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DOI: 10.4230/LIPIcs.STACS.2021.46

Locality Sensitive Hashing for Efficient Similar Polygon Retrieval

Authors: Haim Kaplan and Jay Tenenbaum

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract

Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval. Arkin et al. [Arkin et al., 1991] represent polygons by their "turning function" - a function which follows the angle between the polygon’s tangent and the x-axis while traversing the perimeter of the polygon. They define the distance between polygons to be variations of the L_p (for p = 1,2) distance between their turning functions. This metric is invariant under translation, rotation and scaling (and the selection of the initial point on the perimeter) and therefore models well the intuitive notion of shape resemblance. We develop and analyze LSH near neighbor data structures for several variations of the L_p distance for functions (for p = 1,2). By applying our schemes to the turning functions of a collection of polygons we obtain efficient near neighbor LSH-based structures for polygons. To tune our structures to turning functions of polygons, we prove some new properties of these turning functions that may be of independent interest. As part of our analysis, we address the following problem which is of independent interest. Find the vertical translation of a function f that is closest in L₁ distance to a function g. We prove tight bounds on the approximation guarantee obtained by the translation which is equal to the difference between the averages of g and f.

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Haim Kaplan and Jay Tenenbaum. Locality Sensitive Hashing for Efficient Similar Polygon Retrieval. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kaplan_et_al:LIPIcs.STACS.2021.46,
  author =	{Kaplan, Haim and Tenenbaum, Jay},
  title =	{{Locality Sensitive Hashing for Efficient Similar Polygon Retrieval}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{46:1--46:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.46},
  URN =		{urn:nbn:de:0030-drops-136910},
  doi =		{10.4230/LIPIcs.STACS.2021.46},
  annote =	{Keywords: Locality sensitive hashing, polygons, turning function, L\underlinep distance, nearest neighbors, similarity search}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2020.28

Locality Sensitive Hashing for Set-Queries, Motivated by Group Recommendations

Authors: Haim Kaplan and Jay Tenenbaum

Published in: LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Abstract

Locality Sensitive Hashing (LSH) is an effective method to index a set of points such that we can efficiently find the nearest neighbors of a query point. We extend this method to our novel Set-query LSH (SLSH), such that it can find the nearest neighbors of a set of points, given as a query. Let s(x,y) be the similarity between two points x and y. We define a similarity between a set Q and a point x by aggregating the similarities s(p,x) for all p∈ Q. For example, we can take s(p,x) to be the angular similarity between p and x (i.e., 1-(∠(x,p)/π)), and aggregate by arithmetic or geometric averaging, or taking the lowest similarity. We develop locality sensitive hash families and data structures for a large set of such arithmetic and geometric averaging similarities, and analyze their collision probabilities. We also establish an analogous framework and hash families for distance functions. Specifically, we give a structure for the euclidean distance aggregated by either averaging or taking the maximum. We leverage SLSH to solve a geometric extension of the approximate near neighbors problem. In this version, we consider a metric for which the unit ball is an ellipsoid and its orientation is specified with the query. An important application that motivates our work is group recommendation systems. Such a system embeds movies and users in the same feature space, and the task of recommending a movie for a group to watch together, translates to a set-query Q using an appropriate similarity.

Cite as

Haim Kaplan and Jay Tenenbaum. Locality Sensitive Hashing for Set-Queries, Motivated by Group Recommendations. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaplan_et_al:LIPIcs.SWAT.2020.28,
  author =	{Kaplan, Haim and Tenenbaum, Jay},
  title =	{{Locality Sensitive Hashing for Set-Queries, Motivated by Group Recommendations}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{28:1--28:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-150-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{162},
  editor =	{Albers, Susanne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.28},
  URN =		{urn:nbn:de:0030-drops-122756},
  doi =		{10.4230/LIPIcs.SWAT.2020.28},
  annote =	{Keywords: Locality sensitive hashing, nearest neighbors, similarity search, group recommendations, distance functions, similarity functions, ellipsoid}
}

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Documents authored by Tenenbaum, Jay

Locality Sensitive Hashing for Efficient Similar Polygon Retrieval

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Locality Sensitive Hashing for Set-Queries, Motivated by Group Recommendations

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