Low-rank Bayesian matrix completion via geodesic Hamiltonian Monte Carlo on Stiefel manifolds (2410.20318v1)

Abstract: We present a new sampling-based approach for enabling efficient computation of low-rank Bayesian matrix completion and quantifying the associated uncertainty. Firstly, we design a new prior model based on the singular-value-decomposition (SVD) parametrization of low-rank matrices. Our prior is analogous to the seminal nuclear-norm regularization used in non-Bayesian setting and enforces orthogonality in the factor matrices by constraining them to Stiefel manifolds. Then, we design a geodesic Hamiltonian Monte Carlo (-within-Gibbs) algorithm for generating posterior samples of the SVD factor matrices. We demonstrate that our approach resolves the sampling difficulties encountered by standard Gibbs samplers for the common two-matrix factorization used in matrix completion. More importantly, the geodesic Hamiltonian sampler allows for sampling in cases with more general likelihoods than the typical Gaussian likelihood and Gaussian prior assumptions adopted in most of the existing Bayesian matrix completion literature. We demonstrate an applications of our approach to fit the categorical data of a mice protein dataset and the MovieLens recommendation problem. Numerical examples demonstrate superior sampling performance, including better mixing and faster convergence to a stationary distribution. Moreover, they demonstrate improved accuracy on the two real-world benchmark problems we considered.

• The paper introduces an SVD-based prior model that enforces orthogonality on factor matrices by leveraging Stiefel manifolds.
• It presents a geodesic HMC-within-Gibbs algorithm that efficiently samples from complex, non-Gaussian posterior distributions.
• Empirical results demonstrate enhanced mixing and faster convergence, outperforming traditional methods on real-world datasets.

Low-rank Bayesian Matrix Completion via Geodesic Hamiltonian Monte Carlo on Stiefel Manifolds

The paper "Low-rank Bayesian Matrix Completion via Geodesic Hamiltonian Monte Carlo on Stiefel Manifolds" by Tiangang Cui and Alex A. Gorodetsky presents a novel approach for efficiently addressing low-rank Bayesian matrix completion problems, with an emphasis on robust uncertainty quantification. This approach leverages geodesic Hamiltonian Monte Carlo (HMC) sampling to navigate the complexities inherent in Bayesian formulations, particularly those concerning non-uniqueness and complex posterior distributions.

Core Contributions

1. SVD-based Prior Model: The authors introduce a prior model based on the singular-value-decomposition (SVD) parameterization, which incorporates orthogonality constraints on factor matrices by placing them on Stiefel manifolds. This setup mirrors the nuclear-norm regularization strategies prevalent in non-Bayesian frameworks and aims to address the degeneracies that arise from the common two-matrix factorization approach in matrix completion tasks.
2. Geodesic HMC Algorithm: The development of a geodesic HMC-within-Gibbs sampling algorithm is central to this work. This modification is crucial for sampling from posterior distributions where traditional Gibbs samplers encounter difficulties, especially in the presence of non-Gaussian likelihoods and priors. The proposed method is uniquely effective in generating samples from complex posterior distributions that exhibit challenging geometrical properties.
3. Robust Numerical Performance: Through empirical validations on real-world datasets, including the mice protein categorization and the MovieLens recommendation problem, the methodology demonstrates enhanced sampling performance, as evidenced by better mixing and faster convergence to stationarity than existing methods.

Implications and Future Directions

Practical Applications: The proposed geodesic HMC approach equips researchers with a powerful tool for performing Bayesian inference in matrix factorization settings where the underlying data or assumptions deviate from Gaussianity. Its flexibility in accommodating diverse likelihood structures is particularly advantageous for real-world applications where categorical or bespoke error models are essential.

Theoretical Extensions: The adoption of non-standard geodesic HMC techniques opens pathways for exploring similar advancements in other areas of machine learning and Bayesian inference that involve manifold-embedded parameter spaces. The general principles demonstrated here may have further applications in fields such as manifold learning, statistical shape analysis, and beyond.

Performance Benchmarking: As demonstrated, the methodology outperforms conventional approaches by effectively reducing the inherent uncertainty in factor estimates and enabling more confident predictions on unobserved data elements. This performance, however, comes at the cost of increased per-iteration computational overhead. Continued research could focus on optimizing these computations or exploring hybrid algorithms that balance accuracy and efficiency in real-time settings.

In conclusion, the integration of manifold-based priors and advanced sampling techniques proposed in this paper represents a substantial step forward for Bayesian matrix completion. Its capacity to handle non-trivial challenges posed by complex likelihoods further enriches its potential applicability across an expansive array of data-driven domains.