Exact Nonnegative Matrix Factorization via Cone-Ray Witnesses: Obtuseness Ranking, Saturation Curves, and an Augmented Alt-LP Breakthrough

Abstract

We study exact nonnegative matrix factorization (NMF) of small exact-rank-r matrices via a cone-ray pipeline combining the truncated SVD, the polyhedral cone of nonnegative preimages, the Double Description Method (DDM, via Fukuda's cddlib), and an alternating linear program (alt-LP) for slack minimisation. Under a uniform-support restriction the factorisation constraint Q P^T = I_r reduces to entrywise nonnegativity of an r x r witness matrix M_T = R_T^{-1} (R_K^T)^{-1} for an r-subset pair (T, K) of cone rays; this closed-form witness recovers an exact NMF in microseconds when feasible. We characterise feasibility by ranking r-subsets via geometric near-orthogonality ("obtuseness") and walking the top of each list. A 100-trial Monte Carlo at m=n=10 exposes a clean saturation curve: success 44/32/8, 79/85/58, and 79/87/59 of 100 at top-5/200/400 for r=4,5,6 -- beyond top-200 the failures are structural, not budget-limited. Enlarging m,n at fixed r hurts: at m=n=15 success collapses to 37/7/0/0/0 for r=4..8. On Olivetti faces (400x4096) the DDM step itself times out. Our main contribution is a hybrid that breaks this ceiling: at each pair we first try the closed-form M_T, and when it is infeasible we augment both supports by k=2 maximally angularly-separated rays and solve for mu,nu>=0 by a short slack-LP alternation. On the same m=n=10 Monte Carlo this lifts success from 79/85/58 to 99/95/75 at r=4,5,6, with cone reconstruction error at or near machine precision. We close with the four scaling walls the pipeline faces and concrete next steps.