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Documents authored by Flores-Velazco, Alejandro

Document
DOI: 10.4230/LIPIcs.ESA.2022.54
Improved Search of Relevant Points for Nearest-Neighbor Classification

Authors: Alejandro Flores-Velazco

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Given a training set P ⊂ ℝ^d, the nearest-neighbor classifier assigns any query point q ∈ ℝ^d to the class of its closest point in P. To answer these classification queries, some training points are more relevant than others. We say a training point is relevant if its omission from the training set could induce the misclassification of some query point in ℝ^d. These relevant points are commonly known as border points, as they define the boundaries of the Voronoi diagram of P that separate points of different classes. Being able to compute this set of points efficiently is crucial to reduce the size of the training set without affecting the accuracy of the nearest-neighbor classifier. Improving over a decades-long result by Clarkson (FOCS'94), Eppstein (SOSA’22) recently proposed an output-sensitive algorithm to find the set of border points of P in 𝒪(n² + nk²) time, where k is the size of such set. In this paper, we improve this algorithm to have time complexity equal to 𝒪(nk²) by proving that the first phase of their algorithm, which requires 𝒪(n²) time, are unnecessary.
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Alejandro Flores-Velazco. Improved Search of Relevant Points for Nearest-Neighbor Classification. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 54:1-54:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{floresvelazco:LIPIcs.ESA.2022.54,
  author =	{Flores-Velazco, Alejandro},
  title =	{{Improved Search of Relevant Points for Nearest-Neighbor Classification}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{54:1--54:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.54},
  URN =		{urn:nbn:de:0030-drops-169922},
  doi =		{10.4230/LIPIcs.ESA.2022.54},
  annote =	{Keywords: nearest-neighbor classification, nearest-neighbor rule, decision boundaries, border points, relevant points}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.44
Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification

Authors: Alejandro Flores-Velazco and David M. Mount

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
The problem of nearest-neighbor classification is a fundamental technique in machine-learning. Given a training set P of n labeled points in ℝ^d, and an approximation parameter 0 < ε ≤ 1/2, any unlabeled query point should be classified with the class of any of its ε-approximate nearest-neighbors in P. Answering these queries efficiently has been the focus of extensive research, proposing techniques that are mainly tailored towards resolving the more general problem of ε-approximate nearest-neighbor search. While the latest can only hope to provide query time and space complexities dependent on n, the problem of nearest-neighbor classification accepts other parameters more suitable to its analysis. Such is the number k_ε of ε-border points, which describes the complexity of boundaries between sets of points of different classes. This paper presents a new data structure called Chromatic AVD. This is the first approach for ε-approximate nearest-neighbor classification whose space and query time complexities are only dependent on ε, k_ε and d, while being independent on both n and Δ, the spread of P.
Cite as

Alejandro Flores-Velazco and David M. Mount. Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{floresvelazco_et_al:LIPIcs.ESA.2021.44,
  author =	{Flores-Velazco, Alejandro and Mount, David M.},
  title =	{{Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{44:1--44:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.44},
  URN =		{urn:nbn:de:0030-drops-146252},
  doi =		{10.4230/LIPIcs.ESA.2021.44},
  annote =	{Keywords: approximate nearest-neighbor searching, nearest-neighbor classification, geometric data structures, space-time tradeoffs}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.47
Coresets for the Nearest-Neighbor Rule

Authors: Alejandro Flores-Velazco and David M. Mount

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Given a training set P of labeled points, the nearest-neighbor rule predicts the class of an unlabeled query point as the label of its closest point in the set. To improve the time and space complexity of classification, a natural question is how to reduce the training set without significantly affecting the accuracy of the nearest-neighbor rule. Nearest-neighbor condensation deals with finding a subset R ⊆ P such that for every point p ∈ P, p’s nearest-neighbor in R has the same label as p. This relates to the concept of coresets, which can be broadly defined as subsets of the set, such that an exact result on the coreset corresponds to an approximate result on the original set. However, the guarantees of a coreset hold for any query point, and not only for the points of the training set. This paper introduces the concept of coresets for nearest-neighbor classification. We extend existing criteria used for condensation, and prove sufficient conditions to correctly classify any query point when using these subsets. Additionally, we prove that finding such subsets of minimum cardinality is NP-hard, and propose quadratic-time approximation algorithms with provable upper-bounds on the size of their selected subsets. Moreover, we show how to improve one of these algorithms to have subquadratic runtime, being the first of this kind for condensation.
Cite as

Alejandro Flores-Velazco and David M. Mount. Coresets for the Nearest-Neighbor Rule. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 47:1-47:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{floresvelazco_et_al:LIPIcs.ESA.2020.47,
  author =	{Flores-Velazco, Alejandro and Mount, David M.},
  title =	{{Coresets for the Nearest-Neighbor Rule}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{47:1--47:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.47},
  URN =		{urn:nbn:de:0030-drops-129138},
  doi =		{10.4230/LIPIcs.ESA.2020.47},
  annote =	{Keywords: coresets, nearest-neighbor rule, classification, nearest-neighbor condensation, training-set reduction, approximate nearest-neighbor, approximation algorithms}
}
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