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Simple Deterministically Constructible RIP Matrices with Sublinear Fourier Sampling Requirements

机译：具有Sublinear傅里叶采样要求的简单确定性地结构性RIP矩阵

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摘要

We present a deterministic number theoretic construction for matrices with the Restricted Isometry Property (RIP). Furthermore, we show that the number theoretic properties of our RIP matrices allow their products with Discrete Fourier Transform (DFT) matrices to be well approximated via a few highly sparse matrix multiplications. Hence, our RIP matrices may be approximately multiplied by the DFT of any input vector in sublinear-time by reading only a small fraction of its entries. As a consequence, we obtain small deterministic sample sets which are guaranteed to allow the recovery of near-optimal sparse Fourier representations for all periodic functions having an integrable second derivative over a single period. Explicit bounds are provided for the sizes of our RIP matrices, the sizes of their associated sublinear Fourier sampling sets, and the errors incurred by quickly approximating their products with DFT matrices. The Fourier sampling requirements obtained herein improve on previous deterministic Fourier sampling results in [1], [2].
机译：我们提出了具有受限制等距特性（RIP）的矩阵的确定性数字理论结构。此外，我们表明我们的RIP矩阵的数量理论属性允许其具有离散傅里叶变换（DFT）矩阵的产品，以通过少数高稀疏的矩阵乘法近似地近似。因此，我们的RIP矩阵可以通过仅读取其条目的一小部分来近似乘以Sublinear-Time中的任何输入载体的DFT。因此，我们获得了小确定性样本集，这保证了允许在单个时段内具有可集成的第二导数的所有周期性函数恢复近最佳稀疏傅立叶表示。为我们的RIP矩阵的大小提供显式界限，它们相关的索布林傅立叶采样集的大小以及通过用DFT矩阵快速逼近其产品而产生的错误。本文获得的傅里叶采样要求改善了先前的确定性傅立叶采样结果[1]，[2]。

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