Randomized Dimensionality Reduction for k-means Clustering (1110.2897v3)

Abstract: We study the topic of dimensionality reduction for $k$-means clustering. Dimensionality reduction encompasses the union of two approaches: \emph{feature selection} and \emph{feature extraction}. A feature selection based algorithm for $k$-means clustering selects a small subset of the input features and then applies $k$-means clustering on the selected features. A feature extraction based algorithm for $k$-means clustering constructs a small set of new artificial features and then applies $k$-means clustering on the constructed features. Despite the significance of $k$-means clustering as well as the wealth of heuristic methods addressing it, provably accurate feature selection methods for $k$-means clustering are not known. On the other hand, two provably accurate feature extraction methods for $k$-means clustering are known in the literature; one is based on random projections and the other is based on the singular value decomposition (SVD). This paper makes further progress towards a better understanding of dimensionality reduction for $k$-means clustering. Namely, we present the first provably accurate feature selection method for $k$-means clustering and, in addition, we present two feature extraction methods. The first feature extraction method is based on random projections and it improves upon the existing results in terms of time complexity and number of features needed to be extracted. The second feature extraction method is based on fast approximate SVD factorizations and it also improves upon the existing results in terms of time complexity. The proposed algorithms are randomized and provide constant-factor approximation guarantees with respect to the optimal $k$-means objective value.

• The paper introduces a provably accurate randomized feature selection method achieving a 3+ε approximation for efficient k-means clustering.
• It presents enhanced rapid random projection and approximate SVD-based feature extraction techniques that yield a 2+ε approximation ratio.
• Empirical evaluations demonstrate that drastically reducing the feature space maintains near-optimal clustering accuracy on high-dimensional datasets.

An Analytical Overview of Randomized Dimensionality Reduction for $k$-means Clustering

In "Randomized Dimensionality Reduction for $k$-means Clustering," the authors address a fundamental problem in clustering algorithms, particularly the computational inefficiencies associated with high-dimensional data sets. Dimensionality reduction, a key preprocessing step in clustering, is typically accomplished through either feature selection or feature extraction. While feature selection methods aim to choose a subset of the original features, feature extraction creates new artificial features on which to perform clustering.

Introduction to the Problem and Current Landscape

The paper explores the $k$-means clustering algorithm, the most notable method for clustering due to its simplicity and effectiveness. However, in high dimensional datasets, $k$-means is computationally expensive and potentially ineffective due to irrelevant features obfuscating the actual structure in data.

Prior to this paper, no robust feature selection methods for $k$-means existed with provable accuracy, although two feature extraction methods were known: one based on random projections and another using the singular value decomposition (SVD). These methods offer $1+\varepsilon$ and $2$ approximation ratios, respectively, against the optimal clustering results.

Key Contributions and Novel Approaches

The authors introduce the first provably accurate feature selection method for $k$-means clustering alongside two advanced feature extraction methods:

1. Feature Selection via Randomized Sampling: A novel method uses randomized sampling with approximate SVD to achieve a feature selection process with a $3+\varepsilon$ approximation ratio. This contributes to narrowing down the feature space to $r = O(k \log(k)/\varepsilon^2)$ features and executes in $O(mnk/\varepsilon)$ time, comprehensively improving both the speed and reliability of achieving near-optimal $k$-means clustering.
2. Enhanced Random Projection Method: Further development in this setup achieves a $(2+\varepsilon)$ approximation while considerably reducing the number of projected features to $r=O(k /\varepsilon^2)$. By employing fast matrix multiplication techniques, this method can handle data more efficiently.
3. Approximate SVD-based Feature Extraction: Builds on existing SVD techniques by utilizing an approximate, faster version of SVD for constructing $k$ artificial features. This method also achieves a $2+\varepsilon$ approximation ratio.

Theoretical and Practical Implications

The algorithms offer advantageous trade-offs between computational demands and the accuracy of clustering. They are particularly credible for applications where datasets are emerging at larger scales across diverse fields—from bioinformatics to social sciences to e-commerce platforms—where efficiency matters without compromising clustering accuracy. Notably, they provide constant-factor approximations relative to the optimal clustering, thus reducing error margins prevalent in scaling data analyses.

Experimental Observations

The paper includes empirical evaluations demonstrating that, even with $r$ features significantly lesser than theoretically bounded, the clustering accuracy remains promisingly close to optimal $k$-means results. These empirical findings suggest potential for substantial improvements beyond current theoretical bounds, offering scope for practical enhancements.

Concluding Thoughts and Future Directions

This comprehensive exploration into dimensionality reduction contributes significant algorithmic improvements for clustering high-dimensional datasets. Future research could focus on refining approximation guarantees and scaling the effectiveness of these algorithms across different data distributions and further optimizing computational efficiencies.

Overall, this paper propels the field toward more efficient, theoretically sound applications of clustering, particularly in environments characterized by high dimensionality and data redundancy.