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1 Distributed Nuclear Norm Minimization for Matrix Completion Morteza Mardani, Gonzalo Mateos and Georgios Giannakis ECE Department, University of Minnesota Acknowledgments: MURI (AFOSR FA ) grant Cesme, Turkey June 19, 2012

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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 BIG Fast Productive Revealing Ubiquitous Smart K. Cukier, ``Harnessing the data deluge,'' Nov Messy

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33 Context Imputation of network data Goal: Given few incomplete rows per agent, impute missing entries in a distributed fashion by leveraging low-rank of the data matrix. Preference modeling Network cartography Smart metering

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(as) has low rank Goal: denoise observed entries, impute missing ones 44 Low-rank matrix completion Consider matrix, set Given incomplete (noisy) data Nuclear-norm minimization [Fazel’02],[Candes-Recht’09] Sampling operator Noisy Noise-free s.t. ? ? ? ? ? ? ? ? ? ? ? ?

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55 Problem statement Goal: Given per node and single-hop exchanges, find n Network: undirected, connected graph (P1) ? ? ? ? ? ? ? ? ? ? Challenges Nuclear norm is not separable Global optimization variable

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66 Separable regularization Key result [Recht et al’11] New formulation equivalent to (P1) (P2) Proposition 1. If stationary pt. of (P2) and, then is a global optimum of (P1). Nonconvex; reduces complexity: LxρLxρ ≥rank[X]

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77 Distributed estimator Network connectivity (P2) (P3) (P3) Consensus with neighboring nodes Alternating-directions method of multipliers (ADMM) solver Method [Glowinski-Marrocco’75], [Gabay-Mercier’76] Learning over networks [Schizas et al’07] Primal variables per agent : Message passing: n

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8 8 Distributed iterations

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99 Highly parallelizable with simple recursions Unconstrained QPs per agent No SVD per iteration Low overhead for message exchanges is and is small Comm. cost independent of network size Recap: (P1) (P2) (P3) Centralized Convex Sep. regul. Nonconvex Consensus Nonconvex Stationary (P3) Stationary (P2) Global (P1) Attractive features

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10 Optimality Proposition 2. If converges to and, then: i) ii) is the global optimum of (P1). ADMM can converge even for non-convex problems [Boyd et al’11] Simple distributed algorithm for optimal matrix imputation Centralized performance guarantees e.g., [Candes-Recht’09] carry over

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11 Synthetic data Random network topology N=20, L=66, T=66 Data ,

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12 Real data Abilene network data (Aug 18-22,2011) End-to-end latency matrix N=9, L=T=N 80% missing data Network distance prediction [Liau et al’12] Data: Relative error: 10%

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