"We present a very fast algorithm for general matrix factorization of a data matrix for use in the statistical analysis of high-dimensional data via latent factors. Such data are prevalent across many application areas and generate an ever-increasing demand for methods of dimension reduction in order to undertake the statistical analysis of interest. Our algorithm uses a gradient-based approach which can be used with an arbitrary loss function provided the latter is differentiable. The speed and effectiveness of our algorithm for dimension reduction is demonstrated in the context of supervised classification of some real high-dimensional data sets from the bioinformatics literature."

"We revisit the matrix problems sparse null space and matrix sparsification, and show that they are equivalent. We then proceed to seek algorithms for these problems: We prove the hardness of approximation of these problems, and also give a powerful tool to extend algorithms and heuristics for sparse approximation theory to these problems."

"The well known algorithm of VOLKER STRASSEN for matrix multiplication can only be used for $(m2^k \times m2^k)$ matrices. For arbitrary $(n \times n)$ matrices one has to add zero rows and columns to the given matrices to use STRASSEN's algorithm. … The aim of this work is to give a detailed analysis of the number of additional zero rows and columns and the additional work caused by STRASSEN's bad parameters. STRASSEN used the parameters $m$ and $k$ to show that his matrix multiplication algorithm needs less than $4.7n^{\log_2 7}$ flops. We can show in this paper, that these parameters cause an additional work of approx. 20 % in the worst case in comparison to the optimal strategy for the worst case. This is the main reason for the search for better parameters."

"We consider the problem of developing an efficient multi-threaded implementation of the matrix-vector multiplication algorithm for sparse matrices with structural symmetry. Matrices are stored using the compressed sparse row-column format (CSRC), designed for profiting from the symmetric non-zero pattern observed in global finite element matrices. Unlike classical compressed storage formats, performing the sparse matrix-vector product using the CSRC requires thread-safe access to the destination vector. To avoid race conditions, we have implemented two partitioning strategies. In the first one, each thread allocates an array for storing its contributions, which are later combined in an accumulation step. We analyze how to perform this accumulation in four different ways.…"

"NArray is an Numerical N-dimensional Array class. Supported element types are 1/2/4-byte Integer, single/double-precision Real/Complex, and Ruby Object. This extension library incorporates fast calculation and easy manipulation of large numerical arrays into the Ruby language. NArray has features similar to NumPy, but NArray has vector and matrix subclasses."

numpy review

"Many approaches for retrieving documents from electronic databases depend on the literal matching of words in a user's query to the keywords defining database objects. Since there is great diversity in the words people use to describe the same object, literal- or lexical- based methods can often retrieve irrelevant documents. Another approach to exploit the implicit higher-order structure in the association of terms with text objects is to compute the singular value decomposition (SVD) of large sparse term by text-object matrices. Latent Semantic Indexing (LSI) is a conceptual indexing method which employs the SVD to represent terms and objects by dominant singular subspaces so that user queries can be matched in a lower-rank semantic space. This paper considers a third, intermediate approach to facilitate the immediate detection of document (or term) clusters...."