## Wednesday, April 20, 2016

### Heavy hitters via cluster-preserving clustering

Interesting, clustering the most frequent elementsin a turnstile model:

Heavy hitters via cluster-preserving clustering by Kasper Green Larsen, Jelani Nelson, Huy L. Nguyen, Mikkel Thorup

In turnstile $\ell_p$ $\varepsilon$-heavy hitters, one maintains a high-dimensional $x\in\mathbb{R}^n$ subject to $\texttt{update}(i,\Delta)$ causing $x_i\leftarrow x_i + \Delta$, where $i\in[n]$, $\Delta\in\mathbb{R}$. Upon receiving a query, the goal is to report a small list $L\subset[n]$, $|L| = O(1/\varepsilon^p)$, containing every "heavy hitter" $i\in[n]$ with $|x_i| \ge \varepsilon \|x_{\overline{1/\varepsilon^p}}\|_p$, where $x_{\overline{k}}$ denotes the vector obtained by zeroing out the largest $k$ entries of $x$ in magnitude.
For any $p\in(0,2]$ the CountSketch solves $\ell_p$ heavy hitters using $O(\varepsilon^{-p}\log n)$ words of space with $O(\log n)$ update time, $O(n\log n)$ query time to output $L$, and whose output after any query is correct with high probability (whp) $1 - 1/poly(n)$. Unfortunately the query time is very slow. To remedy this, the work [CM05] proposed for $p=1$ in the strict turnstile model, a whp correct algorithm achieving suboptimal space $O(\varepsilon^{-1}\log^2 n)$, worse update time $O(\log^2 n)$, but much better query time $O(\varepsilon^{-1}poly(\log n))$.
We show this tradeoff between space and update time versus query time is unnecessary. We provide a new algorithm, ExpanderSketch, which in the most general turnstile model achieves optimal $O(\varepsilon^{-p}\log n)$ space, $O(\log n)$ update time, and fast $O(\varepsilon^{-p}poly(\log n))$ query time, and whp correctness. Our main innovation is an efficient reduction from the heavy hitters to a clustering problem in which each heavy hitter is encoded as some form of noisy spectral cluster in a much bigger graph, and the goal is to identify every cluster. Since every heavy hitter must be found, correctness requires that every cluster be found. We then develop a "cluster-preserving clustering" algorithm, partitioning the graph into clusters without destroying any original cluster.

Join the CompressiveSensing subreddit or the Google+ Community or the Facebook page and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.