Tight Lower Bound for Linear Sketches of Moments

The problem of estimating frequency moments of a data stream has attracted a lot of attention since the onset of streaming algorithms [AMS99]. While the space complexity for approximately computing the $p^{\rm th}$ moment, for $p\in(0,2]$ has been settled [KNW10], for $p>2$ the exact complexity remains open. For $p>2$ the current best algorithm uses $O(n^{1-2/p}\log n)$ words of space [AKO11,BO10], whereas the lower bound is of $\Omega(n^{1-2/p})$ [BJKS04]. In this paper, we show a tight lower bound of $\Omega(n^{1-2/p}\log n)$ words for the class of algorithms based on linear sketches, which store only a sketch $Ax$ of input vector $x$ and some (possibly randomized) matrix $A$. We note that all known algorithms for this problem are linear sketches.Comment: In Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP), Riga, Latvia, July 201

Time-frequency detection algorithm for gravitational wave bursts

An efficient algorithm is presented for the identification of short bursts of gravitational radiation in the data from broad-band interferometric detectors. The algorithm consists of three steps: pixels of the time-frequency representation of the data that have power above a fixed threshold are first identified. Clusters of such pixels that conform to a set of rules on their size and their proximity to other clusters are formed, and a final threshold is applied on the power integrated over all pixels in such clusters. Formal arguments are given to support the conjecture that this algorithm is very efficient for a wide class of signals. A precise model for the false alarm rate of this algorithm is presented, and it is shown using a number of representative numerical simulations to be accurate at the 1% level for most values of the parameters, with maximal error around 10%.Comment: 26 pages, 15 figures, to appear in PR

Estimation and detection of functions from weighted tensor product spaces

The problems of estimation and detection of an infinitely-variate signal f observed in the continuous white noise model are studied. It is assumed that f belongs to a certain weighted tensor product space. Several examples of such a space are considered. Special attention is given to the tensor product space of analytic functions with exponential weights. In connection with estimating and detecting unknown signal, the problems of rate and sharp optimality are investigated. In particular, it is shown that the use of a weighted tensor product space makes it possible to avoid the "curse of dimensionality" phenomenon

Model Selection for Classification with a Large Number of Classes

In the present paper, we study the problem of model selection for classification of high-dimensional vectors into a large number of classes. The objective is to construct a model selection procedure and study its asymptotic properties when both, the number of features and the number of classes, are large. Although the problem has been investigated by many authors, we research a more difficult version of a less explored random effect model where, moreover, features are sparse and have only moderate strength. The paper formulates necessary and sufficient conditions for separability of features into the informative and noninformative sets. In particular, the surprising conclusion of the paper is that separation of features becomes easier as the number of classes grows