Mithil Ramteke | Searcharxiv

arXiv2026

Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy

We extend the cone-ray exact-NMF pipeline of Ramteke (arXiv:2606.22451) from the uniform-support regime r_+ = r to the gap regime r_+ > r, and classify recoverable nonnegative factorisations by the rank of the W-factor into a three-regime taxonomy. Regime A (rank(W) = r_+, full column rank): a two-sided SVD-gauge cone-ray pipeline W = U_{r+}(G) Q, H = P V_{r+}(K)^T with G, K on Stiefel manifolds and square consistency Q P = diag(S_r, 0). On 10x10 dense random gap matrices it gives 100/100 recovery at r_+ = 5 and 6. We explain this via two geometric facts: slack enclosure (the data cone has codimension r_+ - r in the outer cone) and NRF-variety thickness (valid gauges form a positive-measure set, so the blind SVD lands on one with probability one). Regime B (rank(W) = r, W a column subset of M): a rank-deficient-W branch enumerating r_+-subsets of M's columns with per-column LP tests. On the block-diagonal family diag(C, J_k), where additivity of nonnegative rank collapses the valid gauges to a single point and the blind SVD pipeline fails, the column-subset branch restores recovery in milliseconds. Regime C (r < rank(W) < r_+, W not a column subset): exposed by the regular octagon's slack matrix. An exact size-6 NRF exists and is reachable by the symmetric formulation at an oracle gauge derived from a known factorisation (residual 1.5e-10), but the blind problem is open: 50 Haar random restarts and Riemannian gradient descent on the Stiefel/Grassmann gauge all stall, because the alt-LP residual is piecewise constant on cells of gauge-space, so local descent cannot cross cell walls. A combined toolkit (Regime B then A) covers regimes A and B with no regression on dense draws; Regime C remains open, with the regular octagon as the cleanest unsolved test case.

math.NA↗

arXiv2026

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

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.

math.NA↗