In the context of sketching for compressive mixture modeling, we revisit existing proofs of the Restricted Isometry Property of sketching operators with respect to certain mixtures models. After examining the shortcomings of existing guarantees, we propose an alternative analysis that circumvents the need to assume importance sampling when drawing random Fourier features to build random sketching operators. Our analysis is based on new deterministic bounds on the restricted isometry constant that depend solely on the set of frequencies used to define the sketching operator; then we leverage these bounds to establish concentration inequalities for random sketching operators that lead to the desired RIP guarantees. Our analysis also opens the door to theoretical guarantees for structured sketching with frequencies associated to fast random linear operators.

翻译：在混合模型压缩素描的背景下，我们重新审视了关于素描算子相对于某些混合模型的受限等距性质（RIP）的现有证明。在考察了现有保证的不足后，我们提出了一种替代分析，该分析绕过了在抽取随机傅里叶特征以构建随机素描算子时需假设重要性抽样的要求。我们的分析基于受限等距常数的新确定性界，该界仅依赖于定义素描算子的频率集；随后我们利用这些界建立了随机素描算子的集中不等式，进而得到所需的RIP保证。我们的分析还为与快速随机线性算子关联的频率的结构化素描打开了理论保证之门。