On Sketching Trimmed Statistics

We present space-efficient linear sketches for estimating trimmed statistics of an $n$-dimensional frequency vector $x$, e.g., the sum of $p$-th powers of the largest $k$ frequencies (i.e., entries) in absolute value, or the $k$-trimmed vector, which excludes the top and bottom $k$ frequencies. This is called the $F_p$ moment of the trimmed vector. Trimmed measures are used in robust estimation, as seen in the R programming language's `trim.var' function and the `trim' parameter in the mean function. Linear sketches improve time and memory efficiency and are applicable to streaming and distributed settings. We initiate the study of sketching these statistics and give a new condition for capturing their space complexity. When $k \ge n/poly\log n$, we give a linear sketch using $poly(1/\varepsilon, \log n)$ space which provides a $(1 \pm \varepsilon)$ approximation to the top-$k$ $F_p$ moment for $p \in [0,2]$. For general $k$, we give a sketch with the same guarantees under a condition relating the $k$-th largest frequency to the tail mass, and show this condition is necessary. For the $k$-trimmed version, our sketch achieves optimal error guarantees under the same condition. We extend our methods to $p > 2$ and also address related problems such as computing the $F_p$ moment of frequencies above a threshold, finding the largest $k$ such that the $F_p$ moment of the top $k$ exceeds $k^{p+1}$, and the $F_p$ moment of the top $k$ frequencies such that each entry is at least $k$. Notably, our algorithm for this third application improves upon the space bounds of the algorithm of Govindan, Monemizadeh, and Muthukrishnan (PODS '17) for computing the $h$-index. We show empirically that our top $k$ algorithm uses much less space compared to Count-Sketch while achieving the same error.