Gonzalo Mateos and Georgios B. Giannakis

Sparsity Control for Robust
Principal Component Analysis
Gonzalo Mateos and Georgios B. Giannakis
ECE Department, University of Minnesota
Acknowledgments: NSF grants no. CCF-1016605, EECS-1002180
Asilomar Conference
November 10, 2010
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Principal Component Analysis
 Motivation: (statistical) learning from high-dimensional data
DNA microarray
Traffic surveillance
 Principal component analysis (PCA) [Pearson’1901]
 Extraction of low-dimensional data structure
 Data compression and reconstruction
 PCA is non-robust to outliers [Jolliffe’86]
 Our goal: robustify PCA by controlling outlier sparsity
2
Our work in context
 Contemporary applications
 Anomaly detection in IP networks [Huang et al’07], [Kim et al’09]
 Video surveillance, e.g., [Oliver et al’99]
Original
Robust PCA
`Outliers’
 Robust PCA
 Robust covariance matrix estimators [Campbell’80], [Huber’81]
 Computer vision [Xu-Yuille’95], [De la Torre-Black’03]
 Low-rank matrix recovery from sparse errors [Wright et al’09]
 Huber’s M-class and sparsity in linear regression [Fuchs’99]
3
PCA formulations
 Training data:
 Minimum reconstruction error:
 Dimensionality reduction operator
 Reconstruction operator
 Maximum variance:
 Factor analysis model:
Solution:
4
Robustifying PCA
 Least-trimmed squares (LTS) regression [Rousseeuw’87]
LTS-based PCA for robustness
(LTS PCA)
is the
-th order statistic among
Trimming constant
determines breakdown point
 Q: How should we go about minimizing
?
(LTS PCA) is nonconvex; existence of minimizer(s)?
A: Try all
subsets of size , solve, and pick the best
 Simple but intractable beyond small problems
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Modeling outliers
 Introduce auxiliary variables
 Inliers obey
 Inlier noise:
s.t.
inlier
outlier
; outliers something else
are zero-mean i.i.d. random vectors
 Remarks

and
are unknown
 If outliers sporadic, then vector is sparse!
 Natural (but intractable) estimator
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LTS PCA as sparse regression
 Lagrangian form
(P0)
 Tuning
Proposition 1: If
, then
controls sparsity in
, thus number of outliers
solves (P0) with
chosen such that
solves (LTS PCA) too.
 Justifies the model and its estimator (P0); ties sparsity with robustness
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Just relax!
 (P0) is NP-hard
relax
(P2)
 Role of sparsity controlling
is central
 Q: Does (P2) yield robust estimates
?
A: Yap! Huber estimator is a special case
8
Entrywise outliers
 Use
-norm regularization
(P1)
Original
Outlier
pixels
Robust PCA (P2)
Robust PCA (P1)
Entire
image
rejected
Outlier
pixels
rejected
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Alternating minimization
(P1)


update: reduced-rank Procrustes rotation
update: coordinatewise soft-thresholding
Proposition 2: Alg. 1’s iterates converge to a stationary point of (P1).
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Refinements
 Nonconvex penalty terms approximate
better in (P0)
 Options: SCAD [Fan-Li’01], or sum-of-logs [Candes etal’08]
 Iterative linearization-minimization of
around
 Iteratively reweighted version of Alg. 1
 Warm start: solution of (P1) or (P2)
 Bias reduction in
(cf. weighted Lasso [Zou’06])
 Discard outliers identified in
 Re-estimate
missing data problem
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Online robust PCA
 Motivation: Real-time data and memory limitations
 Exponentially-weighted robust PCA
 Approximation
[Yang’95]
 At time , do not re-estimate past outlier vectors
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Video surveillance
Original
PCA
Data: http://www.cs.cmu.edu/~ftorre/
Robust PCA
`Outliers’
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Online PCA in action
 Inliers:
 Outliers:
and
Angle between C(n) and C
 Figure of merit: angle between
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Concluding summary
 Sparsity control for robust PCA
 LTS PCA as
-(pseudo)norm regularized regression (NP-hard)
 Relaxation
(group)-Lassoed PCA
 Sparsity controlling role of
central
M-type estimator
 Batch and online robust PCA algorithms
 i) Outlier identification, ii) Robust subspace tracking
 Refinements via nonconvex penalty terms
 Tests on real video surveillance data for anomaly extraction
 Ongoing research
 Preference measurement: conjoint analysis and collaborative filtering
 Robustifying kernel PCA and blind dictionary learning
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