Harmonic Decomposition in Data Sketches

In the turnstile streaming model, a dynamic vector $\mathbf{x}=(\mathbf{x}_1,\ldots,\mathbf{x}_n)\in \mathbb{Z}^n$ is updated by a stream of entry-wise increments/decrements. Let $f\colon\mathbb{Z}\to \mathbb{R}_+$ be a symmetric function with $f(0)=0$. The \emph{$f$-moment} of $\mathbf{x}$ is defined to be $f(\mathbf{x}) := \sum_{v\in[n]}f(\mathbf{x}_v)$. We revisit the problem of constructing a \emph{universal sketch} that can estimate many different $f$-moments. Previous constructions of universal sketches rely on the technique of sampling with respect to the $L_0$-mass (uniform samples) or $L_2$-mass ($L_2$-heavy-hitters), whose universality comes from being able to evaluate the function $f$ over the samples. In this work we take a new approach to constructing a universal sketch that does not use \emph{any} explicit samples but relies on the \emph{harmonic structure} of the target function $f$. The new sketch ($\textsf{SymmetricPoissonTower}$) \emph{embraces} hash collisions instead of avoiding them, which saves multiple $\log n$ factors in space, e.g., when estimating all $L_p$-moments ($f(z) = |z|^p,p\in[0,2]$). For many nearly periodic functions, the new sketch is \emph{exponentially} more efficient than sampling-based methods. We conjecture that the $\textsf{SymmetricPoissonTower}$ sketch is \emph{the} universal sketch that can estimate every tractable function $f$.