A Unified Construction of Streaming Sketches via the Lévy-Khintchine Representation Theorem
This work provides a foundational framework for streaming algorithms, benefiting researchers in data streaming and computational theory by unifying and extending sketch constructions.
The paper tackles the problem of estimating f-moments in high-dimensional streaming data by introducing a generic sketching scheme based on hashing indices to Lévy processes, which unifies existing sketches and extends tractability to previously unclassified functions, achieving a characterization via the Lévy-Khintchine theorem.
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 Lévy processes, from which one can estimate the $f$-moment $f(\mathbf{x})$ where $f$ is the characteristic exponent of the Lévy process. The fundamental Lévy-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 Lévy processes and can be further generalized to estimate heterogeneous moments by projecting different indices with different Lévy processes. We conjecture that the set of tractable functions can be characterized using the Lévy-Khintchine representation theorem via what we called the Fourier-Hahn-Lévy method.