16 research outputs found
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
Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space
A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts
Laplace approximations for sums of independent random vectors
Let X i , i∈ℕ, be i.i.d. B-valued random variables where B is a real separable Banach space, and Φ a mapping B→ℝ. Under some conditions an asymptotic evaluation of Z n =E(exp(nΦ(∑ i=1 n X i /n))) is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums ∑ i=1 n X i under the law transformed by the density exp(nΦ (∑ i=1 n X i /n))/Z n