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Logifold: A Geometrical Foundation of Ensemble Machine Learning

Inkee Jung, Siu-Cheong Lau

TL;DR

A local-to-global and measure-theoretical approach to understanding datasets and to formulate a logifold structure and to interpret network models with restricted domains as local charts of datasets provides a mathematical foundation for ensemble machine learning.

Abstract

We present a local-to-global and measure-theoretical approach to understanding datasets. The core idea is to formulate a logifold structure and to interpret network models with restricted domains as local charts of datasets. In particular, this provides a mathematical foundation for ensemble machine learning. Our experiments demonstrate that logifolds can be implemented to identify fuzzy domains and improve accuracy compared to taking average of model outputs. Additionally, we provide a theoretical example of a logifold, highlighting the importance of restricting to domains of classifiers in an ensemble.

Logifold: A Geometrical Foundation of Ensemble Machine Learning

TL;DR

A local-to-global and measure-theoretical approach to understanding datasets and to formulate a logifold structure and to interpret network models with restricted domains as local charts of datasets provides a mathematical foundation for ensemble machine learning.

Abstract

We present a local-to-global and measure-theoretical approach to understanding datasets. The core idea is to formulate a logifold structure and to interpret network models with restricted domains as local charts of datasets. In particular, this provides a mathematical foundation for ensemble machine learning. Our experiments demonstrate that logifolds can be implemented to identify fuzzy domains and improve accuracy compared to taking average of model outputs. Additionally, we provide a theoretical example of a logifold, highlighting the importance of restricting to domains of classifiers in an ensemble.
Paper Structure (10 sections, 3 theorems, 15 equations, 1 figure, 2 tables)

This paper contains 10 sections, 3 theorems, 15 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Linear Logical function
  3. Universality of linear logical functions and definition of a logifold.
  4. Ensemble machine learning and a theoretical example
  5. Fuzzy linear logical function and Fuzzy logifold
  6. Experiments
  7. Experiment Setting
  8. Experiment 1: Ensemble of six Simple CNN and one ResNet20 on CIFAR10
  9. Experiment 2: union dataset of MNIST, Fashion MNIST and CIFAR10
  10. Conclusion

Key Result

Proposition 2.3

There exists a linear logical graph $(G,L)$ whose function $f_{(G,L)}$ gives the above $f$ composed of affine linear functions, the ReLu function, and the index-max function.

Figures (1)

  • Figure 1: Example of a linear logical graph.

Theorems & Definitions (12)

  • Definition 2.1: Linear Logical Graph
  • Definition 2.2: Linear Logical Function
  • Proposition 2.3
  • proof
  • Theorem 2.4: Universal Approximation Theorem for measurable functions(b1)
  • Definition 2.5: Linear Logifold
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 2 more