLMs to not only demonstrate the effect of scaling behavior
+on our proposed conversational APR approach but also to evaluate how different pre-training and
+model design contribute to the overall effectiveness. Table 1 presents an overview of the studied
+LLMs. Column Model is the model name, #Parameters indicates the number of model parameters,
+Context Window represents the size of the context window, and Training Strategy refers to the
+training strategy used.
+5
+
+Table 1: Evaluation LLM overview
+Model
+#Parameters
+Context Window
+Training Strategy
+CODEGEN-MONO
+350M/2B/6B/16B
+2048
+Unsupervised CLM
+CODEGEN-MULTI
+350M/2B/6B/16B
+2048
+Unsupervised CLM
+Codex
+12B
+4096
+Unsupervised CLM
+ChatGPT
+∼175B
+∼4000
+Reinforcement Learning
+from Human Feedback + CLM
+bitcount.py
+bitcount.java
+fixed line
+fixed line
+testcase
+Figure 2: Example bug in both Python and Java in QuixBugs along with the testcases
+• CODEGEN Nijkamp et al. (2022). A family of autoregressive LLMs trained using Causal Lan-
+guage Modeling (CLM) objective (next-token-prediction) ranging from 350M to 16B in parameter
+size. CODEGEN is first trained on the open-source ThePile Gao et al. (2020), containing 22 diverse
+text-based datasets. The models are then trained on BigQuery BigQuery, a dataset of open-source
+code from 6 programming languages. We refer to these models (trained on ThePile then Big-
+Query) as CODEGEN-MULTI. CODEGEN-MULTI is then further trained on a dataset containing
+large amounts of Python GitHub code to produce CODEGEN-MONO. In our experiments, we
+use CODEGEN-MONO for repair benchmarks in Python and CODEGEN-MULTI for repair bench-
+marks in other programming languages by refer to them both as CODEGEN for simplicity.
+• Codex Chen et al. (2021). A programming language focused autoregressive model based on the
+GPT-3 architecture Brown et al. (2020). Codex is first initialized with GPT-3 weights from training
+on natural language corpus and then fine-tuned using next-token-prediction on a large dataset of
+code files. While Codex also contains a version which can take in suffix tokens (i.e., fill-in code
+in the middle), for our experiments, we only use Codex by providing the prefix context.
+• ChatGPT Schulman et al. (2022). A conversational-based LLM first initialized from GPT-3.5
+model and then fine-tuned using Reinforcement Learning from Human Feedback (RLHF) Ziegler
+et al. (2019). ChatGPT is first fine-tuned based on supervised learning where human provides
+example responses to prompts in the dataset. Using this fine-tuned model, a reward model is
+then trained by sampling multiple outputs of the model from a given prompt and again using a
+human to rank the outputs. The reward model is used in the reinforcement learning step where
+Proximal Policy Optimization Schulman et al. (2017) is used to fine-tune ChatGPT. Different from
+the Codex and CODEGEN, ChatGPT through the usage of RLHF and fine-tuning data is designed
+for conversation where the usage encourages a dialogue format. Note that much of the ChatGPT
+model detail is unknown to the public, therefore, we can only provide an approximate value for
+the number of parameters2 and context window size OpenAI (2022) according to verified sources.
+4.2
+BENCHMARKS
+We use the QuixBugs Lin et al. (2017) repair benchmark to evaluate our proposed conversational
+APR approach.
+QuixBugs has been widely used to evaluate many repair tools including both
+learning-based Ye et al. (2022); Zhu et al. (2021); Jiang et al. (2021); Drain et al. (2021) and LLM for
+APR Xia & Zhang (2022); Xia et al. (2022); Kolak et al. (2022); Prenner et al. (2022) approaches.
+QuixBugs dataset contains the same 40 bugs and it associated correct patch in both Python and
+Java. These bugs are self contained functions based on classic algorithms and it usually only takes
+a single line change to fix the underlying bug. Each bug comes with a set of testcases which the
+buggy function failed to pass and can be used to evaluate any candidate patch generated. Figure 2
+shows an example bug for the bitcount function in QuixBugs for both Java and Python. The bug
+occurs inside the while loop where the code incorrectly uses the ˆ operator instead of & operator. We
+also show the example testcases for bitcount where it contains example inputs and the expected
+outputs when evaluated using the function.
+2As ChatGPT is fine-tuned on GPT-3.5, we assume a similar number of parameters as GPT-3.5
+6
+
+Out of the 40 bugs in QuixBugs, we further filter out 10 bugs which includes testcases that are
+difficult to represent with our validation feedback prompt. For example, testcases for detect cycle
+involves a graph as an input to the function. In total, we use 60 bugs (30 and 30 respectively for Java
+and Python) for our evaluation.
+5
+EXPERIMENTAL SETUP
+In this section, we describe the key research questions that our evaluation seek to answer, the evalu-
+ation metrics used and also describe the implementation details.
+5.1
+RESEARCH QUESTIONS
+We aim to investigate the following research questions:
+• RQ1: What is the effectiveness of applying conversational APR?
+• RQ2: How do different components of conversational APR effect performance?
+In RQ1, we first compare the performance of conversational APR with a baseline approach used
+in prior LLM for APR work where the patches are generated by continuously sampling from the
+same initial prompt. We further evaluate both the scaling effective of LLM as we increase the size
+of the model and also investigate the difference in performance of different pre-training strategies
+(e.g., ChatGPT vs. Codex). In RQ2, we dive deeper into the different parameters of conversational
+APR. Specifically, we evaluate how the length of the conversational chain and different validation
+feedback prompts affect the performance.
+5.2
+EVALUATION METRICS
+Our evaluation metric consist of the standard metric used to evaluate APR tools: number of plausible
+patches: patches which passes all the testcases and correct patches: patches which are semantically
+equivalent to the reference developer patch. Additionally, since we are using sampling LLMs, we
+also define tries as the number of samples needed to obtain a plausible/correct patch. This metric is
+useful when comparing two approaches/models that achieve similar number of bugs fixed, the one
+with fewer number of tries is preferred as we want to limit the number of times we have to sample
+the LLM.
+5.3
+IMPLEMENTATION
+We implemented the LLM generation pipeline in Python using Hugging Face HuggingFace imple-
+mentation of the CODEGEN models. We access Codex through the OpenAI API by querying the
+code-davinci-002 engine. Since ChatGPT is not open-sourced and does not provide an official API
+endpoint (like Codex), we manually input the prompt and extract the outputs. For all models apart
+from ChatGPT, we use a default generation setting of nucleus sampling with top p = 0.95, tempera-
+ture = 1, 50 samples per bug with a maximum chain length of 3. We generate and evaluate patches
+on a 32-Core workstation with AMD Ryzen Threadripper PRO 5975WX CPU, 256 GB RAM and 3
+NVIDIA GeForce RTX 3090 GPUs, running Ubuntu 22.04.1 LTS.
+6
+RESULTS
+6.1
+RQ1: CONVERSATIONAL APR EFFECTIVENESS
+We first evaluate the effectiveness of applying conversational APR using validation feedback com-
+pared to prior method of sampling given the same prompt without any feedback. Table 2 shows the
+results on QuixBugs-Python and QuixBugs-Java. We observe that by applying our feedback driven
+conversational APR, we are able to improve the # of correct and plausible patches for all unsupervis-
+edly trained LLM across all model sizes. Additionally, conversational APR is also able to decrease
+the # of tries (# of samples) needed before obtaining the first plausible/correct patch. Compared
+to traditional sampling method of producing patches, conversational APR is able to leverage the
+7
+
+Table 2: Conversational APR performance on both QuixBugs-Python and QuixBugs-Java
+compared with baseline sampling method. #c/#p refers to the number of correct / plausible
+patches.
+Models
+QuixBugs-Python
+QuixBugs-Java
+Sampling
+Conversational
+Sampling
+Conversational
+#c/#p
+#tries
+#c/#p
+#tries
+#c/#p
+#tries
+#c/#p
+#tries
+CODEGEN-350M
+7 / 10
+20.5
+8 / 11
+18.4
+4 / 4
+24.2
+5 / 5
+23.5
+CODEGEN-2B
+22 / 23
+16.6
+25 / 26
+14.3
+12 / 14
+18.8
+15 / 16
+16.4
+CODEGEN-6B
+22 / 24
+14.0
+27 / 28
+12.1
+18 / 20
+19.8
+22 / 22
+13.5
+CODEGEN-16B
+29 / 29
+5.6
+30 / 30
+4.8
+24 / 25
+14.5
+28 / 29
+13.2
+Codex
+29 / 30
+4.6
+30 / 30
+3.8
+28 / 30
+7.2
+29 / 30
+5.7
+Table 3: ChatGPT and Codex comparison on QuixBugs-Python and QuixBugs-Java where
+each cell indicates the number of correct / plausible patches
+Models
+QuixBugs-Python
+QuixBugs-Java
+one try
+two tries
+three tries
+one try
+two tries
+three tries
+Codex
+16 / 16
+21 / 21
+24 / 24
+11 / 12
+18 / 19
+21 / 22
+ChatGPT
+24 / 24
+27 / 28
+28 / 29
+24 / 24
+26 / 26
+26 / 26
+model’s understanding of natural language feedback to indicate why the patch is incorrect. LLMs
+can use this validation feedback information to generate new patches that try to pass the previ-
+ously failed testcase. Furthermore, conversational APR also helps to reduce the number of repeated
+patches from sampling using the same prompt over and over again. By using the large context size
+of many state-of-the-art LLMs, conversational APR can use recently generated incorrect patches as
+previous context to prompt the model to generate a new patch that is different.
+ChatGPT evaluation. We now evaluate the performance of ChatGPT when using conversational
+APR. Due to the requirement of manually inputting and extracting outputs from ChatGPT, we only
+use a single conversation chain with at most 3 tries (i.e. maximum chain length of 3). We compare
+with the best performing LLM of Codex from previous results under the same setting in Table 3.
+We observe that compared to Codex, which is trained in an unsupervised manner, ChatGPT which
+is fine-tuned using Reinforcement Learning from Human Feedback (RLHF) performed much better
+across the two repair datasets. This improvement in result can be partially attributed to increase
+in model parameter size, but we believe this is also due to the dialogue-based fine-tuning dataset
+used in ChatGPT. Conversational APR relies on the model understanding the validation feedback
+to condition the future generation in trying to generate a patch that passes the testcase. A more
+dialogue-oriented model such as ChatGPT is well suited for this task as both the training data and
+algorithm contain feedback driven loops. As ChatGPT and other dialogue-based LLMs become
+more popular, we believe conversational APR can also be further improved through more usage of
+these LLMs.
+6.2
+RQ2: COMPONENT ANALYSIS
+Maximum chain length. We first investigate the effect of different maximum chain length has on
+the repair performance. Figure 3 shows the number of plausible patches when we vary the maximum
+chain length from 1 to 6 for the 4 CODEGEN models. Recall from Section 3 that chain length refers
+Figure 3: Number of plausible patches for the 4 different CODEGEN models as we vary the
+maximum chain length on QuixBugs-Python
+8
+
+CodeGen-350M
+CodeGen-2B
+CodeGen-6B
+CodeGen-16B
+12
+28
+Patches
+30
+30
+10
+26
+28
+28
+8
+24
+Plausible
+26
+26
+22
+24
+24
+20
+#
+6
+2
+3
+4
+6
+1
+2
+3
+Maximum ChainLengthTable 4: Prompting Style Evaluation on QuixBugs-Python with each cell showing the number
+of plausible patches
+Models
+no testcase
+natural language
+functional
+CODEGEN-350M
+9
+11
+11
+CODEGEN-2B
+20
+25
+26
+CODEGEN-6B
+24
+27
+28
+CODEGEN-16B
+27
+30
+30
+Codex
+29
+30
+30
+to the number of turns (each turn consist of generating and validating a new patch) in a conversation
+chain. A maximum chain length of 1 is the simple sampling from the same initial prompt baseline
+(used in prior LLM for APR tools). As we increase chain length, the model has to take in more
+and more previous context in the form of prior generations and feedbacks. We observe that the
+performance increase as we start from a small chain length and reaches the maximum around 3 or 4
+and then decrease as chain length continue to increase. The decrease in number of plausible patches
+once we reach a high chain length is because the context may be too much for the model to handle
+since it can include multiple previously failed patches. We also observe that this decrease is more
+significant in smaller models, where larger models are less affected by longer chain length, showing
+the ability for larger models to better capture the long term context dependencies. This shows that
+the optimal chain length to use for conversational APR can be dependent on the individual LLM
+used.
+Feedback prompting style.
+We now evaluate the effect of the feedback prompting style
+used in our conversational APR. Table 4 shows the number of plausible patches using differ-
+ent validation prompts in QuixBugs-Python.
+Column no testcase does not include any test-
+case feedback (only states that the patch is not correct), natural language describes the failing
+testcase (e.g., when input is 2, the patch incorrectly returns [] but it should
+return [2]) and functional which is the default prompting style discussed in Section 3. We ob-
+serve that different prompting style does have an effect on the final performance of conversational
+APR. Starting from no testcase prompt, we can improve performance by adding specific testcase
+feedback information on top of telling the LLM that the patch is not correct. We also observe that
+the functional prompting style, using the buggy/patch function name and passing parameters (see
+Figure 1), performs the best. Functional prompting style conveys the failing testcase information in
+a more concise and natural way by phrasing the testcase input and expected output relationship as a
+function call.
+7
+CONCLUSION
+We propose conversational APR, a new paradigm for program repair that interleaves patch gener-
+ation with validation to provide immediate feedback for LLMs to better prompt future generated
+patches. Compared to previous LLM for APR approaches that only sample from the same input,
+conversational APR iteratively builds the input by concatenating previously incorrect patches and
+validation feedback. This allows for the model to avoid generating previously incorrect patches and
+also understand the semantic meaning of the function through validation feedback. Our evaluation
+on 10 different LLMs shows the improvement of conversational APR over the baseline sampling
+method used in prior LLM for APR tools. Furthermore, we demonstrate the promising future of ap-
+plying ChatGPT, a conversational/dialogue driven LLM, for conversational APR, or APR in general
+for the first time.
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+DISCOVERING AND EXPLAINING DRIVER BEHAVIOUR UNDER
+HOS REGULATIONS
+A PREPRINT
+Ignacio Vellido1, Juan Fdez-Olivares1, and Ra´ul P´erez1
+1Department of Computer Science and Artificial Intelligence, University of Granada, Spain
+ignaciovellido@ugr.es, {faro, fgr}@decsai.ugr.es
+January 13, 2023
+ABSTRACT
+World wide transport authorities are imposing complex Hours of Service regulations to drivers,
+which constraint the amount of working, driving and resting time when delivering a service. As a
+consequence, transport companies are responsible not only of scheduling driving plans aligned with
+laws that define the legal behaviour of a driver, but also of monitoring and identifying as soon as
+possible problematic patterns that can incur in costs due to sanctions. Transport experts are fre-
+quently in charge of many drivers and lack time to analyse the vast amount of data recorded by
+the onboard sensors, and companies have grown accustomed to pay sanctions rather than predict
+and forestall wrongdoings. This paper exposes an application for summarising raw driver activity
+logs according to these regulations and for explaining driver behaviour in a human readable format.
+The system employs planning, constraint, and clustering techniques to extract and describe what
+the driver has been doing while identifying infractions and the activities that originate them. Fur-
+thermore, it groups drivers based on similar driving patterns. An experimentation in real world data
+indicates that recurring driving patterns can be clustered from short basic driving sequences to whole
+drivers working days.
+1
+Introduction
+World wide transport authorities are imposing complex Hours of Service (from now on, HoS) regulations to drivers
+(Meyer 2011, Goel and Vidal 2013), which constraint the amount of working, driving and resting time when delivering
+a service. As a consequence, transport companies are responsible not only of scheduling driving plans aligned with
+laws that define the legal behaviour of a driver, but also of monitoring and identifying as soon as possible problematic
+patterns that can incur in costs due to sanctions.
+Fortunately, the widespread adoption of onboard IoT devices in vehicle fleets enables recording of the driver activities
+in event logs, but the large amount of data ingested makes difficult for transport experts to understand what happened
+and to make actions that forestall illegal behaviour. For this reason, an important technical challenge is to come up
+with easily interpretable descriptive models that help understand the huge amount of information stored in such event
+logs. The main objective not only consists of finding out if drivers workplan complies with the HoS regulation, but
+also summarising their activities in a concise but representative way. Additionally, these underlying patterns in the
+event log could be analysed in order to discover driving styles which could make possible the suggestion of routes or
+tasks more aligned to the driver preferences.
+The creation of driver profiles based on driving styles with HoS can be extremely useful for managers, as they could
+assign transport routes to the most appropriate drivers, given the length of the route and the proximity of the deadline.
+For example, drivers who maximise their driving hours could be preferred for long distance routes and drivers who
+tend to take split rest to on-city deliveries.
+arXiv:2301.05082v1  [cs.AI]  12 Jan 2023
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Weekly Driving
+Normal 
+Daily
+Driving
+Extended 
+Daily
+Driving
+Driving
+Sequence
+Driving
+Sequence
+Driving
+Sequence
+Driving
+Sequence
+Uninterrupted
+Split 1
+Split 2
+Activities
+Break 
+Type 1
+Rest 
+Day
+Activities
+Activities
+Break 
+Type 2
+...
+Driving
+Sequence
+Weekly 
+Rest
+...
+Figure 1: Partial example of the HoS tree. At the upmost level, a weekly driving period is formed by several daily
+periods, which must end with a weekly rest. Similarly, daily driving periods are separated by daily rests, and according
+to the accumulative hours of driving in them can be classified as Normal Daily Driving periods (up to 9 hours) or
+Extended Daily Driving periods (more than 9 hours). Because each driving sequence should not surpass 4.5 hours of
+driving time, they can be distinguished by the number of driving sequences in them.
+Therefore, in this paper we present a method that, starting from real event logs extracted from a tachograph device, 1)
+labels driver activities according to the HoS regulation, 2) identifies infractions and their cause, 3) extract summarised
+information about the log while clustering driving sequences based on similar behaviour patterns, and 4) group drivers
+by similarity of those clustered patterns. As a results, experts are provided with an understandable analysis of what
+the driver has been doing in multiple levels of granularity, from a detailed description of the activities and infractions
+under the HoS regulation to a categorisation with similar tendencies.
+The remainder of this paper shows, firstly, a description of the problem addressed and some background concepts to
+it. Then, we present the methodology of the approach, followed by details of experimentation conducted over a proof
+of concept of the application. Finally, we conclude discussing related and future work.
+2
+Problem Description
+We are collaborating with a company which provides decision support based on prediction services to its customers.
+Ultimately they want to help them govern the behaviour of their drivers by predicting whether a driver is close to
+committing an infraction, as well as characterising drivers according to their driving style with respect to the HoS
+regulation.
+They handed us tachograph logs of multiple drivers with thousands of activities and asked us to develop a system to
+analyse driver behaviour. Due to the regulation imposing additional difficulties at interpreting the data, and the high
+volume that is constantly being generated, experts cannot interpret directly the original tachograph logs and require
+summarisation of what a driver has been doing during that period of time to make business decisions. A tachograph
+(Baldini et al. 2018) is an automated recording device fitted into a vehicle that extracts information from the driving
+activities such as speed, duration and distance.
+Our dataset represented an event log where every activity is a tuple (id, start, end, dur, a), each component
+referring to: driver identifier id; start and end timestamps; activity duration dur; and activity identifier a, respectively.
+A value for a is any of the labels [Driving, Other, Break, Idle] meaning that the driver is either Driving, performing
+2
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Tachograph 
+Log
+Activity Recognition  
+and  
+HOS anaylsis
+Labelled 
+Log
+Infringement 
+Analysis
+Extended 
+Labelled 
+Log
+Driver Behaviour 
+Analysis
+Day 
+summary
+Driver 
+Patterns 
+Clustering
+Day 
+summary
+...
+Clustering
+...
+...
+...
+Figure 2: General overview of our approach.
+Another Work, at Break or Idle during dur minutes, between start and end. The semantics of each event is completed
+with the definitions provided by the HoS regulation, which are detailed in the following paragraphs.
+Although the HoS standard is applied in several countries, in this work we focus on the European Union regulation
+(EC) No 561/2006, which has been extensively analysed in (Goel and Vidal 2013, Meyer 2011). The basic terms refer
+to four types of driver activities as break (short period for recuperation), rest (free disposal period with enough time to
+sleep), driving (time during which the driver is operating a vehicle) and other work (time devoted to any work except
+driving, like loading).
+These activities are hierarchically grouped up to weekly intervals, based on the duration of the events contained in
+them. To ease the explanation of this article we are referring at the whole structures as HoS trees. In Figure 1 we
+exemplify a portion of a HoS tree displaying a Normal Daily Driving (NDD) period on the first day and a Extended
+Daily Driving (EDD) period on the last.
+At the lower levels activities are joined in different types of driving sequences. A basic driving sequence is composed
+of a totally ordered set of the elements of [Driving, Other, Break, Idle] constrained so that the duration of any
+Break is less than 15 minutes. More constraints are defined over the duration of the rests and breaks, and over the
+accumulated duration of driving sequences.
+The regulation provides a set of basic and optional rules, should the former not be satisfied, thus allowing more
+flexibility to generate and interpret driving schedules under such constraints. For example, either a break of 45 min
+has to be taken after 4.5 hours of accumulated driving or it can be taken split in two parts of at least 15 min and
+30 min respectively. This feature is good for drivers since it provides flexibility to their work, but complicates the
+interpretability of what they are doing. The regulation also defines additional constraints (for example, the maximum
+number of occurrences of a reduced rest in a weekly driving period), and relationships between the different types of
+sub-sequences, as well as their internal structure.
+3
+Background
+Automated planning (Ghallab et al. 2016) is a branch of A.I. concerned with the study of agent acting techniques.
+However, its uses can be broaden and as we show in this paper planning can also be applied to recognition tasks.
+Two elements are required in a planning environment: (i) the action models existing in the world, referred as domain;
+and (ii) a description of the initial state of the world, the objects involved in it and the desired goals, called problem.
+These two inputs are provided to a planner, a search-based algorithm that determines the plan (sequence of actions)
+that achieve the goals from the starting state.
+Our proposed methodology employs hierarchical planning, more commonly referred as Hierarchical Task Networks
+(HTN). HTNs forms a branch of classical planning where the domain can be decomposed in hierarchical structures
+of tasks/subtasks, with low level tasks representing temporally annotated actions, and compound tasks representing
+temporal ordering strategies between those actions.
+4
+Application Overall Description
+To solve the problem of explaining and summarising a driver’s tachograph log and its compliance with the HoS
+regulation we propose a modular architecture divided in three main components, as seen in Figure 2:
+• First, an initial planning process to label the input tachograph log according to the HoS regulation.
+• Then, a system to identify and explain the causes of driver infractions extending the previous labelled log.
+3
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Tachograph 
+Log
+Transformation 
+to HTN 
+problem
+Labelled 
+Log
+HPDL 
+problem
+HPDL 
+domain
+Planner
+HoS 
+Rules
+Transformation 
+to HTN 
+domain
+Temporal 
+Observations
+Atrribute 
+Grammar
+1
+2
+3
+5
+4
+Figure 3: Labelling process for a tachograph log.
+Table 1: Labelling output for legal activities. This example shows the second (and last) driving sequence in a normal
+daily driving period, where the required break has been taken in two parts, a small break in the first split and a second
+one extended as a daily rest.
+Original Log
+Annotated Labels
+Driver
+Start
+End
+Duration
+Activity
+Week
+Day
+DayType
+Sequence
+BreakType
+Token
+Legal
+driver1
+11/01/2017 17:33
+11/01/2017 17:37
+4
+Driving
+1
+4
+ndd
+second
+split 1
+A
+yes
+driver1
+11/01/2017 17:37
+11/01/2017 18:16
+39
+Break
+B T2
+yes
+driver1
+11/01/2017 18:16
+11/01/2017 18:17
+1
+Driving
+split 2
+A
+yes
+driver1
+11/01/2017 18:17
+11/01/2017 18:25
+8
+Other
+A
+yes
+driver1
+11/01/2017 18:25
+11/01/2017 19:54
+89
+Driving
+A
+yes
+driver1
+11/01/2017 19:54
+11/01/2017 19:57
+3
+Break
+B T0
+yes
+driver1
+11/01/2017 19:57
+11/01/2017 19:58
+1
+Driving
+A
+yes
+driver1
+11/01/2017 19:58
+11/01/2017 20:01
+3
+Other
+A
+yes
+driver1
+11/01/2017 20:01
+12/01/2017 07:06
+665
+Break
+DR T1
+yes
+• Thirdly, a module to analyse driver behaviour via summarisation of driving sequences.
+• Lastly, summarised driving days are used as training data to clusterize drivers by similar driving patterns.
+The following subsections provide a detailed explanation of each component.
+4.1
+Labelling Driver Activities
+To label our logs with HoS terms we employ our previously developed methodology proposed in (Vellido-Exp´osito.
+et al. 2022), where a HTN domain serves to both recognise and tag activities from a tachograph log. We provide a
+brief summary below, but we refer the reader to the original paper for an in depth explanation of the methodology. The
+overall steps of this system, represented in Figure 3, are:
+1. Generate a set of ordered temporal observations from the tachograph activity log, which are part of the initial
+state of a HTN problem.
+2. Represent the recognition of a driver activity as a temporal HTN problem, where an activity is added to
+the plan if (i) the temporal information of the activity is consistent with the domain, and (ii) the temporal
+constraints of the activity are consistent with the rest of temporal constraints of the actions already added to
+the plan.
+3. Codify a HoS tree in an attribute grammar (Knuth 1968) as an intermediate representation, with HoS rules as
+productions.
+4. Translate the grammar into a temporal HTN domain, aimed at representing the parsing of the activity log as
+a HTN problem where (i) terminal symbols are recognised as temporal events and (ii) nonterminal symbols
+are recognised according to grammar rules.
+5. Extend the domain to both recognise and label activities from the log to be easily interpretable. The resulting
+log contains five new labels according to the contexts DayType (Normal or Extended Daily Driving period),
+Sequence (if the activity belongs to the first, second or third sequence in the day), BreakType (if breaks are
+taken in one or two parts), Token (the type of activity at the lowest level of in the HoS tree1) and Legal
+1Many types of categories exist at the lowest level, based on the duration of the action. For example, A indicates a working
+activity, B T0 a break of less than 15 minutes, DR T1 a daily rest of more than 11 hours, and WR T1 a weekly rest with more than
+45 hours.
+4
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Labelled 
+Log
+Labelled Log
+with 
+Infringements
+Test  
+Evaluation
+Labels 
+Comparison
+Test 
+List
+Original 
+Tachograph 
+Log
+Relaxed 
+HPDL 
+domain
+Relaxed 
+Labelled 
+Log
+Labelling 
+Process
+Figure 4: Infringement analysis process for a labelled log.
+(whether the activity complies or not with the regulation), as well as two counter columns for the day and the
+week processed. An output example can be seen in Table 1.
+In summary, the recognition problem is solved with a planning process where the domain walks through an activity log
+and its internal HTN structure simultaneously, the latter codifying the HoS tree. If activities comply with the temporal
+and formal restrictions they are labelled with the appropriate terms, in other case contexts are tagged as unrecognised.
+Nevertheless, the domain is designed to label as many contexts as possible. If a higher (more general) context cannot
+be identified, the domain still attempts to identify lower contexts before ignoring the action. That means that when
+a bigger sequence cannot be grouped and labelled together (e.g. when the driver exceeds the maximum number
+of driving hours and the DayType column cannot be tagged), the domain tries to tag smaller sequences with their
+corresponding label. An example is shown below in Table 3, where although DayType and Sequence tags could not be
+recognised the system still identifies both BreakType splits and includes the appropriate labels, as well as the correct
+Token contexts.
+4.2
+Explaining Infringements
+The previous recognition process labels the tachograph log considering the terms defined by the HoS regulation, its
+compliance with it and details of their position in the HoS tree. However, when drivers commit infractions this system
+by itself cannot provide an explanation of the cause and the exact root activity, due to the fact that planning techniques
+rely on backtracking (that is, the ability to retract while exploring the planning graph) and there is not a simple way to
+distinguish between a genuine backtracking step while walking through the HTN domain or a forced one by an illegal
+activity in the log.
+Therefore we found a need to further analyse the labelled log and explain these information to users without requiring
+them to inspect all activities not recognised in the log. We solved this problem from two perspectives, each one
+concerned with different kinds of violations, which are explained in the following subsections. Figure 4 shows an
+overview of the approaches.
+4.2.1
+Test evaluation
+On one hand we represent rules from the HoS regulation as tests and applied them to the sequences the labelling
+process found unrecognisable events (i.e., those missing at least a label). These tests, as exemplified in the left part
+of Table 2, codify limits and restrictions in the duration of driving sequences and breaks. Whenever a test flags a
+sequence the system marks it and provides an explanation of the infringement, as seen in Table 3.
+5
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Table 2: Tests applied to driving sequences in the log in order to identify infringement causes.
+Test
+Infraction type
+dt seq > 4.5h
+Excessive Driving without breaks
+dt day > 9h
+∧
+EDDs this week > 2
+Excessive Driving in day (NDD)
+dt day > 10h
+Excessive Driving in day (EDD)
+Token day before = DR T3
+∧
+Token = ¬ (DR T4 or WR)
+Missing other half of split daily rest
+Token = DR or WR
+∧
+Legal = No
+∧
+Remaining
+contexts = ¬ Unknown
+Rest past the daily/weekly deadline
+Table 3: Labelling output example for illegal activities and the infraction detected by the tests list.
+Original Log
+Annotated Labels
+Driver
+Start
+End
+Duration
+Activity
+Week
+Day
+DayType
+Sequence
+BreakType
+Token
+Legal
+Infraction
+driver39
+10/01/2017 12:12
+10/01/2017 14:17
+125
+Driving
+1
+5
+unkown
+unkown
+split 1
+A
+no
+Surpassed NDD driving time
+driver39
+10/01/2017 14:17
+10/01/2017 14:40
+23
+Break
+B T2
+no
+driver39
+10/01/2017 14:40
+10/01/2017 16:52
+132
+Driving
+split 2
+A
+no
+driver39
+10/01/2017 16:52
+10/01/2017 17:25
+33
+Break
+B T3
+no
+driver39
+10/01/2017 17:25
+10/01/2017 20:27
+182
+Driving
+ndd
+first
+split 1
+A
+yes
+driver39
+10/01/2017 20:27
+10/01/2017 20:42
+15
+Break
+B T2
+yes
+driver39
+10/01/2017 20:42
+10/01/2017 21:54
+72
+Driving
+split 2
+A
+yes
+driver39
+10/01/2017 21:54
+10/01/2017 21:59
+5
+Break
+B T0
+yes
+driver39
+10/01/2017 21:59
+10/01/2017 22:00
+1
+Driving
+A
+yes
+driver39
+10/01/2017 22:00
+10/01/2017 22:37
+37
+Break
+B T3
+yes
+driver39
+10/01/2017 22:37
+10/01/2017 23:21
+44
+Driving
+second
+uninterrupted
+A
+yes
+driver39
+10/01/2017 23:21
+11/01/2017 08:53
+572
+Break
+DR T2
+yes
+Tests takes the form of logic constraints
+f(astart, aend) o V
+(1)
+being:
+• f a function applied over the sequence defined between activities astart and aend (e.g. sum, context value).
+• o a logic operator.
+• V either the value of a context (e.g. Token, DayType), a scalar or a duration.
+As an example, the first constraint in Table 2 could be rewritten as duration(seqstart, seqend) > 4.5h.
+It is important to note that to correctly identify the infraction some tests may consider not only the illegal activities
+but also prior activities of other days, a situation frequently present in reduced breaks and rests, where sometimes
+compensation breaks are not fulfilled. Therefore the interval of activities checked by the tests depends on the test
+itself.
+Because tests are encoded as logic constraints, it is easy to extend the system with additional expert provided rules or
+modify them if the regulation changes.
+4.2.2
+Re-labelling
+A second approach consists of re-labelling the log using a domain with relaxed duration intervals, that is, the limits
+imposed by the regulation are softened (e.g. maximum driving time or minimum break time are enlarged up and down)
+and the system looks for changes between the new log and the original tagged log.
+This process helps to discover infringements caused by a slightly borderline duration, like the driver surpassing (prob-
+ably unconsciously) the restriction by a small amount. These kind of situations are not easily identified by the tests,
+due to the fact that the activity by itself could still be legal but labelled differently, becoming an infraction later on.
+6
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Table 4: Identifying infringements with a relaxed domain. In this example the fourth activity surpasses by one minute
+the duration limit to be considered B T0, making the whole sequence illegal.
+Original Labelled Log
+Duration
+Activity
+DayType
+Sequence
+BreakType
+Token
+Legal
+57
+Driving
+unknown
+unknown
+split 1
+A
+no
+3
+Break
+B T0
+no
+2
+Driving
+A
+no
+16
+Break
+B T2
+no
+Relaxed Labelled Log
+Duration
+Activity
+DayType
+Sequence
+BreakType
+Token
+Legal
+57
+Driving
+ndd
+first
+split 1
+A
+yes
+3
+Break
+B T0
+yes
+2
+Driving
+A
+yes
+16
+Break
+B T0
+yes
+ 
+Extended 
+Labelled 
+Log
+Clustered 
+Log
+Paragraph  
+Vector 
+Model
+HDBSCAN 
+Model
+Vectors
+Legal 
+Days
+Illegal 
+Days
+Paragraph  
+Vector 
+Model
+Vectors
+HDSBSCAN 
+Model
+Centroids
+Encoding
+Encoding
+Figure 5: Clustering process for a labelled log.
+For example, a driver could surpass the maximum limit for a pause before being considered a break by a few minutes
+without noticing, and proceeding like a break has not been consumed. As a consequence, that action will be valid, but
+after the next breaks infractions may arise because the driver is not following its plan as expected, and such actions
+may not fit correctly under the HoS tree.
+If the violation is related with this type of mistake, the new relabelled log will contain less illegal sequences than the
+original and we can compare the Token contexts (concerning the type of activity at the lowest level in the HoS tree) to
+understand which changes make the sequence legal. Table 4 shows an example with a driver exceeding the break time
+by two minutes.
+Therefore, this method allows us to (a) discover new infringements not considered by the test list, and (b) analyse how
+the activity should have been to avoid infractions.
+4.3
+Analysing Driver Behaviour
+The two previous steps provide a way to understand a driver log and its compliance with the HoS regulation. However,
+experts are usually responsible of dozens of drivers and its not feasible to analyse the substantial logs of each one of
+them in order to detect problematic tendencies.
+Therefore, we developed a module that clusters behaviour patterns in driver activities and summarises each cluster
+with expert knowledge. This method helps to separate standard driving days from unusual ones without inspecting the
+driver log, and let users concentrate their efforts in analysing only the problematic sequences.
+In order to do that, we considered our problem as an NLP (Natural Language Processing) task, where activities from
+the log are treated as words and daily sequences as documents. That way we can employ NLP oriented techniques to
+transform sequences of varying length into fixed dimensions and measure similarity between them.
+Figure 5 shows an overview of the process, consisting of the following steps:
+7
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Table 5: Partial output of the clustering process. The system identifies the most similar centroid to the input sequence
+and the description associated with it.
+Labelled Log
+Activity
+DayType
+Sequence
+BreakType
+Token
+Legal
+Cluster
+Driving
+ndd
+unique
+uninterrupted
+A
+yes
+2
+Other
+A
+yes
+Break
+DR T1
+yes
+Most similar centroid
+Activity
+DayType
+Sequence
+BreakType
+Token
+Legal
+Cluster
+Driving
+ndd
+unique
+uninterrupted
+A
+yes
+2
+Other
+A
+yes
+Break
+DR T3
+yes
+Description
+Legal and standard daily driving formed by a unique and uninterrupted driving sequence
+1. First, a preprocessing step is applied in which the dataset is split in two parts depending on whether the days
+contains or not illegal activities. The reason behind this process is that an infraction recognition process
+is already provided by the previous module, and thus there is no need for our clustering model to learn to
+distinguish between legal and illegal sequences. On the contrary, we are providing a prior separation to help
+the model extract more interesting patterns that are not related with the legality of the sequence.
+2. The subset of labelled columns (i.e. contexts) that describe the action from an overall point of view are
+selected, these are (Activity, DayType, BreakType, Token). For the illegal subset, the Infraction column is also
+included to generate clusters and centroids associated with already identified infringement. Specific details
+about duration and timestamps are not relevant to summarise the days. Nevertheless, some of the information
+they provided is encoded in the labels, as it is used by the labelling process. This step could be consider as
+cleaning a document prior an NLP topic categorisation task.
+3. Because columns contains categorical features not suitable for computation they are transformed into numer-
+ical, and then joined together using a special character as a separator. After this step we can consider each
+entry in our log as a word.
+4. Both previous steps are repeated for each activity in our dataset, and activities of the same day are grouped
+into documents. As a result, we have a collection of documents each one encoding the activities in a driving
+day sequence as words.
+5. We then use Paragraph Vector (Le and Mikolov 2014) (also known as Doc2Vec) models to obtain dense rep-
+resentations of fixed dimensions2. Although one model could be trained for both data splits (and reasonable
+so, as both encodings are subset of the same language), we obtained better results finetuning one model for
+each split, but ultimately both transforming a document into a 200 sized output vector.
+6. The resulting representations are now suitable for clustering techniques. We obtained our best results using
+HDBSCAN (Campello et al. 2013) thanks to its robustness to noise, and choosing the number of clusters
+based on both expert knowledge and metrics results (Silhouette Coefficient, Calinski-Harabasz and Davies-
+Bouldin indexs), setting on a final value of 8 clusters for legal data and 7 for days with infractions. In the next
+section we display a comparative analysis of other techniques under this data.
+7. Lastly, days are clustered and presented with the decoded centroids, which are described by an expert with a
+meaningful description, as shown in Table 5.
+4.4
+Generating Driver Profiles
+Similarly to the working days clustering previously explained, we performed categorisation of drivers based on similar
+behaviour with the idea of extracting driver profiles. With enough data, we saw that the large amount of activities
+contained in event logs can be summarised in different types of driving days as described in the previous sections,
+and such types encode enough information to extract a characterisation of the driver that can be informative for the
+transport company.
+2Other techniques like Word2Vec or Bag of Words could be used, but we considered the paragraph weight extracted by Paragraph
+Vector a useful source of information in our task.
+8
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Table 6: Example input data for extracting driver profiles. Each row encodes how frequently a driver perform one of
+four types of driving days. Given the uneven number of data of each driver values are expressed as percentages and
+all rows sums to one.
+Driver
+Driving day type
+Split Sequences Normal Rest
+Uninterrupted Sequences Normal Rest
+Split Sequences Reduced Rest
+Uninterrupted Sequences Reduced Rest
+. . .
+1
+0.5
+0.1
+0.3
+0.1
+2
+0.2
+0.8
+0.0
+0.0
+3
+0.15
+0.5
+0.3
+0.15
+We performed the following steps to categorise drivers:
+1. Drop days with infractions: Tests introducing violations gave us results who grouped drivers by similar ratio
+of infractions and by day types (e.g., those who tended to excess their break time ended up in the same cluster).
+However, we opted to not include such information as they did not align with the purposes of the application.
+Because the context around the infraction cannot be extracted exclusively from the tachograph data (e.g., was
+the infringement voluntary, due to lack of correct planning or caused by unexpected circumstances on the
+road?) we believe managers should analyse case by case rather than making decisions without understanding
+the real motive behind the infraction.
+2. Create frequency table: For each driver its daily logs are processed following the methodology explained
+in subsections 4.1 and 4.3, keeping the day type predicted by the clustering model. The training dataset is
+created counting the frequencies the driver has performed each type of driving day. Due to the fact that in
+our data the amount of information varies for each driver, these frequencies are transformed into percentages.
+Ultimately, we obtain a table D × C as shown in Table 6, being D the number of drivers and C the number
+of different days categories (i.e., the number of clusters discovered in the previous section).
+3. Training: We perform clustering with the resulting table. From a multiple of techniques our best results were
+obtained with a Gaussian mixture model trained with the Expectation-Maximization algorithm (Fraley and
+Raftery 2002).
+The deciding factor at choosing the best partition was based exclusively on expert knowledge. Because we
+were informed that our data was for drivers who performed similar routes on the same country, we looked for
+a few number of clusters that separated nicely the data.
+As a closing point, we would like to note that driver profiles based only on tachograph data could be misleading, as
+they do not account for the specific routes they perform. We believe a better approach would be to combine the cluster
+information with route details like distance, type of vehicle or type of cargo, and as a result get a categorisation of
+the driver given the type of route. That way, decisions based on these profiles will not be biased, and traffic managers
+could assess their drivers for each particular service. We intend to explore those options in future work.
+5
+Experimentation
+We have validated our methodology with an experimentation using real tachograph logs provided by an industrial
+collaborator. We were provided with a dataset formed by two-weeks-long sequences of activities from 290 different
+drivers.
+Because the architecture is composed of three different components, each one was validated individually. The labelling
+process was verified against multiple driving sequences selected at random, both legal and illegal, manually verifying
+that the output was the appropriate under the HoS regulation. For the infringement analysis system multiple tests for
+each kind of infraction were carried out, confirming that not only the cause, but also the subsequence containing the
+infraction was detected.
+Lastly, due to the fact that the clustering in our problem is an unsupervised task, we experimented with different
+techniques and hyperparametrization to discover the best possible clusters. The quality of each partition was measured
+with the Silhouette Coefficient and both Calinski-Harabasz and Davies-Bouldin indexs. The final clustering result was
+selected between the best performing tests, and after expert inspection of the resulting clusters and centroids.
+Figure 6 shows the performance of multiple algorithms in data with and without infractions, respectively. The algo-
+rithms are: Gaussian Mixture models, having each component its own covariance matrix, and controlling the number
+of mixture components as the number of clusters; HDBSCAN (Campello et al. 2013), a hierarchical clustering model
+employing density based measures; classical agglomerative clustering using average, complete and ward criteria; and
+K-Means with cosine similarity as distance metric.
+9
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Data with infractions
+Data without infractions
+2
+4
+6
+8
+10
+2
+4
+6
+8
+10
+12
+14
+0.2
+0.3
+0.4
+0.5
+0.6
+0.1
+0.2
+0.3
+0.4
+0.5
+Silhouette coefficent
+2
+4
+6
+8
+10
+2
+4
+6
+8
+10
+12
+14
+0
+500
+1000
+1500
+0
+100
+200
+300
+Calinski−Harabarsz score
+2
+4
+6
+8
+10
+2
+4
+6
+8
+10
+12
+14
+1
+2
+3
+4
+5
+0.9
+1.2
+1.5
+Davies−Bouldin score
+Number of Clusters
+Algorithm
+Gaussian Mixture
+HDBSCAN
+Hierarchical (avg)
+Hierarchical (complete)
+Hierarchical (ward)
+KMeans
+Figure 6: Silhouette Coefficient, Calinski-Harabasz and Davies-Bouldin indexs as a function of the number of clusters
+for multiple clustering algorithms. Notice that the y-axis scalings differ among the different panels of this figure.
+Some insights can be extracted from the graphs. HDBSCAN is without doubt the best algorithm under both subsets of
+this data, but we believe that the reason relies mostly due to its robustness to noise points. Furthermore, results on data
+with infringements are, as expected, more variable, as this subset combines multiple types of infractions with driving
+sequences that can be perfectly legal. The runner-up model is not clear, as results vary greatly with the number of
+clusters.
+For fully legal data we see hierarchical clustering with average and complete linkage method vastly underperforming.
+The graphs for the rest of techniques take similar shape, mostly agreeing in that 8 clusters seems an appropriate
+partition for this data. Nevertheless, as the results are intended for human interpretation, it is important to remind that
+the clusters should be reviewed by an expert whenever possible before setting on a final value.
+Finally, we believe is worth mentioning our experimentation with the LDA (Latent Dirichlet Allocation) (Blei et al.
+2003). This technique is frequently used in NLP tasks to summarise a document with a set of topics. Due to a small
+vocabulary size in our data as opposed to an NLP task, most words (i.e. driver activities) are present in many different
+clusters (with the exception of illegal activities), and although the most relevant topics could be ranked and considered
+as centroids there is no assurance that these topics are understandable (e.g. a B T2 break only makes sense if followed
+10
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Table 7: Clustering results for driver profiles and they interpretation.
+Cluster
+Interpretation
+Proportion
+1
+No extended days and mostly
+takes rests uninterrupted
+8.6%
+2
+Usually splits rests as much as possible
+and rarely takes extended days
+51.8%
+3
+Neither takes many extended days
+or splits rests
+20.2%
+4
+No clear tendency,
+driver seems to be flexible
+14.4%
+5
+Tends to split rests as much as possible
+and frequently takes extended days
+5.0%
+by a B T3 break. The presence of only one of them as a topic does not clarify if the driver completed the sequence or
+committed an infraction).
+For our driver clustering experimentation Table 7 shows 5 resulting clusters and their interpretation after training.
+Given that our training data is compromised of mostly event logs of national deliveries in Spain, we can see that more
+than half of our drives prefer to spent their rests split in two. Nevetheless, as mentioned above, the lack of data about
+the routes performed in the tachograph hinders the expressivenes, but experts welcome any information that could help
+them assign the best driver to a service as easily as possible.
+The methodology and experimental results are encapsulated in an web application publicly available at https://
+github.com/IgnacioVellido/Driver-Assistance-System.
+6
+Related Work
+This project is an extension of authors prior work (Fernandez-Olivares and Perez 2020) focused on the recognition
+and labelling of driver activities under the HoS regulation. The novel contributions provided in this paper go a step
+forward in our goal of developing an intelligent assistant to drivers and traffic managers, proposing a planning and
+constraint based analysis of infractions causes and summarisation of driver behaviour with NLP techniques.
+Regarding applications concerned with the HoS regulation, many approaches have been developed aimed to solve
+route planning problems under these rules while minimising transportation costs (Mbiydzenyuy 2015, Omelianenko
+et al. 2019, Goel 2018, Goel and Irnich 2017). Nonetheless, the authors have not found works that extract insights that
+can be useful for experts in analysing and understanding driver activities from a legal perspective.
+As for driver behaviour modelling from tachograph data, proposals like (Zhou and Zhang 2019) employs data mining
+techniques to categorise truck drivers and analyse dangerous tendencies. Their approach is similar to ours in that
+clusters are manually studied and labelled. However, PCA for dimensionally reduction and DBSCAN for clustering
+are directly used instead due to the fact that their data does not contain categorical variables.
+Lastly, word embedding techniques like Paragraph Vector models has been previously applied in non textual data like
+web user activities (Tagami et al. 2015) and server logs (Mimura and Tanaka 2018) as a way to transform sequential
+data of variable length into dimensionally fixed data. Similarly, although oriented to process mining applications,
+the trace2vec model proposed in (De Koninck et al. 2018) uses embedding techniques for discovery, monitoring and
+clustering of sequences of activities. Nevertheless, up to the author’s knowledge there have not been prior research
+with tachograph logs.
+7
+Conclusion
+We have presented a novel planning application that brings the worlds of Data Analytics, IoT and Automated Planning
+and Scheduling together. The approach provides support to experts on the task of interpreting what drivers are or have
+been doing by recognising and summarising their activity recorded in an event log.
+11
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Using as a basis our prior work in driver activity recognition, the main contributions exposed in this paper are an
+infringement analysis process with a planning and constraint based approach, the summarisation of temporal activity
+logs using word embeddings and clustering, and the creation of driver profiles based on such summaries. The over-
+all system provides a human readable summary of the driver behaviour under the HoS regulation while explaining
+infractions and the root cause.
+Regarding future work, it is worth noting that the main interest and the ultimate goal of the company is to build an
+intelligent assistant to provide decision support services to both drivers and companies decision makers. This is a
+research direction aligned with the concept of assistive interaction (Freedman and Zilberstein 2017), that advocates
+for the integration of plan recognition and planning. In this way, the recognition of driver’s intent is a previous stage
+needed to respond with a generated plan adapted to the currently recognised task.
+For our next steps we intent to enrich the driver profiling model adding non-tachograph data about the transport service,
+like type of vehicle and cargo. Additionally, we are focused on integrating descriptive support to the assistant, being
+able to suggest drivers plans of actions in compliance with the HoS regulation and considering preference patterns
+extracted from previous personal behaviour.
+References
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+infrastructure: The case study of the smart tachograph in the european union, International Journal of Critical
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+Blei, D. M., Ng, A. Y. and Jordan, M. I.: 2003, Latent dirichlet allocation, Journal of machine Learning research
+3(Jan), 993–1022.
+Campello, R. J. G. B., Moulavi, D. and Sander, J.: 2013, Density-based clustering based on hierarchical density
+estimates, in J. Pei, V. S. Tseng, L. Cao, H. Motoda and G. Xu (eds), Advances in Knowledge Discovery and Data
+Mining, Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 160–172.
+De Koninck, P., vanden Broucke, S. and De Weerdt, J.: 2018, act2vec, trace2vec, log2vec, and model2vec: Repre-
+sentation learning for business processes, in M. Weske, M. Montali, I. Weber and J. vom Brocke (eds), Business
+Process Management, Springer International Publishing, Cham, pp. 305–321.
+Fernandez-Olivares, J. and Perez, R.: 2020, Driver activity recognition by means of temporal htn planning, Proceed-
+ings of the International Conference on Automated Planning and Scheduling 30(1), 375–383.
+URL: https://ojs.aaai.org/index.php/ICAPS/article/view/6683
+Fraley, C. and Raftery, A. E.: 2002, Model-based clustering, discriminant analysis, and density estimation, Journal of
+the American statistical Association 97(458), 611–631.
+Freedman, R. G. and Zilberstein, S.: 2017, Integration of planning with recognition for responsive interaction using
+classical planners, Thirty-First AAAI Conference on Artificial Intelligence.
+Ghallab, M., Nau, D. and Traverso, P.: 2016, Automated Planning and Acting, 1st edn, Cambridge University Press,
+USA.
+Goel, A.: 2018, Legal aspects in road transport optimization in europe, Transportation research part E: logistics and
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+Goel, A. and Irnich, S.: 2017, An exact method for vehicle routing and truck driver scheduling problems, Transporta-
+tion Science 51(2), 737–754.
+Goel, A. and Vidal, T.: 2013, Hours of service regulations in road freight transport: An optimization-based interna-
+tional assessment, Transportation science 48(3), 391–412.
+Knuth, D. E.: 1968, Semantics of context-free languages, Mathematical systems theory 2(2), 127–145.
+Le, Q. and Mikolov, T.: 2014, Distributed representations of sentences and documents, in E. P. Xing and T. Jebara
+(eds), Proceedings of the 31st International Conference on Machine Learning, Vol. 32 of Proceedings of Machine
+Learning Research, PMLR, Bejing, China, pp. 1188–1196.
+URL: https://proceedings.mlr.press/v32/le14.html
+Mbiydzenyuy, G.: 2015, Arrival times with hours of service regulations for truck drivers-tracks and gaps from current
+research, 2015 IEEE 18th International Conference on Intelligent Transportation Systems, pp. 2631–2636.
+Meyer, C. M.: 2011, European Legislation on Driving and Working Hours in Road Transportation, in C. M. Meyer
+(ed.), Vehicle Routing under Consideration of Driving and Working Hours: A Distributed Decision Making Per-
+spective, Gabler, Wiesbaden, pp. 9–24.
+URL: https: // doi. org/ 10. 1007/ 978-3-8349-6732-9_ 2
+12
+
+Discovering and Explaining Driver Behaviour under HoS Regulations
+A PREPRINT
+Mimura, M. and Tanaka, H.: 2018, Leaving all proxy server logs to paragraph vector, Journal of Information Process-
+ing 26, 804–812.
+Omelianenko, S., Kondratenko, Y., Kondratenko, G. and Sidenko, I.: 2019, Advanced system of planning and opti-
+mization of cargo delivery and its iot application, 2019 3rd International Conference on Advanced Information and
+Communications Technologies (AICT), IEEE, pp. 302–307.
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+Proceedings of the 24th International Conference on World Wide Web, pp. 125–126.
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+ance under hos regulations, Proceedings of the 8th International Conference on Vehicle Technology and Intelligent
+Transport Systems - VEHITS,, INSTICC, SciTePress, pp. 463–470.
+Zhou, T. and Zhang, J.: 2019, Analysis of commercial truck drivers’ potentially dangerous driving behaviors based on
+11-month digital tachograph data and multilevel modeling approach, Accident Analysis & Prevention 132, 105256.
+13
+

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+arXiv:2301.11795v1  [math.AP]  27 Jan 2023
+Higher regularity for weak solutions
+to degenerate parabolic problems
+Andrea Gentile - Antonia Passarelli di Napoli∗
+Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
+Universitá di Napoli “Federico II”, via Cintia - 80126 Napoli
+e-mail: andrea.gentile@unina.it,antpassa@unina.it
+January 30, 2023
+Abstract
+In this paper, we study the regularity of weak solutions to the following strongly degen-
+erate parabolic equation
+ut − div
+�
+(|Du| − 1)p−1
++
+Du
+|Du|
+�
+= f
+in ΩT ,
+where Ω is a bounded domain in Rn for n ≥ 2, p ≥ 2 and ( · )+ stands for the positive
+part. We prove the higher differentiability of a nonlinear function of the spatial gradient
+of the weak solutions, assuming only that f ∈ L2
+loc (ΩT ). This allows us to establish the
+higher integrability of the spatial gradient under the same minimal requirement on the
+datum f.
+Key words. Widely degenerate problems. Second order regularity. Higher integrability.
+AMS Classification. 35B45, 35B65, 35D30, 35K10, 35K65
+1
+Introduction
+In this paper, we study the regularity properties of weak solutions u : ΩT → R to the following
+parabolic equation
+ut − div
+�
+(|Du| − 1)p−1
++
+Du
+|Du|
+�
+= f
+in ΩT = Ω × (0, T ),
+(1.1)
+which appears in gas filtration problems taking into account the initial pressure gradient. For a precise
+description of this motivation we refer to [1] and [3, Section 1.1].
+The main feature of this equation is that it possesses a wide degeneracy, coming from the fact that
+its modulus of ellipticity vanishes at all points where |Du| ≤ 1 and hence its principal part of behaves
+like a p-Laplacian operator only at infinity.
+In this paper we address two interrelated aspects of the regularity theory for solutions to parabolic
+problems, namely the higher differentiability and the higher integrability of the weak solutions to
+(1.1), with the main aim of weakening the assumption on the datum f with respect to the available
+literature.
+∗Aknowledgments. The work of the authors is supported by GNAMPA (Gruppo Nazionale per l’Analisi
+Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). The
+authors have been also supported by the Universitá degli Studi di Napoli “Federico II” through the project
+FRA-000022-ALTRI-CDA-752021-FRA-PASSARELLI.
+1
+
+2
+These questions have been exploited in case of non degenerate parabolic problems with quadratic
+growth by Campanato in [9], by Duzaar et al. in [13] in case of superquadratic growth, while Scheven in
+[17] faced the subquadratic growth case. In the above mentioned papers, the problem have been faced
+or in case of homogeneous equations or considering sufficiently regular datum. It is worth mentioning
+that the higher integrability of the gradient of the solution is achieved through an interpolation
+argument, once its higher differentiability is established.
+This strategy has revealed to be successful also for degenerate equations as in (1.1). Indeed the higher
+integrability of the spatial gradient of weak solutions to equation (1.1), has been proven in [3] , under
+suitable assumptions on the datum f in the scale of Sobolev spaces.
+We’d like to recall that a common feature for nonlinear problems with growth rate p > 2 is that the
+higher differentiability is proven for a nonlinear expression of the gradient which takes into account
+the growth of the principal part of the equation.
+Indeed, already for the non degenerate p-Laplace equation, the higher differentiability refers to the
+function Vp (Du) =
+�
+1 + |Du|2� p−2
+4 Du. In case of widely degenerate problems, this phenomenon
+persists, and higher differentiability results, both for the elliptic and the parabolic problems, hold true
+for the function H p
+2 (Du) = (|Du| − 1)
+p
+2
++
+Du
+|Du|. It is worth noticing that, as it can be expected, this
+function of the gradient doesn’t give information on the second regularity of the solutions in the set
+where the equation degenerates. Actually, since every 1-Lipschitz continuous function is a solution to
+the elliptic equation
+div (Hp−1 (Du)) = 0,
+where Hp−1 (Du) = (|Du| − 1)p−1
++
+Du
+|Du|, no more than Lipschitz regularity can be expected.
+Moreover, it is well known that in case of degenerate problems (already for the degenerate p-Laplace
+equation, with p > 2) a Sobolev regularity is required for the datum f in order to get the higher
+differentiability of the solutions (see, for example [8] for elliptic and [3] for parabolic equations).
+Actually, the sharp assumption for the datum in the elliptic setting has been determined in [8] as a
+fractional Sobolev regularity suitably related to the growth exponent p and the dimension n.
+The main aim of this paper is to show that without assuming any kind of Sobolev regularity for the
+datum, but assuming only f ∈ L2, we are still able to obtain higher differentiability for the weak
+solutions but outside a set larger than the degeneracy set of the problem. It is worth mentioning that,
+while for the p-Laplace equation the degeneracy appears for p > 2, here, even in case p = 2, under
+a L2 integrability assumption on the datum f, the local W 2,2 regularity of the solutions cannot be
+obtained.
+Actually, we shall prove the following
+Theorem 1.1. Let n ≥ 2, p ≥ 2 and f ∈ L2
+loc (ΩT ). Moreover, let us assume that
+u ∈ C0 �
+0, T ; L2 (Ω)
+�
+∩ Lp
+loc
+�
+0, T ; W 1,p
+loc (Ω)
+�
+is a weak solution to (1.1). Then, for any δ ∈ (0, 1), we have
+Gδ
+�
+(|Du| − 1 − δ)+
+�
+∈ L2
+loc
+�
+0, T ; W 1,2
+loc (Ω)
+�
+,
+where
+Gδ(t) :=
+ˆ t
+0
+s(s + δ)
+p−2
+2
+√
+1 + δ + s2 ds,
+for every t ≥ 0.
+Moreover the following estimate
+ˆ
+Q R
+16
+��D
+�
+Gδ
+�
+(|Du| − δ − 1)+
+����2 dz
+≤
+c (n, p)
+R2δ2
+�ˆ
+QR
+(|Du|p + 1) dz + 1
+δp
+ˆ
+QR
+|f|2 dz
+�
+,
+(1.2)
+holds for any R > 0 such that QR = QR (z0) ⋐ ΩT .
+
+3
+As already mentioned, the weak solutions of (1.1) are not twice differentiable, and hence it is not
+possible in general to differentiate the equation to estimate the second derivative of the solutions. We
+overcome this difficulty by introducing a suitable family of approximating problems whose solutions
+are regular enough by the standard theory ([11]). The major effort in the proof of previous Theorem
+is to establish suitable estimates for the solutions of the regularized problems that are uniform with
+respect to the approximation’s parameter. Next, we take advantage from these uniform estimates
+in the use of a comparison argument aimed to bound the difference quotient of a suitable nonlinear
+function of the gradient of the solution that vanishes in the set { |Du| ≤ 1 + δ }, with δ > 0.
+Roughly speaking, due to the weakness of our assumption on the datum, we only get the higher
+differentiability of a nonlinear function of the gradient of the solutions that vanishes in a set which is
+larger with respect to that of the degeneracy of the problem. This is quite predictable, since the same
+kind of phenomenon occurs in the setting of widely degenerate elliptic problems (see, for example
+[10]).
+Anyway, as a consequence of the higher differentiability result in Theorem 1.1, we establish a higher
+integrability result for the spatial gradient of the solution to equation (1.1), which is the following
+Theorem 1.2. Under the assumptions of Theorem 1.1, we have
+Du ∈ L
+p+ 4
+n
+loc
+(ΩT )
+with the following estimate
+ˆ
+Q ρ
+2
+|Du|p + 4
+n dz ≤ c (n, p)
+ρ
+2(n+2)
+n
+�ˆ
+Q2ρ
+�
+1 + |Du|p + |f|2�
+dz
+� 2
+n +1
+,
+(1.3)
+for every parabolic cylinder Q2ρ (z0) ⋐ ΩT , with a constant c = c(n, p).
+The proof of previous Theorem consists in using an interpolation argument with the aim of establishing
+an estimate for the Lp+ 4
+n norm of the gradient of the solutions to the approximating problems that
+is preserved in the passage to the limit.
+We conclude mentioning that the elliptic version of our equation naturally arises in optimal transport
+problems with congestion effects, and the regularity properties of its weak solutions have been widely
+investigated (see e.g. [2, 4, 6, 8]). Moreover, we’d like to stress that, for sake of clarity, we confine
+ourselves to equation (1.1), but we believe that our techniques apply as well to a general class of
+equations with a widely degenerate structure.
+2
+Notations and preliminaries
+In this paper we shall denote by C or c a general positive constant that may vary on different
+occasions. Relevant dependencies on parameters will be properly stressed using parentheses or sub-
+scripts. The norm we use on Rn will be the standard Euclidean one and it will be denoted by | · |. In
+particular, for the vectors ξ, η ∈ Rn, we write ⟨ξ, η⟩ for the usual inner product and |ξ| := ⟨ξ, ξ⟩
+1
+2 for
+the corresponding Euclidean norm.
+For points in space-time, we will use abbreviations like z = (x, t) or z0 = (x0, t0), for spatial variables
+x, x0 ∈ Rn and times t, t0 ∈ R. We also denote by B (x0, ρ) = Bρ (x0) = { x ∈ Rn : |x − x0| < ρ } the
+open ball with radius ρ > 0 and center x0 ∈ Rn; when not important, or clear from the context, we
+shall omit to indicate the center, denoting: Bρ ≡ B (x0, ρ). Unless otherwise stated, different balls in
+the same context will have the same center. Moreover, we use the notation
+Qρ (z0) := Bρ (x0) ×
+�
+t0 − ρ2, t0
+�
+,
+z0 = (x0, t0) ∈ Rn × R,
+ρ > 0,
+for the backward parabolic cylinder with vertex (x0, t0) and width ρ. We shall sometimes omit the
+dependence on the vertex when the cylinders occurring share the same vertex. Finally, for a cylinder
+Q = A × (t1, t2), where A ⊂ Rn and t1 < t2, we denote by
+∂parQ := (A × { t1 }) ∪ (∂A × [t1, t2])
+the usual parabolic boundary of Q, which is nothing but the standard topological boundary without
+the upper cap A × { t2 }.
+
+4
+We now recall some tools that will be useful to prove our results.
+For the auxiliary function Hλ : Rn → Rn defined as
+Hλ(ξ) :=
+
+
+
+
+
+
+
+(|ξ| − 1)λ
++
+ξ
+|ξ|
+if
+ξ ∈ Rn \ {0} ,
+0
+if
+ξ = 0,
+(2.1)
+where λ > 0 is a parameter, we record the following estimates (see [7, Lemma 4.1]):
+Lemma 2.1. If 2 ≤ p < ∞, then for every ξ, η ∈ Rn it holds
+⟨Hp−1(ξ) − Hp−1(η), ξ − η⟩ ≥ 4
+p2
+���H p
+2 (ξ) − H p
+2 (η)
+���
+2
+,
+|Hp−1(ξ) − Hp−1(η)| ≤ (p − 1)
+����H p
+2 (ξ)
+���
+p−2
+p
++
+���H p
+2 (η)
+���
+p−2
+p � ���H p
+2 (ξ) − H p
+2 (η)
+��� .
+we record the following estimates (see [4, Lemma 2.8])
+Lemma 2.2. Let ξ, η ∈ Rk with |ξ| > 1. Then, we have
+|Hp−1(ξ) − Hp−1(η)| ≤ c(p)
+�
+(|ξ| − 1) + (|η| − 1)+
+�p−1
+|ξ| − 1
+|ξ − η|
+and
+⟨Hp−1(η) − Hp−1(ξ), ·η − ξ⟩ ≥ min { 1, p − 1 }
+2p+1
+(|ξ| − 1)p
+|ξ| (|ξ| + |η|) |η − ξ|2 .
+Definition 2.3. With the use of (2.1), a function u ∈ C0 �
+0, T ; L2 (Ω)
+�
+∩Lp �
+0, T ; W 1,p (Ω)
+�
+is a weak
+solution of equation (1.1) if
+ˆ
+ΩT
+(u · ∂tϕ − ⟨Hp−1 (Du) , Dϕ⟩) dz = −
+ˆ
+ΩT
+fϕ dz
+(2.2)
+for every ϕ ∈ C∞
+0 (ΩT ).
+In the following, we shall also use the well known auxiliary function Vp : Rn → Rn defined as
+Vp(ξ) :=
+�
+1 + |ξ|2� p−2
+4 ξ,
+where p ≥ 2. We have the following result.
+Lemma 2.4.
+For every ξ, η ∈ Rn there hold
+1
+c1 (p) |Vp(ξ) − Vp(η)|2
+≤
+�
+1 + |ξ|2 + |η|2� p−2
+2 |ξ − η|2
+≤
+c1(p)
+��
+1 + |ξ|2� p−2
+2 ξ −
+�
+1 + |η|2� p−2
+2 η, ξ − η
+�
+,
+We refer to [16, Chapter 12] or to [15, Lemma 9.2] for a proof of these fundamental inequalities.
+For further needs, we also record the following interpolation inequality whose proof can be found
+in [12, Proposition 3.1]
+Lemma 2.5.
+Assume that the function v : Qr(z0) ∪ ∂parQr(z0) → R satisfies
+v ∈ L∞ �
+t0 − r2, t0; Lq (Br (x0))
+�
+∩ Lp �
+t0 − r2, t0; W 1,p
+0
+(Br (x0))
+�
+for some exponents 1 ≤ p, q < ∞ . Then the following estimate
+ˆ
+Qr(z0)
+|v|p+ pq
+n dz ≤ c
+�
+sup
+s∈(t0−r2,t0)
+ˆ
+Br(x0)
+|v(x, s)|q dx
+� p
+n ˆ
+Qr(z0)
+|Dv|p dz
+holds true for a positive constant c depending at most on n, p and q.
+
+5
+2.1
+Difference quotients
+We recall here the definition and some elementary properties of the difference quotients (see, for ex-
+ample, [15, Chapter 8]).
+Definition 2.6. For every function F : Rn → RN the finite difference operator in the direction xs is
+defined by
+τs,hF(x) = F (x + hes) − F(x),
+where h ∈ R, es is the unit vector in the direction xs and s ∈ {1, . . . , n}.
+The difference quotient of F with respect to xs is defined for h ∈ R \ {0} as
+∆s,hF(x) = τs,hF(x)
+h
+.
+We shall omit the index s when it is not necessary, and simply write τhF(x) = F(x + h) − F(x) and
+|∆hF(x)| = |τhF(x)|
+|h|
+for h ∈ Rn.
+Proposition 2.7. Let F ∈ W 1,p (Ω), with p ≥ 1, and let us set
+Ω|h| := { x ∈ Ω : dist (x, ∂Ω) > |h| } .
+Then:
+(a) ∆hF ∈ W 1,p �
+Ω|h|
+�
+and
+Di(∆hF) = ∆h(DiF),
+for every i ∈ {1, . . . , n} .
+(b) If at least one of the functions F or G has support contained in Ω|h|, then
+ˆ
+Ω
+F ∆hG dx = −
+ˆ
+Ω
+G ∆−hF dx.
+(c) We have
+∆h (FG) (x) = F (x + hes) ∆hG(x) + G(x)∆hF(x).
+The next result about the finite difference operator is a kind of integral version of Lagrange Theorem
+(see [15, Lemma 8.1]).
+Lemma 2.8. If 0 < ρ < R, |h| < R − ρ
+2
+, 1 < p < +∞, and F ∈ W 1,p �
+BR, RN�
+, then
+ˆ
+Bρ
+|τhF(x)|p dx ≤ cp(n) |h|p
+ˆ
+BR
+|DF(x)|p dx.
+Moreover
+ˆ
+Bρ
+|F(x + hes)|p dx ≤
+ˆ
+BR
+|F(x)|p dx.
+We conclude this section with the following fundamental result, whose proof can be found in [15,
+Lemma 8.2]:
+Lemma 2.9. Let F : Rn → RN, F ∈ Lp �
+BR, RN�
+with 1 < p < +∞. Suppose that there exist
+ρ ∈ (0, R) and a constant M > 0 such that
+n
+�
+s=1
+ˆ
+Bρ
+|τs,hF(x)|p dx ≤ M p |h|p
+for every h, with |h| < R − ρ
+2
+. Then F ∈ W 1,p �
+Bρ, RN�
+and
+∥DF∥Lp(Bρ) ≤ M.
+Moreover
+∆s,hF → DsF
+strongly in Lp
+loc (BR) , as h → 0,
+for each s ∈ {1, . . . , n}.
+
+6
+2.2
+Some auxiliary functions and related algebraic inequalities
+In this section we introduce some auxiliary functions and we list some of their properties, that will be
+used in what follows.
+For any k > 1 and for s ∈ [0, +∞), let us consider the function
+gk(s) =
+s2
+k + s2 ,
+(2.3)
+for which we record the following
+Lemma 2.10. Let k > 1, and let gk be the function defined by (2.3). Then for every A, B ≥ 0 the
+following Young’s type inequality
+A · B[s · g′
+k ((s − k)+)] ≤ 2
+√
+2k
+�
+αA2gk ((s − k)+) + ασA2 + cαB2�
+,
+(2.4)
+holds for every parameters α, σ > 0 with a constant cα independent of σ. Moreover, there exists a
+constant ck > 0, depending on k, such that
+sg′
+k
+��
+s2 − k
+�
++
+�
+≤ ck,
+∀s ≥ 0.
+(2.5)
+Proof. Since
+g′
+k(s) =
+2ks
+(k + s2)2 ,
+(2.6)
+both the conclusions trivially hold for s ≤
+√
+k. Now assume that s >
+√
+k and note that Young’s
+inequality implies
+A · B [s · g′
+k ((s − k)+)]
+=
+A · B · s · g′
+k ((s − k)+) [σ + (s − k)+]
+1
+2
+[σ + (s − k)+]
+1
+2
+≤
+αA2s · g′
+k ((s − k)+) [σ + (s − k)+] + cα
+B2s · g′
+k ((s − k)+)
+[σ + (s − k)+]
+=
+αA2s · g′
+k ((s − k)+) (s − k)+ + ασA2s · g′
+k ((s − k)+)
++cα
+B2s · g′
+k ((s − k)+)
+[σ + (s − k)+]
+≤
+αA2
+2ks(s − k)2
++
+�
+k + (s − k)2
++
+�2 + ασA2
+2ks(s − k)+
+�
+k + (s − k)2
++
+�2
++cαB2
+2ks
+�
+k + (s − k)2
++
+�2
+(s − k)+
+[σ + (s − k)+],
+(2.7)
+where we used the explicit expression of g′
+k(s) at (2.6). Recalling (2.3) and since
+t
+k + t2 ≤ 1, from
+(2.7) we deduce
+A · B [s · g′
+k ((s − k)+)]
+≤
+αA2
+2ks
+k + (s − k)2
++
+gk ((s − k)+)
++ασA2
+2ks
+k + (s − k)2
++
++ cαB2
+2ks
+k + (s − k)2
++
+.
+(2.8)
+Setting h(s) =
+s
+k + (s − k)2
++
+, we can easily check that
+h(k) = 1,
+lim
+s→+∞ h(s) = 0,
+max
+s∈[k,+∞) h(s) = h
+��
+k2 + k
+�
+= 1
+2
+�
+1 +
+�
+1 + 1
+k
+�
+<
+√
+2
+
+7
+and so
+2ks
+k + (s − k)2
++
+≤ 2
+√
+2k
+∀s > k.
+Inserting this in (2.8), we get (2.4).
+In order to prove (2.5), let us notice that, recalling (2.6), we have
+sg′ ��
+s2 − k
+�
++
+�
+=
+2ks
+�
+s2 − k
+�
++
+�
+k + (s2 − k)2
++
+�2 .
+So, since the function sg′ ��
+s2 − k
+�
++
+�
+is continuous in the interval
+�
+s ≥ 0
+�� s2 > k
+�
+=
+�√
+k, +∞
+�
+and
+lim
+s→+∞
+2ks
+�
+s2 − k
+�
++
+�
+k + (s2 − k)2
++
+�2 = 0,
+then there exists a constant ck > 0 such that
+sg′ ��
+s2 − k
+�
++
+�
+≤ ck
+for every s ≥ 0,
+which is the conclusion.
+For any δ > 0, let us define
+Gδ(t) :=
+ˆ t
+0
+s(s + δ)
+p−2
+2
+√
+1 + δ + s2 ds,
+for t ≥ 0,
+(2.9)
+and observe that
+G′
+δ(t) = t(t + δ)
+p−2
+2
+√
+1 + δ + t2 .
+(2.10)
+Next Lemma relates the function Gδ (|ξ|) with H p
+2 (ξ).
+Lemma 2.11. Let Gδ be the function defined by (2.9) and H p
+2 be the one defined in (2.1) with λ = p
+2.
+Then we have
+��Gδ
+�
+(|ξ| − δ − 1)+
+�
+− Gδ
+�
+(|η| − δ − 1)+
+���2 ≤ cp
+���H p
+2 (ξ) − H p
+2 (η)
+���
+2
+(2.11)
+for any ξ, η ∈ Rn.
+Proof. If |ξ| < 1 + δ and |η| < 1 + δ there is nothing to prove. So will assume that |ξ| > 1 + δ, and
+without loss of generality we may suppose that |η| ≤ |ξ|. Since Gδ(t) is increasing, we have
+��Gδ (|ξ| − 1 − δ) − Gδ
+�
+(|η| − 1 − δ)+
+���
+=
+Gδ (|ξ| − 1 − δ) − Gδ
+�
+(|η| − 1 − δ)+
+�
+=
+ˆ |ξ|−1−δ
+(|η|−1−δ)+
+s(s + δ)
+p−2
+2
+√
+1 + δ + s2 ds
+≤
+ˆ |ξ|−1−δ
+(|η|−1−δ)+
+(s + δ)
+p−2
+2 ds
+=
+2
+p
+�
+(|ξ| − 1)
+p
+2 −
+�
+(|η| − δ − 1)+ + δ
+� p
+2 �
+.
+Now, it can be easily checked that
+(|ξ| − 1)
+p
+2 −
+�
+(|η| − δ − 1)+ + δ
+� p
+2
+
+8
+=
+
+
+
+
+
+(|ξ| − 1)
+p
+2 − δ
+p
+2
+if
+|ξ| > δ + 1 and |η| ≤ δ + 1
+(|ξ| − 1)
+p
+2 − (|η| − 1)
+p
+2
+if
+|ξ| > δ + 1 and |η| > δ + 1.
+In the first case, we have
+���(|ξ| − 1)
+p
+2 − δ
+p
+2
+���
+=
+(|ξ| − 1)
+p
+2 − δ
+p
+2 ≤ (|ξ| − 1)
+p
+2 − (|η| − 1)
+p
+2
++
+=
+���H p
+2 (ξ)
+��� −
+���H p
+2 (η)
+��� ≤
+���H p
+2 (η) − H p
+2 (ξ)
+��� ,
+while, in the second,
+(|ξ| − 1)
+p
+2 −
+�
+(|η| − δ − 1)+ + δ
+� p
+2 =
+���H p
+2 (ξ)
+��� −
+���H p
+2 (η)
+��� ≤
+���H p
+2 (η) − H p
+2 (ξ)
+��� .
+Therefore,
+��Gδ
+�
+(|ξ| − δ − 1)+
+�
+− Gδ
+�
+(|η| − δ − 1)+
+���2 ≤ cp
+���H p
+2 (ξ) − H p
+2 (η)
+���
+2
+for every ξ, η ∈ Rn, which is (2.11).
+Arguing as in [14, Lemma 2.1], we prove the following.
+Lemma 2.12. Let 0 < δ ≤ 1 and p ≥ 2. Then the following inequalities hold
+cp,δ(t + δ)
+p
+2 − ˜cp,δ ≤ Gδ(t) ≤ 2
+p(t + δ)
+p
+2
+with constants ˜cp,δ and cp,δ < 2
+p depending on p and δ.
+Proof. If p = 2, one can easily calculate
+Gδ(t) =
+ˆ t
+0
+s
+√
+1 + δ + s2 ds =
+��
+1 + δ + s2
+�t
+0 =
+�
+1 + δ + t2 −
+√
+1 + δ,
+from which immediately follows
+1
+2 (t + δ) − 1
+2
+�√
+1 + δ + δ
+�
+≤ Gδ(t) ≤ t + δ.
+Let p > 2. The right inequality is a simple consequence of the trivial bound
+s
+√
+1+δ+s2 < 1. For the
+left inequality we start observing that
+�
+1 + δ + s2 ≤
+√
+1 + δ + s
+=⇒
+Gδ(t) ≥
+ˆ t
+0
+s (s + δ)
+p−2
+2
+√
+1 + δ + s ds.
+Now, we calculate the integral in previous formula. By the change of variable r =
+√
+1 + δ + s, we get
+ˆ t
+0
+s (s + δ)
+p−2
+2
+√
+1 + δ + s ds =
+ˆ t+
+√
+1+δ
+√
+1+δ
+�
+r −
+√
+1 + δ
+� �
+r −
+√
+1 + δ + δ
+� p−2
+2
+r
+ds
+=
+ˆ t+
+√
+1+δ
+√
+1+δ
+�
+r −
+√
+1 + δ + δ
+� p−2
+2
+ds −
+√
+1 + δ
+ˆ t+
+√
+1+δ
+√
+1+δ
+�
+r −
+√
+1 + δ + δ
+� p−2
+2
+r
+ds
+≥
+2
+p
+��
+r −
+√
+1 + δ + δ
+� p
+2 �t+√1+δ
+√1+δ
+−
+√
+1 + δ
+ˆ t+
+√
+1+δ
+√
+1+δ
+�
+r −
+√
+1 + δ + δ
+� p
+2 −2
+ds,
+since 0 < δ ≤ 1, we have δ ≤
+√
+1 + δ and therefore r −
+√
+1 + δ + δ ≤ r. Calculating the last integral
+in previous formula, we get
+ˆ t
+0
+s(s + δ)
+p−2
+2
+√
+1 + δ + s ds
+
+9
+≥
+2
+p
+��
+r −
+√
+1 + δ + δ
+� p
+2 �t+√1+δ
+√1+δ
+− 2
+√
+1 + δ
+p − 2
+��
+r −
+√
+1 + δ + δ
+� p
+2 −1�t+√1+δ
+√1+δ
+=
+2
+p
+�
+(t + δ)
+p
+2 − δ
+p
+2
+�
+− 2
+√
+1 + δ
+p − 2
+�
+(t + δ)
+p
+2 −1 − δ
+p
+2 −1�
+=
+2
+p(t + δ)
+p
+2 − 2
+√
+1 + δ
+p − 2 (t + δ)
+p
+2 −1 + 2
+√
+1 + δ
+p − 2 δ
+p
+2 −1 − 2
+pδ
+p
+2 .
+Therefore the lemma will be proven if there exists a constant cp,δ < 2
+p such that
+cp,δ(t + δ)
+p
+2 ≤ 2
+p(t + δ)
+p
+2 − 2
+√
+1 + δ
+p − 2 (t + δ)
+p
+2 −1 + 2
+√
+1 + δ
+p − 2 δ
+p
+2 −1 − 2
+pδ
+p
+2
+which, setting
+h(t) = 2
+√
+1 + δ
+p − 2 (t + δ)
+p
+2 −1 +
+�
+cp,δ − 2
+p
+�
+(t + δ)
+p
+2 ,
+is equivalent to prove that there exists cp,δ such that
+h(t) ≤ 2
+√
+1 + δ
+p − 2 δ
+p
+2 −1 − 2
+pδ
+p
+2 .
+It is easy to check that h(t) attains his maximum for t + δ = 2
+√
+1 + δ
+2 − pcp,δ
+and so
+h(t) ≤ h
+� 2
+√
+1 + δ
+2 − pcp,δ
+− δ
+�
+=
+�
+2
+√
+1 + δ
+� p
+2 �
+1
+2 − pcp,δ
+� p−2
+2
+2
+p (p − 2)
+Therefore, to complete the proof it’s enough to solve the following equation
+�
+2
+√
+1 + δ
+� p
+2 �
+1
+2 − pcp,δ
+� p−2
+2
+2
+p (p − 2) = 2
+√
+1 + δ
+p − 2 δ
+p
+2 −1 − 2
+pδ
+p
+2
+which is equivalent to
+1
+2 − pcp,δ
+=
+�
+δ
+2
+√
+1 + δ
+�
+p
+p−2 �
+p
+�√
+1 + δ − δ
+�
+δ
++ 2
+�
+2
+p−2
+that, for 0 < δ < 1, admits a unique solution cp,δ < 2
+p.
+3
+The regularization
+For ε > 0, we introduce the sequence of operators
+Aε(ξ) := (|ξ| − 1)p−1
++
+ξ
+|ξ| + ε
+�
+1 + |ξ|2� p−2
+2 ξ
+and by
+uε ∈ C0 �
+t0 − R2, t0; L2 (BR)
+�
+∩ Lp �
+t0 − R2, t0; u + W 1,p
+0
+(BR)
+�
+we denote the unique solution to the corresponding problems
+
+
+
+uε
+t − div (Aε (Duε)) = f ε
+in QR (z0)
+uε = u
+in ∂parQR (z0)
+(3.1)
+where QR (z0) ⋐ ΩT with R < 1, f ε = f ∗ ρε with ρε the usual sequence of mollifiers. One can easily
+check that the operator Aε satisfies p-growth and p-ellipticity assumptions with constants depending
+
+10
+on ε.
+Therefore, by the results in [13], we have
+Vp (Duε) ∈ L2
+loc
+�
+0, T ; W 1,2
+loc (BR (x0) , Rn)
+�
+and
+|Duε| ∈ L
+p+ 4
+n
+loc
+(QR)
+and, by the definition of Vp(ξ), this yields
+DVp (Duε) ≈
+�
+1 + |Duε|2� p−2
+4 D2uε ∈ L2
+loc
+�
+QR; Rn×n�
+=⇒
+��D2uε�� ∈ L2
+loc (QR)
+(3.2)
+By virtue of [3, Theorem 1.1], we also have H p
+2 (Duε) ∈ L2
+loc
+�
+0, T ; W 1,2
+loc (Ω, Rn)
+�
+and, by the definition
+of H p
+2 (ξ), it follows
+���DH p
+2 (Du)
+��� ≤ cp (|Duε| − 1)
+p−2
+2
++
+|D2uε| ∈ L2
+loc
+�
+QR; Rn×n�
+.
+(3.3)
+3.1
+Uniform a priori estimates
+The first step in the proof of Theorem 1.1 is the following estimate for solutions to the regularized
+problem (3.1).
+Lemma 3.1. Let uε ∈ C0 �
+t0 − R2, t0; L2 (BR)
+�
+∩ Lp �
+t0 − R2, t0; u + W 1,p
+0
+(BR)
+�
+be the unique solu-
+tion to (3.1). Then the following estimate
+sup
+τ∈(t0−4ρ2,t0)
+ˆ
+Bρ
+�
+|Duε(x, τ)|2 − 1 − δ
+�
++ dx
++
+ˆ
+Qρ
+��D
+�
+Gδ
+�
+(|Duε| − δ − 1)+
+����2 dz
+≤
+c
+ρ2
+�ˆ
+Q2ρ
+(1 + |Duε|p) dz + δ2−p
+ˆ
+Q2ρ
+|f ε|2 dz
+�
+(3.4)
+holds for any ε ∈ (0, 1] and for every Qρ ⋐ Q2ρ ⋐ QR, with a constant c = c(n, p) independent of ε.
+Proof. The weak formulation of (3.1) reads as
+ˆ
+QR
+(uε · ∂tϕ − ⟨Aε (Duε) , Dϕ⟩) dz = −
+ˆ
+QR
+f ε · ϕ dz
+for any test function ϕ ∈ C∞
+0 (QR).
+Recalling the notation used in (2.2), and replacing ϕ with
+∆−hϕ = τ−hϕ
+h
+for a sufficiently small h ∈ R \ { 0 }, by virtue of the properties of difference quotients,
+we have
+ˆ
+QR
+�
+∆huε · ∂tϕ − ⟨∆hHp−1 (Duε) , Dϕ⟩ − ε
+�
+∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+�
+, Dϕ
+��
+dz
+=
+−
+ˆ
+QR
+f ε · ∆−hϕ dz.
+(3.5)
+Arguing as in [13, Lemma 5.1], from (3.5) we get
+ˆ
+QR
+∂t∆huε · ϕ dz +
+ˆ
+QR
+⟨∆hHp−1 (Duε) , Dϕ⟩ dz
++ε
+ˆ
+QR
+�
+∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+�
+, Dϕ
+�
+dz =
+ˆ
+QR
+f ε · ∆−hϕ dz.
+For Φ ∈ W 1,∞
+0
+(QR) non negative and g ∈ W 1,∞ (R) non negative and non decreasing, we choose
+ϕ = Φ · ∆huε · g
+�
+|∆huε|2�
+in previous identity, thus getting
+ˆ
+QR
+∂t (∆huε) ∆huε · g
+�
+|∆huε|2�
+Φ dz
+
+11
++
+ˆ
+QR
+�
+∆hHp−1 (Duε) , D
+�
+Φ∆huεg
+�
+|∆huε|2���
+dz
++ε
+ˆ
+QR
+�
+∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+�
+, D
+�
+Φ∆hug
+�
+|∆huε|2���
+dz
+=
+ˆ
+QR
+f ε · ∆−h
+�
+Φ∆huε · g
+�
+|∆huε|2��
+dz,
+i.e.
+ˆ
+QR
+∂t (∆huε) ∆huε · g
+�
+|∆huε|2�
+Φ dz
++
+ˆ
+QR
+Φ
+�
+∆hHp−1 (Duε) , ∆hDuε · g
+�
+|∆huε|2��
+dz
++ε
+ˆ
+QR
+Φ
+�
+∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+�
+, ∆hDuε · g
+�
+|∆huε|2��
+dz
++2
+ˆ
+QR
+Φ
+�
+∆hHp−1 (Duε) , |∆huε|2 ∆hDuε · g′ �
+|∆huε|2��
+dz
++2ε
+ˆ
+QR
+Φ
+�
+∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+�
+, |∆huε|2 ∆hDuε · g′ �
+|∆huε|2��
+dz
+=
+−
+ˆ
+QR
+�
+∆hHp−1 (Duε) , DΦ · ∆huε · g
+�
+|∆huε|2��
+dz
+−ε
+ˆ
+QR
+�
+∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+�
+, DΦ · ∆huε · g
+�
+|∆huε|2��
+dz
++
+ˆ
+QR
+f ε · ∆−h
+�
+Φ∆huε · g
+�
+|∆huε|2��
+dz,
+(3.6)
+that we rewrite as follows
+Jh,1 + Jh,2 + Jh,3 + Jh,4 + Jh,5 = −Jh,6 − Jh,7 + Jh,8.
+Arguing as in [5],the first integral in equation (3.6) can be expressed as follows
+Jh,1
+=
+ˆ
+QR
+∂t (∆huε) ∆huε · g
+�
+|∆huε|2�
+Φ dz = 1
+2
+ˆ
+QR
+∂t
+�
+|∆huε|2�
+· g
+�
+|∆huε|2�
+Φ dz
+=
+1
+2
+ˆ
+QR
+∂t
+�ˆ |∆huε|2
+0
+g(s) ds
+�
+Φ dz = −1
+2
+ˆ
+QR
+�ˆ |∆huε|2
+0
+g(s) ds
+�
+∂tΦ dz.
+Using Lemma 2.2, since Φ, g are non negative, we have
+Jh,2 ≥
+ˆ
+QR
+Φ · g
+�
+|∆huε|2�
+|∆hDuε|2
+(|Duε| − 1)p
+|Duε| (|Duε| + |Duε(x + h)|) dz.
+The right inequality in the assertion of Lemma 2.4 yields
+Jh,3 ≥ εcp
+ˆ
+QR
+Φ · g
+�
+|∆huε|2�
+|∆hVp (Duε)|2 dz
+Moreover, again by Lemmas 2.2 and 2.4 and the fact that g′(s) ≥ 0, we infer
+Jh,4 + Jh,5 ≥ 0.
+Therefore (3.6) implies
+−1
+2
+ˆ
+QR
+�ˆ |∆huε|2
+0
+g(s) ds
+�
+∂tΦ dz
++
+ˆ
+QR
+Φ · g
+�
+|∆huε|2�
+|∆hDuε|2
+(|Duε| − 1)p
+|Duε| (|Duε| + |Duε(x + h)|) dz
+
+12
++cpε
+ˆ
+QR
+Φ · g
+�
+|∆huε|2�
+|∆hVp (Duε)|2 dz
+≤
+ˆ
+QR
+|DΦ| |∆hHp−1 (Duε)| |∆huε| · g
+�
+|∆huε|2�
+dz
++ε
+ˆ
+QR
+|DΦ|
+����∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+����� |∆huε| · g
+�
+|∆huε|2�
+dz
++
+ˆ
+QR
+|f ε|
+���∆−h
+�
+Φ∆huε · g
+�
+|∆huε|2����� dz.
+(3.7)
+Now let us consider a parabolic cylinder Qρ (z0) ⋐ Q2ρ (z0) ⋐ QR (z0) with ρ < 2ρ < R and t0 > 0.
+For a fixed time τ ∈
+�
+t0 − 4ρ2, t0
+�
+and θ ∈ (0, t0 − τ), we choose Φ(x, t) = η2(x)χ(t)˜χ(t) with η ∈
+C∞
+0 (B2ρ (x0)), 0 ≤ η ≤ 1, χ ∈ W 1,∞ ([0, T ]) with ∂tχ ≥ 0 and ˜χ a Lipschitz continuous function
+defined, for 0 < τ < τ + θ < T , as follows
+˜χ(t) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+1
+if
+t ≤ τ
+1 − t − τ
+θ
+if
+τ < t ≤ τ + θ
+0
+if
+τ + θ < t ≤ T
+so that (3.7) yields
+Ih,1 + Ih,2 + Ih,3
+:=
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |∆huε(x,τ)|2
+0
+g(s) ds
+�
+dx
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|∆huε|2�
+|∆hDuε|2
+(|Duε| − 1)p
+|Duε| (|Duε| + |Duε(x + h)|) dz
++cpε
+ˆ
+Qτ η2χ(t)g
+�
+|∆huε|2�
+|∆hVp (Duε)|2 dz
+≤
+2
+ˆ
+Qτ ηχ(t) |Dη| |∆hHp−1 (Duε)| |∆huε| · g
+�
+|∆huε|2�
+dz
++2ε
+ˆ
+Qτ ηχ(t) |Dη|
+����∆h
+��
+1 + |Duε|2� p−2
+2 Duε
+����� |∆huε| · g
+�
+|∆huε|2�
+dz
++
+ˆ
+Qτ χ(t) |f ε|
+���∆−h
+�
+η2∆huε · g
+�
+|∆huε|2����� dz
++1
+2
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |∆huε|2
+0
+g(s) ds
+�
+dz
+=:
+Ih,4 + Ih,5 + Ih,6 + Ih,7,
+(3.8)
+where we used the notation Qτ = B2ρ (x0) ×
+�
+t0 − 4ρ2, τ
+�
+.
+Since g ∈ W 1,∞ ([0, ∞)), by (3.2), by the last assertion of Lemma 2.9 and by Fatou’s Lemma, we have
+lim inf
+h→0 (Ih,1 + Ih,2 + Ih,3)
+≤
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� ��D2uε��2 (|Duε| − 1)p
+|Duε|2
+dz
++cpε
+ˆ
+Qτ η2χ(t)g
+�
+|Duε|2�
+|DVp (Duε)|2 dz.
+(3.9)
+and
+lim
+h→0 Ih,7 = 1
+2
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz.
+(3.10)
+Now let us observe that
+|DHp−1 (Duε)| ≤ cp (|Duε| − 1)p−2
++
+��D2u�
+(3.11)
+
+13
+and, using Hölder’s inequality with exponents
+�
+2(p−1)
+p−2 , 2(p−1)
+p
+�
+, we have
+ˆ
+BR
+|DHp−1 (Duε)|
+p
+p−1 dx
+≤
+cp
+ˆ
+BR
+�
+(|Duε| − 1)p−2
++
+��D2u�
+�
+p
+p−1 dx
+≤
+cp
+�ˆ
+BR
+(|Duε| − 1)p
++ dx
+�
+p−2
+2(p−1)
+·
+�ˆ
+BR
+�
+(|Duε| − 1)
+p−2
+2
++
+��D2u�
+�2
+dx
+�
+p
+2(p−1)
+,
+and since, by (3.3), the right hand side of previous inequality is finite again by Lemma 2.9, we have
+∆hHp−1 (Duε) → DHp−1 (Duε)
+strongly in
+L2 �
+0, T ; L
+p
+p−1 (BR)
+�
+as h → 0,
+which, since ∆huε → Duε strongly in L2 (0, T ; Lp (BR)) as h → 0, implies
+lim
+h→0 Ih,4 = 2
+ˆ
+Qτ ηχ(t) |Dη| |DHp−1 (Duε)| |Duε| g
+�
+|Duε|2�
+dz.
+(3.12)
+Using similar arguments, we can check that
+lim
+h→0 Ih,5 = 2ε
+ˆ
+Qτ ηχ(t) |Dη|
+����D
+��
+1 + |Duε|2� p−2
+2 Duε
+����� |Duε| · g
+�
+|Duε|2�
+dz.
+(3.13)
+Now, by Proposition 2.7(c), it holds
+���∆−h
+�
+η2∆huε · g
+�
+|∆huε|2�����
+≤
+c∥Dη∥∞ |∆huε|
+���g
+�
+|∆huε|2����
++c |∆−h (∆huε)|
+���g
+�
+|∆huε|2����
++c |∆huε|2 ���g′ �
+|∆huε|2���� |∆hDuε| .
+and choosing g such that
+sg′ �
+s2�
+≤ M,
+(3.14)
+for a positive constant M, we have
+���∆−h
+�
+η2∆huε · g
+�
+|∆huε|2�����
+≤
+c∥Dη∥∞ |∆huε|
+���g
+�
+|∆huε|2����
++c |∆−h (∆huε)|
+���g
+�
+|∆huε|2����
++cM |∆huε| |∆−hDuε|
+(3.15)
+Since ∆huε → Duε, ∆−h (∆huε) → D2uε, ∆−hDuε → D2uε strongly in L2 �
+0, T ; L2
+loc (Ω)
+�
+as h → 0,
+and f ε ∈ C∞ (ΩT ), thanks to (3.15), we have
+lim
+h→0 Ih,6 =
+ˆ
+Qτ χ(t) |f ε|
+���D
+�
+η2Duε · g
+�
+|Duε|2����� dz.
+(3.16)
+So, collecting (3.9), (3.10), (3.12), (3.13) and (3.16), we can pass to the limit as h → 0 in (3.8), thus
+getting
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� ��D2uε��2 (|Duε| − 1)p
+|Duε|2
+dz
++cpε
+ˆ
+Qτ η2χ(t)g
+�
+|Duε|2�
+|DVp (Duε)|2 dz
+
+14
+≤
+2
+ˆ
+Qτ ηχ(t) |Dη| |DHp−1 (Duε)| |Duε| · g
+�
+|Duε|2�
+dz
++2ε
+ˆ
+Qτ ηχ(t) |Dη|
+����D
+��
+1 + |Duε|2� p−2
+2 Duε
+����� |Duε| · g
+�
+|Duε|2�
+dz
++
+ˆ
+Qτ χ(t) |f ε|
+���D
+�
+η2Duε · g
+�
+|Duε|2����� dz
++1
+2
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz
+=:
+˜I1 + ˜I2 + ˜I3 + ˜I4,
+(3.17)
+for every g ∈ W 1,∞(0, +∞) such that (3.14) holds true. Now, by (3.11) and by Young’s inequality,
+we have
+˜I1 + ˜I2
+≤
+cp
+ˆ
+Qτ ηχ(t) |Dη| (|Duε| − 1)p−2
++
+��D2uε�� |Duε| · g
+�
+|Duε|2�
+dz
++cp · ε
+ˆ
+Qτ ηχ(t) |Dη|
+�
+1 + |Duε|2� p−1
+2 ��D2uε�� · g
+�
+|Duε|2�
+dz
+≤
+σ
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2uε��2 · g
+�
+|Duε|2�
+dz
++σε
+ˆ
+Qτ η2χ(t)
+�
+1 + |Duε|2� p−2
+2 ��D2uε��2 · g
+�
+|Duε|2�
+dz
++cσ
+ˆ
+Qτ χ(t) |Dη|2 (|Duε| − 1)p−4
++
+|Duε|4 · g
+�
+|Duε|2�
+dz
++cp,σ · ε
+ˆ
+Qτ χ(t) |Dη|2 �
+1 + |Duε|2� p
+2 · g
+�
+|Duε|2�
+dz
+≤
+σ
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2uε��2 · g
+�
+|Duε|2�
+dz
++σε
+ˆ
+Qτ η2χ(t) |DVp (Duε)|2 · g
+�
+|Duε|2�
+dz
++cσ,p ∥Dη∥2
+L∞ ∥g∥L∞
+ˆ
+Qτ χ(t) (1 + |Duε|)p dz,
+(3.18)
+where we used (3.2), and where σ > 0 is a parameter that will be chosen later.
+Now, using Young’s Inequality, we estimate the term ˜I3, as follows
+˜I3
+≤
+c
+ˆ
+Qτ χ(t) |f ε| η |Dη| |Duε| · g
+�
+|Duε|2�
+dz
++c
+ˆ
+Qτ χ(t) |f ε| η2 ��D2uε�� · g
+�
+|Duε|2�
+dz
++c
+ˆ
+Qτ χ(t) |f ε| η2 |Duε|2 ��D2uε�� · g′ �
+|Duε|2�
+dz
+≤
+c ∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) |f ε|2 dz
++c ∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) |Duε|2 dz
++c
+ˆ
+Qτ η2χ(t) |f ε|
+��D2uε�� · g
+�
+|Duε|2�
+dz
++c
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ �
+|Duε|2�
+dz.
+(3.19)
+Plugging (3.18) and (3.19) into (3.17), we get
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
+
+15
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
++cpε
+ˆ
+Qτ η2χ(t)g
+�
+|Duε|2�
+|DVp (Duε)|2 dz
+≤
+σ
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2uε��2 · g
+�
+|Duε|2�
+dz
++σε
+ˆ
+Qτ η2χ(t) |DVp (Duε)|2 · g
+�
+|Duε|2�
+dz
++cp,σ ∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) |f ε|2 dz
++cp,σ∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) (1 + |Duε|)p dz
++c
+ˆ
+Qτ η2χ(t) |f ε|
+��D2uε�� · g
+�
+|Duε|2�
+dz
++c
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ �
+|Duε|2�
+dz
++1
+2
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz,
+which, for a sufficiently small σ, gives
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
++cpε
+ˆ
+Qτ η2χ(t)g
+�
+|Duε|2�
+|DVp (Duε)|2 dz
+≤
+cp∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) |f ε|2 dz
++cp∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) (1 + |Duε|)p dz
++c
+ˆ
+Qτ η2χ(t) |f ε|
+��D2uε�� · g
+�
+|Duε|2�
+dz
++c
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ �
+|Duε|2�
+dz
++1
+2
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz,
+that, neglecting the third integral in the left hand side, implies
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+cp∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) |f ε|2 dz
++cp∥Dη∥∞ ∥g∥L∞
+ˆ
+Qτ ηχ(t) (1 + |Duε|)p dz
++c
+ˆ
+Qτ η2χ(t) |f ε|
+��D2uε�� · g
+�
+|Duε|2�
+dz
++c
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ �
+|Duε|2�
+dz
++1
+2
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz,
+(3.20)
+
+16
+Now, for δ ∈ (0, 1), recalling the notation in (2.3), we choose
+g(s) = g1+δ
+�
+(s − 1 − δ)+
+�
+that is
+g(s) =
+(s − 1 − δ)2
++
+1 + δ + (s − 1 − δ)2
++
+,
+that is legitimate since g ∈ W 1,∞([0, +∞)).
+Moreover, with this choice, we have g(s) ∈ [0, 1], for every s ≥ 0, and thanks to (2.5), there exists a
+constant cδ > 0 such that
+sg′ �
+s2�
+≤ cδ
+for every s ≥ 0,
+so that (3.14) holds. Therefore, since g(s) vanishes on the set where s ≤ 1 + δ and g(s) ≤ 1 for every
+s, (3.20) becomes
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+c
+ˆ
+Qτ∩{|Duε|2>1+δ}
+η2χ(t) |f ε|
+��D2uε�� (|Duε| − 1)
+p
+2
++
+|Duε|
+|Duε|
+(|Duε| − 1)
+p
+2
++
+· g
+�
+|Duε|2�
+dz
++c
+ˆ
+Qτ ∩{|Duε|2>1+δ}
+η2χ(t) |f ε| |Duε|2 (|Duε| − 1)
+p
+2
++
+|Duε|
+|Duε|
+(|Duε| − 1)
+p
+2
++
+��D2uε�� g′ �
+|Duε|2�
+dz
++cp∥Dη∥∞ ∥χ∥L∞
+ˆ
+Qτ
+�
+1 + |Duε|p + |f ε|2�
+dz +
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz
+≤
+cp
+δ
+p
+2
+ˆ
+Qτ η2χ(t) |f ε|
+��D2uε�� (|Duε| − 1)
+p
+2
++
+|Duε|
+· g
+�
+|Duε|2�
+dz
++ cp
+δ
+p
+2
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1)
+p
+2
++
+|Duε|
+��D2uε�� g′ �
+|Duε|2�
+dz
++cp∥Dη∥∞ ∥χ∥L∞
+ˆ
+Qτ
+�
+1 + |Duε|p + |f ε|2�
+dz +
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz,
+where we used that
+sup
+x∈(
+√
+1+δ,+∞)
+x
+(x − 1)
+p
+2 =
+√
+1 + δ
+�√
+1 + δ − 1
+� p
+2 =
+√
+1 + δ
+�√
+1 + δ + 1
+� p
+2
+δ
+p
+2
+≤ cp
+δ
+p
+2 ,
+since δ < 1. Using Young’s inequality in the first integral in the right hand, previous estimate yields
+1
+2
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++cp
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+cp(β)
+δp
+ˆ
+Qτ η2χ(t) |f ε|2 · g
+�
+|Duε|2�
+dz
++β
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2uε��2 · g
+�
+|Duε|2�
+dz
++ cp
+δ
+p
+2
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1)
+p
+2
++
+|Duε|
+��D2uε�� g′ �
+|Duε|2�
+dz
++cp∥Dη∥∞ ∥χ∥L∞
+ˆ
+Qτ
+�
+1 + |Duε|p + |f ε|2�
+dz
+
+17
++
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz.
+Choosing β sufficiently small, reabsorbing the second integral in the right hand side by the left hand
+side and using that g(s) ≤ 1, we get
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+c cp
+δ
+p
+2
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1)
+p
+2
++
+|Duε|
+��D2uε�� g′ �
+|Duε|2�
+dz
++
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz
+c ∥Dη∥2
+∞ ∥χ∥∞
+ˆ
+Qτ (1 + |Duε|)p dz
++c ∥χ∥L∞
+�cp
+δp + ∥Dη∥L∞
+� ˆ
+Qτ |f ε|2 dz.
+(3.21)
+We now estimate the first integral in the right side of previous inequality with the use of (2.4) with
+s = |Duε|2, A = (|Duε| − 1)
+p
+2
++
+|Duε|
+��D2u�, B = cp
+δ
+p
+2 |f ε| and k = 1 + δ, thus getting
+cp
+δ
+p
+2
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1)
+p
+2
++
+|Duε|
+��D2uε�� g′ �
+|Duε|2�
+dz
+≤
+2α
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2u�2 g
+�
+|Duε|2�
+dz
++2ασ
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
++cα,p
+δp
+ˆ
+Qτ η2χ(t) |f ε|2 dz,
+with constants c, cα both independent of σ and where we used that δ < 1. By virtue of (3.3), taking
+the limit as σ → 0 in previous inequality, we have
+cp
+δ
+p
+2
+ˆ
+Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1)
+p
+2
++
+|Duε|
+��D2uε�� g′ �
+|Duε|2�
+dz
+≤
+2α
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2u�2 g
+�
+|Duε|2�
+dz
++cα,p
+δp
+ˆ
+Qτ η2χ(t) |f ε|2 dz,
+(3.22)
+Inserting (3.22) in (3.21), we find
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+2α
+ˆ
+Qτ η2χ(t)(|Duε| − 1)p
++
+|Duε|2
+��D2u�2 g
+�
+|Duε|2�
+dz
++cα,p
+δp
+ˆ
+Qτ η2χ(t)|f ε|2 dz
+
+18
++
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz
+c ∥Dη∥2
+∞ ∥χ∥∞
+ˆ
+Qτ (1 + |Duε|)p dz
++c ∥χ∥L∞
+�cp
+δp + ∥Dη∥L∞
+� ˆ
+Qτ |f ε|2 dz.
+Choosing α = 1
+4 , we can reabsorb the first integral in the right hand side by the left hand side, thus
+obtaining
+ˆ
+B2ρ
+η2χ(τ)
+�ˆ |Duε(x,τ)|2
+0
+g(s) ds
+�
+dx
++
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+c ∥Dη∥2
+∞ ∥χ∥∞
+ˆ
+Qτ (1 + |Duε|)p dz
++ c
+δp ∥χ∥L∞ (1 + ∥Dη∥L∞)
+ˆ
+Qτ |f ε|2 dz
++c
+ˆ
+Qτ η2∂tχ(t)
+�ˆ |Duε|2
+0
+g(s) ds
+�
+dz.
+(3.23)
+By the definition of g, we have
+ˆ ζ
+0
+g(s) ds =
+
+
+
+
+
+
+
+0
+if
+0 < ζ ≤ 1 + δ
+ˆ ζ
+1+δ
+(s − 1 − δ)2
+1 + δ + (s − 1 − δ)2 ds
+if
+ζ > 1 + δ,
+and so it is easy to check that
+ˆ ζ
+0
+g(s) ds =
+
+
+
+
+
+
+
+0
+if
+0 < ζ ≤ 1 + δ
+ζ − 1 − δ −
+√
+1 + δ arctan
+�ζ − 1 − δ
+√
+1 + δ
+�
+if
+ζ > 1 + δ,
+that is
+ˆ ζ
+0
+g(s) ds = (ζ − 1 − δ)+ −
+√
+1 + δ arctan
+�(ζ − 1 − δ)+
+√
+1 + δ
+�
+.
+Therefore, by previous equality and the properties of χ and η, (3.23) implies
+ˆ
+B2ρ
+η2χ(τ)
+�
+|Duε(x, τ)|2 − 1 − δ
+�
++ dx
++
+ˆ
+Qτ η2χ(t) · g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+c ∥Dη∥2
+∞ ∥χ∥∞
+ˆ
+Qτ (1 + |Duε|)p dz
++ c
+δp ∥χ∥L∞ (1 + ∥Dη∥L∞)
+ˆ
+Qτ |f ε|2 dz
++c
+ˆ
+Qτ η2∂tχ(t)
+�
+|Duε|2 − 1 − δ
+�
++ dz
++c ∥∂tχ∥∞ |Qτ| + c ∥χ∥∞ |BR| ,
+(3.24)
+which holds for almost every τ ∈
+�
+t0 − 4ρ2, t0
+�
+.
+We now choose a cut-off function η ∈ C∞ (B2ρ (x0)) with η ≡ 1 on Bρ (x0) such that 0 ≤ η ≤ 1 and
+
+19
+|Dη| ≤ c
+ρ. For the cut-off function in time, we choose χ ∈ W 1,∞ �
+t0 − R2, t0, [0, 1]
+�
+such that χ ≡ 0
+on
+�
+t0 − R2, t0 − 4ρ2�
+, χ ≡ 1 on
+�
+t0 − ρ2, t0
+�
+and ∂tχ ≤ c
+ρ2 on
+�
+t0 − 4ρ2, t0 − ρ2�
+. With these choices,
+(3.24) gives
+sup
+τ∈(t0−4ρ2,t0)
+ˆ
+Bρ
+χ(τ)
+�
+|Duε(x, τ)|2 − 1 − δ
+�
++ dx
++
+ˆ
+Qρ
+g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+c
+ρ2
+ˆ
+Q2ρ
+(1 + |Duε|p) dz +
+c
+ρ2δp
+ˆ
+Q2ρ
+|f ε|2 dz
++c |Q2ρ|
+ρ2
++ c |B2ρ| ,
+and since ρ < 2ρ < R < 1, and Q2ρ = Bρ ×
+�
+t0 − 4ρ2, t0
+�
+, we have
+sup
+τ∈(t0−4ρ2,t0)
+ˆ
+Bρ
+�
+|Duε(x, τ)|2 − 1 − δ
+�
++ dx
++
+ˆ
+Qρ
+g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 dz
+≤
+c
+ρ2
+ˆ
+Q2ρ
+(1 + |Duε|p) dz +
+c
+ρ2δp
+ˆ
+Q2ρ
+|f ε|2 dz.
+(3.25)
+Now, with Gδ(t) defined at (2.9), recalling (2.10), we have
+��D
+�
+Gδ
+�
+(|Duε| − δ − 1)+
+����2
+≤
+(|Duε| − δ − 1)2
++
+1 + δ + (|Duε| − δ − 1)2
++
+�
+(|Duε| − δ − 1)+ + δ
+�p−2 ��D2uε��2
+=
+g (|Duε|)
+�
+(|Duε| − δ − 1)+ + δ
+�p−2 ��D2uε��2 .
+Since g(s) is nondecreasing, we have g(s) ≤ g
+�
+s2�
+, and therefore
+��D
+�
+Gδ
+�
+(|Duε| − δ − 1)+
+����2 ≤ g
+�
+|Duε|2�
+(|Duε| − 1)p−2
++
+��D2u�2
+≤
+cp
+δ2 g
+�
+|Duε|2� (|Duε| − 1)p
++
+|Duε|2
+��D2u�2 ,
+(3.26)
+where we also used that g(s) = 0, for 0 < s ≤ 1 + δ. Using (3.26) in the left hand side of (3.25), we
+obtain
+sup
+τ∈(t0−4ρ2,t0)
+ˆ
+Bρ
+�
+|Duε(x, τ)|2 − 1 − δ
+�
++ dx
++
+ˆ
+Qρ
+��D
+�
+Gδ
+�
+(|Duε| − δ − 1)+
+����2 dz
+≤
+c
+ρ2δ2
+�ˆ
+Q2ρ
+(1 + |Duε|p) dz + 1
+δp
+ˆ
+Q2ρ
+|f ε|2 dz
+�
+,
+which is (3.4).
+Combining Lemma 3.1 and Lemma 2.8, we have the following.
+Corollary 3.2. Let uε ∈ C0 �
+t0 − R2, t0; L2 (BR)
+�
+∩ Lp �
+t0 − R2, t0; u + W 1,p
+0
+(BR)
+�
+be the unique
+solution to (3.1). Then the following estimate
+ˆ
+Q ρ
+2
+��τh
+�
+Gδ
+�
+(|Duε| − δ − 1)+
+����2 dz
+
+20
+≤
+c|h|2
+ρ2δ2
+�ˆ
+Q2ρ
+(1 + |Duε|p) dz + 1
+δp
+ˆ
+Q2ρ
+|f ε|2 dz
+�
+(3.27)
+holds for |h| < ρ
+4, for any parabolic cylinder Q2ρ ⋐ QR (z0).
+4
+Proof of Theorem 1.1
+This section is devoted to the proof of Theorem 1.1, that will be divided in two steps.
+In the first one we shall establish an estimate that will allow us to measure the L2-distance between
+H p
+2 (Du) and H p
+2 (Duε) in terms of the L2-distance between f and f ε.
+In the second one, we conclude combining this comparison estimate with the one obtained for the
+difference quotient of the solution to the regularized problem at (3.27).
+Proof of Theorem 1.1. Step 1: the comparison estimate.
+We formally proceed by testing equations (1.1) and (3.1) with the map ϕ = k(t) (uε − u), where
+k ∈ W 1,∞ (R) is chosen such that
+k(t) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+1
+if
+t ≤ t2,
+− 1
+ω (t − t2 − ω)
+if
+t2 < t < t2 + ω,
+0
+if
+t ≥ t2 + ω,
+with t0 − R2 < t2 < t2 + ω < t0, and then letting ω → 0. We observe that, at this stage, it is
+important that uε and u agree on the parabolic boundary ∂parQR (z0).
+Proceeding in a standard way (see for example [13]), for almost every t2 ∈
+�
+t0 − R2, t0
+�
+, we find
+1
+2
+ˆ
+BR(x0)
+|uε (x, t2) − u (x, t2)|2 dx
++
+ˆ
+QR,t2
+⟨Hp−1 (Duε) − Hp−1 (Du) , Duε − Du⟩ dz
++ε
+ˆ
+QR,t2
+��
+1 + |Duε|2� p−2
+2 Duε, Duε − Du
+�
+dz
+=
+ˆ
+QR,t2
+(f − f ε) (uε − u) dz,
+(4.1)
+where we used the abbreviation QR,t2 = BR (x0) ×
+�
+t0 − R2, t2
+�
+.
+Using Lemma 2.1, the Cauchy-
+Schwarz inequality as well as Young’s inequality, from (4.1) we infer
+λp
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
++λp
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz + ε
+ˆ
+QR(z0)
+|Duε|p dz
+≤
+ˆ
+QR
+|f − f ε| |uε − u| dz + ε
+ˆ
+QR
+|Duε|p−1 |Du| dz
+≤
+ˆ
+QR
+|f − f ε| |uε − u| dz + ε · cp
+ˆ
+QR
+|Du|p dz
++1
+2 · ε
+ˆ
+QR
+|Duε|p dz,
+(4.2)
+where we set λp = min
+� 1
+2, 4
+p2
+�
+. Reabsorbing the last integral in the right-hand side of (4.2) by the
+left-hand side, we arrive at
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
+
+21
++
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz +
+ε
+2λp
+ˆ
+QR
+|Duε|p dz
+≤
+ε cp
+ˆ
+QR
+|Du|p dz + cp
+ˆ
+QR
+|f − f ε| |uε − u| dz.
+(4.3)
+Using in turn Hölder’s inequality and Lemma 2.5, we get
+˜I
+:=
+ˆ
+QR
+|f − f ε| |uε − u| dz
+≤
+C (R, n, p) ∥f − f ε∥L2(QR) ·
+�ˆ
+QR
+|uε − u|p+ 2p
+n dz
+�
+n
+p(n+2)
+≤
+c (n, p, R) ∥f − f ε∥L2(QR) ·
+�ˆ
+QR
+|Duε − Du|p dz
+�
+n
+p(n+2)
+·
+�
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
+�
+1
+n+2
+(4.4)
+Now, let us notice that
+ˆ
+QR
+|Duε − Du|p dz
+=
+ˆ
+QR∩{|Duε|≥1}
+(|Duε| − 1 + 1)p dz +
+ˆ
+QR∩{|Duε|<1}
+|Duε|p dz +
+ˆ
+QR
+|Du|p dz
+≤
+cp
+ˆ
+QR
+�
+(|Duε| − 1)p
++
+�
+dz +
+ˆ
+QR
+(|Du|p + 1) dz
+≤
+cp
+ˆ
+QR
+����H p
+2 (Duε) − H p
+2 (Du) + H p
+2 (Du)
+���
+2�
+dz + cp
+ˆ
+QR
+(|Du|p + 1) dz
+≤
+cp
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz + cp
+ˆ
+QR
+(|Du|p + 1) dz.
+(4.5)
+Inserting (4.5) in (4.4), we get
+˜I
+≤
+c (n, p, R) ∥f − f ε∥L2(QR(z0))
+·
+�ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz +
+ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+2)
+·
+�
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
+�
+1
+n+2
+≤
+c (n, p, R) ∥f − f ε∥L2(QR) ·
+�ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz
+�
+n
+p(n+2)
+·
+�
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
+�
+1
+n+2
++c (n, p, R) ∥f − f ε∥L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+2)
+·
+�
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
+�
+1
+n+2
+and, by Young’s inequality, we get
+˜I
+≤
+β
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz + β
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
++c (n, p, R, β) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
+
+22
++c (n, p, R, β) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
+.
+(4.6)
+Inserting (4.6) in (4.3), we obtain
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
++
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz +
+ε
+2λp
+ˆ
+QR
+|Duε|p dz
+≤
+β
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz + β
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0))
++c (n, p, R, β) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
++c (n, p, R, β) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
++ ε cp
+ˆ
+QR
+|Du|p dz.
+(4.7)
+Choosing β = 1
+2 and neglecting the third non negative term in the left hand side of (4.7), we get
+sup
+t∈(t0−R2,t0)
+∥uε(·, t) − u(·, t)∥2
+L2(BR(x0)) +
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz
+≤
+c (n, p, R) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
++c (n, p, R) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
++ ε cp
+ˆ
+QR
+|Du|p dz.
+(4.8)
+For further needs, we also record that, combining (4.5) and (4.8), we have
+ˆ
+QR
+|Duε|p dz
+≤
+c (n, p, R) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
++c (n, p, R) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
++ ε cp
+ˆ
+QR
+|Du|p dz
++cp
+ˆ
+QR
+(|Du|p + 1) dz.
+(4.9)
+Step 2: The conclusion.
+Let us fix ρ > 0 such that Q2ρ ⊂ QR. We start observing that
+ˆ
+Q ρ
+2
+��τh
+�
+Gδ
+�
+(|Du| − δ − 1)+
+����2 dz
+≤
+c
+ˆ
+Q ρ
+2
+��τh
+�
+Gδ
+�
+(|Duε| − δ − 1)+
+����2 dz
++c
+ˆ
+Qρ
+��Gδ
+�
+(|Duε| − δ − 1)+
+�
+− Gδ
+�
+(|Du| − δ − 1)+
+���2 dz.
+We estimate the right hand side of previous inequality using (3.27) and (2.11), as follows
+ˆ
+Q ρ
+2
+��τh
+�
+Gδ
+�
+(|Du| − δ − 1)+
+����2 dz
+≤
+c|h|2
+ρ2
+�ˆ
+Q2ρ
+(1 + |Duε|p) dz + δ2−p
+ˆ
+Q2ρ
+|f ε|2 dz
+�
++cp
+ˆ
+Q2ρ
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz
+that, thanks to (4.8), implies
+ˆ
+Q ρ
+2
+��τh
+�
+Gδ
+�
+(|Du| − δ − 1)+
+����2 dz
+
+23
+≤
+c|h|2
+ρ2
+�ˆ
+Q2ρ
+(1 + |Duε|p) dz + δ2−p
+ˆ
+Q2ρ
+|f ε|2 dz
+�
++c (n, p, R) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
++c (n, p, R) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
++ ε cp
+ˆ
+QR
+|Du|p dz.
+(4.10)
+Now, using (4.9), we get
+ˆ
+Q2ρ
+(1 + |Duε|p) dz
+≤
+c (n, p, R) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
++c (n, p, R) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
++ ε cp
+ˆ
+QR
+|Du|p dz
++cp
+ˆ
+QR
+(|Du|p + 1) dz
+which, combined with (4.10), implies
+ˆ
+Q ρ
+2
+��τh
+�
+Gδ
+�
+(|Du| − δ − 1)+
+����2 dz
+≤
+c (n, p) |h|2
+ρ2
+�
+c(R) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
++c(R) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
++ ε
+ˆ
+QR
+|Du|p dz
++
+ˆ
+QR
+(|Du|p + 1) dz + δ2−p
+ˆ
+QR
+|f ε|2 dz
+�
+.
+Taking the limit as ε → 0, and since f ε → f strongly in L2 (BR), we obtain
+ˆ
+Q ρ
+2
+��τh
+�
+Gδ
+�
+(|Du| − δ − 1)+
+����2 dz
+≤
+c (n, p) |h|2
+ρ2
+�ˆ
+QR
+(|Du|p + 1) dz + δ2−p
+ˆ
+QR
+|f|2 dz
+�
+,
+and thanks to Lemma 2.9, we have Gδ
+�
+(|Du| − δ − 1)+
+�
+∈ L2 �
+t0 − ρ2, t0; W 1,2 (Bρ)
+�
+with the follow-
+ing estimate
+ˆ
+Q ρ
+2
+��D
+�
+Gδ
+�
+(|Du| − δ − 1)+
+����2 dz
+≤
+c (n, p)
+ρ2
+�ˆ
+QR
+(|Du|p + 1) dz + δ2−p
+ˆ
+QR
+|f|2 dz
+�
+.
+Since previous estimate holds true for any ρ > 0 such that 4ρ < R, we may choose ρ = R
+8 thus getting
+(1.2).
+5
+Proof of Theorem 1.2
+The higher differentiability result of Theorem 1.1 allows us to argue as in [13, Lemma 5.3] and [17,
+Lemma 3.2] to obtain the proof of Theorem 1.2.
+Proof of Theorem 1.2. We start observing that
+���D
+��
+Gδ
+�
+(|Duε| − 1 − δ)+
+�� 4
+np + 1����
+
+24
+≤
+c
+��Gδ
+�
+(|Duε| − 1 − δ)+
+���
+4
+np ��D
+�
+Gδ
+�
+(|Duε| − 1 − δ)+
+���� ,
+(5.1)
+where c ≡ c(n, p) > 0 and Gδ(t) is the function defined at (2.9).
+With the notation we used in the previous sections, for B2ρ (x0) ⋐ BR (x0), let ϕ ∈ C∞
+0 (Bρ (x0)) and
+χ ∈ W 1,∞ ((0, T )) be two non-negative cut-off functions with χ(0) = 0 and ∂tχ ≥ 0. Now, we fix a
+time t0 ∈ (0, T ) and apply the Sobolev embedding theorem on the time slices Σt := Bρ(x0) × {t} for
+almost every t ∈ (0, t0), to infer that
+ˆ
+Σt
+ϕ2 ��
+Gδ
+�
+(|Duε| − 1 − δ)+
+�� 4
+np + 1�2
+dx
+≤
+c
+�ˆ
+Σt
+���D
+�
+ϕ
+�
+Gδ
+�
+(|Duε| − 1 − δ)+
+�� 4
+np + 1����
+2n
+n+2 dx
+� n+2
+n
+≤
+c
+�ˆ
+Σt
+���ϕ D
+��
+Gδ
+�
+(|Duε| − 1 − δ)+
+�� 4
+np + 1����
+2n
+n+2 dx
+� n+2
+n
++c
+�ˆ
+Σt
+���
+��Gδ
+�
+(|Duε| − 1 − δ)+
+���
+4
+np + 1 Dϕ
+���
+2n
+n+2 dx
+� n+2
+n
+=:
+c I1(t) + c I2(t),
+where, in the second to last line, we have applied Minkowski’s and Young’s inequalities one after the
+other. We estimate I1(t) and I2(t) separately. Let us first consider I1(t). Using (5.1), Lemma 2.12
+and Hölder’s inequality with exponents
+�n + 2
+n
+, n + 2
+2
+�
+, we deduce
+I1(t)
+≤
+c
+�ˆ
+Σt
+ϕ
+2n
+n+2
+�
+(|Duε| − 1)
+2
+n
++
+��DGδ
+�
+(|Duε| − 1 − δ)+
+���
+� 2n
+n+2 dx
+� n+2
+n
+≤
+c
+ˆ
+Σt
+ϕ2 ��DGδ
+�
+(|Duε| − 1 − δ)+
+���2 dx
+�ˆ
+supp(ϕ)
+(|Duε| − 1)2
++ dx
+� 2
+n
+≤
+c
+ˆ
+Σt
+ϕ2 ��DGδ
+�
+(|Duε| − 1 − δ)+
+���2 dx
+�ˆ
+supp(ϕ)
+|Duε|2 dx
+� 2
+n
+.
+We now turn our attention to I2(t). Lemma 2.12 and Hölder’s inequality yield
+I2(t)
+≤
+c
+�ˆ
+Σt
+(|Duε| − 1)
+np + 4
+n+2
++
+|Dϕ|
+2n
+n+2 dx
+� n+2
+n
+≤
+c
+�ˆ
+Σt
+�
+|Dϕ|2 |Duε|p�
+n
+n+2 |Du|
+4
+n+2 dx
+� n+2
+n
+≤
+c
+ˆ
+Σt
+|Dϕ|2 |Duε|p dx
+�ˆ
+supp(ϕ)
+|Duε|2 dx
+� 2
+n
+.
+Putting together the last three estimates, using Lemma 2.12 in the left hand side, and integrating
+with respect to time, we obtain
+ˆ
+Qt0
+χϕ2 (|Duε| − 1)
+p + 4
+n
++
+dz
+≤
+c
+ˆ t0
+0
+χ
+�ˆ
+supp(ϕ)
+|Duε(x, t)|2 dx
+� 2
+n
+·
+·
+�ˆ
+Σt
+�
+ϕ2 ��DGδ
+�
+(|Duε| − 1 − δ)+
+���2 + |Dϕ|2 |Du|p�
+dx
+�
+dt
+≤
+c
+ˆ
+Qt0
+χ
+�
+ϕ2 ��DGδ
+�
+(|Duε| − 1 − δ)+
+���2 + |Dϕ|2 |Du|p�
+dz
+
+25
+·
+�
+sup
+0<t<t0, χ(t)̸=0
+ˆ
+supp(ϕ)
+|Duε(x, t)|2 dx
+� 2
+n
+,
+(5.2)
+where we have used the abbreviation Qt0 := Bρ (x0) × (0, t0).
+Now we choose χ ∈ W 1,∞ ((0, T )) such that χ ≡ 0 on
+�
+0, t0 − ρ2�
+, χ ≡ 1 on
+�
+t0 −
+�ρ
+2
+�2
+, T
+�
+and
+∂tχ ≥ 0. For ϕ ∈ C∞
+0 (Bρ (x0)), we assume that ϕ ≡ 1 on B ρ
+2 (x0), 0 ≤ ϕ ≤ 1 and |Dϕ| ≤ C
+ρ .
+With these choices (5.2) turns into
+ˆ
+Q ρ
+2
+(|Duε| − 1)
+p + 4
+n
++
+dz
+≤
+c(n, p)
+ˆ
+Qρ
+���DGδ (|Duε| − 1 − δ)+)
+��2 + ρ−2 |Duε|p�
+dz
+·
+�
+sup
+t0−ρ2<t<t0
+ˆ
+Bρ(x0)
+|Duε(x, t)|2 dx
+� 2
+n
+.
+(5.3)
+We now use (3.4), in order to estimate the first and second integral on the right-hand side of (5.3),
+thus getting
+ˆ
+Q ρ
+2
+(|Duε| − 1)
+p + 4
+n
++
+dz ≤
+c
+ρ
+2(n+2)
+n
+�ˆ
+Q2ρ
+(1 + |Duε|p) dz + δ2−p
+ˆ
+Q2ρ
+|f ε|2 dz
+� 2
+n +1
+.
+Now we use (4.9) to deduce that
+ˆ
+Q ρ
+2
+(|Duε| − 1)
+p + 4
+n
++
+dz
+≤
+c (n, p)
+ρ
+2(n+2)
+n
+
+c(ρ) ∥f − f ε∥
+n+2
+n+1
+L2(Q2ρ(z0)) ·
+�ˆ
+Q2ρ
+(|Du|p + 1) dz
+�
+n
+p(n+1) 
+
+2
+n +1
++c (n, p)
+ρ
+2(n+2)
+n
+�
+c(ρ) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(Q2ρ) + ε cp
+ˆ
+Q2ρ
+|Du|p dz
+� 2
+n +1
++c (n, p)
+ρ
+2(n+2)
+n
+�ˆ
+Q2ρ
+(|Du|p + 1) dz + δ2−p
+ˆ
+Q2ρ
+|f ε|2 dz
+� 2
+n +1
+.
+(5.4)
+Let us observe that estimate (4.8) in particular implies that
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz
+≤
+c (n, p, R) ∥f − f ε∥
+n+2
+n+1
+L2(QR) ·
+�ˆ
+QR
+(|Du|p + 1) dz
+�
+n
+p(n+1)
++c (n, p, R) ∥f − f ε∥
+p(n+2)
+n(p−1)+p
+L2(QR)
++ ε cp
+ˆ
+QR
+|Du|p dz.
+By the strong convergence of f ε → f in L2 (QR), passing to the limit as ε → 0, from previous estimate
+we deduce
+lim
+ε→0
+ˆ
+QR
+���H p
+2 (Duε) − H p
+2 (Du)
+���
+2
+dz = 0
+that is H p
+2 (Duε) → H p
+2 (Du), strongly in L2 (QR) . Therefore, up to a not relabelled subsequence, we
+also have H p
+2 (Duε) → H p
+2 (Du), a.e. in QR (z0) and so
+(|Duε| − 1)+ → (|Du| − 1)+
+a.e. in QR (z0)
+By Fatou’s Lemma, taking the limit as ε → 0 in both sides of (5.4)
+ˆ
+Q ρ
+2
+(|Du| − 1)
+p + 4
+n
++
+dz ≤ lim inf
+ε→0
+ˆ
+Q ρ
+2
+(|Duε| − 1)
+p + 4
+n
++
+dz
+
+26
+≤
+c (n, p)
+ρ
+2(n+2)
+n
+�ˆ
+Q2ρ
+(|Du|p + 1) dz + δ2−p
+ˆ
+Q2ρ
+|f|2 dz
+� 2
+n +1
+,
+which holds for any δ ∈ (0, 1), so we can fix δ = 1
+2, to get the conclusion (1.3).
+References
+[1]
+Z. M. Akhmedov, G. I. Barenblatt, V. M. Entov, and A. Kh. Mirzadzhan-Zade. “Nonlin-
+ear effects in gas filtration”. In: Fluid Dynamics 4.5 (1969), pp. 68–72. doi: https://doi.org/10.1007/BF01015960.
+[2]
+P. Ambrosio. “Besov regularity for a class of singular or degenerate elliptic equations”.
+In: Journal of Mathematical Analysis and Applications 505.2 (2022), p. 125636. doi:
+https://doi.org/10.1016/j.jmaa.2021.125636.
+[3]
+P. Ambrosio and A. Passarelli di Napoli. Regularity results for a class of widely degener-
+ate parabolic equations. 2022. arXiv: 2204.05966 [math.AP]. url: https://arxiv.org/abs/2204.05966v3.
+[4]
+V. Bögelein, F. Duzaar, R. Giova, and A. Passarelli di Napoli. “Higher regularity in con-
+gested traffic dynamics”. In: Mathematische Annalen (2022), pp. 1–56. doi: https://doi.org/10.1007/s00208-022-02375-y.
+[5]
+V. Bögelein, F. Duzaar, and P. Marcellini. “Parabolic equations with p, q-growth”.
+In: Journal de Mathématiques Pures et Appliquées 100.4 (2013), pp. 535–563. doi:
+https://doi.org/10.1016/j.matpur.2013.01.012.
+[6]
+L. Brasco. “Global L∞ gradient estimates for solutions to a certain degenerate elliptic
+equation”. In: Nonlinear Analysis: Theory, Methods & Applications 74.2 (2011), pp. 516–
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+[7]
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+[8]
+L. Brasco and F. Santambrogio. “A sharp estimate à la Calderón–Zygmund for the p-
+Laplacian”. In: Communications in Contemporary Mathematics 20.03 (2018), p. 1750030.
+doi: https://doi.org/10.1142/S0219199717500304.
+[9]
+S. Campanato. “Equazioni paraboliche del secondo ordine e spazi L2,θ (Ω, δ)”. In: Annali
+di Matematica Pura ed Applicata 73.1 (1966), pp. 55–102. doi: https://doi.org/10.1007/BF02415082.
+[10]
+G. Cupini, F. Giannetti, R. Giova, and A. Passarelli di Napoli. “Regularity results for
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+[11]
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+850.
+[12]
+E. DiBenedetto. Degenerate parabolic equations. Springer Science & Business Media,
+1993.
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+[15]
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+[16]
+P. Lindqvist. Notes on the stationary p-Laplace equation. Springer, 2019.
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+[17]
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+

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