arXiv subjects

Purushottam Kar

Publications and source records attributed to Purushottam Kar.

39 records · Page 3Linked to original sources

arXiv2010

Estimating small frequency moments of data stream: a characteristic function approach

A data stream is viewed as a sequence of $M$ updates of the form $(\text{index},i,v)$ to an $n$-dimensional integer frequency vector $f$, where the update changes $f_i$ to $f_i + v$, and $v$ is an integer and assumed to be in $\{-m, ..., m\}$. The $p$th frequency moment $F_p$ is defined as $\sum_{i=1}^n \abs{f_i}^p$. We consider the problem of estimating $F_p$ to within a multiplicative approximation factor of $1\pm ε$, for $p \in [0,2]$. Several estimators have been proposed for this problem, including Indyk's median estimator \cite{indy:focs00}, Li's geometric means estimator \cite{pinglib:2006}, an \Hss-based estimator \cite{gc:random07}. The first two estimators require space $\tilde{O}(ε^{-2})$, where the $\tilde{O}$ notation hides polylogarithmic factors in $ε^{-1}, m, n$ and $M$. Recently, Kane, Nelson and Woodruff in \cite{knw:soda10} present a space-optimal and novel estimator, called the log-cosine estimator. In this paper, we present an elementary analysis of the log-cosine estimator in a stand-alone setting. The analysis in \cite{knw:soda10} is more complicated.

cs.DS↗

arXiv2010

On Estimating the First Frequency Moment of Data Streams

Estimating the first moment of a data stream defined as $F_1 = \sum_{i \in \{1, 2, \ldots, n\}} \abs{f_i}$ to within $1 \pm ε$-relative error with high probability is a basic and influential problem in data stream processing. A tight space bound of $O(ε^{-2} \log (mM))$ is known from the work of [Kane-Nelson-Woodruff-SODA10]. However, all known algorithms for this problem require per-update stream processing time of $Ω(ε^{-2})$, with the only exception being the algorithm of [Ganguly-Cormode-RANDOM07] that requires per-update processing time of $O(\log^2(mM)(\log n))$ albeit with sub-optimal space $O(ε^{-3}\log^2(mM))$. In this paper, we present an algorithm for estimating $F_1$ that achieves near-optimality in both space and update processing time. The space requirement is $O(ε^{-2}(\log n + (\log ε^{-1})\log(mM)))$ and the per-update processing time is $O( (\log n)\log (ε^{-1}))$.

cs.DS↗

arXiv2009

On Low Distortion Embeddings of Statistical Distance Measures into Low Dimensional Spaces

Statistical distance measures have found wide applicability in information retrieval tasks that typically involve high dimensional datasets. In order to reduce the storage space and ensure efficient performance of queries, dimensionality reduction while preserving the inter-point similarity is highly desirable. In this paper, we investigate various statistical distance measures from the point of view of discovering low distortion embeddings into low-dimensional spaces. More specifically, we consider the Mahalanobis distance measure, the Bhattacharyya class of divergences and the Kullback-Leibler divergence. We present a dimensionality reduction method based on the Johnson-Lindenstrauss Lemma for the Mahalanobis measure that achieves arbitrarily low distortion. By using the Johnson-Lindenstrauss Lemma again, we further demonstrate that the Bhattacharyya distance admits dimensionality reduction with arbitrarily low additive error. We also examine the question of embeddability into metric spaces for these distance measures due to the availability of efficient indexing schemes on metric spaces. We provide explicit constructions of point sets under the Bhattacharyya and the Kullback-Leibler divergences whose embeddings into any metric space incur arbitrarily large distortions. We show that the lower bound presented for Bhattacharyya distance is nearly tight by providing an embedding that approaches the lower bound for relatively small dimensional datasets.

cs.CG↗