$k$-MLE: A fast algorithm for learning statistical mixture models (1203.5181v1)

Abstract: We describe $k$-MLE, a fast and efficient local search algorithm for learning finite statistical mixtures of exponential families such as Gaussian mixture models. Mixture models are traditionally learned using the expectation-maximization (EM) soft clustering technique that monotonically increases the incomplete (expected complete) likelihood. Given prescribed mixture weights, the hard clustering $k$-MLE algorithm iteratively assigns data to the most likely weighted component and update the component models using Maximum Likelihood Estimators (MLEs). Using the duality between exponential families and Bregman divergences, we prove that the local convergence of the complete likelihood of $k$-MLE follows directly from the convergence of a dual additively weighted Bregman hard clustering. The inner loop of $k$-MLE can be implemented using any $k$-means heuristic like the celebrated Lloyd's batched or Hartigan's greedy swap updates. We then show how to update the mixture weights by minimizing a cross-entropy criterion that implies to update weights by taking the relative proportion of cluster points, and reiterate the mixture parameter update and mixture weight update processes until convergence. Hard EM is interpreted as a special case of $k$-MLE when both the component update and the weight update are performed successively in the inner loop. To initialize $k$-MLE, we propose $k$-MLE++, a careful initialization of $k$-MLE guaranteeing probabilistically a global bound on the best possible complete likelihood.

• The paper presents k-MLE as a fast hard-clustering variant of EM that leverages local search and Bregman divergences to optimize the complete likelihood.
• The algorithm incorporates a novel k-MLE++ initializer inspired by k-means++ to enhance convergence quality through probabilistic approximations.
• The method significantly reduces runtime while maintaining clustering quality in high-dimensional data, offering practical benefits for image and signal processing.

Insightful Overview of "$k$-MLE: A fast algorithm for learning statistical mixture models"

The paper introduces $k$-MLE, a computationally efficient algorithm for learning finite statistical mixtures derived from exponential families, with a focal application on Gaussian mixture models (GMMs). Primarily, this work seeks a fast reconciliation between the inherently soft clustering nature of Expectation-Maximization (EM) and the hard clustering philosophy intrinsic to $k$-means. $k$-MLE positions itself as a hard clustering variant of the EM, endowed with faster local convergence while maintaining analytical equivalency to certain soft clustering schemes via Bregman divergences.

Core Algorithmic Contributions

1. Algorithmic Design: $k$-MLE operates through a local search mechanism, iteratively aligning data points to the most probable mixture component given their weights, akin to $k$-means. This registration leverages Bregman divergence properties to induce convergence in terms of complete likelihood.
2. Weight Updating: The mixture weights are refined using a cross-entropy minimization objective, relying on the relative size of clusters. This introduces a hard-assignment aspect where components' updates are followed by a recalibration of weights in a loop until a stable solution is reached.
3. Initialization Strategy: On the initialization front, the paper proposes $k$-MLE++, an initializer inspired by $k$-means++ which guarantees a probabilistic approximation on the convergence quality concerning the complete likelihood.
4. Convergence Analysis: An added theoretical underpinning is provided by demonstrating that the convergence of $k$-MLE is inherently linked to the duality between exponential families and Bregman divergences, ensuring that the local search mechanism strictly reduces the loss function, albeit potentially converging to local optima due to the heuristic nature of the search.

Numerical and Methodological Implications

The algorithm exhibits significant reductions in runtime while maintaining comparable clustering quality to EM when benchmarked over datasets modeled by mixtures of various exponential families. This positions $k$-MLE as a robust alternative for applications demanding quick convergence, particularly in high-dimensional scenarios like those experienced in image and video processing.

Theoretical and Practical Implications

Theoretically, the amalgamation of Bregman divergences with mixture modeling frameworks enriches the understanding of mixture learning under bounded parameter spaces. Practically, it paves the path for augmenting the efficacy of machine learning methods required to decompose complex datasets into interpretable components. By fortifying clustering tasks under statistical mixture models, $k$-MLE effectively adapts to contexts ranging from signal processing to advanced image analysis.

Future Prospects and Extensions

Potential future extensions revolve around easing the initialization complexity and exploring the utility of adaptive heuristic methods to surmount the local optima entrapment issues. Refining weight dynamics and incorporating advanced probabilistic methods to handle missing data as seen in EM could further enhance its applicability. Further scrutiny in the lens of information theory could bolster the development of mixture models with more precise characteristics and improved generative capabilities.

In summary, the contribution of this paper is rooted in advancing mixture model learning towards faster yet theoretically grounded methodologies, showcasing the importance of bridging classical clustering with modern probabilistic paradigms.