Turning Big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering (1807.04518v1)

Abstract: We develop and analyze a method to reduce the size of a very large set of data points in a high dimensional Euclidean space R d to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set. For example, computing the first k principal components of the reduced set will return approximately the first k principal components of the original set or computing the centers of a k-means clustering on the reduced set will return an approximation for the original set. Such a reduced set is also known as a coreset. The main new feature of our construction is that the cardinality of the reduced set is independent of the dimension d of the input space and that the sets are mergable. The latter property means that the union of two reduced sets is a reduced set for the union of the two original sets (this property has recently also been called composability, see Indyk et. al., PODS 2014). It allows us to turn our methods into streaming or distributed algorithms using standard approaches. For problems such as k-means and subspace approximation the coreset sizes are also independent of the number of input points. Our method is based on projecting the points on a low dimensional subspace and reducing the cardinality of the points inside this subspace using known methods. The proposed approach works for a wide range of data analysis techniques including k-means clustering, principal component analysis and subspace clustering. The main conceptual contribution is a new coreset definition that allows to charge costs that appear for every solution to an additive constant.

• The paper presents coreset constructions independent of data dimension, significantly reducing complexity in clustering tasks.
• It employs projections into low-dimensional subspaces combined with reduction techniques to preserve essential structure for k-means and PCA.
• The scalable, mergeable coresets support streaming and distributed computing, enhancing practical applications in AI and data analysis.

Coresets for Clustering and Dimensionality Reduction

This paper explores the construction of coresets and dimensionality reductions for various clustering problems, particularly focusing on methods such as $k$-means, PCA, and projective clustering. The authors present a technique to handle high-dimensional data by reducing it to a manageable size, maintaining the essential structure for these clustering tasks. Such reduced datasets are known as coresets.

Overview and Contributions

The paper introduces a novel approach wherein the cardinality of coresets is independent of the dimension of the data space, $d$. This key feature, along with the mergable property of the corest sets, allows their application in streaming and distributed computing paradigms.

The authors build their method on projections into low-dimensional subspaces, followed by application of known reduction techniques. For $k$-means, PCA, and subspace clustering, the proposed coreset sizes do not depend on the number of input points, $n$. The paper claims to provide coresets of $O(j/)$ for linear and affine $j$-subspace queries, $\tilde{O}(k^3/^4)$ for $k$-means, and other sizes for more complex clustering scenarios.

Theoretical and Practical Implications

The implications of this research span both theoretical and practical realms:

• Efficiency in High Dimensions: Addressing the computational challenges posed by high-dimensional data, the coresets offer a significant reduction in complexity, providing near-optimal solutions rapidly.
• Scalability: The independence from data dimension $d$ and straightforward composability make these coresets robust for large-scale data scenarios, including applications in cloud computing and streaming data.
• Application Diversification: By extending the coreset construction to several clustering methodologies, the paper broadens the applicability of these techniques to diverse domains requiring substantial statistical or geometric insights.

Numerical Results and Claims

The paper substantiates its contributions through theoretical proofs rather than exhaustive empirical validation, ensuring rigor in the bounds claimed. The coreset sizes effectively demonstrate independence from $n$ and $d$ under the given constraints.

Speculation on Future Developments in AI

The results provided for PCA and $k$-means set a promising direction for enhanced data processing in AI systems, especially in areas like data compression and anomaly detection. The work could inspire future studies focusing on coresets for complex AI models, such as deep learning architectures that require effective model reduction strategies.

Conclusion

This research marks a substantial step in addressing the dimensionality and scale challenges commonly encountered in clustering-related data analysis. By focusing on coresets with cardinality independent of both data dimension and size, the work lays a foundation for more efficient processing pipelines in both traditional and emerging AI applications. Future endeavors can potentially enhance these methods, integrating them further into real-time and distributed systems, enriching AI's capability to handle vast and complex datasets.