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**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 1 1

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**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 Ubiquitous Revealing Productive Smart Messy 2 K. Cukier, ``Harnessing the data deluge,'' Nov 2

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**Context Imputation of network data**

Preference modeling Imputation of network data Smart metering Network cartography Goal: Given few incomplete rows per agent, impute missing entries in a distributed fashion by leveraging low-rank of the data matrix. 3 3 3 3

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**Low-rank matrix completion**

Consider matrix , set Sampling operator ? ? ? ? Given incomplete (noisy) data ? ? ? ? ? ? (as) has low rank Goal: denoise observed entries, impute missing ones ? ? Nuclear-norm minimization [Fazel’02],[Candes-Recht’09] Noisy Noise-free s.t. 4 4

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**Problem statement Network: undirected, connected graph**

? ? ? ? ? ? n ? ? ? ? Goal: Given per node and single-hop exchanges, find (P1) Challenges Nuclear norm is not separable Global optimization variable 5 5

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**Separable regularization**

Key result [Recht et al’11] Lxρ ≥rank[X] New formulation equivalent to (P1) (P2) Nonconvex; reduces complexity: Proposition 1. If stationary pt. of (P2) and , then is a global optimum of (P1). 6 6

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**Distributed estimator**

(P3) Consensus with neighboring nodes Network connectivity (P2) (P3) 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 : n Message passing: 7 7

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**Distributed iterations**

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**Attractive features 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) 9 9

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**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 10 10

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

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**Real data Network distance prediction [Liau et al’12]**

Abilene network data (Aug 18-22,2011) End-to-end latency matrix N=9, L=T=N 80% missing data Figures: 1) ROC 2) 3D plot of the detected anomalies like the proposal Relative error: 10% 12 Data: 12

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