Low rank approximation and decomposition of large matrices using error correcting codes

Low rank approximation is an important tool used in many applications of signal processing and machine learning. Recently, randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate singular value decompositions of large matrices. Similar ideas were used to solve least squares regression problems. In this paper, we show how matrices from error correcting codes can be used to find such low rank approximations and matrix decompositions, and extend the framework to linear least squares regression problems. The benefits of using these code matrices are the following: (i) They are easy to generate and they reduce randomness significantly. (ii) Code matrices with mild properties satisfy the subspace embedding property, and have a better chance of preserving the geometry of an entire subspace of vectors. (iii) For parallel and distributed applications, code matrices have significant advantages over structured random matrices and Gaussian random matrices. (iv) Unlike Fourier or Hadamard transform matrices, which require sampling $O(k\log k)$ columns for a rank-$k$ approximation, the log factor is not necessary for certain types of code matrices. That is, $(1+\epsilon)$ optimal Frobenius norm error can be achieved for a rank-$k$ approximation with $O(k/\epsilon)$ samples. (v) Fast multiplication is possible with structured code matrices, so fast approximations can be achieved for general dense input matrices. (vi) For least squares regression problem $\min\|Ax-b\|_2$ where $A\in \mathbb{R}^{n\times d}$, the $(1+\epsilon)$ relative error approximation can be achieved with $O(d/\epsilon)$ samples, with high probability, when certain code matrices are used

Sampling and multilevel coarsening algorithms for fast matrix approximations

This paper addresses matrix approximation problems for matrices that are large, sparse and/or that are representations of large graphs. To tackle these problems, we consider algorithms that are based primarily on coarsening techniques, possibly combined with random sampling. A multilevel coarsening technique is proposed which utilizes a hypergraph associated with the data matrix and a graph coarsening strategy based on column matching. Theoretical results are established that characterize the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. We consider a number of standard applications of this technique as well as a few new ones. Among the standard applications we first consider the problem of computing the partial SVD for which a combination of sampling and coarsening yields significantly improved SVD results relative to sampling alone. We also consider the Column subset selection problem, a popular low rank approximation method used in data related applications, and show how multilevel coarsening can be adapted for this problem. Similarly, we consider the problem of graph sparsification and show how coarsening techniques can be employed to solve it. Numerical experiments illustrate the performances of the methods in various applications