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.
Author(s): Julio C. Urenda, Vladik Kreinovich
Series: Studies in Big Data, 115
Publisher: Springer
Year: 2022
Language: English
Pages: 245
City: Cham
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
