arXiv subjects

Vladimir Braverman

Publications and source records attributed to Vladimir Braverman.

113 records · Page 7Linked to original sources

arXiv2010

Recursive Sketching For Frequency Moments

In a ground-breaking paper, Indyk and Woodruff (STOC 05) showed how to compute $F_k$ (for $k>2$) in space complexity $O(\mbox{\em poly-log}(n,m)\cdot n^{1-\frac2k})$, which is optimal up to (large) poly-logarithmic factors in $n$ and $m$, where $m$ is the length of the stream and $n$ is the upper bound on the number of distinct elements in a stream. The best known lower bound for large moments is $Ω(\log(n)n^{1-\frac2k})$. A follow-up work of Bhuvanagiri, Ganguly, Kesh and Saha (SODA 2006) reduced the poly-logarithmic factors of Indyk and Woodruff to $O(\log^2(m)\cdot (\log n+ \log m)\cdot n^{1-{2\over k}})$. Further reduction of poly-log factors has been an elusive goal since 2006, when Indyk and Woodruff method seemed to hit a natural "barrier." Using our simple recursive sketch, we provide a different yet simple approach to obtain a $O(\log(m)\log(nm)\cdot (\log\log n)^4\cdot n^{1-{2\over k}})$ algorithm for constant $ε$ (our bound is, in fact, somewhat stronger, where the $(\log\log n)$ term can be replaced by any constant number of $\log $ iterations instead of just two or three, thus approaching $log^*n$. Our bound also works for non-constant $ε$ (for details see the body of the paper). Further, our algorithm requires only $4$-wise independence, in contrast to existing methods that use pseudo-random generators for computing large frequency moments.

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arXiv2010

Rademacher Chaos, Random Eulerian Graphs and The Sparse Johnson-Lindenstrauss Transform

The celebrated dimension reduction lemma of Johnson and Lindenstrauss has numerous computational and other applications. Due to its application in practice, speeding up the computation of a Johnson-Lindenstrauss style dimension reduction is an important question. Recently, Dasgupta, Kumar, and Sarlos (STOC 2010) constructed such a transform that uses a sparse matrix. This is motivated by the desire to speed up the computation when applied to sparse input vectors, a scenario that comes up in applications. The sparsity of their construction was further improved by Kane and Nelson (ArXiv 2010). We improve the previous bound on the number of non-zero entries per column of Kane and Nelson from $O(1/ε\log(1/δ)\log(k/δ))$ (where the target dimension is $k$, the distortion is $1\pm ε$, and the failure probability is $δ$) to $$ O\left({1\overε} \left({\log(1/δ)\log\log\log(1/δ) \over \log\log(1/δ)}\right)^2\right). $$ We also improve the amount of randomness needed to generate the matrix. Our results are obtained by connecting the moments of an order 2 Rademacher chaos to the combinatorial properties of random Eulerian multigraphs. Estimating the chance that a random multigraph is composed of a given number of node-disjoint Eulerian components leads to a new tail bound on the chaos. Our estimates may be of independent interest, and as this part of the argument is decoupled from the analysis of the coefficients of the chaos, we believe that our methods can be useful in the analysis of other chaoses.

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arXiv2010

AMS Without 4-Wise Independence on Product Domains

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that 4-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of 4-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1 \pm ε)$ approximation (with probability $1-δ$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.

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arXiv2009

Measuring Independence of Datasets

A data stream model represents setting where approximating pairwise, or $k$-wise, independence with sublinear memory is of considerable importance. In the streaming model the joint distribution is given by a stream of $k$-tuples, with the goal of testing correlations among the components measured over the entire stream. In the streaming model, Indyk and McGregor (SODA 08) recently gave exciting new results for measuring pairwise independence. The Indyk and McGregor methods provide $\log{n}$-approximation under statistical distance between the joint and product distributions in the streaming model. Indyk and McGregor leave, as their main open question, the problem of improving their $\log n$-approximation for the statistical distance metric. In this paper we solve the main open problem posed by of Indyk and McGregor for the statistical distance for pairwise independence and extend this result to any constant $k$. In particular, we present an algorithm that computes an $(ε, δ)$-approximation of the statistical distance between the joint and product distributions defined by a stream of $k$-tuples. Our algorithm requires $O(({1\over ε}\log({nm\over δ}))^{(30+k)^k})$ memory and a single pass over the data stream.

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arXiv2008

Succinct Sampling on Streams

A streaming model is one where data items arrive over long period of time, either one item at a time or in bursts. Typical tasks include computing various statistics over a sliding window of some fixed time-horizon. What makes the streaming model interesting is that as the time progresses, old items expire and new ones arrive. One of the simplest and central tasks in this model is sampling. That is, the task of maintaining up to $k$ uniformly distributed items from a current time-window as old items expire and new ones arrive. We call sampling algorithms {\bf succinct} if they use provably optimal (up to constant factors) {\bf worst-case} memory to maintain $k$ items (either with or without replacement). We stress that in many applications structures that have {\em expected} succinct representation as the time progresses are not sufficient, as small probability events eventually happen with probability 1. Thus, in this paper we ask the following question: are Succinct Sampling on Streams (or $S^3$-algorithms)possible, and if so for what models? Perhaps somewhat surprisingly, we show that $S^3$-algorithms are possible for {\em all} variants of the problem mentioned above, i.e. both with and without replacement and both for one-at-a-time and bursty arrival models. Finally, we use $S^3$ algorithms to solve various problems in sliding windows model, including frequency moments, counting triangles, entropy and density estimations. For these problems we present \emph{first} solutions with provable worst-case memory guarantees.

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