Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Distributed Sensing and Perception via Sparse Representation Allen Y. Yang Department of EECS, UCB yang@eecs.berkeley.edu University of Texas, Austin, 2011 Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Distributed Sensing and Perception Centralized Perception Distributed Perception Up: powerful processors Down: mobile processors Up: unlimited memory Down: limited onboard memory Up: unlimited bandwidth Down: band-limited communications Down: single modality Up: distributed, multi-modality Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Distributed Sensing and Perception Centralized Perception Distributed Perception Up: powerful processors Down: mobile processors Up: unlimited memory Down: limited onboard memory Up: unlimited bandwidth Down: band-limited communications Down: single modality Up: distributed, multi-modality When the sensing resources are limited or scarce: What is the optimal strategy to deploy these agents? 1 How to effectively take measurements of the events? 2 How to properly tally the local observations to reach a global consensus? 3 An intelligent system over a sensor network shall perform better than the sum of its parts? Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Challenges Making real-time decisions on mobile devices is difficult. 1 Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Challenges Making real-time decisions on mobile devices is difficult. 1 Applications demand extremely high accuracy: 99% Precision, 99% Recall? 2 Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Challenges Making real-time decisions on mobile devices is difficult. 1 Applications demand extremely high accuracy: 99% Precision, 99% Recall? 2 Scenarios demand the ability to reconstruct 3-D models. 3 Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Outline Robust Face Recognition 1 A sparse representation framework via ℓ 1 -min . x ∗ = arg min � x � 1 subj. to b = A x . x Accelerate ℓ 1 -min algorithms towards a semi-real time face recognition system. Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Informative Feature Selection for Object Recognition 2 Informative feature selection via Sparse PCA x ∗ = arg max x T Σ A x subj. to � x � 2 = 1 , � x � 1 ≤ k . x Accelerate Sparse PCA algorithms. Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Reconstruct large-scale 3-D objects by large-baseline feature matching 3 Extract a new class of low-rank texture regions using Robust PCA A ∗ = arg min A , E ,τ � A � ∗ + λ � E � 1 subj. to I ◦ τ = A + E . Complete pipeline from low-rank texture in single views to 3-D model in multiple views. Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Face Recognition Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Classification via Sparse Representation Face-subspace model [Belhumeur et al. ’97, Basri & Jacobs ’03] 1 Assume b belongs to Class i in K classes. b = α i , 1 v i , 1 + α i , 2 v i , 2 + · · · + α i , n 1 v i , n i , = A i α i . Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Classification via Sparse Representation Face-subspace model [Belhumeur et al. ’97, Basri & Jacobs ’03] 1 Assume b belongs to Class i in K classes. b = α i , 1 v i , 1 + α i , 2 v i , 2 + · · · + α i , n 1 v i , n i , = A i α i . Nevertheless, Class i is the unknown label we need to solve: 2 α 1 2 3 α 2 5 = A x . . Sparse representation b = [ A 1 , A 2 , · · · , A K ] . 4 . α K Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Classification via Sparse Representation Face-subspace model [Belhumeur et al. ’97, Basri & Jacobs ’03] 1 Assume b belongs to Class i in K classes. b = α i , 1 v i , 1 + α i , 2 v i , 2 + · · · + α i , n 1 v i , n i , = A i α i . Nevertheless, Class i is the unknown label we need to solve: 2 α 1 2 3 α 2 5 = A x . . Sparse representation b = [ A 1 , A 2 , · · · , A K ] . 4 . α K x ∗ = [ 0 ··· 0 α T 0 ··· 0 ] T ∈ R n . 3 i Sparse representation x ∗ encodes membership! Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Image Corruption Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Image Corruption Sparse representation + sparse error 1 b = A x + e Cross-and-bouquet model [Wright et al. ’09, ’10] 2 I ´ „ x « ` A b = | = B w e Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Image Corruption Sparse representation + sparse error 1 b = A x + e Cross-and-bouquet model [Wright et al. ’09, ’10] 2 I ´ „ x « ` A b = | = B w e When size of A grows proportionally with the sparsity in x , asymptotically CAB can correct 100% noise. Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Performance on the AR database ( ℓ 1 -min): min � x � 1 + � e � 1 subj. to b = A x + e Reference: AY, et al. Robust face recognition via sparse representation . IEEE PAMI, 2009. Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion Question: How to effectively estimate HD sparse signals? “Black gold” age [Claerbout & Muir 1973, Taylor, Banks & McCoy 1979] Figure: Deconvolution of spike train. Basis pursuit / ℓ 1 -minimization [Chen-Donoho 1999] : x ∗ = arg min � x � 1 , subject to b = A x ( P 1 ) : The Lasso (least absolute shrinkage and selection operator) [Tibshirani 1996] x ∗ = arg min � b − A x � 2 , subject to � x � 1 ≤ k ( P 1 , 2 ) : Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
Introduction Sparsity-based Classification Sparse Feature Selection Low-Rank Texture Conclusion ℓ 1 -Minimization via Linear Programming Using interior-point methods [Karmarkar ’84] n 1 T x − µ X Log-Barrier: min log x i , subj. to A x = b , x ≥ 0 . (1) x i =1 Using the Karush-Kuhn-Tucker (KKT) conditions 1 − µ X − 1 1 − A T y = 0 . (2) where x ≥ 0 are the primal variables, and y are the dual variables. Update by solving a linear system with O ( n 3 ) [Monteiro & Adler ’89] Z ( k ) ∆ x + X ( k ) ∆ z µ 1 − X ( k ) z ( k ) , = ˆ A ∆ x = 0 , (3) A T ∆ y + ∆ z = 0 , Distributed Sensing and Perception via Sparse Representation http://www.eecs.berkeley.edu/~yang
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