Signal-Dependent Performance Analysis of Orthogonal Matching Pursuit for Exact Sparse Recovery
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Bibliographic record
Abstract
Exact recovery of K-sparse signals x ∈ ℝ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> from linear measurements y = Ax, where A ∈ ℝ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m×n</sup> is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is widely used for reconstructing x based on y and A due to its excellent recovery performance and high efficiency. A fundamental question in the performance analysis of OMP is the characterizations of the probability of exact recovery of x for random matrix A and the minimal m to guarantee a target recovery performance. In many practical applications, in addition to sparsity, x also has some additional properties (for example, the nonzero entries of x independently and identically follow a Gaussian distribution, or x has exponentially decaying property). This paper shows that these properties can be used to refine the answer to the above question. Toward this end, we first show that the prior information of the nonzero entries of x can be used to provide an upper bound on ||x|| <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /||x|| <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Then, we use this upper bound to develop a lower bound on the probability of exact recovery of x using OMP in K iterations. Furthermore, we develop a lower bound on the number of measurements m to guarantee that the exact recovery probability using K iterations of OMP is no smaller than a given target probability. Finally, we show that when K = O(√ln n), as both n and K go to infinity, sufficient to ensure that the probability of exact recovering any K-for any 0 <; ζ ≤ 1/√π, m = 2K ln(n/ζ) measurements are ln n)sparse x is no lower than 1 - ζ with K iterations of OMP. This improves the m = 4K ln(2n/ζ) result of Tropp et al. For K-sparse α-strongly decaying signals and for K-sparse x whose nonzero entries independently and identically follow the Gaussian distribution, the number of measurements sufficient for exact recovery with probability no lower than 1 - ζ reduces further to m = (√K + 4√α+1/α-1 ln(n/ζ)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and to asymptotically m ≈ 1.9K ln(n/ζ), respectively.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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