Algebraic Approach to Data Processing: Techniques and Applications

Description

The book explores a new general approach to selecting―and designing―data processing techniques. Symmetry and invariance ideas behind this algebraic approach have been successful in physics, where many new theories are formulated in symmetry terms. The book explains this approach and expands it to new application areas ranging from engineering, medicine, education to social sciences. In many cases, this approach leads to optimal techniques and optimal solutions. That the same data processing techniques help us better analyze wooden structures, lung dysfunctions, and deep learning algorithms is a good indication that these techniques can be used in many other applications as well. The book is recommended to researchers and practitioners who need to select a data processing technique―or who want to design a new technique when the existing techniques do not work. It is also recommended to students who want to learn the state-of-the-art data processing.   Preface Contents 1 Introduction 1.1 What Is Data Processing and Why Do We Need It? 1.2 Why Algebraic Approach? 1.3 What We Do in This Book: An Overview 1.4 Thanks References 2 What Are the Most Natural and the Most Frequent Transformations 2.1 Main Idea: Numerical Values Change When We Change a Measuring Unit and/or Starting Point 2.2 Scaling Transformations 2.3 Shifts 2.4 Linear Transformations 2.5 Geometric Transformations 2.6 Beyond Linear Transformations 2.7 Permutations References 3 Which Functions and Which Families of Functions Are Invariant 3.1 Why Do We Need Invariant Functions 3.2 What Does It Mean for a Function to Be Invariant 3.3 Example: Scale-Invariant Functions of One Variable 3.4 What If We Have Both Shift- and Scale-Invariance? 3.5 Which Families of Functions Are Invariant: Case of Shift-Invariance 3.6 Which Families of Functions Are Invariant: Case of Scale-Invariance 3.7 What If We Have Both Shift- and Scale-Invariance 3.8 Which Linear Transformations Are Shift-Invariant References 4 What Is the General Relation Between Invariance and Optimality 4.1 What Is an Optimality Criterion 4.2 We Need a Final Optimality Criterion 4.3 It Is Often Reasonable to Require That the Optimality Criterion Be Invariant 4.4 Main Result of This Chapter 5 General Application: Dynamical Systems 5.1 Problem: Why a Linear-Based Classification Often Works in Nonlinear Cases 5.2 Our Explanation References 6 First Application to Physics: Why Liquids? 6.1 Two Applications to Physics: Summary 6.2 Problem: Why Liquids? 6.3 Towards a Formulation of the Problem in Precise Terms 6.4 Main Result of This Chapter References 7 Second Application to Physics: Warping of Our Galaxy 7.1 Formulation of the Problem 7.2 Analysis of the Problem and the Resulting Explanation References 8 Application to Electrical Engineering: Class-D Audio Amplifiers 8.1 Applications to Engineering: Summary 8.2 Problem: Why Class-D Audio Amplifiers Work Well? 8.3 Why Pulses 8.4 Why the Pulse's Duration Should Linearly Depend … References 9 Application to Mechanical Engineering: Wood Structures 9.1 Problem: Need for a Theoretical Explanation of an Empirical Fact 9.2 Our Explanation: Main Idea 9.3 Our Explanation: Details 9.4 Proof References 10 Medical Application: Prevention 10.1 Problem: How to Best Maintain Social Distance 10.2 Towards Formulating This Problem in Precise Terms 10.3 Solution Reference 11 Medical Application: Testing 11.1 Problem: Optimal Group Testing 11.2 What Was Proposed 11.3 Resulting Problem 11.4 Let Us Formulate This Problem in Precise Terms 11.5 Solution References 12 Medical Application: Diagnostics, Part 1 12.1 Problem: Diagnosing Lung Disfunctions in Children 12.2 First Pre-processing Stage: Scale-Invariant Smoothing 12.3 Which Order Polynomials Should We Use? 12.4 Second Pre-processing Stage: Using the Approximating Polynomials to Distinguish Between Different Diseases 12.5 Third Pre-processing Stage: Scale-Invariant Similarity/Dissimilarity Measures 12.6 How to Select α: Need to Have Efficient and Robust Estimates 12.7 Scale-Invariance Helps to Take Into Account That Signal Informativeness Decreases with Time 12.8 Pre-processing Summarized: What Information Serves as An Input to a Neural Network 12.9 The Results of Training Neural Networks on These Pre-processed Data References 13 Medical Application: Diagnostics, Part 2 13.1 Problem: Why Hierarchical Multiclass Classification Works Better Than Direct Classification 13.2 Our Explanation References 14 Medical Application: Diagnostics, Part 3 14.1 Problem: Which Fourier Components Are Most Informative 14.2 Main Idea 14.3 First Case Study: Human Color Vision 14.4 Second Case Study: Classifying Lung Dysfunctions References 15 Medical Application: Treatment 15.1 Problem: Geometric Aspects of Wound Healing 15.2 What Are Natural Symmetries Here and What Are the Resulting Cell Shapes: Case of Undamaged Skin 15.3 What If the Skin Is Damaged: Resulting Symmetries and Cell Shapes 15.4 Geometric Symmetries Also Explain Observed Cell Motions References 16 Applications to Economics: How Do People Make Decisions, Part 1 References 17 Application to Economics: How Do People Make Decisions, Part 2 17.1 Problem: Need to Consider Multiple Scenarios 17.2 Our Explanation References 18 Application to Economics: How Do People Make Decisions, Part 3 18.1 Problem: Using Experts 18.2 Towards an Explanation References 19 Application to Economics: How Do People Make Decisions, Part 4 19.1 Why Should We Play Down Emotions 19.2 Towards Explanation References 20 Application to Economics: Stimuli, Part 1 20.1 Problem: Why Rewards Work Better Than Punishment 20.2 Analysis of the Problem 20.3 Our Explanation References 21 Application to Economics: Stimuli, Part 2 21.1 Problem: Why Top Experts Are Paid So Much 21.2 Our Explanation References 22 Application to Economics: Investment 22.1 1/n Investment: Formulation of the Problem 22.2 Our Explanation 22.3 Discussion References 23 Application to Social Sciences: When Revolutions Happen 23.1 Formulation of the Problem 23.2 Analysis of the Problem References 24 Application to Education: General 24.1 Problem: Is Immediate Repetition Good for Learning? 24.2 Analysis of the Problem and the Resulting Explanation References 25 Application to Education: Specific 25.1 Problem: Why Derivative 25.2 Invariance Naturally Leads to the Derivative Reference 26 Application to Mathematics: Why Necessary Conditions Are Often Sufficient 26.1 Formulation of the Problem 26.2 Analysis of the Problem 26.3 How Can We Formalize What Is Not Abnormal 26.4 Resulting Explanation of the TONCAS Phenomenon References 27 Data Processing: Neural Techniques, Part 1 27.1 Machine Learning Is Needed to Analyze Complex Systems 27.2 Neural Networks and Deep Learning: A Brief Reminder 27.3 Why Traditional Neural Networks 27.4 Why Sigmoid Activation Function: Idea 27.5 Why Sigmoid—Derivation 27.6 Limit Cases 27.7 We Need Multi-layer Neural Networks 27.8 Which Activation Function Should We Use 27.9 This Leads Exactly to Squashing Functions 27.10 Why Rectified Linear Functions References 28 Data Processing: Neural Techniques, Part 2 28.1 Problem: Spiking Neural Networks 28.2 Analysis of the Problem and the First Result 28.3 Main Result: Spikes Are, in Some Reasonable Sense, Optimal References 29 Data Processing: Fuzzy Techniques, Part 1 29.1 Why Fuzzy Techniques 29.2 Fuzzy Techniques: Main Ideas 29.3 Fuzzy Techniques: Logic References 30 Data Processing: Neural and Fuzzy Techniques 30.1 Problem: Computations Should Be Fast and Understandable 30.2 Definitions and the Main Results 30.3 Auxiliary Result: What Can We Do with Two-Layer Networks References 31 Data Processing: Fuzzy Techniques, Part 2 31.1 Problem: Which Fuzzy Techniques to Use? 31.2 Analysis of the Problem 31.3 Which Symmetric Membership Functions Should We … 31.4 Which Hedge Operations and Negation Operations Should We Select 31.5 Proofs References 32 Data Processing: Fuzzy Techniques, Part 3 32.1 Problem: Which Fuzzy Degrees to Use? 32.2 Definitions and the Main Result 32.3 How General Is This Result? 32.4 What If We Allow Unlimited Number of ``And''-Operations and Negations: Case Study References 33 Data Processing: Fuzzy Techniques, Part 4 33.1 Problem: How to Explain Commonsense Reasoning 33.2 Our Explanation 33.3 Auxiliary Result: Why the Usual Quantifiers? References 34 Data Processing: Probabilistic Techniques, Part 1 34.1 Problem: How to Represent Interval Uncertainty 34.2 Analysis of the Problem 34.3 Our Results References 35 Data Processing: Probabilistic Techniques, Part 2 35.1 Problem: How to Represent General Uncertainty 35.2 Definitions and the Main Result 35.3 Consequence References 36 Data Processing: Probabilistic Techniques, Part 3 36.1 Problem: Experts Don't Perform Well in Unusual Situations 36.2 Our Explanation References 37 Data Processing: Beyond Traditional Techniques 37.1 DNA Computing: Introduction 37.2 Computing Without Computing—Quantum Version: A Brief Reminder 37.3 Computing Without Computing—Version Involving Acausal Processes: A Reminder 37.4 Computing Without Computing—DNA Version 37.5 DNA Computing Without Computing Is Somewhat Less … 37.6 First Related Result: Security Is More Difficult to Achieve than Privacy 37.7 Second Related Result: Data Storage Is More Difficult Than Data Transmission References Appendix References Index