Practical Sparse Matrices in C++ with Hybrid Storage and Template-Based Expression Optimisation

Despite the importance of sparse matrices in numerous fields of science, software implementations remain difficult to use for non-expert users, generally requiring the understanding of underlying details of the chosen sparse matrix storage format. In addition, to achieve good performance, several formats may need to be used in one program, requiring explicit selection and conversion between the formats. This can be both tedious and error-prone, especially for non-expert users. Motivated by these issues, we present a user-friendly and open-source sparse matrix class for the C++ language, with a high-level application programming interface deliberately similar to the widely used MATLAB language. This facilitates prototyping directly in C++ and aids the conversion of research code into production environments. The class internally uses two main approaches to achieve efficient execution: (i) a hybrid storage framework, which automatically and seamlessly switches between three underlying storage formats (compressed sparse column, Red-Black tree, coordinate list) depending on which format is best suited and/or available for specific operations, and (ii) a template-based meta-programming framework to automatically detect and optimise execution of common expression patterns. Empirical evaluations on large sparse matrices with various densities of non-zero elements demonstrate the advantages of the hybrid storage framework and the expression optimisation mechanism.Comment: extended and revised version of an earlier conference paper arXiv:1805.0338

Practical Sparse Matrices in C++ with Hybrid Storage and Template-Based Expression Optimisation

Despite the importance of sparse matrices in numerous fields of science, software implementations remain difficult to use for non-expert users, generally requiring the understanding of underlying details of the chosen sparse matrix storage format. In addition, to achieve good performance, several formats may need to be used in one program, requiring explicit selection and conversion between the formats. This can be both tedious and error-prone, especially for non-expert users. Motivated by these issues, we present a user-friendly and open-source sparse matrix class for the C++ language, with a high-level application programming interface deliberately similar to the widely used MATLAB language. This facilitates prototyping directly in C++ and aids the conversion of research code into production environments. The class internally uses two main approaches to achieve efficient execution: (i) a hybrid storage framework, which automatically and seamlessly switches between three underlying storage formats (compressed sparse column, Red-Black tree, coordinate list) depending on which format is best suited and/or available for specific operations, and (ii) a template-based meta-programming framework to automatically detect and optimise execution of common expression patterns. Empirical evaluations on large sparse matrices with various densities of non-zero elements demonstrate the advantages of the hybrid storage framework and the expression optimisation mechanism.Comment: extended and revised version of an earlier conference paper arXiv:1805.0338

Compressing Word Embeddings

Recent methods for learning vector space representations of words have succeeded in capturing fine-grained semantic and syntactic regularities using vector arithmetic. However, these vector space representations (created through large-scale text analysis) are typically stored verbatim, since their internal structure is opaque. Using word-analogy tests to monitor the level of detail stored in compressed re-representations of the same vector space, the trade-offs between the reduction in memory usage and expressiveness are investigated. A simple scheme is outlined that can reduce the memory footprint of a state-of-the-art embedding by a factor of 10, with only minimal impact on performance. Then, using the same `bit budget', a binary (approximate) factorisation of the same space is also explored, with the aim of creating an equivalent representation with better interpretability.Comment: 10 pages, 0 figures, submitted to ICONIP-2016. Previous experimental results were submitted to ICLR-2016, but the paper has been significantly updated, since a new experimental set-up worked much bette

Exact Sparse Matrix-Vector Multiplication on GPU's and Multicore Architectures

We propose different implementations of the sparse matrix--dense vector multiplication (\spmv{}) for finite fields and rings \Zb/m\Zb. We take advantage of graphic card processors (GPU) and multi-core architectures. Our aim is to improve the speed of \spmv{} in the \linbox library, and henceforth the speed of its black box algorithms. Besides, we use this and a new parallelization of the sigma-basis algorithm in a parallel block Wiedemann rank implementation over finite fields

Communication Cost for Updating Linear Functions when Message Updates are Sparse: Connections to Maximally Recoverable Codes

We consider a communication problem in which an update of the source message needs to be conveyed to one or more distant receivers that are interested in maintaining specific linear functions of the source message. The setting is one in which the updates are sparse in nature, and where neither the source nor the receiver(s) is aware of the exact {\em difference vector}, but only know the amount of sparsity that is present in the difference-vector. Under this setting, we are interested in devising linear encoding and decoding schemes that minimize the communication cost involved. We show that the optimal solution to this problem is closely related to the notion of maximally recoverable codes (MRCs), which were originally introduced in the context of coding for storage systems. In the context of storage, MRCs guarantee optimal erasure protection when the system is partially constrained to have local parity relations among the storage nodes. In our problem, we show that optimal solutions exist if and only if MRCs of certain kind (identified by the desired linear functions) exist. We consider point-to-point and broadcast versions of the problem, and identify connections to MRCs under both these settings. For the point-to-point setting, we show that our linear-encoder based achievable scheme is optimal even when non-linear encoding is permitted. The theory is illustrated in the context of updating erasure coded storage nodes. We present examples based on modern storage codes such as the minimum bandwidth regenerating codes.Comment: To Appear in IEEE Transactions on Information Theor

Linear Optimal Power Flow Using Cycle Flows

Linear optimal power flow (LOPF) algorithms use a linearization of the alternating current (AC) load flow equations to optimize generator dispatch in a network subject to the loading constraints of the network branches. Common algorithms use the voltage angles at the buses as optimization variables, but alternatives can be computationally advantageous. In this article we provide a review of existing methods and describe a new formulation that expresses the loading constraints directly in terms of the flows themselves, using a decomposition of the network graph into a spanning tree and closed cycles. We provide a comprehensive study of the computational performance of the various formulations, in settings that include computationally challenging applications such as multi-period LOPF with storage dispatch and generation capacity expansion. We show that the new formulation of the LOPF solves up to 7 times faster than the angle formulation using a commercial linear programming solver, while another existing cycle-based formulation solves up to 20 times faster, with an average speed-up of factor 3 for the standard networks considered here. If generation capacities are also optimized, the average speed-up rises to a factor of 12, reaching up to factor 213 in a particular instance. The speed-up is largest for networks with many buses and decentral generators throughout the network, which is highly relevant given the rise of distributed renewable generation and the computational challenge of operation and planning in such networks.Comment: 11 pages, 5 figures; version 2 includes results for generation capacity optimization; version 3 is the final accepted journal versio

Fast Matlab compatible sparse assembly on multicore computers

We develop and implement in this paper a fast sparse assembly algorithm, the fundamental operation which creates a compressed matrix from raw index data. Since it is often a quite demanding and sometimes critical operation, it is of interest to design a highly efficient implementation. We show how to do this, and moreover, we show how our implementation can be parallelized to utilize the power of modern multicore computers. Our freely available code, fully Matlab compatible, achieves about a factor of 5 times in speedup on a typical 6-core machine and 10 times on a dual-socket 16 core machine compared to the built-in serial implementation