Singular Learning Theory

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Singularities of Non-negative Matrix Factorization and their application to Bayesian inference
Sep 28, 2026Naoki Hayashi, Yota Maeda, Yasushi EsakiNon-Negative Matrix Factorization Low-Rank Matrix Decomposition
Toyota Central R&D Labs., Inc.
Patterning in Practice: Debiasing Reward Models with Susceptibilities
Sep 1, 2026George Wang, Elizabeth Donoway, Daniel MurfetReward Modeling Debiased ML
Resolution
The Off-Support Barrier: Why Semantic Safety Constraints Are Not Learning-Problem Invariants, and What Follows for Prior Design, Containment, and Verification
Aug 1, 2026Yoshinori WatanabeSingular Learning Theory AI Safety
Measuring Dead Directions: Decomposing and Classifying Singular Structure off Canonical Alignment
Jul 1, 2026Tejas Pradeep ShirodkarSingular Learning Theory Statistical Learning Theory
IIIT, Hyderabad
Singular Learning and Occam's Razor in Deep Monomial Networks
Jun 26, 2026Kathlén Kohn, Giovanni Luca Marchetti, Farhan Shabir +2Neural Network Optimization Singular Learning Theory
KTH Stockholm · No Affiliation · Umeå University +1
Noise-Debiased Thermodynamic Variance for Local Learning Coefficient Probes
Jun 21, 2026Yingjia CaiSingular Learning Theory Neural Network Training Dynamics
Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity
Jun 19, 2026Tejas Pradeep Shirodkar, P. J. NarayananSingular Learning Theory Spectral Methods
IIIT, Hyderabad
Dead Directions: Geometric Singular Learning
Jun 4, 2026Tejas Pradeep ShirodkarInformation Geometry Singular Learning Theory
IIIT, Hyderabad
PAC-Bayes Bounds for Gibbs Posteriors via Singular Learning Theory
Apr 19, 2026Chenyang Wang, Yun YangNeural Network Generalization PAC-Bayesian Generalization Bounds
Department of Statistics, University of Illinois Urbana-Champaign · Department of Mathematics, University of Maryland, College Park

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Singularities of Non-negative Matrix Factorization and their application to Bayesian inference

Naoki Hayashi, Yota Maeda, Yasushi Esaki

Toyota Central R&D Labs., Inc.

Non-Negative Matrix Factorization Low-Rank Matrix Decomposition Singular Learning Theory Bayesian Inference

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Non-negative matrix factorization (NMF) is a singular statistical model whose Bayesian asymptotics are governed by the real log canonical threshold (RLCT). We study the local geometry of the factorization map and derive an upper bound for the RLCT of NMF. Let

H

be the model inner dimension and

H_0

the non-negative rank of the true

M\times N

matrix. Assuming that the true matrix admits a strictly positive factorization of inner dimension

H_0

in the interior of the parameter domain, we prove, for smooth positive priors, that

λ\leq \{(H-H_0)\min(M,N)+H_0(M+N-H_0)\}/2

. This bound strictly improves the previous bound when

H_0\geq3

. The proof uses a local analytic normal form that separates independent linear coordinates from a residual matrix product. When

H=H_0

also equals the ordinary rank of the true matrix, we obtain the exact value

λ=H_0(M+N-H_0)/2

. Under the standard assumptions of singular learning theory, these results bound the leading coefficients of the expected Bayesian generalization error and the Bayesian free energy.

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