Subspace Learning and Imputation for Streaming Big Data Matrices and Tensors (1404.4667v1)

Abstract: Extracting latent low-dimensional structure from high-dimensional data is of paramount importance in timely inference tasks encountered with `Big Data' analytics. However, increasingly noisy, heterogeneous, and incomplete datasets as well as the need for {\em real-time} processing of streaming data pose major challenges to this end. In this context, the present paper permeates benefits from rank minimization to scalable imputation of missing data, via tracking low-dimensional subspaces and unraveling latent (possibly multi-way) structure from \emph{incomplete streaming} data. For low-rank matrix data, a subspace estimator is proposed based on an exponentially-weighted least-squares criterion regularized with the nuclear norm. After recasting the non-separable nuclear norm into a form amenable to online optimization, real-time algorithms with complementary strengths are developed and their convergence is established under simplifying technical assumptions. In a stationary setting, the asymptotic estimates obtained offer the well-documented performance guarantees of the {\em batch} nuclear-norm regularized estimator. Under the same unifying framework, a novel online (adaptive) algorithm is developed to obtain multi-way decompositions of \emph{low-rank tensors} with missing entries, and perform imputation as a byproduct. Simulated tests with both synthetic as well as real Internet and cardiac magnetic resonance imagery (MRI) data confirm the efficacy of the proposed algorithms, and their superior performance relative to state-of-the-art alternatives.

• The paper introduces a scalable nuclear norm regularization framework that efficiently performs subspace learning and missing data imputation.
• The methodology employs bilinear factorization for matrices and PARAFAC for tensors, enabling online processing of high-dimensional streaming data.
• Numerical tests demonstrate robust performance, successfully imputing up to 75% missing data in applications like MRI imaging and network traffic analysis.

Subspace Learning and Imputation for Streaming Big Data Matrices and Tensors

This paper presents a sophisticated approach for extracting low-dimensional structures from high-dimensional streaming data, crucial in processing large and incomplete datasets typical in today's Big Data age. The authors propose methods leveraging rank minimization for efficient subspace learning and missing data imputation, focusing on both matrices and tensors.

Methodological Innovations

The core methodology is grounded in the use of nuclear norm regularization, which serves as a convex surrogate for rank minimization. This technique is pivotal in deriving scalable solutions, as it allows for formulations that lead to efficient optimization. The paper introduces a separable formulation of the nuclear norm, which breaks from traditional non-separable models, thus enabling online processing—a significant advancement for Big Data applications.

Matrix Completion: For matrix data, the authors present an exponentially-weighted least squares (EWLS) criterion, regularized with the nuclear norm. They employ a bilinear factorization framework where the matrix is expressed as the product of two low-rank matrices, reducing computational complexity significantly compared to direct nuclear norm minimization.

Tensor Completion: Extending their approach to tensors, the authors propose an online algorithm for multi-way data arrays, employing a similar rank-minimization framework. This is achieved through the parallel factor analysis (PARAFAC) model, offering a novel mechanism to handle missing data in higher-dimensional arrays.

Numerical Results and Practical Implications

Numerical experiments underscore the robustness and efficacy of the proposed algorithms. Both synthetic and real data tests reveal superior performance in imputation tasks compared to existing methods. For instance, the algorithms effectively impute cardiac MRI images with missing data up to 75% and accurately estimate traffic anomalies in network data, demonstrating potential applications in medical imaging and cybersecurity.

Convergence and Optimality

An important aspect discussed is the convergence properties of the proposed algorithms. The paper provides theoretical guarantees under certain conditions, ensuring that the methods asymptotically approach the stationary points of the batch formulation and the nuclear-norm regularized estimator. This convergence to globally optimal solutions is crucial for practical applications where computational resources are constrained.

Speculation on Future Developments in AI

The research presented opens avenues for future exploration, particularly in integrating these methods with more advanced machine learning models, such as deep learning frameworks, to further enhance their predictive capabilities. The tensor decomposition technique could significantly impact real-time analytics in dynamic environments, fostering advancements in fields like real-time anomaly detection and automated decision-making systems.

Overall, the paper provides substantial contributions to the landscape of data processing, particularly in navigating the complexities of streaming big data matrices and tensors. Its methodological insights and robust numerical results establish a foundation for future research in scalable, real-time data analytics.