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稀疏

优化

数值操作

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稀疏与低秩表示：优化与运算——解决凸（非凸）函数，光滑（不光滑）函数问题。

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Introduction IST APG ALM ADMM Extensions Conclusion

Variations of Sparse Optimization and Numerical Implementation

Allen Y. Yang

Department of EECS, UC Berkeley

yang@eecs.berkeley.edu

with

Yi Ma & John Wright

ECCV 2012 Tutorial

Sparse Representation and Low-Rank Representation in Computer Vision

Lecture Two

http://www.eecs.berkeley.edu/

yang Sparse Optimization

Introduction IST APG ALM ADMM Extensions Conclusion

Two Previous Sparse Optimization Problems

-min seeks sparse solution in underdetermined system (A in general is full rank):

min kxk

subj. to b = Ax where A ∈ R

d×n

, (d < n)

A x

Robust PCA seeks sparse and low-rank decomposition:

min kAk

∗

+ λkEk

subj. to D = A + E ∈ R

m×n

= +

http://www.eecs.berkeley.edu/

yang Sparse Optimization

Introduction IST APG ALM ADMM Extensions Conclusion

Eﬃcient sparse optimization is challenging

Generic second-order method toolboxes do exist: CVX, SparseLab.

However, standard interior-point methods are very expensive in HD space.

min

subj. to Ax = b

x ≥ 0

The KKT condition:

∇f (x

∗

)+µ∇g(x

∗

)+λ∇h(x

∗

) = 0

Complexity: O(n

)

Robust PCA: CVX can solve smaller than 80 × 80 matrices on typical PC

Complexity bound: O(n

http://www.eecs.berkeley.edu/

yang Sparse Optimization

Introduction IST APG ALM ADMM Extensions Conclusion

Eﬃcient sparse optimization is challenging

Generic second-order method toolboxes do exist: CVX, SparseLab.

However, standard interior-point methods are very expensive in HD space.

min

subj. to Ax = b

x ≥ 0

The KKT condition:

∇f (x

∗

)+µ∇g(x

∗

)+λ∇h(x

∗

) = 0

Complexity: O(n

)

Robust PCA: CVX can solve smaller than 80 × 80 matrices on typical PC

Complexity bound: O(n

http://www.eecs.berkeley.edu/

yang Sparse Optimization

Introduction IST APG ALM ADMM Extensions Conclusion

Sparse Optimization Literature: `

-Minimization

Interior-point methods

Log-Barrier [Frisch ’55, Karmarkar ’84, Megiddo ’89, Monteiro-Adler ’89, Kojima-Megiddo-Mizuno

’93]

Homotopy

Homotopy [Osborne-Presnell-Turlach ’00, Malioutov-Cetin-Willsky ’05, Donoho-Tsaig ’06]

Polytope Faces Pursuit (PFP) [Plumbley ’06]

Least Angle Regression (LARS) [Efron-Hastie-Johnstone-Tibshirani ’04]

Gradient Projection

Gradient Projection Sparse Representation (GPSR) [Figueiredo-Nowak-Wright ’07]

Truncated Newton Interior-Point Method (TNIPM) [Kim-Koh-Lustig-Boyd-Gorinevsky ’07]

Iterative Soft-Thresholding

Soft Thresholding [Donoho ’95]

Sparse Reconstruction by Separable Approximation (SpaRSA) [Wright-Nowak-Figueiredo ’08]

Proximal Gradient [Nesterov ’83, Nesterov ’07]

FISTA [Beck-Teboulle ’09]

Nesterov’s Method (NESTA) [Becker-Bobin-Cand´es ’09]

Augmented Lagrangian Methods [Bertsekas ’82]

Bergman [Yin et al. ’08]

SALSA [Figueiredo et al. ’09]

Primal ALM, Dual ALM [AY et al ’10]

Alternating Direction Method of Multipliers

YALL1 [Yang-Zhang ’09]

http://www.eecs.berkeley.edu/

yang Sparse Optimization

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