[Paper Review] Tensor denoising and completion based on ordinal observations

Chan‐Woo Lee, Miaoyan Wang|arXiv (Cornell University)|Feb 16, 2020

Tensor decomposition and applications40 references4 citations

TL;DR

This paper proposes a multi-linear cumulative link model for low-rank tensor denoising and completion from incomplete, ordinal-valued observations. It introduces a rank-constrained M-estimator that achieves minimax optimal mean squared error with a convergence rate of $\mathcal{O}(d^{-(K-1)})$, enabling consistent recovery of an order-$K$ tensor using only $\tilde{\mathcal{O}}(Kd)$ quantized observations.

ABSTRACT

Higher-order tensors arise frequently in applications such as neuroimaging, recommendation system, social network analysis, and psychological studies. We consider the problem of low-rank tensor estimation from possibly incomplete, ordinal-valued observations. Two related problems are studied, one on tensor denoising and the other on tensor completion. We propose a multi-linear cumulative link model, develop a rank-constrained M-estimator, and obtain theoretical accuracy guarantees. Our mean squared error bound enjoys a faster convergence rate than previous results, and we show that the proposed estimator is minimax optimal under the class of low-rank models. Furthermore, the procedure developed serves as an efficient completion method which guarantees consistent recovery of an order-$K$ $(d,\ldots,d)$-dimensional low-rank tensor using only $ ilde{\mathcal{O}}(Kd)$ noisy, quantized observations. We demonstrate the outperformance of our approach over previous methods on the tasks of clustering and collaborative filtering.

Motivation & Objective

• Address the challenge of low-rank tensor estimation from incomplete, discrete, ordinal-valued observations, which are common in applications like collaborative filtering and neuroimaging.
• Overcome limitations of existing models that fail to preserve invariance under category reversal or are inconsistent under category merging/splitting.
• Develop a statistically optimal method for tensor denoising and completion under low-rank structure with ordinal data.
• Establish theoretical convergence rates and sample complexity bounds that match the information-theoretic limits for such problems.
• Provide a practical and consistent algorithm for tensor recovery from highly quantized, noisy observations.

Proposed method

• Propose a multi-linear cumulative link model that ensures palindromic invariance, making it robust to reversal of ordinal category labels.
• Formulate a rank-constrained M-estimator to estimate the low-rank signal tensor from noisy, quantized observations.
• Use an alternating non-convex optimization algorithm to solve the estimation problem, avoiding the computational and statistical drawbacks of convex relaxation.
• Leverage the Tucker decomposition to extract principal components from the estimated tensor for downstream tasks like clustering.
• Apply the Varshamov-Gilbert and Le Cam’s inequality-based testing frameworks to derive minimax lower bounds.
• Establish theoretical guarantees using concentration inequalities and uniform bounds on the Frobenius norm over low-rank tensors.

Experimental results

Research questions

• RQ1 Can a tensor estimation method achieve minimax optimal error rates when observations are ordinal and highly quantized?
• RQ2 What is the minimal number of noisy, ordinal observations required for consistent recovery of a low-rank tensor?
• RQ3 How does the proposed method compare to existing approaches in terms of statistical efficiency and convergence rate?
• RQ4 Can the model preserve invariance under category reversal while remaining consistent under category merging or splitting?
• RQ5 Does the proposed method outperform existing binary or continuous tensor models in collaborative filtering and clustering tasks?

Key findings

• The proposed estimator achieves a mean squared error bound of $\mathcal{O}(d^{-(K-1)})$, which is faster than the $\mathcal{O}(d^{-(K-1)/2})$ rate in prior work and is minimax optimal under the low-rank model.
• The method guarantees consistent recovery of an order-$K$ $(d,\ldots,d)$-dimensional low-rank tensor using only $\tilde{\mathcal{O}}(Kd)$ noisy, quantized observations.
• The cumulative link model ensures palindromic invariance and consistent parameter interpretation under category merging/splitting, unlike previous models.
• Numerical experiments show superior performance over existing methods in collaborative filtering and clustering tasks, particularly on real neuroimaging data.
• The theoretical analysis confirms that quantized ordinal observations can achieve the same information-theoretic convergence rate as continuous observations under the proposed model.
• The alternating non-convex algorithm outperforms convex relaxation methods in both error rate and computational efficiency, especially for higher-order tensors.