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Provable Imbalanced Point Clustering

David Denisov, Dan Feldman, Shlomi Dolev, Michael Segal

TL;DR

This work addresses imbalanced clustering for the $k$-center problem in $\mathbb{R}^d$ without requiring labels. It introduces a relaxed mean-cluster loss $\tilde{\ell}$ and develops a coreset-based compression framework that yields provable guarantees, including a $\alpha=2\log^2(1+n)$-approximation and an $\varepsilon$-coreset of size $|C|=O\left(\frac{kd^3\log(k)\log^4(n)}{\varepsilon^2}\left(\log(\log(k)\log(n)) + \log\left(\frac{1}{\delta}\right)\right)\right)$ with runtime $O(ndk\log(1/\delta))$. The paper further introduces Bi-Criteria approximations and a Coreset construction method, and proposes Choice clustering to select the best clustering among multiple candidates. Empirically, the approach is validated on image quantization tasks and synthetic/real datasets, showing competitive performance and meaningful compression advantages, while operating without labeled data. Limitations include the current restriction to Euclidean distance and weighted-coreset outputs, with future work aimed at generalizing to other metrics and robust estimators.

Abstract

We suggest efficient and provable methods to compute an approximation for imbalanced point clustering, that is, fitting $k$-centers to a set of points in $\mathbb{R}^d$, for any $d,k\geq 1$. To this end, we utilize \emph{coresets}, which, in the context of the paper, are essentially weighted sets of points in $\mathbb{R}^d$ that approximate the fitting loss for every model in a given set, up to a multiplicative factor of $1\pm\varepsilon$. We provide [Section 3 and Section E in the appendix] experiments that show the empirical contribution of our suggested methods for real images (novel and reference), synthetic data, and real-world data. We also propose choice clustering, which by combining clustering algorithms yields better performance than each one separately.

Provable Imbalanced Point Clustering

TL;DR

This work addresses imbalanced clustering for the -center problem in without requiring labels. It introduces a relaxed mean-cluster loss and develops a coreset-based compression framework that yields provable guarantees, including a -approximation and an -coreset of size with runtime . The paper further introduces Bi-Criteria approximations and a Coreset construction method, and proposes Choice clustering to select the best clustering among multiple candidates. Empirically, the approach is validated on image quantization tasks and synthetic/real datasets, showing competitive performance and meaningful compression advantages, while operating without labeled data. Limitations include the current restriction to Euclidean distance and weighted-coreset outputs, with future work aimed at generalizing to other metrics and robust estimators.

Abstract

We suggest efficient and provable methods to compute an approximation for imbalanced point clustering, that is, fitting -centers to a set of points in , for any . To this end, we utilize \emph{coresets}, which, in the context of the paper, are essentially weighted sets of points in that approximate the fitting loss for every model in a given set, up to a multiplicative factor of . We provide [Section 3 and Section E in the appendix] experiments that show the empirical contribution of our suggested methods for real images (novel and reference), synthetic data, and real-world data. We also propose choice clustering, which by combining clustering algorithms yields better performance than each one separately.
Paper Structure (33 sections, 12 theorems, 37 equations, 10 figures, 2 algorithms)

This paper contains 33 sections, 12 theorems, 37 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Paper structure
  3. Objective functions
  4. Motivation
  5. Theoretical results
  6. Main results
  7. Experimental results
  8. Real world motivation: image quantization
  9. Comparison to common clustering algorithms
  10. "Choice" clustering
  11. Examples
  12. Conclusion
  13. Future work and limitations
  14. Algorithms
  15. $(\alpha,\beta)$-approximation
  16. ...and 18 more sections

Key Result

lemma 1

Let $(P,w)$ be a weighted set of size $n\geq k$ where $w:P\to [0,\infty)$. That is $P\subset\mathbb{R}^d$ and $|P|=n$. Suppose that $\min_{C\subset\mathbb{R}^d,|C|=k} \tilde{\ell}((P,w),C)$ exists. Let $Q:=\textsc{Approx}((P,w),k)$; see Definition alg: approx. Then $Q$ is a $2 \log^2(1+n)$-approxima

Figures (10)

  • Figure 1: Motivation for minimizing the loss function of Definition \ref{['def: loss function']}. (left) The left figure is the data generated for $n=1250$ "inliers" points and 25 "outliers", the color of each point corresponds to the set from which it was generated (outlier or inlier). (middle) the black dots are the two optimal centers according to our approximation to the minimizer of the loss function in Definition Definition \ref{['def: loss function']}. The red points are closest to the first center, while the blue points are closest to the second center. (middle) the black dots are the two optimal centers according to our approximation to the minimizer of the loss function in Definition Definition \ref{['def: loss function']}. The red points are closest to the first center, while the blue points are closest to the second center. ((right) the black dots are the two centers resulting by applying $k$-means++ kmeans++ with $k=2$ for all the points. Again, the red points are closest to the first center, while the blue points are closest to the second center.
  • Figure 2: Motivation for the problem suggested in Definition \ref{['def: loss function 2']}. The top row corresponds to $n=1250$ "inliers" points along $25$ "outliers" points, and the bottom row to $n=625$ ''inliers" points along $25$ "outliers" points. The left columns are the data generated for the value of $n$, the color of each point corresponds to the set from which it was generated. In the right and middle columns, the black dots are the $2$ centers computed, and the points of each cluster are colored depending on which center they are closest to (red or blue). The right figure demonstrates the output of $k$-means++ kmeans++. The middle figure demonstrates our approximations of the problem suggested in Definition \ref{['def: loss function 2']}.
  • Figure 3: Results for Section \ref{['mot: image']}. The images are (left to right): the input image to cluster, the result of our quantization, and the result of the Scikit-Learn quantization. Note that the right image contains a small near-gray rectangular in place of the blue rectangle of the original image.
  • Figure 4: Results for the comparison at Section \ref{['sec: Sk-learn comp']}. The rightmost column is our method and the bottom row is our motivation data from Section \ref{['sec: motivation']} for $x:=625$. The leftmost column is the ground truth clustering. The other rows were copied from the comparison at Scikit-learn scikit-learn, which this comparison is based on.
  • Figure 5: Results for our original cat (Gray) image of Section \ref{['mot: image']}. The rows correspond to (top): The full images. (bottom): a "zoom in" on a section of the cat's fur for easier comparison. The images (left to right) are: the image we attempted to cluster (ground truth), the result of our quantization, the result of the Scikit-Learn quantization, and the result of the Choice quantization. Observe the cat's fur at the ground truth image that has a blue hue in all the quantizations besides the Choice quantization.
  • ...and 5 more figures

Theorems & Definitions (32)

  • definition 1: Loss function
  • definition 2: Relaxed loss function
  • definition 3
  • definition 4: $\alpha$-approximation
  • definition 5: $\varepsilon$-coreset
  • lemma 1
  • theorem 1
  • definition 6: $(\alpha,\beta)$-approximation
  • theorem 2
  • theorem 3
  • ...and 22 more