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Deep Neural Networks in a Mathematical Framework

Deep Neural Networks in a Mathematical Framework
Author: Anthony L. Caterini
Publisher: Springer
Total Pages: 95
Release: 2018-03-22
Genre: Computers
ISBN: 3319753045

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This SpringerBrief describes how to build a rigorous end-to-end mathematical framework for deep neural networks. The authors provide tools to represent and describe neural networks, casting previous results in the field in a more natural light. In particular, the authors derive gradient descent algorithms in a unified way for several neural network structures, including multilayer perceptrons, convolutional neural networks, deep autoencoders and recurrent neural networks. Furthermore, the authors developed framework is both more concise and mathematically intuitive than previous representations of neural networks. This SpringerBrief is one step towards unlocking the black box of Deep Learning. The authors believe that this framework will help catalyze further discoveries regarding the mathematical properties of neural networks.This SpringerBrief is accessible not only to researchers, professionals and students working and studying in the field of deep learning, but also to those outside of the neutral network community.


A Novel Mathematical Framework for the Analysis of Neural Networks

A Novel Mathematical Framework for the Analysis of Neural Networks
Author: Anthony L. Caterini
Publisher:
Total Pages: 89
Release: 2017
Genre: Convolutions (Mathematics)
ISBN:

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Over the past decade, Deep Neural Networks (DNNs) have become very popular models for processing large amounts of data because of their successful application in a wide variety of fields. These models are layered, often containing parametrized linear and non-linear transformations at each layer in the network. At this point, however, we do not rigorously understand why DNNs are so effective. In this thesis, we explore one way to approach this problem: we develop a generic mathematical framework for representing neural networks, and demonstrate how this framework can be used to represent specific neural network architectures. In chapter 1, we start by exploring mathematical contributions to neural networks. We can rigorously explain some properties of DNNs, but these results fail to fully describe the mechanics of a generic neural network. We also note that most approaches to describing neural networks rely upon breaking down the parameters and inputs into scalars, as opposed to referencing their underlying vector spaces, which adds some awkwardness into their analysis. Our framework strictly operates over these spaces, affording a more natural description of DNNs once the mathematical objects that we use are well-defined and understood. We then develop the generic framework in chapter 3. We are able to describe an algorithm for calculating one step of gradient descent directly over the inner product space in which the parameters are defined. Also, we can represent the error backpropagation step in a concise and compact form. Besides a standard squared loss or cross-entropy loss, we also demonstrate that our framework, including gradient calculation, extends to a more complex loss function involving the first derivative of the network. After developing the generic framework, we apply it to three specific network examples in chapter 4. We start with the Multilayer Perceptron, the simplest type of DNN, and show how to generate a gradient descent step for it. We then represent the Convolutional Neural Network (CNN), which contains more complicated input spaces, parameter spaces, and transformations at each layer. The CNN, however, still fits into the generic framework. The last structure that we consider is the Deep Auto-Encoder, which has parameters that are not completely independent at each layer. We are able to extend the generic framework to handle this case as well. In chapter 5, we use some of the results from the previous chapters to develop a framework for Recurrent Neural Networks (RNNs), the sequence-parsing DNN architecture. The parameters are shared across all layers of the network, and thus we require some additional machinery to describe RNNs. We describe a generic RNN first, and then the specific case of the vanilla RNN. We again compute gradients directly over inner product spaces.


Hands-On Mathematics for Deep Learning

Hands-On Mathematics for Deep Learning
Author: Jay Dawani
Publisher: Packt Publishing Ltd
Total Pages: 347
Release: 2020-06-12
Genre: Computers
ISBN: 183864184X

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A comprehensive guide to getting well-versed with the mathematical techniques for building modern deep learning architectures Key FeaturesUnderstand linear algebra, calculus, gradient algorithms, and other concepts essential for training deep neural networksLearn the mathematical concepts needed to understand how deep learning models functionUse deep learning for solving problems related to vision, image, text, and sequence applicationsBook Description Most programmers and data scientists struggle with mathematics, having either overlooked or forgotten core mathematical concepts. This book uses Python libraries to help you understand the math required to build deep learning (DL) models. You'll begin by learning about core mathematical and modern computational techniques used to design and implement DL algorithms. This book will cover essential topics, such as linear algebra, eigenvalues and eigenvectors, the singular value decomposition concept, and gradient algorithms, to help you understand how to train deep neural networks. Later chapters focus on important neural networks, such as the linear neural network and multilayer perceptrons, with a primary focus on helping you learn how each model works. As you advance, you will delve into the math used for regularization, multi-layered DL, forward propagation, optimization, and backpropagation techniques to understand what it takes to build full-fledged DL models. Finally, you’ll explore CNN, recurrent neural network (RNN), and GAN models and their application. By the end of this book, you'll have built a strong foundation in neural networks and DL mathematical concepts, which will help you to confidently research and build custom models in DL. What you will learnUnderstand the key mathematical concepts for building neural network modelsDiscover core multivariable calculus conceptsImprove the performance of deep learning models using optimization techniquesCover optimization algorithms, from basic stochastic gradient descent (SGD) to the advanced Adam optimizerUnderstand computational graphs and their importance in DLExplore the backpropagation algorithm to reduce output errorCover DL algorithms such as convolutional neural networks (CNNs), sequence models, and generative adversarial networks (GANs)Who this book is for This book is for data scientists, machine learning developers, aspiring deep learning developers, or anyone who wants to understand the foundation of deep learning by learning the math behind it. Working knowledge of the Python programming language and machine learning basics is required.


Math for Deep Learning

Math for Deep Learning
Author: Ronald T. Kneusel
Publisher: No Starch Press
Total Pages: 346
Release: 2021-11-23
Genre: Computers
ISBN: 1718501919

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Math for Deep Learning provides the essential math you need to understand deep learning discussions, explore more complex implementations, and better use the deep learning toolkits. With Math for Deep Learning, you'll learn the essential mathematics used by and as a background for deep learning. You’ll work through Python examples to learn key deep learning related topics in probability, statistics, linear algebra, differential calculus, and matrix calculus as well as how to implement data flow in a neural network, backpropagation, and gradient descent. You’ll also use Python to work through the mathematics that underlies those algorithms and even build a fully-functional neural network. In addition you’ll find coverage of gradient descent including variations commonly used by the deep learning community: SGD, Adam, RMSprop, and Adagrad/Adadelta.


The Principles of Deep Learning Theory

The Principles of Deep Learning Theory
Author: Daniel A. Roberts
Publisher: Cambridge University Press
Total Pages: 473
Release: 2022-05-26
Genre: Computers
ISBN: 1316519333

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This volume develops an effective theory approach to understanding deep neural networks of practical relevance.


Algorithms for Verifying Deep Neural Networks

Algorithms for Verifying Deep Neural Networks
Author: Changliu Liu
Publisher:
Total Pages:
Release: 2021-02-11
Genre:
ISBN: 9781680837865

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Neural networks have been widely used in many applications, such as image classification and understanding, language processing, and control of autonomous systems. These networks work by mapping inputs to outputs through a sequence of layers. At each layer, the input to that layer undergoes an affine transformation followed by a simple nonlinear transformation before being passed to the next layer. Neural networks are being used for increasingly important tasks, and in some cases, incorrect outputs can lead to costly consequences, hence validation of correctness at each layer is vital. The sheer size of the networks makes this not feasible using traditional methods. In this monograph, the authors survey a class of methods that are capable of formally verifying properties of deep neural networks. In doing so, they introduce a unified mathematical framework for verifying neural networks, classify existing methods under this framework, provide pedagogical implementations of existing methods, and compare those methods on a set of benchmark problems. Algorithms for Verifying Deep Neural Networks serves as a tutorial for students and professionals interested in this emerging field as well as a benchmark to facilitate the design of new verification algorithms.


Multi-faceted Deep Learning

Multi-faceted Deep Learning
Author: Jenny Benois-Pineau
Publisher: Springer Nature
Total Pages: 321
Release: 2021-10-20
Genre: Computers
ISBN: 3030744787

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This book covers a large set of methods in the field of Artificial Intelligence - Deep Learning applied to real-world problems. The fundamentals of the Deep Learning approach and different types of Deep Neural Networks (DNNs) are first summarized in this book, which offers a comprehensive preamble for further problem–oriented chapters. The most interesting and open problems of machine learning in the framework of Deep Learning are discussed in this book and solutions are proposed. This book illustrates how to implement the zero-shot learning with Deep Neural Network Classifiers, which require a large amount of training data. The lack of annotated training data naturally pushes the researchers to implement low supervision algorithms. Metric learning is a long-term research but in the framework of Deep Learning approaches, it gets freshness and originality. Fine-grained classification with a low inter-class variability is a difficult problem for any classification tasks. This book presents how it is solved, by using different modalities and attention mechanisms in 3D convolutional networks. Researchers focused on Machine Learning, Deep learning, Multimedia and Computer Vision will want to buy this book. Advanced level students studying computer science within these topic areas will also find this book useful.


Math and Architectures of Deep Learning

Math and Architectures of Deep Learning
Author: Krishnendu Chaudhury
Publisher: Simon and Schuster
Total Pages: 550
Release: 2024-05-21
Genre: Computers
ISBN: 1638350809

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Shine a spotlight into the deep learning “black box”. This comprehensive and detailed guide reveals the mathematical and architectural concepts behind deep learning models, so you can customize, maintain, and explain them more effectively. Inside Math and Architectures of Deep Learning you will find: Math, theory, and programming principles side by side Linear algebra, vector calculus and multivariate statistics for deep learning The structure of neural networks Implementing deep learning architectures with Python and PyTorch Troubleshooting underperforming models Working code samples in downloadable Jupyter notebooks The mathematical paradigms behind deep learning models typically begin as hard-to-read academic papers that leave engineers in the dark about how those models actually function. Math and Architectures of Deep Learning bridges the gap between theory and practice, laying out the math of deep learning side by side with practical implementations in Python and PyTorch. Written by deep learning expert Krishnendu Chaudhury, you’ll peer inside the “black box” to understand how your code is working, and learn to comprehend cutting-edge research you can turn into practical applications. Foreword by Prith Banerjee. About the technology Discover what’s going on inside the black box! To work with deep learning you’ll have to choose the right model, train it, preprocess your data, evaluate performance and accuracy, and deal with uncertainty and variability in the outputs of a deployed solution. This book takes you systematically through the core mathematical concepts you’ll need as a working data scientist: vector calculus, linear algebra, and Bayesian inference, all from a deep learning perspective. About the book Math and Architectures of Deep Learning teaches the math, theory, and programming principles of deep learning models laid out side by side, and then puts them into practice with well-annotated Python code. You’ll progress from algebra, calculus, and statistics all the way to state-of-the-art DL architectures taken from the latest research. What's inside The core design principles of neural networks Implementing deep learning with Python and PyTorch Regularizing and optimizing underperforming models About the reader Readers need to know Python and the basics of algebra and calculus. About the author Krishnendu Chaudhury is co-founder and CTO of the AI startup Drishti Technologies. He previously spent a decade each at Google and Adobe. Table of Contents 1 An overview of machine learning and deep learning 2 Vectors, matrices, and tensors in machine learning 3 Classifiers and vector calculus 4 Linear algebraic tools in machine learning 5 Probability distributions in machine learning 6 Bayesian tools for machine learning 7 Function approximation: How neural networks model the world 8 Training neural networks: Forward propagation and backpropagation 9 Loss, optimization, and regularization 10 Convolutions in neural networks 11 Neural networks for image classification and object detection 12 Manifolds, homeomorphism, and neural networks 13 Fully Bayes model parameter estimation 14 Latent space and generative modeling, autoencoders, and variational autoencoders A Appendix


Mathematical Methods for Neural Network Analysis and Design

Mathematical Methods for Neural Network Analysis and Design
Author: Richard M. Golden
Publisher: MIT Press
Total Pages: 452
Release: 1996
Genre: Computers
ISBN: 9780262071741

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For convenience, many of the proofs of the key theorems have been rewritten so that the entire book uses a relatively uniform notion.


Deep Learning Architectures

Deep Learning Architectures
Author: Ovidiu Calin
Publisher: Springer Nature
Total Pages: 760
Release: 2020-02-13
Genre: Mathematics
ISBN: 3030367215

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This book describes how neural networks operate from the mathematical point of view. As a result, neural networks can be interpreted both as function universal approximators and information processors. The book bridges the gap between ideas and concepts of neural networks, which are used nowadays at an intuitive level, and the precise modern mathematical language, presenting the best practices of the former and enjoying the robustness and elegance of the latter. This book can be used in a graduate course in deep learning, with the first few parts being accessible to senior undergraduates. In addition, the book will be of wide interest to machine learning researchers who are interested in a theoretical understanding of the subject.