Sparsity-Free Compressed Sensing With Applications to Generative Priors
Bibliographic record
Abstract
We study the problem of reconstructing a high-dimensional signal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {x} \in \mathbb {R}^{n}$ </tex-math></inline-formula> from a low-dimensional noisy linear measurement <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {y}=\mathrm {M}\mathrm {x}+\mathrm {e} \in \mathbb {R}^{\ell }$ </tex-math></inline-formula> , assuming x admits a certain structure. We model the measurement matrix as M = BA, with arbitrary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {B} \in \mathbb {R}^{\ell \times m}$ </tex-math></inline-formula> and sub-gaussian <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {A} \in \mathbb {R}^{m \times n}$ </tex-math></inline-formula> ; therefore allowing for a family of random measurement matrices which may have heavy tails, dependent rows and columns, and a large dynamic range for the singular values. The structure is either given as a non-convex cone <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T \subset \mathbb {R}^{n}$ </tex-math></inline-formula> , or is induced via minimizing a given convex function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f(\cdot)$ </tex-math></inline-formula> , hence our study is sparsity-free. We prove, in both cases, that an approximate empirical risk minimizer robustly recovers the signal if the effective number of measurements is sufficient, even in the presence of a model mismatch, i.e., the signal not exactly admitting the model’s structure. While in classical compressed sensing the number of independent (sub)-gaussian measurements regulates the possibility of a robust reconstruction, in our setting the effective number of measurements depends on the properties of B. We show that, in this model, the stable rank of B indicates the effective number of measurements, and an accurate recovery is guaranteed whenever it exceeds, to within a constant factor, the effective dimension of the structure set. We apply our results to the special case of generative priors, i.e., when x is close to the range of a Generative Neural Network (GNN) with ReLU activation functions. Also, if the GNN has random weights in the last layer, our theory allows a partial Fourier measurement matrix, thus taking the first step towards a theoretical analysis of compressed sensing MRI with GNN. Our work relies on a recent result in random matrix theory by Jeong et al. (2020).
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.
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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".