Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery (1307.5870v2)

Abstract: Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a $K$-way tensor of length $n$ and Tucker rank $r$ from Gaussian measurements requires $\Omega(r n^{K-1})$ observations. In contrast, a certain (intractable) nonconvex formulation needs only $O(r^K + nrK)$ observations. We introduce a very simple, new convex relaxation, which partially bridges this gap. Our new formulation succeeds with $O(r^{\lfloor K/2 \rfloor}n^{\lceil K/2 \rceil})$ observations. While these results pertain to Gaussian measurements, simulations strongly suggest that the new norm also outperforms the sum of nuclear norms for tensor completion from a random subset of entries. Our lower bound for the sum-of-nuclear-norms model follows from a new result on recovering signals with multiple sparse structures (e.g. sparse, low rank), which perhaps surprisingly demonstrates the significant suboptimality of the commonly used recovery approach via minimizing the sum of individual sparsity inducing norms (e.g. $l_1$, nuclear norm). Our new formulation for low-rank tensor recovery however opens the possibility in reducing the sample complexity by exploiting several structures jointly.

• The paper reveals the suboptimality of the traditional sum-of-nuclear-norms model for low-rank tensor recovery in certain cases.
• It introduces the 'square norm' convex relaxation, which requires significantly fewer observations for recovering higher-order tensors compared to the sum-of-nuclear-norms model.
• Empirical evaluations show the square norm outperforms the sum-of-nuclear-norms in tensor completion simulations, offering enhanced strategies for processing high-dimensional data.

An Examination of "Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery"

The paper "Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery" addresses the problem of recovering low-rank tensors from incomplete data, a pivotal challenge in signal processing and machine learning. The established methodology, employing the sum of nuclear norms for the unfoldings of a tensor, is scrutinized for its efficiency. The authors reveal that this widely used approach can be substantially suboptimal when compared with a newly proposed convex relaxation.

Problem Context and Motivation

Tensors are multidimensional arrays that naturally arise in contexts such as video data processing and hyperspectral imaging, where these arrays are indexed by several variables. Despite residing in high-dimensional spaces, these tensors often have low-rank structures, making their recovery from partial observations a problem of significant interest. Traditional approaches have leveraged convex optimization techniques to recover low-dimensional structures like sparsity or low-rankness in vectors or matrices. However, extending these methods to higher-order tensors poses complexities, due to computational intractability of grid search over rank and nuclear norm associated with tensors.

Key Insights and Methodological Contributions

The authors critically evaluate the traditional sum-of-nuclear-norms (SNN) model and introduce a potentially advantageous alternative. Their endeavors focus on two primary goals:

1. Establish Bounds for the SNN Model: The paper demonstrates, through theoretical lower bounds, the inefficiency of the SNN model in certain cases. It is shown that recovering a $K$-way tensor using Gaussian measurements and the SNN formulation requires $\Omega(r n^{K-1})$ observations, which is an underperforming requirement when considering the degrees of freedom for low-rank tensors. This aligns with broader understandings of simultaneous structured recovery observed in previous research.
2. Introduction of the Square Norm: Seeking improved tensor recovery, the authors propose the "square norm" model. This formulation essentially optimizes a reshaping of the tensor's matricization that endeavors to balance the tensor's dimensions prior to applying the nuclear norm. The theoretical implications of this model are substantial, requiring $O(r^{\lfloor K/2 \rfloor} n^{\lceil K/2 \rceil})$ observations— a significant reduction compared to the traditional SNN approach for $K > 3$.

Theoretical and Empirical Evaluation

The authors anchor their theoretical enhancements with rigorous mathematical proofs and compare against nonconvex baselines traditionally deemed intractable due to NP-hardness. The remarkable facet of their findings shows that although the newly proposed square norm model doesn’t achieve the theoretical lower bound of a nonconvex method, it does bring the complexity closer, offering a practical convex relaxation feasible with fewer observations for higher-order tensors.

Empirical results fortify these theoretical claims, particularly in the application of tensor completion. Simulations of tensor completion with four-way tensors revealed superior performance of the square-norm approach over the SNN model in terms of required observation ratios for successful recovery.

Implications and Future Speculation

The paper's findings illuminate potential advancements in tensor recovery methodologies, highlighting the suboptimality of the SNN model in certain scenarios while providing a scalable alternative with the square norm. Practically, this research suggests enhanced strategies for processing high-dimensional datasets, reducing data requirements, and improving the accuracy of models reliant on tensor completion.

For future developments in artificial intelligence and data science, this method provides a refined toolkit not only for tensor recovery but also opens the exploration of similar relaxations for other structured problem domains featuring simultaneous constraints. Meanwhile, the challenge of achieving the minimal measurement requirements known only in nonconvex settings remains an open research avenue. Addressing this gap could transform theoretical insights into broader applicability and efficiency gains across varied applications in machine learning and computational mathematics.

Overall, this paper contributes significant theoretical advancements to the field of tensor recovery, offering both a critical examination of existing methodologies and a path forward with improved algorithms.