English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Matrix factorization is an inference problem that has acquired importance dueto its vast range of applications that go from dictionary learning torecommendation systems and machine learning with deep networks. The study ofits fundamental statistical limits represents a true challenge, and despite adecade-long history of efforts in the community, there is still no closedformula able to describe its optimal performances in the case where the rank ofthe matrix scales linearly with its size. In the present paper, we study thisextensive rank problem, extending the alternative 'decimation' procedure thatwe recently introduced, and carry out a thorough study of its performance.Decimation aims at recovering one column/line of the factors at a time, bymapping the problem into a sequence of neural network models of associativememory at a tunable temperature. Though being sub-optimal, decimation has theadvantage of being theoretically analyzable. We extend its scope and analysisto two families of matrices. For a large class of compactly supported priors,we show that the replica symmetric free entropy of the neural network modelstakes a universal form in the low temperature limit. For sparse Ising prior, weshow that the storage capacity of the neural network models diverges assparsity in the patterns increases, and we introduce a simple algorithm basedon a ground state search that implements decimation and performs matrixfactorization, with no need of an informative initialization.

The Decimation Scheme for Symmetric Matrix Factorization