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Efficient Coresets for Kernel Clustering with Improved Computational Guarantees


Core Concepts
We devise coresets for kernel k-Means and the more general kernel (k,z)-Clustering problems, which significantly improve upon previous results in terms of coreset size and construction time. Our coresets have size poly(kϵ^-1) and can be constructed in near-linear time, enabling efficient algorithms for kernel clustering.
Abstract
The paper addresses the computational challenges of kernel k-Means, which has superior clustering capability compared to classical k-Means but introduces significant computational overhead. To tackle this, the authors adapt the notion of coresets to kernel clustering. Key highlights: The authors devise a coreset for kernel (k,z)-Clustering that works for a general kernel function and has size poly(kϵ^-1), vastly improving upon previous results. The coreset can be constructed in near-linear time, ˜O(nk), using a black-box application of recent coreset constructions for Euclidean spaces. The authors show that their coreset implies new efficient algorithms for kernel k-Means, including a (1+ϵ)-approximation in time near-linear in n, and a streaming algorithm using space and update time poly(kϵ^-1 log n). Experimental results validate the efficiency and accuracy of the coresets, and demonstrate significant speedups in applications like kernel k-Means++ and spectral clustering.
Stats
The paper does not contain any explicit numerical data or statistics to support the key claims. The focus is on the theoretical construction and guarantees of the coresets.
Quotes
None.

Key Insights Distilled From

by Shaofeng H.-... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2110.02898.pdf
Coresets for Kernel Clustering

Deeper Inquiries

How can the coreset construction be extended or adapted to handle dynamic datasets, where points are added or removed over time

To adapt the coreset construction for dynamic datasets, where points are added or removed over time, we can employ a technique known as "merge-and-reduce." This method involves updating the existing coreset with the new points added or removing points that are no longer part of the dataset. When a new point is added to the dataset, we can update the coreset by incorporating this point into the existing coreset. This can be done by recalculating the importance scores for the points in the coreset and adjusting the weights accordingly to reflect the new data point. Similarly, when a point is removed from the dataset, we can update the coreset by removing the corresponding point from the coreset and readjusting the weights of the remaining points. By implementing these update mechanisms efficiently, we can ensure that the coreset remains representative of the dataset even as it evolves over time. This adaptability to dynamic datasets enhances the scalability and applicability of the coreset construction in real-world scenarios where data is constantly changing.

Can the coreset ideas be applied to other kernel-based clustering or learning problems beyond k-Means and (k,z)-Clustering

The concept of coresets can indeed be extended to various other kernel-based clustering or learning problems beyond k-Means and (k,z)-Clustering. Some potential applications include: Spectral Clustering: Coresets can be utilized to speed up spectral clustering algorithms by reducing the computational complexity of kernel-based spectral clustering methods. By constructing coresets that accurately represent the dataset in the feature space, spectral clustering can be performed more efficiently without compromising the quality of the clustering results. Support Vector Machines (SVM): Coresets can be applied to SVMs to improve the efficiency of training large-scale kernel SVM models. By constructing coresets that capture the essential information of the dataset, the training process of SVMs can be accelerated without sacrificing the accuracy of the model. Kernel Principal Component Analysis (PCA): Coresets can be used to speed up kernel PCA, a nonlinear dimensionality reduction technique. By constructing coresets that preserve the kernel similarities between data points, the computational complexity of kernel PCA can be reduced while retaining the ability to capture complex data structures. By extending the coreset framework to these and other kernel-based learning problems, we can enhance the scalability and efficiency of various machine learning algorithms that rely on kernel methods.

What are the potential implications of the improved computational guarantees for kernel clustering on real-world applications in areas like computer vision, natural language processing, or bioinformatics

The improved computational guarantees for kernel clustering provided by the coreset construction have significant implications for real-world applications in fields such as computer vision, natural language processing, and bioinformatics. Some potential implications include: Efficient Image Clustering: In computer vision, where clustering is used for image segmentation and object recognition, the use of coresets for kernel clustering can lead to faster and more accurate clustering of image data. This can improve the efficiency of image processing tasks and enhance the performance of computer vision systems. Enhanced Text Clustering: In natural language processing, clustering is employed for text categorization, document clustering, and sentiment analysis. By leveraging coresets for kernel clustering, text data can be clustered more efficiently, enabling quicker analysis of large text datasets and improving the accuracy of text classification tasks. Biological Data Analysis: In bioinformatics, clustering is utilized for gene expression analysis, protein classification, and drug discovery. The application of coresets for kernel clustering in bioinformatics can streamline the analysis of biological data, leading to faster identification of patterns and relationships in complex biological datasets. Overall, the improved computational guarantees offered by coresets for kernel clustering can revolutionize the way clustering algorithms are applied in these domains, enabling faster processing, more accurate results, and enhanced insights from large and high-dimensional datasets.
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