Subspace learning and imputation for streaming big data matrices and tensors

Morteza Mardani, Gonzalo Mateos, Georgios B. Giannakis

Research output: Contribution to journalArticlepeer-review

119 Scopus citations


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 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 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 nonseparable 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 batch nuclear-norm regularized estimator. Under the same unifying framework, a novel online (adaptive) algorithm is developed to obtain multi-way decompositions of 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.

Original languageEnglish (US)
Article number7072498
Pages (from-to)2663-2677
Number of pages15
JournalIEEE Transactions on Signal Processing
Issue number10
StatePublished - May 15 2015

Bibliographical note

Publisher Copyright:
© 2015 IEEE.


  • Low rank
  • matrix and tensor completion
  • missing data
  • streaming analytics
  • subspace tracking


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