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A Novel Gaussian Min-Max Theorem and its Applications

Danil Akhtiamov, David Bosch, Reza Ghane, K Nithin Varma, Babak Hassibi

TL;DR

This paper identifies a new pair of Gaussian processes satisfying Gordon’s comparison inequalities, and extends the classical GMT and CGMT Theorems from the case where the underlying Gaussian matrix in the primary process has iid rows to where it has independent but non-identically-distributed ones.

Abstract

A celebrated result by Gordon allows one to compare the min-max behavior of two Gaussian processes if certain inequality conditions are met. The consequences of this result include the Gaussian min-max (GMT) and convex Gaussian min-max (CGMT) theorems which have had far-reaching implications in high-dimensional statistics, machine learning, non-smooth optimization, and signal processing. Both theorems rely on a pair of Gaussian processes, first identified by Slepian, that satisfy Gordon's comparison inequalities. In this paper, we identify such a new pair. The resulting theorems extend the classical GMT and CGMT Theorems from the case where the underlying Gaussian matrix in the primary process has iid rows to where it has independent but non-identically-distributed ones. The new CGMT is applied to the problems of multi-source Gaussian regression, as well as to binary classification of general Gaussian mixture models.

A Novel Gaussian Min-Max Theorem and its Applications

TL;DR

This paper identifies a new pair of Gaussian processes satisfying Gordon’s comparison inequalities, and extends the classical GMT and CGMT Theorems from the case where the underlying Gaussian matrix in the primary process has iid rows to where it has independent but non-identically-distributed ones.

Abstract

A celebrated result by Gordon allows one to compare the min-max behavior of two Gaussian processes if certain inequality conditions are met. The consequences of this result include the Gaussian min-max (GMT) and convex Gaussian min-max (CGMT) theorems which have had far-reaching implications in high-dimensional statistics, machine learning, non-smooth optimization, and signal processing. Both theorems rely on a pair of Gaussian processes, first identified by Slepian, that satisfy Gordon's comparison inequalities. In this paper, we identify such a new pair. The resulting theorems extend the classical GMT and CGMT Theorems from the case where the underlying Gaussian matrix in the primary process has iid rows to where it has independent but non-identically-distributed ones. The new CGMT is applied to the problems of multi-source Gaussian regression, as well as to binary classification of general Gaussian mixture models.
Paper Structure (30 sections, 14 theorems, 158 equations, 2 figures)

This paper contains 30 sections, 14 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction and Motivation
  2. Related works
  3. Preliminaries
  4. Gordon's comparison inequality for Gaussian Processes
  5. Standard CGMT
  6. Multi-source Gaussian regression
  7. Binary classification for Gaussian Mixture Models
  8. Main results
  9. Applications
  10. Multi-source regression
  11. Binary classification for general GMMs
  12. Numerical experiments
  13. Multi-source regression plots
  14. Binary classification for general GMMs
  15. Conclusion and future work
  16. ...and 15 more sections

Key Result

Theorem 1

Let $I$ and $J$ be two finite sets and $\{X_{ij}\}_{i \in I, j \in J}$, $\{Y_{ij}\}_{i \in I, j \in J}$ be two real valued centered Gaussian processes, which satisfy the following conditions: Let $\{t_{ij}\}_{i \in I, j \in J}$ be an arbitrary sequence of real numbers indexed by $(i, j) \in I \times J$. Then the following inequality holds:

Figures (2)

  • Figure 1: Training and Generalization Error for the Multi-source Gaussian Regression
  • Figure 2: The error for binary classification for GMMs

Theorems & Definitions (28)

  • Theorem 1: Gordon's comparison inequality Gordon1985SomeIF
  • Theorem 2
  • Corollary 1: Gaussian Min-Max Theorem
  • Theorem 3: Convex Gaussian Min-Max Theorem
  • Lemma 1
  • Theorem 4: Generalized CGMT
  • Remark 1
  • Remark 2
  • Theorem 5: Multi-source Generalization Error
  • Theorem 6: Binary Classification Error
  • ...and 18 more