Provable Tensor Factorization with Missing Data (1406.2784v1)

Abstract: We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We show that under certain standard assumptions, our method can recover a three-mode $n\times n\times n$ dimensional rank-$r$ tensor exactly from $O(n^{3/2} r^5 \log^4 n)$ randomly sampled entries. In the process of proving this result, we solve two challenging sub-problems for tensors with missing data. First, in the process of analyzing the initialization step, we prove a generalization of a celebrated result by Szemer\'edie et al. on the spectrum of random graphs. Next, we prove global convergence of alternating minimization with a good initialization. Simulations suggest that the dependence of the sample size on dimensionality $n$ is indeed tight.

Provable Tensor Factorization with Missing Data: A Detailed Examination

This paper addresses a critical challenge in data analysis: the factorization of low-rank tensors with missing entries—a situation common in real-world applications. The research provides both a novel algorithm and the theoretical grounds to assure exact recovery under defined conditions.

Problem Overview and Approach

Tensors are multidimensional arrays that extend the concept of matrices to more dimensions, capturing intricate relationships in data. However, missing data in tensors poses significant difficulties in analyzing or reconstructing complete data structures. This research focuses on tensors having a low-rank orthogonal decomposition, aimed at reconstructing them efficiently and precisely despite incomplete data availability.

The paper introduces an innovative alternating minimization-based approach. This method iteratively refines estimates of singular vectors to recover an $n \times n \times n$ dimensional rank-$r$ tensor. The remarkable aspect of this algorithm is its demonstrated capability to achieve exact recovery from $O(n^{3/2} r^5 \log^4 n)$ randomly sampled tensor entries.

Theoretical Contributions and Results

This work establishes a rigorous theoretical basis for tensor recovery in a sparse data environment. Under specific assumptions, it guarantees global convergence of the alternating minimization process starting from a well-chosen initialization. The assumptions involve the incoherence of the true tensor, which refers to how the tensor's mass is distributed across its elements, helping ensure that essential information isn't lost in unsampled entries.

Main Findings

1. Sample Complexity: The method guarantees exact tensor recovery from a number of samples significantly less than the total number of entries, specifically $O(n^{3/2} r^5 \log^4 n)$. This sample requirement is a substantial improvement, given that tensor completion remains challenging with increasing dimensions.
2. Algorithm Convergence: The paper proves that their alternating minimization algorithm, provided a good initial guess, convergently reaches the precise tensor decomposition. This aspect is crucial since naive initialization might lead conventional algorithms into poor convergence due to local minima.
3. Simulation Results: Numerical experiments affirm that the theoretical bounds on sample size are tight in terms of the dependence on dimensionality $n$. Yet, the dependency on rank $r$ could be optimized further, suggesting room for future exploration.

Practical and Theoretical Implications

The implications are substantial for disciplines where multi-dimensional data is prevalent, such as signal processing, neuroscience, or any high-dimensional data context. The algorithm's application could power technologies in areas like recommendation systems, image reconstruction, or even quantum chemistry where tensors naturally describe complex phenomena.

Moreover, the paper pushes theoretical boundaries, advancing understanding of higher-order tensor completion. Unlike the well-understood limitations for matrix completion, fundamental limits for tensors remain largely uncharted, thus this paper contributes valuable insights into where those limits may lie.

Future Directions

While groundbreaking, the paper acknowledges existing limitations in handling non-orthogonal decompositions and reaching optimal dependency concerning the rank $r$. Subsequent research should address these challenges and explore more robust tensor factorization techniques.

Additionally, connecting the structural properties of the sampling pattern to factors such as hypergraph theory could illuminate paths to more efficient and generalizable algorithms capable of yielding strong guarantees across different data structures and applications.

In conclusion, this paper presents a significant step forward in tensor factorization amidst missing data, coupling mathematical rigor with practical utility, thus paving the way for enhanced data processing technologies.