We study a sketch-and-solve approach to speed up the Peng–Wei semidefinite relaxation of $k$-means clustering. When the data are appropriately separated we identify the $k$-means optimal clustering. Otherwise, our approach provides a high-confidence lower bound on the optimal $k$-means value. This lower bound is data-driven; it does not make any assumption on the data nor how they are generated. We provide code and an extensive set of numerical experiments where we use this approach to certify approximate optimality of clustering solutions obtained by k-means++.
Sketch-and-Lift: Scalable Subsampled Semidefinite Program for K-means Clustering
Semidefinite programming (SDP) is a powerful tool for tackling a wide range of computationally hard problems such as clustering. Despite the high accuracy, semidefinite programs are often too slow in practice with poor scalability on large (or even moderate) datasets. In this paper, we introduce a linear time complexity algorithm for approximating an SDP relaxed K-means clustering. The proposed sketch-and-lift (SL) approach solves an SDP on a subsampled dataset and then propagates the solution to all data points by a nearest-centroid rounding procedure. It is shown that the SL approach enjoys a similar exact recovery threshold as the K-means SDP on the full dataset, which is known to be information-theoretically tight under the Gaussian mixture model. The SL method can be made adaptive with enhanced theoretic properties when the cluster sizes are unbalanced. Our simulation experiments demonstrate that the statistical accuracy of the proposed method outperforms state-of-the-art fast clustering algorithms without sacrificing too much computational efficiency, and is comparable to the original K-means SDP with substantially reduced runtime.
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- Award ID(s):
- 1752614
- PAR ID:
- 10321218
- Date Published:
- Journal Name:
- International Conference on Artificial Intelligence and Statistics
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Clustering continues to be an important tool for data engineering and analysis. While advances in deep learning tend to be at the forefront of machine learning, it is only useful for the supervised classification of data sets. Clustering is an essential tool for problems where labeling data sets is either too labor intensive or where there is no agreed upon ground truth. The well studied k-means problem partitions groups of similar vectors into k clusters by iteratively updating the cluster assignment such that it minimizes the within cluster sum of squares metric. Unfortunately k-means can become prohibitive for very large high dimensional data sets as iterative methods often rely on random access to, or multiple passes over, the data set — a requirement that is not often possible for large and potentially unbounded data sets. In this work we explore an randomized, approximate method for clustering called Tree-Walk Random Projection Clustering (TWRP) that is a fast, memory efficient method for finding cluster embedding in high dimensional spaces. TWRP combines random projection with a tree based partitioner to achieve a clustering method that forgoes storing the exhaustive representation of all vectors in the data space and instead performs a bounded search over the implied cluster bifurcation tree represented as approximate vector and count values. The TWRP algorithm is described and experimentally evaluated for scalability and accuracy in the presence of noise against several other well-known algorithms.more » « less
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