mjphayes/textbook_dataset · Datasets at Hugging Face

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MATHEMATICS FOR MACHINE LEARNING
Marc Peter Deisenroth
A. Aldo Faisal
Cheng Soon Ong
Contents
Foreword 1
Part I Mathematical Foundations 9
1 Introduction and Motivation 11 1.1 Finding Words for Intuitions 12 1.2 Two Ways to Read This Book 13 1.3 Exercises and Feedback 16
2 Linear Algebra 17 2.1 Systems of Linear Equations 19 2.2 Matrices 22 2.3 Solving Systems of Linear Equations 27 2.4 Vector Spaces 35 2.5 Linear Independence 40 2.6 Basis and Rank 44 2.7 Linear Mappings 48 2.8 Affine Spaces 61 2.9 Further Reading 63 Exercises 64
3 Analytic Geometry 70 3.1 Norms 71 3.2 Inner Products 72 3.3 Lengths and Distances 75 3.4 Angles and Orthogonality 76 3.5 Orthonormal Basis 78 3.6 Orthogonal Complement 79 3.7 Inner Product of Functions 80 3.8 Orthogonal Projections 81 3.9 Rotations 91 3.10 Further Reading 94 Exercises 96
4 Matrix Decompositions 98 4.1 Determinant and Trace 99
i
This material is published by Cambridge University Press as Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong (2020). This version is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. ©by M. P. Deisenroth, A. A. Faisal, and C. S. Ong, 2023. https://mml-book.com.
ii Contents
4.2 Eigenvalues and Eigenvectors 105 4.3 Cholesky Decomposition 114 4.4 Eigendecomposition and Diagonalization 115 4.5 Singular Value Decomposition 119 4.6 Matrix Approximation 129 4.7 Matrix Phylogeny 134 4.8 Further Reading 135 Exercises 137
5 Vector Calculus 139 5.1 Differentiation of Univariate Functions 141 5.2 Partial Differentiation and Gradients 146 5.3 Gradients of Vector-Valued Functions 149 5.4 Gradients of Matrices 155 5.5 Useful Identities for Computing Gradients 158 5.6 Backpropagation and Automatic Differentiation 159 5.7 Higher-Order Derivatives 164 5.8 Linearization and Multivariate Taylor Series 165 5.9 Further Reading 170 Exercises 170
6 Probability and Distributions 172 6.1 Construction of a Probability Space 172 6.2 Discrete and Continuous Probabilities 178 6.3 Sum Rule, Product Rule, and Bayes’ Theorem 183 6.4 Summary Statistics and Independence 186 6.5 Gaussian Distribution 197 6.6 Conjugacy and the Exponential Family 205 6.7 Change of Variables/Inverse Transform 214 6.8 Further Reading 221 Exercises 222
7 Continuous Optimization 225 7.1 Optimization Using Gradient Descent 227 7.2 Constrained Optimization and Lagrange Multipliers 233 7.3 Convex Optimization 236 7.4 Further Reading 246 Exercises 247
Part II Central Machine Learning Problems 249
8 When Models Meet Data 251 8.1 Data, Models, and Learning 251 8.2 Empirical Risk Minimization 258 8.3 Parameter Estimation 265 8.4 Probabilistic Modeling and Inference 272 8.5 Directed Graphical Models 278
Draft (2023-10-18) of “Mathematics for Machine Learning”. Feedback: https://mml-book.com.
Contents iii 8.6 Model Selection 283
9 Linear Regression 289 9.1 Problem Formulation 291 9.2 Parameter Estimation 292 9.3 Bayesian Linear Regression 303 9.4 Maximum Likelihood as Orthogonal Projection 313 9.5 Further Reading 315
10 Dimensionality Reduction with Principal Component Analysis 317 10.1 Problem Setting 318 10.2 Maximum Variance Perspective 320 10.3 Projection Perspective 325 10.4 Eigenvector Computation and Low-Rank Approximations 333 10.5 PCA in High Dimensions 335 10.6 Key Steps of PCA in Practice 336 10.7 Latent Variable Perspective 339 10.8 Further Reading 343
11 Density Estimation with Gaussian Mixture Models 348 11.1 Gaussian Mixture Model 349 11.2 Parameter Learning via Maximum Likelihood 350 11.3 EM Algorithm 360 11.4 Latent-Variable Perspective 363 11.5 Further Reading 368
12 Classification with Support Vector Machines 370 12.1 Separating Hyperplanes 372 12.2 Primal Support Vector Machine 374 12.3 Dual Support Vector Machine 383 12.4 Kernels 388 12.5 Numerical Solution 390 12.6 Further Reading 392
References 395 Index 407
©2023 M. P. Deisenroth, A. A. Faisal, C. S. Ong. Published by Cambridge University Press (2020).
Foreword
Machine learning is the latest in a long line of attempts to distill human knowledge and reasoning into a form that is suitable for constructing ma chines and engineering automated systems. As machine learning becomes more ubiquitous and its software packages become easier to use, it is nat ural and desirable that the low-level technical details are abstracted away and hidden from the practitioner. However, this brings with it the danger that a practitioner becomes unaware of the design decisions and, hence, the limits of machine learning algorithms.
The enthusiastic practitioner who is interested to learn more about the magic behind successful machine learning algorithms currently faces a daunting set of pre-requisite knowledge:
Programming languages and data analysis tools
Large-scale computation and the associated frameworks Mathematics and statistics and how machine learning builds on it
At universities, introductory courses on machine learning tend to spend early parts of the course covering some of these pre-requisites. For histori cal reasons, courses in machine learning tend to be taught in the computer science department, where students are often trained in the first two areas of knowledge, but not so much in mathematics and statistics.
Current machine learning textbooks primarily focus on machine learn ing algorithms and methodologies and assume that the reader is com petent in mathematics and statistics. Therefore, these books only spend one or two chapters on background mathematics, either at the beginning of the book or as appendices. We have found many people who want to delve into the foundations of basic machine learning methods who strug gle with the mathematical knowledge required to read a machine learning textbook. Having taught undergraduate and graduate courses at universi ties, we find that the gap between high school mathematics and the math ematics level required to read a standard machine learning textbook is too big for many people.
This book brings the mathematical foundations of basic machine learn ing concepts to the fore and collects the information in a single place so that this skills gap is narrowed or even closed.
1
This material is published by Cambridge University Press as Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong (2020). This version is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. ©by M. P. Deisenroth, A. A. Faisal, and C. S. Ong, 2023. https://mml-book.com.
2 Foreword
Why Another Book on Machine Learning?
Machine learning builds upon the language of mathematics to express concepts that seem intuitively obvious but that are surprisingly difficult to formalize. Once formalized properly, we can gain insights into the task we want to solve. One common complaint of students of mathematics around the globe is that the topics covered seem to have little relevance to practical problems. We believe that machine learning is an obvious and direct motivation for people to learn mathematics.
This book is intended to be a guidebook to the vast mathematical lit- “Math is linked in erature that forms the foundations of modern machine learning. We mo
the popular mind with phobia and anxiety. You’d think we’re discussing spiders.” (Strogatz, 2014, page 281)
tivate the need for mathematical concepts by directly pointing out their usefulness in the context of fundamental machine learning problems. In the interest of keeping the book short, many details and more advanced concepts have been left out. Equipped with the basic concepts presented here, and how they fit into the larger context of machine learning, the reader can find numerous resources for further study, which we provide at the end of the respective chapters. For readers with a mathematical back ground, this book provides a brief but precisely stated glimpse of machine learning. In contrast to other books that focus on methods and models of machine learning (MacKay, 2003; Bishop, 2006; Alpaydin, 2010; Bar ber, 2012; Murphy, 2012; Shalev-Shwartz and Ben-David, 2014; Rogers and Girolami, 2016) or programmatic aspects of machine learning (Muller ¨ and Guido, 2016; Raschka and Mirjalili, 2017; Chollet and Allaire, 2018), we provide only four representative examples of machine learning algo rithms. Instead, we focus on the mathematical concepts behind the models themselves. We hope that readers will be able to gain a deeper understand ing of the basic questions in machine learning and connect practical ques tions arising from the use of machine learning with fundamental choices in the mathematical model.
We do not aim to write a classical machine learning book. Instead, our intention is to provide the mathematical background, applied to four cen tral machine learning problems, to make it easier to read other machine learning textbooks.
Who Is the Target Audience?
As applications of machine learning become widespread in society, we believe that everybody should have some understanding of its underlying principles. This book is written in an academic mathematical style, which enables us to be precise about the concepts behind machine learning. We encourage readers unfamiliar with this seemingly terse style to persevere and to keep the goals of each topic in mind. We sprinkle comments and remarks throughout the text, in the hope that it provides useful guidance with respect to the big picture.
The book assumes the reader to have mathematical knowledge commonly Draft (2023-10-18) of “Mathematics for Machine Learning”. Feedback: https://mml-book.com.
Foreword 3
covered in high school mathematics and physics. For example, the reader should have seen derivatives and integrals before, and geometric vectors in two or three dimensions. Starting from there, we generalize these con cepts. Therefore, the target audience of the book includes undergraduate university students, evening learners and learners participating in online machine learning courses.
In analogy to music, there are three types of interaction that people have with machine learning:
Astute Listener The democratization of machine learning by the pro vision of open-source software, online tutorials and cloud-based tools al lows users to not worry about the specifics of pipelines. Users can focus on extracting insights from data using off-the-shelf tools. This enables non tech-savvy domain experts to benefit from machine learning. This is sim ilar to listening to music; the user is able to choose and discern between different types of machine learning, and benefits from it. More experi enced users are like music critics, asking important questions about the application of machine learning in society such as ethics, fairness, and pri vacy of the individual. We hope that this book provides a foundation for thinking about the certification and risk management of machine learning systems, and allows them to use their domain expertise to build better machine learning systems.
Experienced Artist Skilled practitioners of machine learning can plug and play different tools and libraries into an analysis pipeline. The stereo typical practitioner would be a data scientist or engineer who understands machine learning interfaces and their use cases, and is able to perform wonderful feats of prediction from data. This is similar to a virtuoso play ing music, where highly skilled practitioners can bring existing instru ments to life and bring enjoyment to their audience. Using the mathe matics presented here as a primer, practitioners would be able to under stand the benefits and limits of their favorite method, and to extend and generalize existing machine learning algorithms. We hope that this book provides the impetus for more rigorous and principled development of machine learning methods.
Fledgling Composer As machine learning is applied to new domains, developers of machine learning need to develop new methods and extend existing algorithms. They are often researchers who need to understand the mathematical basis of machine learning and uncover relationships be tween different tasks. This is similar to composers of music who, within the rules and structure of musical theory, create new and amazing pieces. We hope this book provides a high-level overview of other technical books for people who want to become composers of machine learning. There is a great need in society for new researchers who are able to propose and explore novel approaches for attacking the many challenges of learning from data.
©2023 M. P. Deisenroth, A. A. Faisal, C. S. Ong. Published by Cambridge University Press (2020).
4 Foreword
Acknowledgments
We are grateful to many people who looked at early drafts of the book and suffered through painful expositions of concepts. We tried to imple ment their ideas that we did not vehemently disagree with. We would like to especially acknowledge Christfried Webers for his careful reading of many parts of the book, and his detailed suggestions on structure and presentation. Many friends and colleagues have also been kind enough to provide their time and energy on different versions of each chapter. We have been lucky to benefit from the generosity of the online commu nity, who have suggested improvements via https://github.com, which greatly improved the book.
The following people have found bugs, proposed clarifications and sug gested relevant literature, either via https://github.com or personal communication. Their names are sorted alphabetically.
Abdul-Ganiy Usman Adam Gaier
Adele Jackson
Aditya Menon
Alasdair Tran
Aleksandar Krnjaic Alexander Makrigiorgos Alfredo Canziani
Ali Shafti
Amr Khalifa
Andrew Tanggara
Angus Gruen
Antal A. Buss
Antoine Toisoul Le Cann Areg Sarvazyan
Artem Artemev
Artyom Stepanov
Bill Kromydas
Bob Williamson
Boon Ping Lim
Chao Qu
Cheng Li
Chris Sherlock
Christopher Gray
Daniel McNamara
Daniel Wood
Darren Siegel
David Johnston
Dawei Chen
Ellen Broad
Fengkuangtian Zhu Fiona Condon
Georgios Theodorou He Xin
Irene Raissa Kameni Jakub Nabaglo
James Hensman
Jamie Liu
Jean Kaddour
Jean-Paul Ebejer
Jerry Qiang
Jitesh Sindhare
John Lloyd
Jonas Ngnawe
Jon Martin
Justin Hsi
Kai Arulkumaran
Kamil Dreczkowski Lily Wang

MATHEMATICS FOR MACHINE LEARNING

Marc Peter Deisenroth

A. Aldo Faisal

Cheng Soon Ong

Contents

Foreword 1

Part I Mathematical Foundations 9

1 Introduction and Motivation 11 1.1 Finding Words for Intuitions 12 1.2 Two Ways to Read This Book 13 1.3 Exercises and Feedback 16

2 Linear Algebra 17 2.1 Systems of Linear Equations 19 2.2 Matrices 22 2.3 Solving Systems of Linear Equations 27 2.4 Vector Spaces 35 2.5 Linear Independence 40 2.6 Basis and Rank 44 2.7 Linear Mappings 48 2.8 Affine Spaces 61 2.9 Further Reading 63 Exercises 64

3 Analytic Geometry 70 3.1 Norms 71 3.2 Inner Products 72 3.3 Lengths and Distances 75 3.4 Angles and Orthogonality 76 3.5 Orthonormal Basis 78 3.6 Orthogonal Complement 79 3.7 Inner Product of Functions 80 3.8 Orthogonal Projections 81 3.9 Rotations 91 3.10 Further Reading 94 Exercises 96

4 Matrix Decompositions 98 4.1 Determinant and Trace 99

i

This material is published by Cambridge University Press as Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong (2020). This version is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. ©by M. P. Deisenroth, A. A. Faisal, and C. S. Ong, 2023. https://mml-book.com.

ii Contents

4.2 Eigenvalues and Eigenvectors 105 4.3 Cholesky Decomposition 114 4.4 Eigendecomposition and Diagonalization 115 4.5 Singular Value Decomposition 119 4.6 Matrix Approximation 129 4.7 Matrix Phylogeny 134 4.8 Further Reading 135 Exercises 137

5 Vector Calculus 139 5.1 Differentiation of Univariate Functions 141 5.2 Partial Differentiation and Gradients 146 5.3 Gradients of Vector-Valued Functions 149 5.4 Gradients of Matrices 155 5.5 Useful Identities for Computing Gradients 158 5.6 Backpropagation and Automatic Differentiation 159 5.7 Higher-Order Derivatives 164 5.8 Linearization and Multivariate Taylor Series 165 5.9 Further Reading 170 Exercises 170

6 Probability and Distributions 172 6.1 Construction of a Probability Space 172 6.2 Discrete and Continuous Probabilities 178 6.3 Sum Rule, Product Rule, and Bayes’ Theorem 183 6.4 Summary Statistics and Independence 186 6.5 Gaussian Distribution 197 6.6 Conjugacy and the Exponential Family 205 6.7 Change of Variables/Inverse Transform 214 6.8 Further Reading 221 Exercises 222

7 Continuous Optimization 225 7.1 Optimization Using Gradient Descent 227 7.2 Constrained Optimization and Lagrange Multipliers 233 7.3 Convex Optimization 236 7.4 Further Reading 246 Exercises 247

Part II Central Machine Learning Problems 249

8 When Models Meet Data 251 8.1 Data, Models, and Learning 251 8.2 Empirical Risk Minimization 258 8.3 Parameter Estimation 265 8.4 Probabilistic Modeling and Inference 272 8.5 Directed Graphical Models 278

Draft (2023-10-18) of “Mathematics for Machine Learning”. Feedback: https://mml-book.com.

Contents iii 8.6 Model Selection 283

9 Linear Regression 289 9.1 Problem Formulation 291 9.2 Parameter Estimation 292 9.3 Bayesian Linear Regression 303 9.4 Maximum Likelihood as Orthogonal Projection 313 9.5 Further Reading 315

10 Dimensionality Reduction with Principal Component Analysis 317 10.1 Problem Setting 318 10.2 Maximum Variance Perspective 320 10.3 Projection Perspective 325 10.4 Eigenvector Computation and Low-Rank Approximations 333 10.5 PCA in High Dimensions 335 10.6 Key Steps of PCA in Practice 336 10.7 Latent Variable Perspective 339 10.8 Further Reading 343

11 Density Estimation with Gaussian Mixture Models 348 11.1 Gaussian Mixture Model 349 11.2 Parameter Learning via Maximum Likelihood 350 11.3 EM Algorithm 360 11.4 Latent-Variable Perspective 363 11.5 Further Reading 368

12 Classification with Support Vector Machines 370 12.1 Separating Hyperplanes 372 12.2 Primal Support Vector Machine 374 12.3 Dual Support Vector Machine 383 12.4 Kernels 388 12.5 Numerical Solution 390 12.6 Further Reading 392

References 395 Index 407

Foreword

Machine learning is the latest in a long line of attempts to distill human knowledge and reasoning into a form that is suitable for constructing ma chines and engineering automated systems. As machine learning becomes more ubiquitous and its software packages become easier to use, it is nat ural and desirable that the low-level technical details are abstracted away and hidden from the practitioner. However, this brings with it the danger that a practitioner becomes unaware of the design decisions and, hence, the limits of machine learning algorithms.

The enthusiastic practitioner who is interested to learn more about the magic behind successful machine learning algorithms currently faces a daunting set of pre-requisite knowledge:

Programming languages and data analysis tools

Large-scale computation and the associated frameworks Mathematics and statistics and how machine learning builds on it

At universities, introductory courses on machine learning tend to spend early parts of the course covering some of these pre-requisites. For histori cal reasons, courses in machine learning tend to be taught in the computer science department, where students are often trained in the first two areas of knowledge, but not so much in mathematics and statistics.

Current machine learning textbooks primarily focus on machine learn ing algorithms and methodologies and assume that the reader is com petent in mathematics and statistics. Therefore, these books only spend one or two chapters on background mathematics, either at the beginning of the book or as appendices. We have found many people who want to delve into the foundations of basic machine learning methods who strug gle with the mathematical knowledge required to read a machine learning textbook. Having taught undergraduate and graduate courses at universi ties, we find that the gap between high school mathematics and the math ematics level required to read a standard machine learning textbook is too big for many people.

This book brings the mathematical foundations of basic machine learn ing concepts to the fore and collects the information in a single place so that this skills gap is narrowed or even closed.

1

This material is published by Cambridge University Press as Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong (2020). This version is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. ©by M. P. Deisenroth, A. A. Faisal, and C. S. Ong, 2023. https://mml-book.com.

2 Foreword

Why Another Book on Machine Learning?

Machine learning builds upon the language of mathematics to express concepts that seem intuitively obvious but that are surprisingly difficult to formalize. Once formalized properly, we can gain insights into the task we want to solve. One common complaint of students of mathematics around the globe is that the topics covered seem to have little relevance to practical problems. We believe that machine learning is an obvious and direct motivation for people to learn mathematics.

This book is intended to be a guidebook to the vast mathematical lit- “Math is linked in erature that forms the foundations of modern machine learning. We mo

the popular mind with phobia and anxiety. You’d think we’re discussing spiders.” (Strogatz, 2014, page 281)

tivate the need for mathematical concepts by directly pointing out their usefulness in the context of fundamental machine learning problems. In the interest of keeping the book short, many details and more advanced concepts have been left out. Equipped with the basic concepts presented here, and how they fit into the larger context of machine learning, the reader can find numerous resources for further study, which we provide at the end of the respective chapters. For readers with a mathematical back ground, this book provides a brief but precisely stated glimpse of machine learning. In contrast to other books that focus on methods and models of machine learning (MacKay, 2003; Bishop, 2006; Alpaydin, 2010; Bar ber, 2012; Murphy, 2012; Shalev-Shwartz and Ben-David, 2014; Rogers and Girolami, 2016) or programmatic aspects of machine learning (Muller ¨ and Guido, 2016; Raschka and Mirjalili, 2017; Chollet and Allaire, 2018), we provide only four representative examples of machine learning algo rithms. Instead, we focus on the mathematical concepts behind the models themselves. We hope that readers will be able to gain a deeper understand ing of the basic questions in machine learning and connect practical ques tions arising from the use of machine learning with fundamental choices in the mathematical model.

We do not aim to write a classical machine learning book. Instead, our intention is to provide the mathematical background, applied to four cen tral machine learning problems, to make it easier to read other machine learning textbooks.

Who Is the Target Audience?

As applications of machine learning become widespread in society, we believe that everybody should have some understanding of its underlying principles. This book is written in an academic mathematical style, which enables us to be precise about the concepts behind machine learning. We encourage readers unfamiliar with this seemingly terse style to persevere and to keep the goals of each topic in mind. We sprinkle comments and remarks throughout the text, in the hope that it provides useful guidance with respect to the big picture.

The book assumes the reader to have mathematical knowledge commonly Draft (2023-10-18) of “Mathematics for Machine Learning”. Feedback: https://mml-book.com.

Foreword 3

covered in high school mathematics and physics. For example, the reader should have seen derivatives and integrals before, and geometric vectors in two or three dimensions. Starting from there, we generalize these con cepts. Therefore, the target audience of the book includes undergraduate university students, evening learners and learners participating in online machine learning courses.

In analogy to music, there are three types of interaction that people have with machine learning:

Astute Listener The democratization of machine learning by the pro vision of open-source software, online tutorials and cloud-based tools al lows users to not worry about the specifics of pipelines. Users can focus on extracting insights from data using off-the-shelf tools. This enables non tech-savvy domain experts to benefit from machine learning. This is sim ilar to listening to music; the user is able to choose and discern between different types of machine learning, and benefits from it. More experi enced users are like music critics, asking important questions about the application of machine learning in society such as ethics, fairness, and pri vacy of the individual. We hope that this book provides a foundation for thinking about the certification and risk management of machine learning systems, and allows them to use their domain expertise to build better machine learning systems.

Experienced Artist Skilled practitioners of machine learning can plug and play different tools and libraries into an analysis pipeline. The stereo typical practitioner would be a data scientist or engineer who understands machine learning interfaces and their use cases, and is able to perform wonderful feats of prediction from data. This is similar to a virtuoso play ing music, where highly skilled practitioners can bring existing instru ments to life and bring enjoyment to their audience. Using the mathe matics presented here as a primer, practitioners would be able to under stand the benefits and limits of their favorite method, and to extend and generalize existing machine learning algorithms. We hope that this book provides the impetus for more rigorous and principled development of machine learning methods.

Fledgling Composer As machine learning is applied to new domains, developers of machine learning need to develop new methods and extend existing algorithms. They are often researchers who need to understand the mathematical basis of machine learning and uncover relationships be tween different tasks. This is similar to composers of music who, within the rules and structure of musical theory, create new and amazing pieces. We hope this book provides a high-level overview of other technical books for people who want to become composers of machine learning. There is a great need in society for new researchers who are able to propose and explore novel approaches for attacking the many challenges of learning from data.

4 Foreword

Acknowledgments

We are grateful to many people who looked at early drafts of the book and suffered through painful expositions of concepts. We tried to imple ment their ideas that we did not vehemently disagree with. We would like to especially acknowledge Christfried Webers for his careful reading of many parts of the book, and his detailed suggestions on structure and presentation. Many friends and colleagues have also been kind enough to provide their time and energy on different versions of each chapter. We have been lucky to benefit from the generosity of the online commu nity, who have suggested improvements via https://github.com, which greatly improved the book.

The following people have found bugs, proposed clarifications and sug gested relevant literature, either via https://github.com or personal communication. Their names are sorted alphabetically.

Abdul-Ganiy Usman Adam Gaier

Adele Jackson

Aditya Menon

Alasdair Tran

Aleksandar Krnjaic Alexander Makrigiorgos Alfredo Canziani

Ali Shafti

Amr Khalifa

Andrew Tanggara

Angus Gruen

Antal A. Buss

Antoine Toisoul Le Cann Areg Sarvazyan

Artem Artemev

Artyom Stepanov

Bill Kromydas

Bob Williamson

Boon Ping Lim

Chao Qu

Cheng Li

Chris Sherlock

Christopher Gray

Daniel McNamara

Daniel Wood

Darren Siegel

David Johnston

Dawei Chen

Ellen Broad

Fengkuangtian Zhu Fiona Condon

Georgios Theodorou He Xin

Irene Raissa Kameni Jakub Nabaglo

James Hensman

Jamie Liu

Jean Kaddour

Jean-Paul Ebejer

Jerry Qiang

Jitesh Sindhare

John Lloyd

Jonas Ngnawe

Jon Martin

Justin Hsi

Kai Arulkumaran

Kamil Dreczkowski Lily Wang

Datasets:

mjphayes
/

textbook_dataset

Dataset Card for "textbook_dataset"