We consider stable signal recovery in `q quasi-norm for 0 < q ≤ 1. In this problem, given a measurement vector y = Ax for some unknown signal vector x ∈ R and a known matrix A ∈ Rm×n, we want to recover z ∈ R with ‖x − z‖q = O(‖x − x‖q) from a measurement vector, where x∗ is the s-sparse vector closest to x in `q quasi-norm. Although a small value of q is favorable for measuring the distance to...

Vector quantization (VQ) is an efficient technique for image compression. However, one of the most serious problems of VQ is the heavily computational load for searching the closest codeword to the input vector in the codebook. To overcome this problem, we propose a simple and fast encoding algorithm for vector quantization. The algorithm uses three significant features of a vector, that is, th...

Abstract. The security of lattice-based cryptosystems such as NTRU, GGH and Ajtai-Dwork essentially relies upon the intractability of computing a shortest non-zero lattice vector and a closest lattice vector to a given target vector in high dimensions. The best algorithms for these tasks are due to Kannan, and, though remarkably simple, their complexity estimates have not been improved since ov...

In this paper, we propose a strongly convergent variant of Robinson’s subgradient algorithm for solving a system of vector convex inequalities in Hilbert spaces. The advantage of the proposed method is that it converges strongly, when the problem has solutions, under mild assumptions. The proposed algorithm also has the following desirable property: the sequence converges to the solution of the...

In this paper, we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps. We provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path. Our method only requires O(log N ) arithmetic operations and the total bit complexity is O(log2 N log log N log log log N ), where ...

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost vectors, determine a cost vector such that the corresponding optimal objective value of the linear program is closest to the desired value. The above problem, referred here as the inverse optimal value problem, is significantly different from sta...

We address the problem of designing data structures that allow e cient search for approximate nearest neighbors. More speci cally, given a database consisting of a set of vectors in some high dimensional Euclidean space, we want to construct a space-e cient data structure that would allow us to search, given a query vector, for the closest or nearly closest vector in the database. We also addre...

In this paper we present a new NTRU-Like public key cryptosystem with security provably based on the worst case hardness of the approximate both Shortest Vector Problem (SVP) and Closest Vector Problem (CVP) in some structured lattices, called ideal lattices. We show how to modify the ETRU cryptosystem, an NTRU-Like public key cryptosystem based on the Eisenstein integers 3 [ ]  where 3  is a...