The Johnson-Lindenstrauss lemma is a fundamental result in probability with several applications in the design and analysis of algorithms in high dimensional geometry. Most known constructions of linear embeddings that satisfy the Johnson-Lindenstrauss property involve randomness. We address the question of explicitly constructing such embedding families and provide a construction with an almost optimal use of randomness: we use O(log(n delta)log(log(n delta) epsilon)) random bits for embedding n dimensions to O(log(1 delta) epsilon^2) dimensions with error probability at most delta, and distortion at most epsilon. In particular, for delta = 1 poly(n) and fixed epsilon, we use O(log n loglog n) random bits. Previous constructions required at least O(log^2 n) random bits to get polynomially small error.
The best known explicit JL family is the construction of Clarkson and Woodruff @cite_16 who show that a random scaled Bernoulli matrix with @math -wise independent entries satisfies the JL lemma. We make use of their result in our construction.
@cite_16
2059867647
We give near-optimal space bounds in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank. In the streaming model, sketches of input matrices are maintained under updates of matrix entries; we prove results for turnstile updates, given in an arbitrary order. We give the first lower bounds known for the space needed by the sketches, for a given estimation error e. We sharpen prior upper bounds, with respect to combinations of space, failure probability, and number of passes. The sketch we use for matrix A is simply STA, where S is a sign matrix. Our results include the following upper and lower bounds on the bits of space needed for 1-pass algorithms. Here A is an n x d matrix, B is an n x d' matrix, and c := d+d'. These results are given for fixed failure probability; for failure probability δ>0, the upper bounds require a factor of log(1 δ) more space. We assume the inputs have integer entries specified by O(log(nc)) bits, or O(log(nd)) bits. (Matrix Product) Output matrix C with F(ATB-C) ≤ e F(A) F(B). We show that Θ(ce-2log(nc)) space is needed. (Linear Regression) For d'=1, so that B is a vector b, find x so that Ax-b ≤ (1+e) minx' ∈ Reald Ax'-b. We show that Θ(d2e-1 log(nd)) space is needed. (Rank-k Approximation) Find matrix tAk of rank no more than k, so that F(A-tAk) ≤ (1+e) F A-Ak , where Ak is the best rank-k approximation to A. Our lower bound is Ω(ke-1(n+d)log(nd)) space, and we give a one-pass algorithm matching this when A is given row-wise or column-wise. For general updates, we give a one-pass algorithm needing [O(ke-2(n + d e2)log(nd))] space. We also give upper and lower bounds for algorithms using multiple passes, and a sketching analog of the CUR decomposition.