This paper addresses the nearest neighbor search problem under inner product similarity and introduces a compact code-based approach. The idea is to approximate a vector using the composition of several elements selected from a source dictionary and to represent this vector by a short code composed of the indices of the selected elements. The inner product between a query vector and a database ...

2 A denote the transpose, Moore-Penrose generalized inverse, Frobenius norm and Euclid norm, respectively. For any , m n A B  R ,   , 0 T A B trace B A   denotes the inner product of A and B . Therefore, m n R  is a complete inner product space endowed with 2 , A A A  . For any non-zero matrices 1 2 , , , m n k A A A  R  , if  , T j i i A A trace A     0 j A i j   , then it is...

In the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. In particular, it isshown that the Cauchy-Schwarz inequality holds. Moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy Hilbert space has a complementary subspace.Finally, the notions of fuzzy bo...

Working in the framework of the Simplicial Topology, a method for calculating the p-local homology of a twisted cartesian product X(π,m, τ, π′, n) = K(π,m)×τ K(π′, n) of Eilenberg-Mac Lane spaces is given. The chief technique is the construction of an explicit homotopy equivalence between the normalized chain complex of X and a free DGA-module of finite type M , via homological perturbation. If...

We present an introduction to Hilbert space frames. A frame is a generalization of a basis that is useful, for example, in signal processing. It allows us to expand Hilbert space vectors in terms of a set of other vectors that satisfy a certain “energy equivalence” condition. This condition guarantees that any vector in the Hilbert space can be reconstructed in a numerically stable way from its...

Let T be any normed linear space [l, p. S3]. Then an inner product is defined in T if to each pair of elements x and y there is associated a real number (x, y) in such a way that (#, y) » (y, x), \\x\\ = (#, #), (x, y+z) = (#,y) + (x, 2), and (/#,y) = /(#, y) for all real numbers /and elements x and y. An inner product can be defined in T if and only if any two-dimensional subspace is equivalen...

This paper is an investigation of $L$-dual frames with respect to a function-valued inner product, the so called $L$-bracket product on $L^{2}(G)$, where G is a locally compact abelian group with a uniform lattice $L$. We show that several well known theorems for dual frames and dual Riesz bases in a Hilbert space remain valid for $L$-dual frames and $L$-dual Riesz bases in $L^{2}(G)$.

let a be a unital algebra over a field of characteristic zero. we show that every derivation from( ) n m a into its dual ( ) n m a ∗ is the sum of an inner derivation and a derivation induced by a derivationfrom a into a∗