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Support Size Estimation: The Power of Conditioning

Authors Diptarka Chakraborty, Gunjan Kumar, Kuldeep S. Meel

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Author Details

Diptarka Chakraborty
  • National University of Singapore, Singapore
Gunjan Kumar
  • National University of Singapore, Singapore
Kuldeep S. Meel
  • National University of Singapore, Singapore

Funding

  • Chakraborty, Diptarka: Supported in part by an MoE AcRF Tier 2 grant (MOE-T2EP20221-0009).
  • Kumar, Gunjan: Supported in part by National Research Foundation Singapore under its NRF Fellowship Programme[NRF-NRFFAI1-2019-0004].
  • Meel, Kuldeep S.: Supported in part by National Research Foundation Singapore under its NRF Fellowship Programme[NRF-NRFFAI1-2019-0004] , Ministry of Education Singapore Tier 2 grant MOE-T2EP20121-0011, and Ministry of Education Singapore Tier 1 Grant [R-252-000-B59-114].

Acknowledgements

The authors would like to thank anonymous reviewers for their useful suggestions and comments on an earlier version of this paper.

Cite AsGet BibTex

Diptarka Chakraborty, Gunjan Kumar, and Kuldeep S. Meel. Support Size Estimation: The Power of Conditioning. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 33:1-33:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.MFCS.2023.33

Abstract

We consider the problem of estimating the support size of a distribution D. Our investigations are pursued through the lens of distribution testing and seek to understand the power of conditional sampling (denoted as COND), wherein one is allowed to query the given distribution conditioned on an arbitrary subset S. The primary contribution of this work is to introduce a new approach to lower bounds for the COND model that relies on using powerful tools from information theory and communication complexity. Our approach allows us to obtain surprisingly strong lower bounds for the COND model and its extensions. - We bridge the longstanding gap between the upper bound O(log log n + 1/ε²) and the lower bound Ω(√{log log n}) for the COND model by providing a nearly matching lower bound. Surprisingly, we show that even if we get to know the actual probabilities along with COND samples, still Ω(log log n + 1/{ε² log (1/ε)}) queries are necessary. - We obtain the first non-trivial lower bound for the COND equipped with an additional oracle that reveals the actual as well as the conditional probabilities of the samples (to the best of our knowledge, this subsumes all of the models previously studied): in particular, we demonstrate that Ω(log log log n + 1/{ε² log (1/ε)}) queries are necessary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Support-size estimation
  • Distribution testing
  • Conditional sampling
  • Lower bound

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