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arXiv · 2606.22451

Exact Nonnegative Matrix Factorization via Cone-Ray Witnesses: Certificates, a One-Sided Solver, and a Findability Phase Transition

Mithil Ramteke
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Abstract

We study exact nonnegative matrix factorization (NMF) of small exact-rank-r matrices through the polyhedral cones of nonnegative preimages of the truncated SVD. Restricting each factor to an r-subset of a cone's extreme rays collapses the factorization constraint to the entrywise nonnegativity of a single r x r witness matrix; witness feasibility is a certificate that an exact size-r NMF exists, decided in one matrix inverse. A single-coupling completeness theorem shows every size-r NMF is representable this way, for every m, and a one-sided relaxation gives a closed-form solver that provably dominates the two-sided witness. Our main result concerns findability: at a fixed search budget, witness recoverability undergoes a sharp conic phase transition whose width collapses to a step as r grows. We rule out a universal-constant explanation (Goemans-Williamson) and a statistical-dimension one: the statistical dimension is flat across the transition, while the intrinsic-volume profile's variance tracks it. The transition is not an existence boundary; by completeness a ray-economical witness always exists and persists past the threshold, so what decays with m is its density among r-subsets. The threshold is thus budget-relative, moving logarithmically as the pool grows, and combinatorial rather than smooth-conic. A two-sided union of one-sided relaxations returns exact machine-precision factorizations; being budget-limited, its recovery obeys the transition we study, and a well-initialized coordinate-descent solver, not subject to this threshold, recovers more at larger sizes. Beyond the exact-rank case, the same coupling certifies the gap regime (nonnegative rank > r): infeasibility of its linear relaxation decidably proves no size-r factorization exists, exhibiting a dual witness. Our contribution is these certificates and the transition they expose, not a faster general-purpose solver.

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Mithil Ramteke. 2026-08-21. Exact Nonnegative Matrix Factorization via Cone-Ray Witnesses: Certificates, a One-Sided Solver, and a Findability Phase Transition. https://arxiv.org/abs/2606.22451

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