Real log canonical threshold of non-negative matrix factorization (Conjecture V.1)

Let $A\in\mathbb{R}_{\geq 0}^{M\times N}$ have non-negative rank $H_0$ , and suppose that $A=U_0V_0$ for some strictly positive matrices $U_0\in\mathbb{R}_{>0}^{M\times H_0}$ and $V_0\in\mathbb{R}_{>0}^{H_0\times N}$ . For an NMF model with inner dimension $H\geq H_0$ , let $\lambda$ denote the real log canonical threshold at $A$ for the factorization map $(U,V)\mapsto UV$ , equivalently for the squared-error function $\|UV-A\|_F^2$ with a smooth positive prior. The conjecture is that

\lambda=\frac{(H-H_0)\min(M,N)+H_0(M+N-H_0)}{2}.

References

Primary source

arXiv

Additional references

Singularities of Non-negative Matrix Factorization and their application to Bayesian inference — arXiv — Naoki Hayashi, Yota Maeda, Yasushi Esaki

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new unrefereed paper improves the known estimate and settles special cases, but does not settle the full conjecture.

The conjecture asks for exact real log canonical thresholds governing singular behavior in non-negative matrix factorization. Its full scope remains unresolved.

Known results

Hayashi and Watanabe: an upper bound for the NMF threshold, exact when $H=H_0=1$ .
Related work derives another upper bound for Poisson NMF and likewise identifies exactness when $H=H_0=1$ .
Numerical evidence suggests the threshold can exceed the reduced-rank-regression value, reflecting the distinction between ordinary and non-negative rank.

September 2026 improved bounds

A September 2026 preprint by Naoki Hayashi, Yota Maeda, and Yasushi Esaki gives an improved upper bound for strictly positive factorizations when the true non-negative rank is at least $3$ , and proves the exact value when the inner dimension equals both the non-negative rank and ordinary rank. The broader Conjecture V.1 is only partially addressed, and the manuscript is unrefereed.

Current status (as of September 2026): The full conjecture remains open; improved bounds and exact values in the stated special cases are claimed in an unrefereed preprint.

Sources

No solutions have been posted yet.