Universal Harmonic Sketching over Turnstile Streams

In this work we take a new approach to constructing a universal sketch that focuses on a class of \emph{basis functions} $\{f_s(x)=1-\cos(sx)\mid s>0\}$, so that any $f$-moment can be estimated if $f$ can be expressed as a linear combination of basis functions. We construct and analyze the $\mathsf{SymmetricPoissonTower}$ sketch, which occupies $O(\epsilon^{-2}\log^2(nM\epsilon^{-1}))$ bits and is $\mathcal{F}$-universal for the function class $$\mathcal{F}= \left\{f(x)=cx^2+\int_0^\infty (1-\cos (xs))\,\nu(ds) \mid c\geq 0, \text{$\nu$ is a positive measure}\right\},$$ i.e., given any $f\in \mathcal{F}$, the new sketch $(1\pm\epsilon)$-estimates the $f$-moment with probability 2/3. The family $\mathcal{F}$ includes all the classic frequency moments ($f(z)=|z|^p$, $p\in [0,2]$) as well as a large family of nearly-periodic functions that cannot be estimated with $L_2$-heavy hitter machinery. This new approach to universality requires significantly less space in comparison to previous universal schemes and sheds new light on the full characterization of the class $\mathcal{T}$ of tractable functions.