An Efficient Gauss-Newton Algorithm for Symmetric Low-Rank Product Matrix Approximatins
Bibliographic record
Abstract
We derive and study a Gauss-Newton method for computing the symmetric low-rank product (SLRP) XXT, where X / Rnkfor k<n, that is the closest approximation to a given symmetric matrix A / Rnn in Frobenius norm. When A=BTB (or BBT), this problem essentially reduces to finding a truncated singular value decomposition of B. Our Gauss-Newton method, which has a particularly simple form, shares the same order of iteration-complexity as a gradient method when k << n, but can be significantly faster on a wide range of problems. In this paper, we prove global convergence and a Q-linear convergence rate for this algorithm, and perform numerical experiments on various test problems, including those from recently active areas of matrix completion and robust principal component analysis. Numerical results show that the proposed algorithm is capable of providing considerable speed advantages over Krylov subspace methods on suitable application problems. Moreover, the algorithm possesses a higher degree of concurrency than Krylov subspace methods, thus offering better scalability on modern multi/many-core computers.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".