We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix @math one row at a time. It performed @math operations per row and maintains a sketch matrix @math such that for any @math @math and @math . Here, @math stands for the minimizer of @math over all rank @math matrices (similarly @math ) and @math is the rank @math matrix resulting from projecting @math on the row span of @math . We show both of these bounds are the best possible for the space allowed. The summary is mergeable, and hence trivially parallelizable. Moreover, Frequent Directions outperforms exemplar implementations of existing streaming algorithms in the space-error tradeoff.
@cite_38: Given a matrix A, it is often desirable to find a good approximation to A that has low rank. We introduce a simple technique for accelerating the computation of such approximations when A has strong spectral features, that is, when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to A. Our technique amounts to independently sampling and or quantizing the entries of A, thus speeding up computation by reducing the number of nonzero entries and or the length of their representation. Our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix N to A, whose entries are independent random variables with zero-mean and bounded variance. Since, with high probability, N has very weak spectral features, we can prove that the effect of sampling and quantization nearly vanishes when a low-rank approximation to A p N is computed. We give high probability bounds on the quality of our approximation both in the Frobenius and the 2-norm.
@cite_29: We describe a simple random-sampling based procedure for producing sparse matrix approximations. Our procedure and analysis are extremely simple: the analysis uses nothing more than the Chernoff-Hoeffding bounds. Despite the simplicity, the approximation is comparable and sometimes better than previous work. Our algorithm computes the sparse matrix approximation in a single pass over the data. Further, most of the entries in the output matrix are quantized, and can be succinctly represented by a bit vector, thus leading to much savings in space.
@cite_25: 
Given these models, there are several types of matrix sketches. We describe them here with a bit more specificity than in the Introduction, with particular attention to those that can operate in the row-update model we focus on. Specific exemplars are described which are used in out empirical study to follow. The first approach is to the matrix @cite_29 @cite_38 @cite_25 , by retaining a small number of non-zero. These algorithms typically assume to know the @math dimensions of @math , and are thus not directly applicable in out model.