AGT: Symmetric Nonnegative Trifactorization Rank

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  • เผยแพร่เมื่อ 9 ม.ค. 2025
  • Talk by Helena Šmigoc.
    Factorizations of matrices where the factors are required to be entry-wise nonnegative provide a powerful tool in analyzing nonnegative data. In this talk we will consider Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization), a factorization of nonnegative symmetric matrices that takes into account symmetry, non-negativity, and low rank of a matrix.
    SN-Trifactorization is a factorization of an n × n (entry-wise) nonnegative symmetric matrix A of the form BCBT, where C is a k × k symmetric matrix, and both B and C are required to be nonnegative. The SNT-rank of A is the minimal k for which such factorization exists. The zero-nonzero pattern of A poses constraints on the zero-nonzero pattern of B and C satisfying A = BCBT. Those constraints can be studied through the SNT-rank of simple graphs that allow loops, defined to be the minimal possible SNT-rank of all symmetric nonnegative matrices whose zero-nonzero pattern is prescribed by a given graph.
    After introducing the SNT-rank of nonnegative symmetric matrices and exploring its basic properties, we will turn our discussion to the SNT-rank of graphs. We will define set-join covers of graphs and show that finding the SNT-rank of a graph G is equivalent to finding the minimal order of a set-join cover of G. Using this insight, we will develop basic properties of the SNT-rank for graphs and compute it for trees and cycles without loops. We will show the equivalence between the SNT-rank for complete graphs and the Katona problem and discuss the uniqueness of patterns of matrices in the factorization.

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