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版本更新 · v4arXiv:2605.14058

利用非凸优化求解器计算非负秩的下界

Computing Lower Bounds on the Nonnegative Rank via Non-Convex Optimization Solvers

Timothy Baeckelant, Arnaud Vandaele, Nicolas Gillis

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作者提出四种非负秩下界（FSB、RCB、HSB、SSB）的非凸优化建模与实现，并报告了SSB的首次可计算算法及基准测试表现。

math.OC · cs.DM

arXiv ↗PDF ↗
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The nonnegative rank of a nonnegative matrix $X$ is the smallest number of nonnegative rank-one factors that sum to $X$. Since computing the nonnegative rank is NP-hard, it is common to circumvent this issue by computing lower and upper bounds. In this paper, we propose non-convex formulations and practical implementations for four important lower bounds for the nonnegative rank, namely the fooling set bound (FSB), the rectangle covering bound (RCB), the hyperplane separation bound (HSB), and the self-scaled bound (SSB). In particular, to the best of our knowledge, our algorithm for computing the SSB is the first available in the literature. We compute the SSB for several matrices for which its value had not previously been reported. In some cases, the SSB provides the best known upper bound, while the other three bounds do not. Moreover, on canonical benchmarks, we show that our non-convex approaches are often competitive with existing methods. The paper also provides a consolidated reference for the current state of several classical lower bounds on a large number of benchmark matrices.

版本更新 · v4arXiv:2512.17473

Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions

Atharva Awari, Nicolas Gillis, Arnaud Vandaele

原文摘要

We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \in \mathbb{R}^{m \times n}$ and a factorization rank $r \ll \min(m, n)$, NMD seeks matrices $W \in \mathbb{R}^{m \times r}$ and $H \in \mathbb{R}^{r \times n}$ such that $X \approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \min(b, \max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications.

eess.SP · cs.LG · math.OC · stat.ML

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2601.06262

Matrix Factorization Framework for Community Detection under the Degree-Corrected Block Model

Alexandra Dache, Arnaud Vandaele, Nicolas Gillis

原文摘要

Community detection is a fundamental task in data analysis, and block models provide an approach for identifying diverse community structures. The degree-corrected block model (DCBM) is an established model that captures node degree heterogeneity. However, inference methods are computationally costly and sensitive to initialization, while cheaper alternatives, such as spectral or modularity-based approaches, may be less effective at recovering certain non-assortative structures. In this work, we show that DCBM inference can be reformulated as a constrained nonnegative matrix trifactorization problem. Leveraging this insight, we propose a scalable community detection method with a theoretically grounded initialization providing an initial partition for inference algorithms. Our approach is agnostic to network structure and applies to any graph representable by a DCBM. Experiments on synthetic and real-world benchmark networks show that our method achieves community recovery comparable to, and sometimes better than, that achieved by the DCBM inference methods, while being faster: it processes a graph with 100,000 nodes and 500,000 edges in about 30 seconds. Moreover, the initialization significantly improves solution quality. Overall, this work provides a scalable and robust framework for community detection and highlights the benefits of a matrix-factorization perspective for the DCBM.

cs.SI · math.OC · stat.ML

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arXiv ↗PDF ↗