Typical and generic ranks in matrix completion

Abstract

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks. We give a combinatorial description of the patterns of specified entires of n × n symmetric matrices that have n as a typical rank. Moreover, we describe exactly when such a generic partial matrix is minimally completable to rank n. We also characterize the typical ranks for patterns of entries with low maximal typical rank.

Multiple typical ranks in matrix completion

“…Our second major contribution is the full determination of the typical ranks for unspecified entry set G(n, 1) for n < 9, see Proposition 3.9, Proposition 3.12, and Corollary 3.14. In the course of these results we successfully answer several questions raised in [BBS20]. Moreover, we present results that bound the growth of the typical ranks of G(n, 1) (Proposition 4.1) and of G(n, k) (Proposition 4.3) as n grows.…”

“…Building on ideas from rigidity theory Király, Theran, and Tomioka [KTT15] developed in 2015 a new approach using tools from algebraic geometry, graph theory, and matroid theory. This novel direction entails several new works in recent years [BBS20,DIBL19,Tsa20].…”

“…Inspired by the results in [BBS20], we study here the exact low-rank matrix completion problem. In more detail, we are concerned with the following task: Given a partially filled matrix A with fixed positions of specified and unspecified entries, we want to find the unspecified entries of A such that the completed matrix has the lowest possible rank, called the completion rank of A.…”

“…These graphs have shown to be particularly challenging to complete over the real numbers to particular ranks. The smallest graph with multiple typical ranks is G(4, 1), and in [BBS20] it was posited that other graphs in the family might be examples with larger numbers of typical ranks. Our second major contribution is the full determination of the typical ranks for unspecified entry set G(n, 1) for n < 9, see Proposition 3.9, Proposition 3.12, and Corollary 3.14.…”

The algebraic matroid of the finite unit norm tight frame (funtf) variety

“…So thus far, we only need to consider (n , r ) = (2, 2), (2, 3), (2,4), (3,2), (3,3), (3,4), (3,5). Among these, the only (n , r )-pairs such that there even exists such a bipartite graph with the correct number of edges are (2,4), (3,3), (3,4), (3,5). For these values of (n , r ), we may compute all the connected bipartite graphs on partite sets of size n and r with minimum degree 2 and exactly n + r + 2 edges using the genbg command of Nauty and Traces [19].…”