Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries

Marco Bressan, Nicoló Cesa-Bianchi, Silvio Lattanzi, Andrea Paudice
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:775-803, 2021.

Abstract

We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural characterization of clusters, called $(\beta,\gamma)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not needed). Using $(\beta,\gamma)$-convexity, we can translate natural density properties of clusters (which include, for instance, clusters that are strongly non-convex in $R^d$) into a graph-theoretic notion of convexity. By exploiting this convexity notion, we design a deterministic algorithm that recovers $(\beta,\gamma)$-convex clusters using $O(k^2 \log n + k^2 (\frac{6}{\beta\gamma})^{dens(X)})$ same-cluster queries, where $k$ is the number of clusters and $dens(X)$ is the density dimension of the semimetric. We show that an exponential dependence on the density dimension is necessary, and we also show that, if we are allowed to make $O(k^2 + k \log n)$ additional queries to a "cluster separation" oracle, then we can recover clusters that have different and arbitrary scales, even when the scale of each cluster is unknown.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-bressan21a, title = {Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries}, author = {Bressan, Marco and Cesa-Bianchi, Nicol\'o and Lattanzi, Silvio and Paudice, Andrea}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {775--803}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/bressan21a/bressan21a.pdf}, url = {https://proceedings.mlr.press/v134/bressan21a.html}, abstract = {We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural characterization of clusters, called $(\beta,\gamma)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not needed). Using $(\beta,\gamma)$-convexity, we can translate natural density properties of clusters (which include, for instance, clusters that are strongly non-convex in $R^d$) into a graph-theoretic notion of convexity. By exploiting this convexity notion, we design a deterministic algorithm that recovers $(\beta,\gamma)$-convex clusters using $O(k^2 \log n + k^2 (\frac{6}{\beta\gamma})^{dens(X)})$ same-cluster queries, where $k$ is the number of clusters and $dens(X)$ is the density dimension of the semimetric. We show that an exponential dependence on the density dimension is necessary, and we also show that, if we are allowed to make $O(k^2 + k \log n)$ additional queries to a "cluster separation" oracle, then we can recover clusters that have different and arbitrary scales, even when the scale of each cluster is unknown.} }
Endnote
%0 Conference Paper %T Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries %A Marco Bressan %A Nicoló Cesa-Bianchi %A Silvio Lattanzi %A Andrea Paudice %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-bressan21a %I PMLR %P 775--803 %U https://proceedings.mlr.press/v134/bressan21a.html %V 134 %X We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural characterization of clusters, called $(\beta,\gamma)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not needed). Using $(\beta,\gamma)$-convexity, we can translate natural density properties of clusters (which include, for instance, clusters that are strongly non-convex in $R^d$) into a graph-theoretic notion of convexity. By exploiting this convexity notion, we design a deterministic algorithm that recovers $(\beta,\gamma)$-convex clusters using $O(k^2 \log n + k^2 (\frac{6}{\beta\gamma})^{dens(X)})$ same-cluster queries, where $k$ is the number of clusters and $dens(X)$ is the density dimension of the semimetric. We show that an exponential dependence on the density dimension is necessary, and we also show that, if we are allowed to make $O(k^2 + k \log n)$ additional queries to a "cluster separation" oracle, then we can recover clusters that have different and arbitrary scales, even when the scale of each cluster is unknown.
APA
Bressan, M., Cesa-Bianchi, N., Lattanzi, S. & Paudice, A.. (2021). Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:775-803 Available from https://proceedings.mlr.press/v134/bressan21a.html.

Related Material