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：在压缩混合建模的素描背景下，我们重新审视了关于特定混合模型的素描算子受限等距性质的现有证明。在检视现有保证的不足之处后，我们提出了一种替代分析方案，该方案避免了在构建随机素描算子时绘制随机傅里叶特征需假设重要性采样的要求。我们的分析基于对受限等距常数的新确定性界，该界仅取决于用于定义素描算子的频率集合；随后我们利用这些界建立随机素描算子的集中不等式，从而导出所需的RIP保证。我们的分析也为使用与快速随机线性算子相关联的频率进行结构化素描的理论保证开辟了新的可能性。