On the Complexity of Robust PCA and <i>ℓ</i><sub>1</sub>-Norm Low-Rank Matrix Approximation
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
The low-rank matrix approximation problem with respect to the component-wise ℓ 1 -norm (ℓ 1 -LRA), which is closely related to robust principal component analysis (PCA), has become a very popular tool in data mining and machine learning. Robust PCA aims to recover a low-rank matrix that was perturbed with sparse noise, with applications for example in foreground-background video separation. Although ℓ 1 -LRA is strongly believed to be NP-hard, there is, to our knowledge, no formal proof of this fact. In this paper, we prove that ℓ 1 -LRA is NP-hard, already in the rank-one case, using a reduction from MAX CUT. Our derivations draw interesting connections between ℓ 1 -LRA and several other well-known problems, i.e., robust PCA, ℓ 0 -LRA, binary matrix factorization, a particular densest bipartite subgraph problem, the computation of the cut norm of {−1, + 1} matrices, and the discrete basis problem, all of which we prove to be NP-hard.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| 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 it