基于非线性观测的实例最优稀疏恢复：一个统一框架

中文摘要 AI 辅助

本文提出了一个用于从非线性观测中进行实例最优稀疏恢复的统一框架。其核心要素是某种梯度的信号相关受限近似可逆性条件（RAIC），该条件可保证迭代硬阈值法的实例最优性。在高斯设计下，将该框架应用于无相位、1比特和ReLU测量，分别对应稀疏相位检索、1比特压缩感知和稀疏ReLU回归问题。针对稀疏相位检索，提出了阈值振幅流的变体，证明其在O(s³)次测量（对数因子除外）下具有实例最优性，其中s为稀疏度水平。据所知，这是首个针对稀疏相位检索的实例最优高效算法，补充了Gao、Wang和Xu（2016）通过计算不可行程序实现的相关结果。在1比特压缩感知中，确立了归一化二元迭代硬阈值法的实例最优性，并强化了Matsumoto和Mazumdar（2024）的近期结果。在稀疏ReLU回归中，证明Soltanolkotabi（2017）算法的轻微变体具有实例最优性。此外，还为这些问题获得了(ℓ₂,ℓ₂)非均匀实例最优保证。分析基于多个高维集中界，包括受限特征值界和新的实例相关超平面镶嵌结果。

英文摘要

This paper develops a unified framework for instance optimal sparse recovery from nonlinear observations. The main ingredient is a signal-dependent restricted approximate invertibility condition (RAIC) of some gradient, which leads to the instance optimality of iterative hard thresholding. Under Gaussian designs, we apply the proposed framework to phaseless, one-bit, and ReLU measurements, which correspond to the problems of sparse phase retrieval, one-bit compressed sensing, and sparse ReLU regression, respectively. For sparse phase retrieval, we propose a variant of thresholded amplitude flow and show its instance optimality under $O(s^3)$ measurements (up to logarithmic factors), where $s$ is the sparsity level. To our best knowledge, this is the first instance optimal efficient algorithm for sparse phase retrieval and complements Gao, Wang and Xu (2016) that achieved this via a computationally intractable program. In one-bit compressed sensing, we establish the instance optimality of normalized binary iterative hard thresholding and strengthen the recent result of Matsumoto and Mazumdar (2024). In sparse ReLU regression, it is shown that a slight variant of the algorithm in Soltanolkotabi (2017) is instance optimal. Moreover, $(\ell_2,\ell_2)$ non-uniform instance optimal guarantees are obtained for these problems. The analysis is built upon a number of high-dimensional concentration bounds, including bounds on restricted eigenvalues and a novel instance-dependent hyperplane tessellation result.

发表机构

• Columbia University（哥伦比亚大学）

机构由 AI 辅助整理，请以论文原文为准。