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Learning Read-Once Determinants and the Principal Minor Assignment Problem

Abhiram Aravind, Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, Roshan Raj, Chandan Saha

Abstract

A symbolic determinant under rank-one restriction computes a polynomial of the form $\det(A_0+A_1y_1+\ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are square matrices over a field $\mathbb{F}$ and $rank(A_i)=1$ for each $i\in[n]$. This class of polynomials has been studied extensively, since the work of Edmonds (1967), in the context of linear matroids, matching, matrix completion and polynomial identity testing. We study the following learning problem for this class: Given black-box access to an $n$-variate polynomial $f=\det(A_0+A_1y_1+ \ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are unknown square matrices over $\mathbb{F}$ and rank$(A_i)=1$ for each $i\in[n]$, find a square matrix $B_0$ and rank-one square matrices $B_1,\ldots,B_n$ over $\mathbb{F}$ such that $f=\det(B_0+B_1y_1+\ldots+B_ny_n)$. In this work, we give a randomized poly(n) time algorithm to solve this problem. As the above-mentioned class is known to be equivalent to the class of read-once determinants (RODs), we will refer to the problem as learning RODs. The algorithm for learning RODs is obtained by connecting with a well-known open problem in linear algebra, namely the Principal Minor Assignment Problem (PMAP), which asks to find (if possible) a matrix having prescribed principal minors. PMAP has also been studied in machine learning to learn the kernel matrix of a determinantal point process. Here, we study a natural black-box version of PMAP: Given black-box access to an $n$-variate polynomial $f = \det(A + Y)$, where $A \in \mathbb{F}^{n \times n}$ is unknown and $Y = diag(y_1,\ldots,y_n)$, find a $B\in\mathbb{F}^{n\times n}$ such that $f=det(B+Y)$. We show that black-box PMAP can be solved in randomized poly(n) time, and further, it is randomized polynomial-time equivalent to learning RODs. We resolve black-box PMAP by investigating a property of dense matrices that we call the rank-one extension property.

Learning Read-Once Determinants and the Principal Minor Assignment Problem

Abstract

A symbolic determinant under rank-one restriction computes a polynomial of the form , where are square matrices over a field and for each . This class of polynomials has been studied extensively, since the work of Edmonds (1967), in the context of linear matroids, matching, matrix completion and polynomial identity testing. We study the following learning problem for this class: Given black-box access to an -variate polynomial , where are unknown square matrices over and rank for each , find a square matrix and rank-one square matrices over such that . In this work, we give a randomized poly(n) time algorithm to solve this problem. As the above-mentioned class is known to be equivalent to the class of read-once determinants (RODs), we will refer to the problem as learning RODs. The algorithm for learning RODs is obtained by connecting with a well-known open problem in linear algebra, namely the Principal Minor Assignment Problem (PMAP), which asks to find (if possible) a matrix having prescribed principal minors. PMAP has also been studied in machine learning to learn the kernel matrix of a determinantal point process. Here, we study a natural black-box version of PMAP: Given black-box access to an -variate polynomial , where is unknown and , find a such that . We show that black-box PMAP can be solved in randomized poly(n) time, and further, it is randomized polynomial-time equivalent to learning RODs. We resolve black-box PMAP by investigating a property of dense matrices that we call the rank-one extension property.
Paper Structure (54 sections, 45 theorems, 151 equations, 9 algorithms)

This paper contains 54 sections, 45 theorems, 151 equations, 9 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Proof ideas
  4. Proof sketch for Theorem \ref{['thm:main']} and Theorem \ref{['thm:main-reconstruction-PM-with-Q-property']}
  5. Proof sketch for Theorem \ref{['thm:main-characterization']}
  6. Proof sketch for Theorem \ref{['thm:main-NC-algorithm']}
  7. Comparison with previous works
  8. Other related works
  9. Preliminaries
  10. Reducible and Irreducible matrix
  11. Diagonal similarity
  12. Principal minor equivalence
  13. Cut of a matrix
  14. Cut-transpose operation
  15. Black-box evaluation with inverted inputs
  16. ...and 39 more sections

Key Result

Theorem 1.1

Let $n \in \mathbb{N}$ and $|\mathbb{F}| > n^6$.

Theorems & Definitions (114)

  • Theorem 1.1: Main theorem
  • Theorem 1.2: PME is in NC
  • Definition 1.1: Rank-one extension and property $\mathcal{R}$
  • Theorem 1.3: Sufficiency of PME up to order $4$
  • Theorem 1.4: PMAP for matrices with property $\mathcal{R}$
  • Definition 2.1: Reducible and Irreducible matrix
  • Lemma 2.1
  • Lemma 2.2: Hartfiell84, Lemma 1
  • Lemma 2.3
  • Definition 2.2: Cut of a matrix
  • ...and 104 more