[论文解读] Sparse Recovery for Orthogonal Polynomial Transforms

Anna C. Gilbert, Albert Gu+3 more|arXiv (Cornell University)|Jul 19, 2019

Sparse and Compressive Sensing Techniques参考文献 24被引用 2

一句话总结

本文提出了首个针对由雅可比多项式导出的正交多项式变换的可证明亚线性时间稀疏恢复算法。通过将k-稀疏恢复问题一般化地约化为1-稀疏恢复问题，并利用雅可比多项式的余弦逼近求解1-稀疏情形，该方法在有界噪声和特定结构假设下，对支持分布较开的近似k-稀疏信号实现了O(poly(k log N))的运行时间和样本复杂度。

ABSTRACT

In this paper we consider the following sparse recovery problem. We have query access to a vector 𝐱 ∈ ℝ^N such that x̂ = 𝐅 𝐱 is k-sparse (or nearly k-sparse) for some orthogonal transform 𝐅. The goal is to output an approximation (in an 𝓁₂ sense) to x̂ in sublinear time. This problem has been well-studied in the special case that 𝐅 is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time O(k ⋅ polylog N). However, for transforms 𝐅 other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled. In this paper we give sublinear-time algorithms - running in time poly(k log(N)) - for solving the sparse recovery problem for orthogonal transforms 𝐅 that arise from orthogonal polynomials. More precisely, our algorithm works for any 𝐅 that is an orthogonal polynomial transform derived from Jacobi polynomials. The Jacobi polynomials are a large class of classical orthogonal polynomials (and include Chebyshev and Legendre polynomials as special cases), and show up extensively in applications like numerical analysis and signal processing. One caveat of our work is that we require an assumption on the sparsity structure of the sparse vector, although we note that vectors with random support have this property with high probability. Our approach is to give a very general reduction from the k-sparse sparse recovery problem to the 1-sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this one-sparse recovery problem for transforms derived from Jacobi polynomials. Frequently, sparse FFT algorithms are described as implementing such a reduction; however, the technical details of such works are quite specific to the Fourier transform and moreover the actual implementations of these algorithms do not use the 1-sparse algorithm as a black box. In this work we give a reduction that works for a broad class of orthogonal polynomial families, and which uses any 1-sparse recovery algorithm as a black box.

研究动机与目标

• 开发适用于傅里叶变换以外的正交多项式变换的高效亚线性时间稀疏恢复算法。
• 解决如雅可比多项式等一般正交多项式变换缺乏亚线性时间解法的问题。
• 在稀疏性具有特定结构假设的条件下，为近似k-稀疏信号的恢复提供可证明的保证。
• 通过黑箱1-稀疏恢复子程序，将稀疏FFT技术推广至更广泛的正交多项式类别。

提出的方法

• 提出一种适用于平坦正交多项式变换的k-稀疏到1-稀疏稀疏恢复的一般约化方法。
• 将该约化方法应用于雅可比多项式变换，其包含切比雪夫、勒让德和盖根鲍尔多项式作为特例。
• 利用已知的雅可比多项式求值对余弦函数的逼近，求解1-稀疏恢复问题。
• 采用递归精化过程（Refine）逐步提高信号分量定位的精度。
• 利用角度环绕和周期性性质，控制迭代精化过程中的误差传播。
• 设计了一种查询高效的算法，仅需O(poly(k log N))次查询，并在O(poly(k log N))时间内运行。

实验结果

研究问题

• RQ1 能否在离散傅里叶变换以外的正交多项式变换中实现亚线性时间稀疏恢复？
• RQ2 是否存在一种适用于广泛正交多项式类别的通用约化方法，将k-稀疏恢复约化为1-稀疏恢复？
• RQ3 能否利用余弦逼近高效求解雅可比多项式对应的1-稀疏恢复问题？
• RQ4 为实现非傅里叶变换的亚线性时间恢复，稀疏性需要满足何种结构假设？
• RQ5 该算法如何处理近似k-稀疏信号中的噪声？

主要发现

• 该算法运行时间为O(poly(k log N))，仅需O(poly(k log N))次查询，实现了亚线性复杂度。
• 对于支持分布较开的近似k-稀疏信号，该算法以高概率保证∥ẑ − ˆx∥₂ ≤ 0.01∥ˆx∥₂。
• 该方法适用于由雅可比多项式导出的任意正交多项式变换，包括切比雪夫和勒让德多项式作为特例。
• 从k-稀疏到1-稀疏恢复的约化方法具有通用性，适用于任意平坦正交多项式变换。
• 该算法的正确性依赖于一个结构假设：稀疏信号必须具有‘分布较开’的支持，该假设在随机支持下以高概率成立。
• 该方法对具有足够小ℓ₂范数的对抗性噪声具有鲁棒性，确保在现实条件下实现稳定恢复。