Sum-of-norms regularized Nonnegative Matrix Factorization
Bibliographic record
Abstract
When applying nonnegative matrix factorization (NMF), the rank parameter is generally unknown. This rank, called the nonnegative rank, is usually estimated heuristically since computing its exact value is NP-hard. In this work, we propose an approximation method to estimate the rank on-the-fly while solving NMF. We use the sum-of-norm (SON), a group-lasso structure that encourages pairwise similarity, to reduce the rank of a factor matrix when the initial rank is overestimated. On various datasets, SON-NMF can reveal the correct nonnegative rank of the data without prior knowledge or parameter tuning. SON-NMF is a nonconvex, nonsmooth, non-separable, and non-proximable problem, making it nontrivial to solve. First, since rank estimation in NMF is NP-hard, the proposed approach does not benefit from lower computational complexity. Using a graph-theoretic argument, we prove that the complexity of SON-NMF is essentially irreducible. Second, the per-iteration cost of algorithms for SON-NMF can be high. This motivates us to propose a first-order BCD algorithm that approximately solves SON-NMF with low per-iteration cost via the proximal average operator. SON-NMF exhibits favorable features for applications. Besides the ability to automatically estimate the rank from data, SON-NMF can handle rank-deficient data matrices and detect weak components with small energy. Furthermore, in hyperspectral imaging, SON-NMF naturally addresses the issue of spectral variability.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".