We consider processing an n x d matrix A in a stream with row-wise updates according to a recent algorithm called Frequent Directions (Liberty, KDD 2013). This algorithm maintains an l x d matrix Q deterministically, processing each row in O(d l^2) time; the processing time can be decreased to O(d l) with a slight modification in the algorithm and a constant increase in space. We show that if one sets l = k+ k eps and returns Q_k, a k x d matrix that is the best rank k approximation to Q, then we achieve the following properties: ||A - A_k||_F^2 <= ||A||_F^2 - ||Q_k||_F^2 <= (1+eps) ||A - A_k||_F^2 and where pi_ Q_k (A) is the projection of A onto the rowspace of Q_k then ||A - pi_ Q_k (A)||_F^2 <= (1+eps) ||A - A_k||_F^2. We also show that Frequent Directions cannot be adapted to a sparse version in an obvious way that retains the l original rows of the matrix, as opposed to a linear combination or sketch of the rows.
There is a wealth of literature on this problem; most recently two algorithms @cite_0 @cite_6 showed how to construct a decomposition of @math that has rank @math with error bound @math with constant probability in approximately @math time. We refer to these papers for a more thorough survey of the history of the area, many other results, and other similar approximate linear algebra applications. But we attempt to report many of the most important related results in Appendix .
@cite_0 @cite_6
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We design a new distribution over poly(r e-1) x n matrices S so that for any fixed n x d matrix A of rank r, with probability at least 9 10, SAx2 = (1 pm e)Ax2 simultaneously for all x ∈ Rd. Such a matrix S is called a subspace embedding. Furthermore, SA can be computed in O(nnz(A)) + O(r2e-2) time, where nnz(A) is the number of non-zero entries of A. This improves over all previous subspace embeddings, which required at least Ω(nd log d) time to achieve this property. We call our matrices S sparse embedding matrices. Using our sparse embedding matrices, we obtain the fastest known algorithms for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and lp-regression: to output an x' for which Ax'-b2 ≤ (1+e)minx Ax-b2 for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A)) + O(d3e-2) time, and another in O(nnz(A)log(1 e)) + O(d3log(1 e)) time. (Here O(f) = f ⋅ logO(1)(f).) to obtain a decomposition of an n x n matrix A into a product of an n x k matrix L, a k x k diagonal matrix D, and a n x k matrix W, for which F A - L D W ≤ (1+e)F A-Ak , where Ak is the best rank-k approximation, our algorithm runs in O(nnz(A)) + O(nk2 e-4log n + k3e-5log2n) time. to output an approximation to all leverage scores of an n x d input matrix A simultaneously, with constant relative error, our algorithms run in O(nnz(A) log n) + O(r3) time. to output an x' for which Ax'-bp ≤ (1+e)minx Ax-bp for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A) log n) + poly(r e-1) time, for any constant 1 ≤ p An "oblivious subspace embedding (OSE)" given some parameters eps,d is a distribution D over matrices B in R^ m x n such that for any linear subspace W in R^n with dim(W) = d it holds that Pr_ B D (forall x in W ||B x||_2 in (1 + - eps)||x||_2) > 2 3 We show an OSE exists with m = O(d^2 eps^2) and where every B in the support of D has exactly s=1 non-zero entries per column. This improves previously best known bound in [Clarkson-Woodruff, arXiv:1207.6365]. Our quadratic dependence on d is optimal for any OSE with s=1 [Nelson-Nguyen, 2012]. We also give two OSE's, which we call Oblivious Sparse Norm-Approximating Projections (OSNAPs), that both allow the parameter settings m = O(d eps^2) and s = polylog(d) eps, or m = O(d^ 1+gamma eps^2) and s=O(1 eps) for any constant gamma>0. This m is nearly optimal since m >= d is required simply to no non-zero vector of W lands in the kernel of B. These are the first constructions with m=o(d^2) to have s=o(d). In fact, our OSNAPs are nothing more than the sparse Johnson-Lindenstrauss matrices of [Kane-Nelson, SODA 2012]. Our analyses all yield OSE's that are sampled using either O(1)-wise or O(log d)-wise independent hash functions, which provides some efficiency advantages over previous work for turnstile streaming applications. Our main result is essentially a Bai-Yin type theorem in random matrix theory and is likely to be of independent interest: i.e. we show that for any U in R^ n x d with orthonormal columns and random sparse B, all singular values of BU lie in [1-eps, 1+eps] with good probability. Plugging OSNAPs into known algorithms for numerical linear algebra problems such as approximate least squares regression, low rank approximation, and approximating leverage scores implies faster algorithms for all these problems.