DROPS

Document

DOI: 10.4230/LIPIcs.ITCS.2025.77

Sketching, Moment Estimation, and the Lévy-Khintchine Representation Theorem

Authors: Seth Pettie and Dingyu Wang

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

In the d-dimensional turnstile streaming model, a frequency vector 𝐱 = (𝐱(1),…,𝐱(n)) ∈ (ℝ^d)ⁿ is updated entry-wisely over a stream. We consider the problem of f-moment estimation for which one wants to estimate f(𝐱)=∑_{v ∈ [n]}f(𝐱(v)) with a small-space sketch. A function f is tractable if the f-moment can be estimated to within a constant factor using polylog(n) space. The f-moment estimation problem has been intensively studied in the d = 1 case. Flajolet and Martin estimate the F₀-moment (f(x) = 1 (x > 0), incremental stream); Alon, Matias, and Szegedy estimate the L₂-moment (f(x) = x²); Indyk estimates the L_α-moment (f(x) = |x|^α), α ∈ (0,2]. For d ≥ 2, Ganguly, Bansal, and Dube estimate the L_{p,q} hybrid moment (f:ℝ^d → ℝ,f(x) = (∑_{j = 1}^d |x_j|^p)^q), p ∈ (0,2],q ∈ (0,1). For tractability, Bar-Yossef, Jayram, Kumar, and Sivakumar show that f(x) = |x|^α is not tractable for α > 2. Braverman, Chestnut, Woodruff, and Yang characterize the class of tractable one-variable functions except for a class of nearly periodic functions. 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(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 (F₀, L₀, L₂, L_α, L_{p,q}, etc.) 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 ≥ 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.

Cite as

Seth Pettie and Dingyu Wang. Sketching, Moment Estimation, and the Lévy-Khintchine Representation Theorem. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 77:1-77:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{pettie_et_al:LIPIcs.ITCS.2025.77,
  author =	{Pettie, Seth and Wang, Dingyu},
  title =	{{Sketching, Moment Estimation, and the L\'{e}vy-Khintchine Representation Theorem}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{77:1--77:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.77},
  URN =		{urn:nbn:de:0030-drops-227057},
  doi =		{10.4230/LIPIcs.ITCS.2025.77},
  annote =	{Keywords: Streaming Sketches, L\'{e}vy Processes}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2021.104

Non-Mergeable Sketching for Cardinality Estimation

Authors: Seth Pettie, Dingyu Wang, and Longhui Yin

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract

Cardinality estimation is perhaps the simplest non-trivial statistical problem that can be solved via sketching. Industrially-deployed sketches like HyperLogLog, MinHash, and PCSA are mergeable, which means that large data sets can be sketched in a distributed environment, and then merged into a single sketch of the whole data set. In the last decade a variety of sketches have been developed that are non-mergeable, but attractive for other reasons. They are simpler, their cardinality estimates are strictly unbiased, and they have substantially lower variance. We evaluate sketching schemes on a reasonably level playing field, in terms of their memory-variance product (MVP). E.g., a sketch that occupies 5m bits and whose relative variance is 2/m (standard error √{2/m}) has an MVP of 10. Our contributions are as follows. - Cohen [Edith Cohen, 2015] and Ting [Daniel Ting, 2014] independently discovered what we call the {Martingale transform} for converting a mergeable sketch into a non-mergeable sketch. We present a simpler way to analyze the limiting MVP of Martingale-type sketches. - Pettie and Wang proved that the Fishmonger sketch [Seth Pettie and Dingyu Wang, 2021] has the best MVP, H₀/I₀ ≈ 1.98, among a class of mergeable sketches called "linearizable" sketches. (H₀ and I₀ are precisely defined constants.) We prove that the Martingale transform is optimal in the non-mergeable world, and that Martingale Fishmonger in particular is optimal among linearizable sketches, with an MVP of H₀/2 ≈ 1.63. E.g., this is circumstantial evidence that to achieve 1% standard error, we cannot do better than a 2 kilobyte sketch. - Martingale Fishmonger is neither simple nor practical. We develop a new mergeable sketch called Curtain that strikes a nice balance between simplicity and efficiency, and prove that Martingale Curtain has limiting MVP≈ 2.31. It can be updated with O(1) memory accesses and it has lower empirical variance than Martingale LogLog, a practical non-mergeable version of HyperLogLog.

Cite as

Seth Pettie, Dingyu Wang, and Longhui Yin. Non-Mergeable Sketching for Cardinality Estimation. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 104:1-104:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{pettie_et_al:LIPIcs.ICALP.2021.104,
  author =	{Pettie, Seth and Wang, Dingyu and Yin, Longhui},
  title =	{{Non-Mergeable Sketching for Cardinality Estimation}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{104:1--104:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.104},
  URN =		{urn:nbn:de:0030-drops-141731},
  doi =		{10.4230/LIPIcs.ICALP.2021.104},
  annote =	{Keywords: Cardinality Estimation, Sketching}
}

Search Results

Documents authored by Wang, Dingyu

Sketching, Moment Estimation, and the Lévy-Khintchine Representation Theorem

Abstract

Cite as

Non-Mergeable Sketching for Cardinality Estimation

Abstract

Cite as

Thanks for your feedback!

Could not send message