E$^2$M: Double Bounded $\alpha$-Divergence Optimization for Tensor-based Discrete Density Estimation

arXiv:2405.18220v4 Announce Type: replace-cross Abstract: Tensor-based discrete density estimation requires flexible modeling and proper divergence criteria to enable effective learning; however, traditional approaches using $\alpha$-divergence face analytical challenges due to the $\alpha$-power terms in the objective function, which hinder the derivation of closed-form update rules. We present a generalization of the expectation-maximization (EM) algorithm, called the E$^2$M algorithm. It circumvents this issue by first relaxing the optimization into the minimization of a surrogate objective based on the Kullback-Leibler (KL) divergence, which is tractable via the standard EM algorithm, and subsequently applying a tensor many-body approximation in the M-step to enable simultaneous closed-form updates of all parameters. Our approach offers flexible modeling for not only a variety of low-rank structures, including the CP, Tucker, and Tensor Train formats, but also their mixtures, thus allowing us to leverage the strengths of different low-rank structures. We evaluate the effectiveness of our approach on synthetic and real datasets, highlighting its comparable convergence to gradient-based procedures, robustness to outliers, and favorable density estimation performance compared to prominent existing tensor-based methods.

Abstract:Tensor-based discrete density estimation requires flexible modeling and proper divergence criteria to enable effective learning; however, traditional approaches using $\alpha$-divergence face analytical challenges due to the $\alpha$-power terms in the objective function, which hinder the derivation of closed-form update rules. We present a generalization of the expectation-maximization (EM) algorithm, called the E$^2$M algorithm. It circumvents this issue by first relaxing the optimization into the minimization of a surrogate objective based on the Kullback-Leibler (KL) divergence, which is tractable via the standard EM algorithm, and subsequently applying a tensor many-body approximation in the M-step to enable simultaneous closed-form updates of all parameters. Our approach offers flexible modeling for not only a variety of low-rank structures, including the CP, Tucker, and Tensor Train formats, but also their mixtures, thus allowing us to leverage the strengths of different low-rank structures. We evaluate the effectiveness of our approach on synthetic and real datasets, highlighting its comparable convergence to gradient-based procedures, robustness to outliers, and favorable density estimation performance compared to prominent existing tensor-based methods.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2405.18220 [stat.ML]

(or arXiv:2405.18220v4 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2405.18220 Focus to learn more arXiv-issued DOI via DataCite

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E$^2$M: Double Bounded $\alpha$-Divergence Optimization for Tensor-based Discrete Density Estimation

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