Estimation of low-rank tensors via convex optimization

Abstract: In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is guaranteed to be unique. The proposed approaches can automatically estimate the number of factors (rank) through the optimization. Thus, there is no need to specify the rank beforehand. The key technique we employ is the trace norm regularization, which is a popular approach for the estimation of low-rank matrices. In addition, we propose a simple heuristic to improve the interpretability of the obtained factorization. The advantages and disadvantages of three proposed approaches are demonstrated through numerical experiments on both synthetic and real world datasets. We show that the proposed convex optimization based approaches are more accurate in predictive performance, faster, and more reliable in recovering a known multilinear structure than conventional approaches.

• The paper introduces three convex optimization frameworks for estimating low-rank tensors from partial observations, addressing challenges of non-convex methods.
• The proposed methods include matrix unfolding with trace norm, simultaneous trace norm regularization on all modes, and a mixture approach relaxing joint low-rankness.
• Numerical results show these convex methods outperform traditional approaches in accuracy and efficiency and automatically estimate tensor rank.

Estimation of Low-Rank Tensors via Convex Optimization

The paper in question introduces a novel framework for estimating low-rank tensors from partial observations through convex optimization methods. Unlike traditional approaches, which often involve non-convex formulations with inherent challenges such as local minima and convergence to suboptimal solutions, this framework offers three convex formulations that guarantee unique minimization outcomes.

Summary of Approaches

The tensor decomposition problem is addressed through three distinct methods:

1. As a Matrix Approach: This method unfolds the tensor into a matrix along a particular mode and applies trace norm regularization to estimate its low-rank structure. It effectively transforms the tensor estimation problem into a more tractable matrix estimation problem, which can be solved efficiently.
2. Constraint Approach: Extending the idea from the first method, this approach introduces simultaneous trace norm regularization on all modes of the tensor. By enforcing low-rank constraints across every mode, this approach potentially exploits more rank deficiencies but could be restrictive if the tensor is not truly low-rank in all modes.
3. Mixture Approach: This technique relaxes the assumption of joint low-rankness by leveraging a mixture of tensors. Each tensor in the mixture is regularized to be low-rank in one mode, allowing for more flexibility and potentially improved performance when the tensor is low-rank in only a subset of its modes.

Numerical Results and Implications

The paper presents comprehensive numerical experiments demonstrating the effectiveness of these methods on both synthetic and real-world datasets. The results indicate that the proposed convex optimization approaches significantly outperform conventional EM-based Tucker decompositions in terms of predictive accuracy and computational efficiency. Notably, the "Constraint" approach exhibits a sharp transition from poor to almost perfect reconstruction performance when the fraction of observed data increases beyond a certain threshold. This threshold behavior suggests a critical sampling density proportional to the sum of the tensor's mode-k ranks, although a rigorous theoretical underpinning for this observation is yet to be provided.

The applicability of the proposed methods extends to various domains, including signal processing, neuroscience, and data mining, where multi-way data analysis is routinely employed. Specifically, the estimation of low-rank tensors facilitates interpretable decompositions of complex datasets, enabling better insights into underlying structures.

Theoretical and Practical Implications

From a theoretical standpoint, these convex formulations are significant as they extend the concept of trace norm regularization beyond matrices, providing a robust alternative to non-convex optimization in tensor decomposition. Practically, the automatic estimation of tensor rank, integral to these approaches, simplifies the decomposition process by obviating the need for prior specification, which is often difficult in large-scale data scenarios.

Future Directions

Future research could explore several avenues for extension and improvement, such as developing frameworks for non-Gaussian noise models, enabling robust tensor recovery in the face of outlier noise, akin to robust PCA. Furthermore, scaling these methods to handle large datasets efficiently and integrating them into real-time applications present significant challenges and opportunities. The analytical understanding of the sharp threshold behavior observed in the experiments remains an open question that warrants further exploration.

In conclusion, the paper presents a compelling case for the integration of convex optimization techniques in tensor decomposition, offering substantial improvements in addressing multi-way data analysis challenges. Such innovations are poised to enhance the capacity of AI systems to process and interpret complex, high-dimensional data efficiently.