Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F24%3A00370382" target="_blank" >RIV/68407700:21230/24:00370382 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s10107-023-01993-x" target="_blank" >https://doi.org/10.1007/s10107-023-01993-x</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s10107-023-01993-x" target="_blank" >10.1007/s10107-023-01993-x</a>

Result language

angličtina

Original language name

Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

Original language description

We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. In this setting, one optimizes over a measure restricted to be supported on the variety of an ideal generated by quadratic bilinear monomials. We show that this restriction enables an equivalent sparse reformulation of the GMP, where the single (high dimensional) measure variable is replaced by several (lower dimensional) measure variables supported on the maximal cliques of the graph corresponding to the quadratic bilinear constraints. We explore the resulting hierarchies of moment-based relaxations for the original dense formulation of GMP and this new, equivalent ideal-sparse reformulation, when applied to the problem of bounding nonnegative- and completely positive matrix factorization ranks. We show that the ideal-sparse hierarchies provide bounds that are at least as good (and often tighter) as those obtained from the dense hierarchy. This is in sharp contrast to the situation when exploiting correlative sparsity, as is most common in the literature, where the resulting bounds are weaker than the dense bounds. Moreover, while correlative sparsity requires the underlying graph to be chordal, no such assumption is needed for ideal-sparsity. Numerical results show that the ideal-sparse bounds are often tighter and much faster to compute than their dense analogs.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10102 - Applied mathematics

Project

<a href="/en/project/GJ20-11626Y" target="_blank" >GJ20-11626Y: Koopman operator framework for control of complex nonlinear dynamical systems</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Mathematical Programming

ISSN

0025-5610

e-ISSN

1436-4646

Volume of the periodical

205

Issue of the periodical within the volume

1-2

Country of publishing house

DE - GERMANY

Number of pages

42

Pages from-to

703-744

UT code for WoS article

001021517900001

EID of the result in the Scopus database

2-s2.0-85164161074