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First-order Optimization Methods with Generalized Smoothness for Low-rank Matrix Factorization — arXiv:2610.19059

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.19059
License (as archived): CC0-1.0
Archived: 2026-10-15T14:25:32.026Z
SHA-256 of the archived text: f9ab4021ffe433baa106db5349ef94050e8106ea49960d38a354f36b29630b60
Stable URL of this page: https://metraipolis.com/documento/dc_ffc847cb2990

How to cite

First-order Optimization Methods with Generalized Smoothness for Low-rank Matrix Factorization — arXiv:2610.19059 — arXiv (en), https://arxiv.org/abs/2610.19059 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ffc847cb2990 (Metraipolis archive dc_ffc847cb2990, sha256 f9ab4021ffe433baa106db5349ef94050e8106ea49960d38a354f36b29630b60)

BibTeX

@misc{dcffc847cb2990,
  title = {First-order Optimization Methods with Generalized Smoothness for Low-rank Matrix Factorization — arXiv:2610.19059},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.19059},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ffc847cb2990; sha256 f9ab4021ffe433baa106db5349ef94050e8106ea49960d38a354f36b29630b60},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.19059. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

This paper studies several first-order methods for low-rank matrix minimization in the Burer--Monteiro form, where the target matrix is represented as the product of a nonnegative factor matrix and its transpose. Such a factorization destroys convexity of the objective function and global Lipschitz continuity of the gradient. We prove that for a quadratic objective function that is strongly convex only on the subspace of symmetric matrices, the Burer-Monteiro factorized objective possesses generalized smoothness with a fractional exponent 2/3. For a quadratic objective, the generalized smoothness property requires positive definiteness of the operator A only on the subspace of symmetric matrices, which relaxes the assumptions of Dragomir et al. On this basis, three methods are proposed: an adaptive gradient descent method with Armijo backtracking; an unconstrained SafeAdam method combining the Adam direction with a descent-angle certificate and a fallback to the normalized gradient; and a certified projected version for the nonnegativity constraint. Global convergence and explicit convergence rates for the proposed methods have been proved. The obtained convergence rate is better than the convergence rate of ordinary unaccelerated gradient descent in the case of a sufficiently large gradient. In the small-gradient regime, the convergence guarantee is of square-root type, and the corresponding complexity scales as the inverse square of the desired accuracy. In the large-gradient regime, the convergence guarantee is of three-quarters type, and the corresponding complexity is proportional to the inverse of the accuracy raised to the power of 4/3. Experiments on SymNMF using the TDT2, Reuters and CBCL datasets, which are standard benchmarks for document and image clustering, show that the methods match or outperform Dyn-NoLips without tuning the relative-smoothness kernel parameters.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.OC, nuovi lavori del giorno).
Categorie arXiv: Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2610.19059 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-15T13:55:20.317Z dalla pagina https://arxiv.org/list/math.OC/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).