A Unified Construction of Streaming Sketches via the L\'evy-Khintchine Representation Theorem

Abstract

In the $d$-dimensional turnstile streaming model, a frequency vector $\mathbf{x}=(\mathbf{x}(1),\ldots,\mathbf{x}(n))\in (\mathbb{R}^d)^n$ is updated entry-wisely over a stream. We consider the problem of $f$-moment estimation, where one wants to estimate $$f(\mathbf{x})=\sum_{v\in[n]}f(\mathbf{x}(v))$$ with a small-space sketch. In this work we present a simple and generic scheme to construct sketches with the novel idea of hashing indices to L\'evy processes, from which one can estimate the $f$-moment $f(\mathbf{x})$ where $f$ is the characteristic exponent of the L\'evy process. The fundamental L\'evy-Khintchine representation theorem completely characterizes the space of all possible characteristic exponents, which in turn characterizes the set of $f$-moments that can be estimated by this generic scheme. The new scheme has strong explanatory power. It unifies the construction of many existing sketches and it implies the tractability of many nearly periodic functions that were previously unclassified. Furthermore, the scheme can be conveniently generalized to multidimensional cases ($d\geq 2$) by considering multidimensional L\'evy processes and can be further generalized to estimate heterogeneous moments by projecting different indices with different L\'evy processes. We conjecture that the set of tractable functions can be characterized using the L\'evy-Khintchine representation theorem via what we called the Fourier-Hahn-L\'evy method.