Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have considered the recovery of a sparse signal through Φ and y = Φx + b, where is a matrix with more columns than rows and denotes the measurement noise. In this paper, based on Restricted Isometry Property (RIP), the performance of OMP is analyzed under general perturbations, which means both y and Φ are perturbed. Though the exact recovery of an almost sparse signal x is no longer feasible, the main contribution reveals that the support set of the best k-term approximation of x can be recovered under reasonable conditions. The error bound between x and the estimation of OMP is also derived. By constructing an example it is also demonstrated that the sufficient conditions for support recovery of the best k-term approximation of are rather tight. When x is strong-decaying, it is proved that the sufficient conditions for support recovery of the best k-term approximation of x can be relaxed, and the support can even be recovered in the order of the entries' magnitude. Our results are also compared in detail with some related previous ones.
First, let @math satisfy @math , i.e. @math Consider the ratio of the required upper bound of @math in the result of @cite_3 to that in Corollary @math : @math It can be concluded from @math that @math , which means that the requirement of @math in Corollary @math is more relaxed.
@cite_3
2952621790 30524954 1981891714 2949121808
Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. In this paper, we prove tight lower bounds on the smallest degree of a nonzero polynomial in the ideal generated by @math or @math in the polynomial ring @math , @math are coprime, which is called over @math . The immunity of @math is lower bounded by @math , which is achievable when @math is a multiple of @math ; the immunity of @math is exactly @math for every @math and @math . Our result improves the previous bound @math by Green. We observe how immunity over @math is related to @math circuit lower bound. For example, if the immunity of @math over @math is lower bounded by @math , and @math , then @math requires @math circuit of exponential size to compute. Let @math and @math be standard normal random variables with covariance matrices @math and @math , respectively. Slepian's lemma says that if @math for @math , the lower bound @math is at least @math . In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdős and Revesz, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues. Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz.