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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Consider the geometric range space (X, H_d) where X ⊂ ℝ^d and H_d is the set of ranges defined by d-dimensional halfspaces. In this setting we consider that X is the disjoint union of a red and blue set. For each halfspace h ∈ H_d define a function Φ(h) that measures the "difference" between the fraction of red and fraction of blue points which fall in the range h. In this context the maximum discrepancy problem is to find the h^* = arg max_{h ∈ (X, H_d)} Φ(h). We aim to instead find an ĥ such that Φ(h^*) - Φ(ĥ) ≤ ε. This is the central problem in linear classification for machine learning, in spatial scan statistics for spatial anomaly detection, and shows up in many other areas. We provide a solution for this problem in O(|X| + (1/ε^d) log⁴ (1/ε)) time, for constant d, which improves polynomially over the previous best solutions. For d = 2 we show that this is nearly tight through conditional lower bounds. For different classes of Φ we can either provide a Ω(|X|^{3/2 - o(1)}) time lower bound for the exact solution with a reduction to APSP, or an Ω(|X| + 1/ε^{2-o(1)}) lower bound for the approximate solution with a reduction to 3Sum.
A key technical result is a ε-approximate halfspace range counting data structure of size O(1/ε^d) with O(log (1/ε)) query time, which we can build in O(|X| + (1/ε^d) log⁴ (1/ε)) time.

Michael Matheny and Jeff M. Phillips. Approximate Maximum Halfspace Discrepancy. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{matheny_et_al:LIPIcs.ISAAC.2021.4, author = {Matheny, Michael and Phillips, Jeff M.}, title = {{Approximate Maximum Halfspace Discrepancy}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {4:1--4:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.4}, URN = {urn:nbn:de:0030-drops-154377}, doi = {10.4230/LIPIcs.ISAAC.2021.4}, annote = {Keywords: range spaces, halfspaces, scan statistics, fine-grained complexity} }

Document

**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Consider a geometric range space (X,A) where X is comprised of the union of a red set R and blue set B. Let Phi(A) define the absolute difference between the fraction of red and fraction of blue points which fall in the range A. The maximum discrepancy range A^* = arg max_{A in (X,A)} Phi(A). Our goal is to find some A^ in (X,A) such that Phi(A^*) - Phi(A^) <= epsilon. We develop general algorithms for this approximation problem for range spaces with bounded VC-dimension, as well as significant improvements for specific geometric range spaces defined by balls, halfspaces, and axis-aligned rectangles. This problem has direct applications in discrepancy evaluation and classification, and we also show an improved reduction to a class of problems in spatial scan statistics.

Michael Matheny and Jeff M. Phillips. Computing Approximate Statistical Discrepancy. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{matheny_et_al:LIPIcs.ISAAC.2018.32, author = {Matheny, Michael and Phillips, Jeff M.}, title = {{Computing Approximate Statistical Discrepancy}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {32:1--32:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.32}, URN = {urn:nbn:de:0030-drops-99800}, doi = {10.4230/LIPIcs.ISAAC.2018.32}, annote = {Keywords: Scan Statistics, Discrepancy, Rectangles} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We develop, analyze, implement, and compare new algorithms for creating epsilon-samples of range spaces defined by halfspaces which have size sub-quadratic in 1/epsilon, and have runtime linear in the input size and near-quadratic in 1/epsilon. The key to our solution is an efficient construction of partition trees. Despite not requiring any techniques developed after the early 1990s, apparently such a result was never explicitly described. We demonstrate that our implementations, including new implementations of several variants of partition trees, do indeed run in time linear in the input, appear to run linear in output size, and observe smaller error for the same size sample compared to the ubiquitous random sample (which requires size quadratic in 1/epsilon). This result has direct applications in speeding up discrepancy evaluation, approximate range counting, and spatial anomaly detection.

Michael Matheny and Jeff M. Phillips. Practical Low-Dimensional Halfspace Range Space Sampling. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{matheny_et_al:LIPIcs.ESA.2018.62, author = {Matheny, Michael and Phillips, Jeff M.}, title = {{Practical Low-Dimensional Halfspace Range Space Sampling}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {62:1--62:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.62}, URN = {urn:nbn:de:0030-drops-95250}, doi = {10.4230/LIPIcs.ESA.2018.62}, annote = {Keywords: Partitions, Range Spaces, Sampling, Halfspaces} }

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