Logical matrix

About: Logical matrix is a research topic. Over the lifetime, 1623 publications have been published within this topic receiving 25395 citations. The topic is also known as: binary matrix & Boolean matrix.

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

Abstract: Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the $k$ dominant components of the singular value decomposition of an $m \times n$ matrix. (i) For a dense input matrix, randomized algorithms require $\bigO(mn \log(k))$ floating-point operations (flops) in contrast to $ \bigO(mnk)$ for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to $\bigO(k)$ passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

A Linear Representation of Dynamics of Boolean Networks

Abstract: A new matrix product, called semi-tensor product of matrices, is reviewed Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor The corresponding algorithms are developed and used to some examples

OCNet: Object Context Network for Scene Parsing

Abstract: In this paper, we address the semantic segmentation task with a new context aggregation scheme named \emph{object context}, which focuses on enhancing the role of object information. Motivated by the fact that the category of each pixel is inherited from the object it belongs to, we define the object context for each pixel as the set of pixels that belong to the same category as the given pixel in the image. We use a binary relation matrix to represent the relationship between all pixels, where the value one indicates the two selected pixels belong to the same category and zero otherwise. We propose to use a dense relation matrix to serve as a surrogate for the binary relation matrix. The dense relation matrix is capable to emphasize the contribution of object information as the relation scores tend to be larger on the object pixels than the other pixels. Considering that the dense relation matrix estimation requires quadratic computation overhead and memory consumption w.r.t. the input size, we propose an efficient interlaced sparse self-attention scheme to model the dense relations between any two of all pixels via the combination of two sparse relation matrices. To capture richer context information, we further combine our interlaced sparse self-attention scheme with the conventional multi-scale context schemes including pyramid pooling~\citep{zhao2017pyramid} and atrous spatial pyramid pooling~\citep{chen2018deeplab}. We empirically show the advantages of our approach with competitive performances on five challenging benchmarks including: Cityscapes, ADE20K, LIP, PASCAL-Context and COCO-Stuff

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate

Abstract: Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that ran- domization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k)) floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multi- processor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

Multi-Terminal Binary Decision Diagrams: An Efficient DataStructure for Matrix Representation

Abstract: In this paper, we discuss the use of binary decision diagrams to represent general matrices. We demonstrate that binary decision diagrams are an efficient representation for every special-case matrix in common use, notably sparse matrices. In particular, we demonstrate that for any matrix, the BDD representation can be no larger than the corresponding sparse-matrix representation. Further, the BDD representation is often smaller than any other conventional special-case representation: for the n×n Walsh matrix, for example, the BDD representation is of size O(log n). No other special-case representation in common use represents this matrix in space less than O(n²). We describe termwise, row, column, block, and diagonal selection over these matrices, standard an Strassen matrix multiplication, and LU factorization. We demonstrate that the complexity of each of these operations over the BDD representation is no greater than that over any standard representation. Further, we demonstrate that complete pivoting is no more difficult over these matrices than partial pivoting. Finally, we consider an example, the Walsh Spectrum of a Boolean function.