Sparse recovery with partial support knowledge

14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011, Princeton, NJ, USA, August 17-19, 2011. ProceedingsThe goal of sparse recovery is to recover the (approximately) best k-sparse approximation [ˆ over x] of an n-dimensional vector x from linear measurements Ax of x. We consider a variant of the problem which takes into account partial knowledge about the signal. In particular, we focus on the scenario where, after the measurements are taken, we are given a set S of size s that is supposed to contain most of the “large” coefficients of x. The goal is then to find [ˆ over x] such that [ ||x-[ˆ over x]|| [subscript p] ≤ C min ||x-x'||[subscript q]. [over] k-sparse x' [over] supp (x') [c over _] S] We refer to this formulation as the sparse recovery with partial support knowledge problem ( SRPSK ). We show that SRPSK can be solved, up to an approximation factor of C = 1 + ε, using O( (k/ε) log(s/k)) measurements, for p = q = 2. Moreover, this bound is tight as long as s = O(εn / log(n/ε)). This completely resolves the asymptotic measurement complexity of the problem except for a very small range of the parameter s. To the best of our knowledge, this is the first variant of (1 + ε)-approximate sparse recovery for which the asymptotic measurement complexity has been determined.Space and Naval Warfare Systems Center San Diego (U.S.) (Contract N66001-11-C-4092)David & Lucile Packard Foundation (Fellowship)Center for Massive Data Algorithmics (MADALGO)National Science Foundation (U.S.) (Grant CCF-0728645)National Science Foundation (U.S.) (Grant CCF-1065125

Dvoretzky type theorems for multivariate polynomials and sections of convex bodies

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even $d$ and any $k$). Corollaries for the John ellipsoid of projections or sections of a convex body are deduced from the case $d=2$ of the polynomial field conjecture

Almost-Euclidean subspaces of ℓ1N\ell_1^N via tensor products: a simple approach to randomness reduction

It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\Omega(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $\Omega(N)$-dimensional subspaces of $\ell_1^N$ while using only $N^a$ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding "almost Euclidean" subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor change

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans