Abstract:The constrained $\ell_p^p/\ell_q^p$ ratio model is scale invariant and is therefore attractive for sparse signal recovery. However, its nonconvex, nonsmooth, and fractional structure makes a unified theoretical and algorithmic analysis challenging for $0<p\le1$ and $q>1$. This paper develops a unified framework for this general model, covering deterministic exact recovery, stable recovery for sparse and compressible signals, and convergence analysis of a fractional algorithm. We first establish two deterministic sufficient conditions for exact recovery: a local optimality criterion and a null-space condition ensuring uniform recovery. For the $\ell_1/\ell_q$ subfamily, this null-space condition is further converted into high-probability sample-complexity bounds for isotropic sub-Gaussian matrix. We then study noisy recovery. Under the $k$-sparsity assumption, we improve the RIP-based stable recovery theory by relaxing the required sufficient condition and deriving sharper reconstruction-error bounds. For compressible signals, we establish RIP--ROP based error estimates whose constants are independent of the ambient dimension, improving prior bounds with explicit dimension-dependent factors [1]. An RIP-only variant is also derived. On the algorithmic side, we propose a prox-linear Dinkelbach framework that directly handles the fractional structure of the constrained problem and prove its convergence. Numerical experiments demonstrate that suitable choices of $(p,q)$ are effective for high-dynamic-range sparse signals and coherent sensing matrices.
| Subjects: | Optimization and Control (math.OC) |
| Cite as: | arXiv:2605.25397 [math.OC] |
| (or arXiv:2605.25397v1 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25397 arXiv-issued DOI via DataCite (pending registration) |
From: Nan-Jing Huang [view email]
[v1]
Mon, 25 May 2026 03:44:37 UTC (598 KB)
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