Sliced Wasserstein Distance for Learning Gaussian Mixture Models
This work addresses the issue of suboptimal convergence in GMM estimation for machine learning and computer vision applications, though it appears incremental as it modifies an existing approach rather than introducing a new paradigm.
The authors tackled the problem of Gaussian mixture model (GMM) parameter estimation by proposing a new formulation using the sliced Wasserstein distance instead of the traditional KL-divergence, resulting in more robust parameter estimates and better performance in high-dimensional data compared to the EM algorithm.
Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm.