Random Projections for $k$-means Clustering (1011.4632v1)

Abstract: This paper discusses the topic of dimensionality reduction for $k$-means clustering. We prove that any set of $n$ points in $d$ dimensions (rows in a matrix $A \in \RR^{n \times d}$) can be projected into $t = \Omega(k / \eps^2)$ dimensions, for any $\eps \in (0,1/3)$, in $O(n d \lceil \eps^{-2} k/ \log(d) \rceil )$ time, such that with constant probability the optimal $k$-partition of the point set is preserved within a factor of $2+\eps$. The projection is done by post-multiplying $A$ with a $d \times t$ random matrix $R$ having entries $+1/\sqrt{t}$ or $-1/\sqrt{t}$ with equal probability. A numerical implementation of our technique and experiments on a large face images dataset verify the speed and the accuracy of our theoretical results.

Random Projections for $k$-Means Clustering

This paper introduces an innovative approach to dimensionality reduction for the $k$-means clustering problem by utilizing random projections (RPs). The authors offer substantial theoretical contributions as well as empirical evaluations, focused on improving both the efficiency and accuracy of clustering large high-dimensional datasets.

Theoretical Framework and Contributions

The authors start by laying the groundwork for their method, acknowledging the relevance and ubiquity of $k$-means clustering in data mining applications. They harness the concept of RPs, a dimensionality reduction technique that has gained prominence through the Johnson-Lindenstrauss lemma. They show that any set of $n$ points in $d$ dimensions can be projected into $t = \Omega(k /\epsilon^2)$ dimensions, where $\epsilon$ is an error parameter. The projection preserves the optimal $k$-partition within a factor of $2+\epsilon$ with constant probability.

The dimensionality reduction is achieved by post-multiplying the data matrix $A$ with a random matrix $R$ with entries $+1/\sqrt{t}$ or $-1/\sqrt{t}$. This choice of matrix $R$ ensures fast computation and preserves the structure of the data in terms of clustering.

Numerical Results and Comparisons

An important aspect of this paper is the numerical verification of the theoretical results using a large face images dataset. The authors demonstrate compelling empirical results, showcasing the speed and clustering accuracy of their method compared to existing dimensionality reduction techniques such as Singular Value Decomposition (SVD), Local Linear Embedding (LLE), and Laplacian score feature selection.

The authors provide detailed experimentation, revealing that their RP-based algorithm can achieve significant computational savings while maintaining comparable or superior accuracy in clustering tasks. Notably, when applied to high-dimensional datasets, their method shows a marked reduction in running time compared to traditional approaches, without sacrificing the quality of the clustering output.

Implications for Future Developments

The integration of random projections into the $k$-means clustering process poses potential implications for future developments in both theoretical and practical realms of AI. The theoretical contributions provide a foundation for exploring more efficient cluster analysis algorithms, particularly in the context of big data scenarios where dimensionality effects are pronounced. This approach could trigger more advancements in RP-based methods, potentially impacting areas such as unsupervised learning, anomaly detection, and data compression.

Conclusion

Overall, this paper makes a significant contribution to the literature on clustering by presenting an efficient, theoretically sound, and practically viable technique for dimensionality reduction. By leveraging the strengths of random projections, the authors have offered a method that enhances the scalability of $k$-means clustering, making it more applicable to modern data-intensive applications. Future work could explore extensions of this approach to alternative clustering algorithms and assess its efficacy across different data types and contexts.