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Compressed sensing (CS) is a signal acquisition paradigm to simultaneouslyacquire and reduce dimension of signals that admit sparse representations. Thisis achieved by collecting linear, non-adaptive measurements of a signal, whichcan be formalized as multiplying the signal with a "measurement matrix". If themeasurement satisfies the so-called restricted isometry property (RIP), then itwill be appropriate to be used in compressed sensing. While a wide class ofrandom matrices provably satisfy the RIP with high probability, explicit anddeterministic constructions have been shown (so far) to satisfy the RIP only ina significantly suboptimal regime.  In this paper, we propose two novel approaches for improving the RIP constantestimates based on Gershgorin circle theorem for a specific deterministicconstruction based on Paley tight frames, obtaining an improvement over theGershgorin bound by a multiplicative constant. In one approach we use a recentresult on the spectra of the skew-adjacency matrices of oriented graphs. In theother approach, we use the so-called Dembo bounds on the extreme eigenvalues ofa positive semidefinite Hermitian matrix. We also generalize these bounds andwe combine the new bounds with a conjecture we make regarding the distributionof quadratic residues in a finite field to provide a potential path to breakthe so-called "square-root barrier"-we provide a proof based on the assumptionthat the conjecture holds.

RIP constants for deterministic compressed sensing matrices-beyond Gershgorin