Clustering with Spectral Norm and the k-means Algorithm (1004.1823v1)

Abstract: There has been much progress on efficient algorithms for clustering data points generated by a mixture of $k$ probability distributions under the assumption that the means of the distributions are well-separated, i.e., the distance between the means of any two distributions is at least $\Omega(k)$ standard deviations. These results generally make heavy use of the generative model and particular properties of the distributions. In this paper, we show that a simple clustering algorithm works without assuming any generative (probabilistic) model. Our only assumption is what we call a "proximity condition": the projection of any data point onto the line joining its cluster center to any other cluster center is $\Omega(k)$ standard deviations closer to its own center than the other center. Here the notion of standard deviations is based on the spectral norm of the matrix whose rows represent the difference between a point and the mean of the cluster to which it belongs. We show that in the generative models studied, our proximity condition is satisfied and so we are able to derive most known results for generative models as corollaries of our main result. We also prove some new results for generative models - e.g., we can cluster all but a small fraction of points only assuming a bound on the variance. Our algorithm relies on the well known $k$-means algorithm, and along the way, we prove a result of independent interest -- that the $k$-means algorithm converges to the "true centers" even in the presence of spurious points provided the initial (estimated) centers are close enough to the corresponding actual centers and all but a small fraction of the points satisfy the proximity condition. Finally, we present a new technique for boosting the ratio of inter-center separation to standard deviation.

• The paper introduces a proximate condition leveraging the spectral norm to guarantee k-means convergence with minimal deviation.
• It demonstrates that, with sufficiently accurate initialization, classic k-means converges to true centers even in the presence of outliers.
• A novel separation amplification technique relaxes strict assumptions, broadening the applicability of clustering in high-dimensional, non-probabilistic datasets.

Overview of "Clustering with Spectral Norm and the $k$-means Algorithm"

The paper "Clustering with Spectral Norm and the $k$-means Algorithm" by Amit Kumar and Ravindran Kannan presents a significant advancement in data clustering methodologies. The authors introduce a novel approach to clustering, circumventing the need for a generative model assumption, which is commonly relied upon in traditional clustering algorithms. Instead, they introduce the proximate condition that sufficiently separates clustering centers, ensuring stability without assuming specific probabilistic distribution characteristics.

The primary innovation lies in this proximate condition, which requires the projection of a data point onto the line between its cluster center and any other cluster center to be a certain number of standard deviations closer to its own center. The standard deviation here is gauged using the spectral norm of the data matrix. This approach allows the authors to generalize and derive known results for generative models and claim new results where only variance bounds exist.

The algorithm leverages the classical $k$-means technique, demonstrating its convergence to true centers even amidst outliers, contingent on reasonably accurate initial center estimations. Additionally, a new method to amplify the separation between cluster centers relative to the standard deviation is introduced, enabling the deduction of results under less stringent separation constraints.

Core Results and Claims

The paper proposes practical and theoretical implications through its algorithm underpinned by strong numerical results. It suggests:

• Convergence Assurance: The $k$-means algorithm converges to accurate centers if the initialization is sufficiently close, and only a small fraction of points deviate from the proximate condition.
• Generative Model Corollaries: The proposed proximity condition is validated under generative models, deriving most classic results as corollaries while extending new guarantees when variance bounds alone exist.
• Algorithmic Efficiency: Clustering is assured for all but a negligible fraction of points in polynomial time, even under deterministic settings, settling a longstanding query about clustering's computational feasibility without relying on randomness or probabilities.
• Separation Amplification: By employing a novel boosting technique, the ratio of inter-center separation enhances, relaxing the need for stringent mixing component weight dependency, which has substantial implications for learning distributions with heavy tails or when clusters have variances.

Implications and Future Directions

Practically, this research broadens the applicability of clustering algorithms across datasets without clear probabilistic assumptions, which is particularly relevant for real-world data exhibiting complex, non-standard distributions. Theoretically, it bridges a gap between provable algorithmic performance and operational clustering in high-dimensional spaces, potentially catalyzing advancements in machine learning, data mining, and bioinformatics.

The methodologies introduced can ignite further exploration into determining optimal initializations for the $k$-means algorithm, enhancing its robustness and efficiency. Moreover, the boosting technique employed to augment weak separation conditions paves the way for scalable algorithms applicable to scenarios with intricate distributional properties.

Future developments in AI research might include integrating these techniques into deep learning frameworks, potentially enhancing clustering performance in tasks like feature learning and unsupervised image classification. Investigating these algorithms' behavior in adversarial settings or data with noise can also extend their utility and reliability in real-world applications.