Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

Abstract: We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.

Fast Monte Carlo Algorithms for Matrices III: Computing a Compressed Approximate Matrix Decomposition

Abstract: In many applications, the data consist of (or may be naturally formulated as) an $m \times n$ matrix $A$ which may be stored on disk but which is too large to be read into random access memory (RAM) or to practically perform superlinear polynomial time computations on it. Two algorithms are presented which, when given an $m \times n$ matrix $A$, compute approximations to $A$ which are the product of three smaller matrices, $C$, $U$, and $R$, each of which may be computed rapidly. Let $A' = CUR$ be the computed approximate decomposition; both algorithms have provable bounds for the error matrix $A-A'$. In the first algorithm, $c$ columns of $A$ and $r$ rows of $A$ are randomly chosen. If the $m \times c$ matrix $C$ consists of those $c$ columns of $A$ (after appropriate rescaling) and the $r \times n$ matrix $R$ consists of those $r$ rows of $A$ (also after appropriate rescaling), then the $c \times r$ matrix $U$ may be calculated from $C$ and $R$. For any matrix $X$, let $\|X\|_F$ and $\|X\|_2$ denote its Frobenius norm and its spectral norm, respectively. It is proven that $$ \left\|A-A'\right\|_\xi \le \min_{D:\mathrm{rank}(D)\le k} \left\|A-D\right\|_\xi + poly(k,1/c) \left\|A\right\|_F $$ holds in expectation and with high probability for both $\xi = 2,F$ and for all $k=1,\ldots,\mbox{rank}(A)$; thus by appropriate choice of $k$ $$ \left\|A-A'\right\|_2 \le \epsilon \left\|A\right\|_F $$ also holds in expectation and with high probability. This algorithm may be implemented without storing the matrix $A$ in RAM, provided it can make two passes over the matrix stored in external memory and use $O(m+n)$ additional RAM (assuming that $c$ and $r$ are constants, independent of the size of the input). The second algorithm is similar except that it approximates the matrix $C$ by randomly sampling a constant number of rows of $C$. Thus, it has additional error but it can be implemented in three passes over the matrix using only constant additional RAM. To achieve an additional error (beyond the best rank-$k$ approximation) that is at most $\epsilon \|A\|_F$, both algorithms take time which is a low-degree polynomial in $k$, $1/\epsilon$, and $1/\delta$, where $\delta>0$ is a failure probability; the first takes time linear in $\mbox{max}(m,n)$ and the second takes time independent of $m$ and $n$. The proofs for the error bounds make important use of matrix perturbation theory and previous work on approximating matrix multiplication and computing low-rank approximations to a matrix. The probability distribution over columns and rows and the rescaling are crucial features of the algorithms and must be chosen judiciously.

A sum-product estimate in finite fields, and applications

Abstract: Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.

Detection of stock out conditions based on image processing

Abstract: Image analysis techniques, including object recognition analysis, are applied to images obtained by one or more image capture devices deployed within inventory environments. The object recognition analysis provides object recognition data (that may include one or more recognized product instances) based on stored product (training) images. In turn, a variety of functionalities may be enabled based on the object recognition data. For example, a planogram may be extracted and compared to a target planogram, or at least one product display parameter for a product can be determined and used to assess presence of the product within the inventory environment, or to determine compliance of display of the product with a promotional objective. In yet another embodiment, comparisons may be made within a single image or between multiple images over time to detect potential conditions requiring response. In this manner, efficiency and effectiveness of many previously manually-implemented tasks may be improved.

Product formulas and numerical algorithms

Abstract: Product formulas constitute one of several bridges between numerical and functional analysis. In numerical analysis, they represent algorithms intended to approximate some evolution equation and, in functional analysis, they are used to prove estimates, existence and representation theorems. Our aim is to survey the setting for product formulas and to discuss some recent results. Needless to say, we do not attempt to accommodate all the complex variations which occur in practical algorithms, nor the sharpest possible theoretical results. Nevertheless, we hope that our middle ground approach and some of the examples will be of interest to both groups. Because of its survey nature, we have not hesitated to include some well-known examples which are important for understanding the ideas. The general idea of product formulas is the following. Suppose one is interested in an initial value problem