The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

A Physical Interpretation for Finite Tight Frames

Though finite tight frames arise in many applications, they have often proved difficult to understand and construct. We investigate the nonlinear problem of finding a tight frame for which the lengths of the frame elements have been prescribed in advance. Borrowing several ideas from Classical Mechanics, we show that this problem has a natural, intuitive interpretation. In particular, we show that such frames may be characterized as the minimizers of a potential energy function, and justify their interpretation as “maximally orthogonal” sequences. By exploiting this idea, we are able to show that such frames always exist, provided the requisite lengths satisfy a “fundamental inequality.” In so doing, we characterize those sequences of nonnegative numbers which arise as the lengths of a tight frame’s elements

Single-nucleotide polymorphisms in soybean.

Single-nucleotide polymorphisms (SNPs) provide an abundant source of DNA polymorphisms in a number of eukaryotic species. Information on the frequency, nature, and distribution of SNPs in plant genomes is limited. Thus, our objectives were (1) to determine SNP frequency in coding and noncoding soybean (Glycine max L. Merr.) DNA sequence amplified from genomic DNA using PCR primers designed to complete genes, cDNAs, and random genomic sequence; (2) to characterize haplotype variation in these sequences; and (3) to provide initial estimates of linkage disequilibrium (LD) in soybean. Approximately 28.7 kbp of coding sequence, 37.9 kbp of noncoding perigenic DNA, and 9.7 kbp of random noncoding genomic DNA were sequenced in each of 25 diverse soybean genotypes. Over the >76 kbp, mean nucleotide diversity expressed as Watterson's theta was 0.00097. Nucleotide diversity was 0.00053 and 0.00111 in coding and in noncoding perigenic DNA, respectively, lower than estimates in the autogamous model species Arabidopsis thaliana. Haplotype analysis of SNP-containing fragments revealed a deficiency of haplotypes vs. the number that would be anticipated at linkage equilibrium. In 49 fragments with three or more SNPs, five haplotypes were present in one fragment while four or less were present in the remaining 48, thereby supporting the suggestion of relatively limited genetic variation in cultivated soybean. Squared allele-frequency correlations (r(2)) among haplotypes at 54 loci with two or more SNPs indicated low genome-wide LD. The low level of LD and the limited haplotype diversity suggested that the genome of any given soybean accession is a mosaic of three or four haplotypes. To facilitate SNP discovery and the development of a transcript map, subsets of four to six diverse genotypes, whose sequence analysis would permit the discovery of at least 75% of all SNPs present in the 25 genotypes as well as 90% of the common (frequency >0.10) SNPs, were identified