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We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p_1, p_2, …, p_s, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the p_i’s are uniform or ε-far from being uniform in ℓ_1-distance (2) Identity Testing: Testing whether all the p_i’s are equal to an explicitly given distribution q or ε-far from q in ℓ_1-distance, and (3) Closeness Testing: Testing whether all the p_i’s are equal to a distribution q which we have sample access to, or ε-far from q in ℓ_1-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.
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This content will become publicly available on December 31, 2026
Signal and Image Recovery with Scale and Signed Permutation Invariant Sparsity-Promoting Functions
Sparse signal recovery has been a cornerstone of advancements in data processing and imaging. Recently, the squared ratio of $$ℓ_1$$ to $$ℓ_2$$ norms, $$(ℓ_1/ℓ_2)^2,$$ has been introduced as a sparsity-prompting function, showing superior performance compared to traditional $$ℓ_1$$ minimization, particularly in challenging scenarios with high coherence and dynamic range. This paper explores the integration of the proximity operator of $$(ℓ_1/ℓ_2)^2$$ and $$ℓ_1/ℓ_2$$ into efficient optimization frameworks, including the Accelerated Proximal Gradient (APG) and Alternating Direction Method of Multipliers (ADMM). We rigorously analyze the convergence properties of these algorithms and demonstrate their effectiveness in compressed sensing and image restoration applications. Numerical experiments highlight the advantages of our proposed methods in terms of recovery accuracy and computational efficiency, particularly under noise and high-coherence conditions.
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- Award ID(s):
- 2208385
- PAR ID:
- 10703157
- Publisher / Repository:
- Global Science Press
- Date Published:
- Journal Name:
- Analysis in Theory and Applications
- Volume:
- 42
- Issue:
- 1
- ISSN:
- 1672-4070
- Page Range / eLocation ID:
- 62 to 89
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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