Learning Read-Once Determinants and the Principal Minor Assignment Problem
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.