Learning mixtures of spherical Gaussians: moment methods and spectral decompositions (1206.5766v4)

Abstract: This work provides a computationally efficient and statistically consistent moment-based estimator for mixtures of spherical Gaussians. Under the condition that component means are in general position, a simple spectral decomposition technique yields consistent parameter estimates from low-order observable moments, without additional minimum separation assumptions needed by previous computationally efficient estimation procedures. Thus computational and information-theoretic barriers to efficient estimation in mixture models are precluded when the mixture components have means in general position and spherical covariances. Some connections are made to estimation problems related to independent component analysis.

• The paper introduces a moment-based estimator for spherical Gaussian mixtures that avoids stringent separation assumptions by leveraging low-order observable moments.
• It demonstrates statistical consistency with finite-sample bounds and polynomial computational complexity for efficient parameter recovery.
• It leverages spectral decompositions to ensure robust parameter recovery, extending its applicability to high-dimensional statistical models.

Learning Mixtures of Spherical Gaussians: Moment Methods and Spectral Decompositions

This paper explores a method for efficiently estimating parameters of mixtures of spherical Gaussians using moment-based techniques and spectral decompositions. The work addresses a fundamental issue in estimating Gaussian mixture models (GMMs)—specifically, models where the covariance matrices of the components are spherical, a scenario closely related to $k$-means clustering problems.

The authors present a moment-based estimator that does not require additional separation assumptions that other estimators often depend on. Instead, their approach relies on the natural arrangement of the means (component means in general position) and uses the structure of low-order observable moments. They demonstrate that their method is both computationally and statistically efficient, circumventing some of the barriers present in earlier methodologies.

Main Contributions

1. Non-Degeneracy Condition: The procedure efficiently recovers model parameters with a simple spectral decomposition under the condition that component means span a $k$-dimensional subspace, and the probability vector is strictly positive. This condition is less restrictive than assumptions used in other methods requiring well-separated means.
2. Statistical Consistency: The proposed estimator is statistically consistent. The paper shows that accurate parameter estimates can be derived using empirical moments, which converge to true moments at a known rate as per the central limit theorem.
3. Efficiency: The paper illustrates that the computational complexity of their estimation technique is polynomial relative to problem parameters—contrasting with some other moment-based estimators which require higher computational resources.
4. Finite Sample Bounds: The authors provide bounds on the sample complexity needed for accurate estimation, demonstrating its practical feasibility.
5. Robust Parameter Recovery: By employing orthogonal tensor decompositions, they ensure robust recovery of parameters, which is advantageous in practical scenarios subject to sampling errors.

Implications

The implications of this research are significant: it removes some historical constraints, such as the need for minimum separation between component means, thereby broadening the applicability of Gaussian mixture models in high-dimensional statistics and machine learning. It also provides a theoretically sound algorithm that can be used in real-world problems where spherical covariances are a fair assumption.

The methodology has connections to and implications for independent component analysis (ICA), particularly in how algebraic techniques can be utilized across different types of mixture models and noise distributions.

In terms of future directions, this approach could be extended to broader classes of noise models and covariance structures. Furthermore, exploring this estimation methodology in conjunction with more general forms of mixture models could unveil additional insights and utility in both theory and application.

The techniques discussed in this paper offer a prominent alternative to existing methods and raise possibilities for advancing the understanding and practical application of mixture models in complex data environments.