Recovering a Magnitude-Symmetric Matrix from its Principal Minors
arxiv(2024)
摘要
We consider the inverse problem of finding a magnitude-symmetric matrix
(matrix with opposing off-diagonal entries equal in magnitude) with a
prescribed set of principal minors. This problem is closely related to the
theory of recognizing and learning signed determinantal point processes in
machine learning, as kernels of these point processes are magnitude-symmetric
matrices. In this work, we prove a number of properties regarding sparse and
generic magnitude-symmetric matrices. We show that principal minors of order at
most ℓ, for some invariant ℓ depending only on principal minors of
order at most two, uniquely determines principal minors of all orders. In
addition, we produce a polynomial-time algorithm that, given access to
principal minors, recovers a matrix with those principal minors using only a
quadratic number of queries. Furthermore, when principal minors are known only
approximately, we present an algorithm that approximately recovers a matrix,
and show that the approximation guarantee of this algorithm cannot be improved
in general.
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