RIP矩阵的显式构造及相关问题
Explicit constructions of RIP matrices and related problems
摘要：
给出了满足限制等距性（RIP）的$n\times N$矩阵的一种新的显式构造.也就是说，对于某个c>0，大N和任意满足N^{1-c} < n < N的n，我们构造了k^{1/2+c}阶RIP矩阵。这克服了自然障碍k=O（n^{1/2}）的证明的基础上小的一致性，这是使用在所有以前的显式结构的RIP矩阵。在我们的证明中的关键成分是新的估计和产品集的产品具有特殊的加法结构的指数和。我们还给出了一个构造n个复数的集合，其k阶矩对于1 \le k \le N（Turan的幂和问题）是一致小的，这改进了已知的显式构造当（\log N）^{1+o（1）} \le n \le（\log N）^{4+o（1）}。后一种构造产生满足RIP的n × N矩阵的初等显式例子，其列构成新的球形码;对于这些问题，参数在范围（\log N）^{1+o（1）} \le n \le（\log N）^{5/2+o（1）}内与现有构造的参数紧密匹配。
英文摘要：
We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose k-th moments are uniformly small for 1\le k\le N (Turan's power sum problem), which improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le (\log N)^{4+o(1)}. This latter construction produces elementary explicit examples of n by N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log N)^{5/2+o(1)}.
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