Applications of Stochastic Gradient Descent to Nonnegative Matrix Factorization
Bibliographic record
Abstract
We consider the application of stochastic gradient descent (SGD) to the nonnegative matrix factorization (NMF) problem and the unconstrained low-rank matrix factorization problem. While the literature on the SGD algorithm is rich, the application of this specific algorithm to the field of matrix factorization problems is an unexplored area. We develop a series of results for the unconstrained problem, beginning with an analysis of standard gradient descent with a known zero-loss solution, and culminating with results for SGD in the general case where no zero-loss solution is assumed. We show that, with initialization close to a minimizer, there exist linear rate convergence guarantees. \nWe explore these results further with numerical experiments, and examine how the matrix factorization solutions found by SGD can be used as machine learning classifiers in two specific applications. In the first application, handwritten digit recognition, we show that our approach produces classification performance competitive with existing matrix factorization algorithms. In the second application, document topic classification, we examine how well SGD can recover an unknown words-to-topics matrix when the topics-to-document matrix is generated using the Latent Dirichlet Allocation model. This approach allows us to simulate two regimes for SGD: a fixed-sample regime where a large set of data is iterated over to train the model, and a generated-sample regime where a new data point is generated at each training iteration. In both regimes, we show that SGD can be an effective tool for recovering the hidden words-to-topic matrix. We conclude with some suggestions for further expansion of this work.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".