Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform
Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform
复制标题
稀疏 Walsh-Hadamard 变换的近乎最优确定性算法
DOI：
10.1145/3029050
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
P. Indyk
中科院分区：
文献类型：
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作者：
Mahdi Cheraghchi;P. Indyk
对于每个固定常数α> 0，我们设计了一种用于计算n二维矢量x∈Rn的k-sparse walsh-hadamard变换（即，在布尔图中的离散傅立叶变换）的算法（log n（log n）） ）o（1）。具体来说，给出了算法的查询访问X并计算k-sparse x〜∈Rn满足“ x〜- x〜”1≤c” x〜- x〜- hk（xˆ）”””””””””””””””” ” 1绝对常数c> 0，其中xˆ是x和hk（xˆ）的转换是其最佳的k-sparse近似值。算法启动）。此外，我们设计了一种基于一般无损冷凝器的确定性和非适应性ℓ1/ℓ1压缩敏感性方案，该方案配备了在时间K1 +α（log n）O（1）（用于GUV基的冷凝器）和基于GUV的冷凝器）和的快速重建算法和我们的计划显着简化了贝林德，吉尔伯特，indyk，karloff和strauss的早期基于扩展的结构[Berinde等人方法以黑匣子方式使用线性无损冷凝器；在多同源因素中（k log3 n）。
For every fixed constant α > 0, we design an algorithm for computing the k-sparse Walsh-Hadamard transform (i.e., Discrete Fourier Transform over the Boolean cube) of an N-dimensional vector x ∈ RN in time k1 + α(log N)O(1). Specifically, the algorithm is given query access to x and computes a k-sparse x˜ ∈ RN satisfying ‖ x˜− xˆ‖1 ≤ c ‖ xˆ− Hk(xˆ)‖‖‖‖‖‖‖‖1 for an absolute constant c > 0, where xˆ is the transform of x and Hk(xˆ) is its best k-sparse approximation. Our algorithm is fully deterministic and only uses nonadaptive queries to x (i.e., all queries are determined and performed in parallel when the algorithm starts). An important technical tool that we use is a construction of nearly optimal and linear lossless condensers, which is a careful instantiation of the GUV condenser (Guruswami et al. [2009]). Moreover, we design a deterministic and nonadaptive ℓ1/ℓ1 compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time k1 + α(log N)O(1) (for the GUV-based condenser) and is of independent interest. Our scheme significantly simplifies and improves an earlier expander-based construction due to Berinde, Gilbert, Indyk, Karloff, and Strauss [Berinde et al. 2008]. Our methods use linear lossless condensers in a black box fashion; therefore, any future improvement on explicit constructions of such condensers would immediately translate to improved parameters in our framework (potentially leading to k(log N)O(1) reconstruction time with a reduced exponent in the poly-logarithmic factor, and eliminating the extra parameter α). By allowing the algorithm to use randomness while still using nonadaptive queries, the runtime of the algorithm can be improved to õ(k log3 N).
有限向量空间中的显式通用采样集
DOI：
10.1016/j.acha.2016.06.001
发表时间：
2017
期刊：
arXiv: Numerical Analysis
影响因子：
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作者：
Morotti
通讯作者：
Morotti