Tensor Completion Algorithms in Big Data Analytics (1711.10105v2)

Abstract: Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in areas like data mining, computer vision, signal processing, and neuroscience. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data analytics characterized by diverse variety, large volume, and high velocity. We characterize these advances from four perspectives: general tensor completion algorithms, tensor completion with auxiliary information (variety), scalable tensor completion algorithms (volume), and dynamic tensor completion algorithms (velocity). Further, we identify several tensor completion applications on real-world data-driven problems and present some common experimental frameworks popularized in the literature. Our goal is to summarize these popular methods and introduce them to researchers and practitioners for promoting future research and applications. We conclude with a discussion of key challenges and promising research directions in this community for future exploration.

• The paper surveys tensor completion algorithms in big data analytics, detailing frameworks based on tensor decompositions, auxiliary information, scalable, and dynamic methods.
• It presents hybrid methodologies that combine decomposition-based and trace-norm approaches to address computational complexity and missing data challenges.
• The work outlines future directions, emphasizing integration of domain knowledge and theoretical guarantees for evolving, high-dimensional datasets.

Tensor Completion Algorithms in Big Data Analytics

The paper "Tensor Completion Algorithms in Big Data Analytics" by Qingquan Song, Hancheng Ge, James Caverlee, and Xia Hu provides a comprehensive survey of the advancements in tensor completion methods, with an emphasis on their application in the broader context of big data analytics. The authors explore various aspects of tensor completion algorithms under the characterization of big data's 3Vs: variety, volume, and velocity. These three dimensions serve as a framework to analyze and categorize the development of tensor completion techniques.

The multidimensional nature of tensors makes them ideal for representing complex datasets spanning various domains like data mining, computer vision, signal processing, and neuroscience. However, the application of tensor completion is challenged by the inherent missing or unobserved entries due to mal-operations, limited permissions, or random data loss. This paper explores the methods, application cases, and emerging challenges in tensor completion.

General Developments

The paper categorizes tensor completion advancements into four primary frameworks:

1. General Tensor Completion Algorithms:
   • These include decomposition-based methods where tensor decompositions like CP and Tucker are utilized to handle missing entries. Trace-norm based methods such as SNN leverage norms derived from the tensor unfoldings to facilitate a low-rank approximation.
   • One significant challenge is the scalability and computational complexity, especially when handling high-dimensional data. Recent methods attempt hybrid approaches and exploit stochastic methods for more efficient computation.
2. Tensor Completion with Auxiliary Information (Variety):
   • Leveraging auxiliary information, such as related matrices or inherent structure, improves completion when data is sparse. This includes integrating spatial or temporal relationships and coupling tensors with other matrices.
   • Methods such as coupling with auxiliary matrices or similarity-based regularizations help incorporate external, correlated data to increase the robustness of tensor completion.
3. Scalable Tensor Completion Algorithms (Volume):
   • Scalable methods address the burgeoning volume of datasets by optimizing computational frameworks, often using parallel and distributed computing systems.
   • Techniques such as scalable versions of ALS, stochastic gradient descent, and advanced frameworks like GigaTensor and HaTen2 are highlighted for mitigating issues such as intermediate data explosion and memory constraints.
4. Dynamic Tensor Completion Algorithms (Velocity):
   • The paper addresses dynamic, time-varying data analytics using streaming and online tensor completion methods. These approaches constantly update models with incoming data, maintaining effectiveness without reprocessing the entire dataset.
   • Methods like online CP decomposition and multi-aspect streaming tensor completion represent efforts to tackle evolving datasets with temporal and multi-aspect expansion.

Theoretical Implications and Future Directions

A significant portion of the paper is dedicated to theoretical underpinnings such as assumptions regarding sampling and incoherence, which govern the feasibility and accuracy of tensor completions. The paper asserts that achieving effective and provably accurate completion often relies on conditions like the incoherence of the tensor data, structured sampling, and rank constraints.

The authors also ponder potential future research directions:

• Addressing the Big Data Challenges: Techniques need to evolve to better handle the scale and velocity of big data while ensuring veracity and value.
• Enhanced Interaction with Domain Knowledge: Integrating domain-specific knowledge into tensor completion may yield more meaningful and interpretable results, especially in areas like healthcare, social networks, and scientific data analysis.
• Theoretical Exploration in Dynamic Settings: Providing mathematical and statistical guarantees for dynamic, high-velocity tensor completions remains an open challenge.

In summary, this paper provides an in-depth look into tensor completion algorithms from both a methodological and application standpoint amidst the landscape of big data. The work not only synthesizes current advancements and practical applications but also lays down a roadmap for ongoing research and development driven by the complexities of modern data environments.