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Imputation of Streaming Low-Rank Tensor Data

Morteza Mardani , Gonzalo Mateos and Georgios Giannakis ECE Department, University of Minnesota Acknowledgment : AFOSR MURI grant no. FA9550-10-1-0567. Imputation of Streaming Low-Rank Tensor Data. A Coruna, Spain June 25, 2013. 1. Learning from “Big Data”.

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Imputation of Streaming Low-Rank Tensor Data

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  1. MortezaMardani, Gonzalo Mateos and GeorgiosGiannakis ECE Department, University of Minnesota Acknowledgment: AFOSR MURI grant no. FA9550-10-1-0567 Imputation of Streaming Low-Rank Tensor Data A Coruna, Spain June 25, 2013 1

  2. Learning from “Big Data” `Data are widely available, what is scarce is the ability to extract wisdom from them’ Hal Varian, Google’s chief economist Fast BIG Productive Ubiquitous Revealing Messy Smart 2 K. Cukier, ``Harnessing the data deluge,'' Nov. 2011.

  3. Tensor model • Data cube • PARAFAC decomposition cr br ar B= C= A= βi αi γi

  4. Streaming tensor data • Streaming data • Tensor subspace comprises R rank-one matrices Goal: given the streaming data , at time t learn the subspace matrices (At,Bt) and impute the missing entries of Yt?

  5. Prior art • Matrix/tensor subspace tracking • Projection approximation (PAST) [Yang’95] • Misses: rank regularization [Mardani et al’13], GROUSE [Balzano et al’10] • Outliers: [Mateos et al’10], GRASTA [He et al’11] • Adaptive LS tensor tracking [Nion et al’09] with full data; tensor slices treated as long vectors • Batch tensor completion [Juan et al’13], [Gandy et al’11] • Novelty:Online rank regularization with misses • Tensor decomposition/imputation • Scalable and provably convergent iterates

  6. Batch tensor completion • Rank-regularized formulation [Juan et al’13] (P1) • Tikhonovregularizerpromotes low rank Proposition 1[Juan et al’13]: Let , then

  7. Tensor subspace tracking • Exponentially-weighted LS estimator (P2) ft(A,B) • ``on-the-fly’’ imputation • Alternating minimization with stochastic gradient iterations (at time t) • Step1:Projection coefficientupdates • Step2: Subspace update • O(|Ωt|R2) operations per iteration M. Mardani, G. Mateos, and G. B. Giannakis, “Subspace learning and imputation for streaming Big Data matrices and tensors,"IEEE Trans. Signal Process., Apr. 2014 (submitted).

  8. Convergence • As1) Invariant subspace and • As2) Infinite memory β= 1 Proposition 2: If and are i.i.d., and c1) is uniformly bounded; c2) is in a compact set; and c3) is strongly convex w.r.t. hold, then almost surely (a. s.) • asymptotically converges to a st.point of batch(P1)

  9. Cardiac MRI • FOURDIX dataset • 263 images of 512 x 512 • Y: 32 x 32 x 67,328 • 75% misses • R=10  ex=0.14 • R=50  ex=0.046 (b) (a) (c) (d) Ground truth, (b) acquired image; reconstructed for R=10 (c), R=50 (d) http://www.osirix-viewer.com/datasets.

  10. Tracking traffic anomalies • Link load measurements • Internet-2 backbone network • Yt: weighted adjacency matrix • Available data Y: 11x11x6,048 • 75% misses, R=18 http://internet2.edu/observatory/archive/data-collections.html

  11. Conclusions • Real-time subspace trackers for decomposition/imputation • Streaming big and incomplete tensor data • Provably convergent scalable algorithms • Applications • Reducing the MRI acquisition time • Unveiling network traffic anomalies for Internet backbone networks • Ongoing research • Incorporating spatiotemporal correlation information via kernels • Accelerated stochastic-gradient for subspace update

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