Variational Bayesian Orthogonal Nonnegative Matrix Factorization Over the Stiefel Manifold
Bibliographic record
Abstract
Nonnegative matrix factorization (NMF) is one of the best-known multivariate data analysis techniques. The NMF uniqueness and its rank selection are two major open problems in this field. The solutions uniqueness issue can be addressed by imposing the orthogonality condition on NMF. This constraint yields sparser part-based representations and improved performance in clustering and source separation tasks. However, existing orthogonal NMF algorithms rely mainly on non-probabilistic frameworks that ignore the noise inherent in real-life data and lack variable uncertainties. Thus, in this work, we investigate a new probabilistic formulation of orthogonal NMF (ONMF). In the proposed model, we impose the orthogonality through a directional prior distribution defined on the Stiefel manifold called von Mises-Fisher distribution. This manifold consists of a set of directions that comply with the orthogonality condition that arises in many applications. Moreover, our model involves an automatic relevance determination (ARD) prior to address the model order selection issue. We devised an efficient variational Bayesian inference algorithm to solve the proposed ONMF model, which allows fast processing of large datasets. We evaluated the proposed model, called VBONMF, on the task of blind decomposition of real-world multispectral images of ancient documents. The numerical experiments demonstrate its efficiency and competitiveness compared to the state-of-the-art approaches.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| 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 | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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".