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Motion Planning for Identification of Linear Classifiers

Aneesh Raghavan, Karl Henrik Johansson

TL;DR

This work presents the classifier identification problem formulated as a control problem, geometric interpretation of the control problem resulting in one step modified control problems, and control algorithms that result in data sets which are used to identify the true classifier with accuracy.

Abstract

A given region in 2-D Euclidean space is divided by a unknown linear classifier in to two sets each carrying a label. The objective of an agent with known dynamics traversing the region is to identify the true classifier while paying a control cost across its trajectory. We consider two scenarios: (i) the agent is able to measure the true label perfectly; (ii) the observed label is the true label multiplied by noise. We present the following: (i) the classifier identification problem formulated as a control problem; (ii) geometric interpretation of the control problem resulting in one step modified control problems; (iii) control algorithms that result in data sets which are used to identify the true classifier with accuracy; (iv) convergence of estimated classifier to the true classifier when the observed label is not corrupted by noise; (iv) numerical example demonstrating the utility of the control algorithms.

Motion Planning for Identification of Linear Classifiers

TL;DR

This work presents the classifier identification problem formulated as a control problem, geometric interpretation of the control problem resulting in one step modified control problems, and control algorithms that result in data sets which are used to identify the true classifier with accuracy.

Abstract

A given region in 2-D Euclidean space is divided by a unknown linear classifier in to two sets each carrying a label. The objective of an agent with known dynamics traversing the region is to identify the true classifier while paying a control cost across its trajectory. We consider two scenarios: (i) the agent is able to measure the true label perfectly; (ii) the observed label is the true label multiplied by noise. We present the following: (i) the classifier identification problem formulated as a control problem; (ii) geometric interpretation of the control problem resulting in one step modified control problems; (iii) control algorithms that result in data sets which are used to identify the true classifier with accuracy; (iv) convergence of estimated classifier to the true classifier when the observed label is not corrupted by noise; (iv) numerical example demonstrating the utility of the control algorithms.
Paper Structure (22 sections, 1 theorem, 28 equations, 8 figures, 2 algorithms)

This paper contains 22 sections, 1 theorem, 28 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem Considered
  4. Contributions
  5. Outline
  6. Problem Formulation
  7. Deterministic Scenario: Abstraction
  8. Identification of the Classifier
  9. Control Problem
  10. Stochastic Scenario: Abstraction
  11. The Noise Model
  12. The Probability Space
  13. Identification of the Classifier
  14. The Control Problem
  15. Analysis of the problem
  16. ...and 7 more sections

Key Result

Proposition 4.1

As the number of data points increases, estimated classifier converges to the true classifier, i.e., $\underset{m \to \infty} \lim \rho_{m} = \rho^*$ and $\underset{m \to \infty} \lim c_{m} = c^*$.

Figures (8)

  • Figure 1: Schematic for the motion planning problem
  • Figure 2: Schematic for the motion planning problem with noisy data
  • Figure 3: Region of certainty from the four given points
  • Figure 4: Region of certainty from new four points obtained by Agent
  • Figure 5: Possible classifiers with noisy data set
  • ...and 3 more figures

Theorems & Definitions (2)

  • Proposition 4.1
  • proof