Learning with Limited Supervision by Input and Output Coding

1
Learning with Limited Supervision by Input and
Output Coding

• Yi Zhang
• Machine Learning Department
• Carnegie Mellon University
• April 30th, 2012

2
Thesis Committee

• Jeff Schneider, Chair
• Geoff Gordon
• Tom Mitchell
• Xiaojin (Jerry) Zhu, University of
  Wisconsin-Madison

3
Introduction
(x1,y1) (xn,yn)

• Learning a prediction system, usually based on
  examples
• Training examples are usually limited
• Cost of obtaining high-quality examples
• Complexity of the prediction problem

Learning
Y
X
4
Introduction
(x1,y1) (xn,yn)

• Solution exploit extra information about the
  input and output space
• Improve the prediction performance
• Reduce the cost for collecting training examples

Learning
Y
X
5
Introduction
(x1,y1) (xn,yn)
?
?

• Solution exploit extra information about the
  input and output space
• Representation and discovery?
• Incorporation?

Learning
Y
X
6
Outline
Part I Encoding Input Information by
Regularization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
7
Regularization

• The general formulation
• Ridge regression
• Lasso

8
Outline
Part I Encoding Input Information by
Regularization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
9
Learning with unlabeled text

• For a text classification task
• ? plenty of unlabeled text on the Web
• ? seemingly unrelated to the task
• What can we gain from such unlabeled text?

Yi Zhang, Jeff Schneider and Artur Dubrawski.
Learning the Semantic Correlation An Alternative
Way to Gain from Unlabeled Text. NIPS 2008
10
A motivating example for text learning

• Humans learn text classification effectively!
• Two training examples
• gasoline, truck
• - vote, election
• Query
• gallon, vehicle
• Seems very easy! But why?

11
A motivating example for text learning

• Humans learn text classification effectively!
• Two training examples
• gasoline, truck
• - vote, election
• Query
• gallon, vehicle
• Seems very easy! But why?
• Gasoline gallon, truck vehicle

12
A covariance operator for regularization

• Covariance structure of model coefficients
• Usually unknown -- learn from unlabeled text?

13
Learning with unlabeled text

• Infer the covariance operator
• Extract latent topics from unlabeled text (with
  resampling)
• Observe the contribution of words in each topic
• gas 0.3, gallon 0.2, truck 0.2, safety
  0.2,
• Estimate the correlation (covariance) of
  words

14
Learning with unlabeled text

• Infer the covariance operator
• Extract latent topics from unlabeled text (with
  resampling)
• Observe the contribution of words in each topic
• gas 0.3, gallon 0.2, truck 0.2, safety
  0.2,
• Estimate the correlation (covariance) of
  words
• For a new task, we learn with regularization

15
Experiments

• Empirical results on 20 newsgroups
• 190 1-vs-1 classification tasks, 2 labeled
  examples
• For any task, majority of unlabeled text (18/20)
  is irrelevant
• Similar results on logistic regression and least
  squares

1 V. Sindhwani and S. Keerthi. Large scale
semi-supervised linear svms. In SIGIR, 2006
16
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
17
Multi-task learning

• Different but related prediction tasks
• An example
• Landmine detection using radar images
• Multiple tasks different landmine fields
• Geographic conditions
• Landmine types
• Goal information sharing among tasks

18
Regularization for multi-task learning

• Our approach view MTL as estimating a parameter
  matrix

W
19
Regularization for multi-task learning

• Our approach view MTL as estimating a parameter
  matrix
• A covariance operator for regularizing a matrix?
• Vector w
• Matrix W

W
(Gaussian prior)
?
Yi Zhang and Jeff Schneider. Learning Multiple
Tasks with a Sparse Matrix-Normal Penalty. NIPS
2010
20
Matrix-normal distributions

• Consider a 2 by 3 matrix W
• The full covariance Kronecker product of
  and

full covariance
row covariance
column covariance

21
Matrix-normal distributions

• Consider a 2 by 3 matrix W
• The full covariance Kronecker product of
  and
• The matrix-normal density offers a compact form
  for

full covariance
row covariance
column covariance

22
Learning with a matrix-normal penalty

• Joint learning of multiple tasks
• Alternating optimization

Matrix-normal prior
23
Learning with a matrix-normal penalty

• Joint learning of multiple tasks
• Alternating optimization
• Other recent work as variants of special cases
• Multi-task feature learning Argyriou et al, NIPS
  06 learning with the feature covariance
• Clustered multi-task learning Jacob et al, NIPS
  08 learning with the task covariance and
  spectral constraints
• Multi-task relationship learning Zhang et al,
  UAI 10 learning with the task covariance

Matrix-normal prior
24
Sparse covariance selection

• Sparse covariance selection in matrix-normal
  penalties
• Sparsity of
• Conditional independence of rows (tasks) and
  columns (feature dimensions) of W

25
Sparse covariance selection

• Sparse covariance selection in matrix-normal
  penalties
• Sparsity of
• Conditional independence of rows (tasks) and
  columns (feature dimensions) of W
• Alternating optimization
• Estimating W same as before
• Estimating and L-1 penalized
  covariance estimation

26
Results on multi-task learning

• Landmine detection multiple landmine
  fields
• Face recognition multiple 1-vs-1 tasks

1 Jacob, Bach, and Vert. Clustered multi-task
learning A convex formulation. NIPS, 2008
2 Argyriou, Evgeniou, and Pontil. Multi-task
feature learning, NIPS 2006
27
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
28
Learning compressible models

• Learning compressible models
• A compression operator P instead of
• Bias model compressibility

Yi Zhang, Jeff Schneider and Artur Dubrawski.
Learning Compressible Models. SDM 2010
29
Energy compaction

• Image energy is concentrated at a few frequencies

JPEG (2D-DCT), 46 1 compression
30
Energy compaction

• Image energy is concentrated at a few frequencies
• Models need to operate at relevant frequencies

JPEG (2D-DCT), 46 1 compression
2D-DCT
31
Digit recognition

• Sparse vs. compressible
• Model coefficients w

sparse vs compressible
sparse vs compressible
sparse vs compressible
coefficients w
compressed coefficients Pw
coefficients w as an image
32
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Encode a dimension reduction
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
33
Dimension reduction

• Dimension reduction conveys information about the
  input space
• Feature selection ? importance
• Feature clustering ? granularity
• Feature extraction ? more general structures

34
How to use a dimension reduction?

• However, any reduction loses certain information
• May be relevant to a prediction task
• Goal of projection penalties
• Encode useful information from a dimension
  reduction
• Control the risk of potential information loss

Yi Zhang and Jeff Schneider. Projection Penalty
Dimension Reduction without Loss. ICML 2010
35
Projection penalties the basic idea

• The basic idea
• Observation reduce the feature space ? restrict
  the model search to a model subspace MP
• Solution still search in the full model space M,
  and penalize the projection distance to the model
  subspace MP

36
Projection penalties linear cases

• Learn with a (linear) dimension reduction P

37
Projection penalties linear cases

• Learn with projection penalties
• Optimization

projection distance
38
Projection penalties nonlinear cases
w
MP
M
P
wP
Rd
Rp
P
?
F
F
X
P
?
F
F
Yi Zhang and Jeff Schneider. Projection Penalty
Dimension Reduction without Loss. ICML 2010
39
Projection penalties nonlinear cases
w
MP
M
P
wP
Rd
Rp
M
w
MP
P
wP
F
F
X
w
MP
M
P
wP
F
F
Yi Zhang and Jeff Schneider. Projection Penalty
Dimension Reduction without Loss. ICML 2010
40
Empirical results

• Text classification (20 newsgroups), using
  logistic regression
• Dimension reduction latent Dirichlet allocation

Classification Errors
Projection Penalty
Projection Penalty
Original
Original
Reduction
Reduction
41
Empirical results

• Text classification (20 newsgroups), using
  logistic regression
• Dimension reduction latent Dirichlet allocation

Classification Errors
Similar results on face recognition, using SVM
(poly-2) Dimension reduction KPCA, KDA,
OLaplacian Face Similar results on house price
prediction, using regression Dimension reduction
PCA and partial least squares
42
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Encode a dimension reduction
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
43
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Encode a dimension reduction
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
44
Multi-label classification

• Multi-label classification
• Existence of certain label dependency
• Example classify an image into scenes (deserts,
  river, forest, etc)
• Multi-class problem is a special case only one
  class is true

Label dependency
Learn to predict

x
yq
y2
y1
45
Output coding

• d lt q compression, i.e., source coding
• d gt q error-correcting codes, i.e., channel
  coding
• Use the redundancy to correct prediction
  (transmission) errors

Learn to predict

x
z
z1
zd
z2
z3
encoding
decoding

yq
y2
y1
y
46
Error-correcting output codes (ECOCs)

• Multi-class ECOCs Dietterich Bakiri, 1994
  Allwein, Schapire Singer 2001
• Encode into a (redundant) set of binary problems
• Learn to predict the code
• Decode the predictions
• Our goal design ECOCs for multi-label
  classification

y1
y2 vs. y3
y3,y4 vs. y7
Learn to predict


x
zt
z2
z1
encoding
decoding

yq
y2
y1
47
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Encode a dimension reduction
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
48
Composite likelihood

• The composite likelihood (CL) a partial
  specification of the likelihood as the product of
  simple component likelihoods
• e.g., pairwise likelihood
• e.g., full conditional likelihood
• Estimation using composite likelihoods
• Computational and statistical efficiency
• Robustness under model misspecification

49
Multi-label problem decomposition

• Problem decomposition methods
• Decomposition into subproblems (encoding)
• Decision making by combining subproblem
  predictions (decoding)
• Examples 1-vs-all, 1-vs-1, 1-vs-1 1-vs-all,
  etc

Learn to predict
x

yq
y2
y1
50
1-vs-All (Binary Relevance)
Independently

• Classify each label independently
• The composite likelihood view

Learn to predict

x
yq
y2
y1
51
Pairwise label ranking 1

y1 vs. y2
yq-1 vs. yq
y1 vs. y3
Learn to predict
x

• 1-vs-1 method (a.k.a. pairwise label ranking)
• Subproblems pairwise label comparisons
• Decision making label ranking by counting the
  number winning comparisons, and thresholding

yq
y2
y1
1 Hullermeier et. al. Artif. Intell., 2008
52
Pairwise label ranking 1

y1 vs. y2
yq-1 vs. yq
y1 vs. y3
Learn to predict
x

• 1-vs-1 method (a.k.a. pairwise label ranking)
• Subproblems pairwise label comparisons
• Decision making label ranking by counting the
  number winning comparisons, and thresholding
• The composite likelihood view

yq
y2
y1
1 Hullermeier et. al. Artif. Intell., 2008
53
Calibrated label ranking 2

y1 vs. y2
yq-1 vs. yq
y1 vs. y3
Learn to predict

• 1-vs-1 1-vs-all (a.k.a. calibrated label
  ranking)
• Subproblems 1-vs-1 1-vs-all
• Decision making label ranking, and a smart
  thresholding based on 1-vs-1 and 1-vs-all
  predictions

x
Learn to predict

yq
y2
y1
2 Furnkranz et. al. MLJ, 2008
54
Calibrated label ranking 2

y1 vs. y2
yq-1 vs. yq
y1 vs. y3
Learn to predict

• 1-vs-1 1-vs-all (a.k.a. calibrated label
  ranking)
• Subproblems 1-vs-1 1-vs-all
• Decision making label ranking, and a smart
  thresholding based on 1-vs-1 and 1-vs-all
  predictions
• The composite likelihood view

x
Learn to predict

yq
y2
y1
2 Furnkranz et. al. MLJ, 2008
55
A composite likelihood view

• A composite likelihood view for problem
  decomposition
• Choice of subproblems ? specification of a
  composite likelihood?
• Decision making ? inference on the composite
  likelihood?

Learn to predict
x

yq
y2
y1
56
A composite pairwise coding

• Subproblems individual and pairwise label
  densities
• conveys more information
  than

yi0, yj0 yi0, yj1
yi1, yj0 yi1, yj1
yi0, yj0 yi0, yj1
yi1, yj0 yi1, yj1
Yi Zhang and Jeff Schneider. A Composite
Likelihood View for Multi-Label Classification.
AISTATS 2012
57
A composite pairwise coding

• Decision making a robust mean-field
  approximation
• is not robust to
  underestimation of label densities

Yi Zhang and Jeff Schneider. A Composite
Likelihood View for Multi-Label Classification.
AISTATS 2012
58
A composite pairwise coding

• Decision making a robust mean-field
  approximation
• is not robust to
  underestimation of label densities
• A composite divergence, robust and efficient to
  optimize

Yi Zhang and Jeff Schneider. A Composite
Likelihood View for Multi-Label Classification.
AISTATS 2012
59
Data sets

• The Scene data
• Image ? scenes (beach, sunset, fall foliage,
  field, mountain and urban)
• 
  ? beach, urban

Boutell et. al., Pattern Recognition 2004
60
Data sets

• The Emotion data
• Music ? emotions (amazed, happy, relaxed, sad,
  etc)
• The Medical data
• Clinical text ? medical categories (ICD-9-CM
  codes)
• The Yeast data
• Gene ? functional categories
• The Enron data
• Email ? tags on topics, attachment types, and
  emotional tones

61
Empirical results

• Similar results on other data sets (emotions,
  medical, etc)

1 Hullermeier et. al. Label ranking by learning
pairwise preferences. Artif. Intell., 2008
2 Furnkranz et. al. Multi-label classification
via calibrated label ranking. MLJ, 2008
3 Read et. al. Classifier chains for
multi-label classification. ECML, 2009
4 Tsoumak et. al. Random k-labelsets an
ensemble method for multilabel classification.
ECML, 2007
5 Zhang et. al. Multi-label learning by
exploiting label dependency. KDD, 2010
62
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Encode a dimension reduction
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
problem-dependent coding and code predictability
Composite likelihood for pairwise coding
Multi-label output codes with CCA
Maximum-margin output coding
63
Multi-label output coding

• Design output coding to multi-label problems
• Problem-dependent encodings to exploit label
  dependency
• Code predictability
• Propose multi-label ECOCs via CCA

?
?
?
Learn to predict


x
zt
z2
z1
encoding
decoding

yq
y2
y1
64
Canonical correlation analysis

• Given , CCA finds
  projection directions
• with maximum correlation

65
Canonical correlation analysis

• Given , CCA finds
  projection directions
• with maximum correlation
• Also known as the most predictable criterion
• CCA finds most predictable directions v in the
  label space

66
Multi-label ECOCs using CCA

• Encoding and learning
• Perform CCA
• Code includes both original labels and label
  projections

Learn to predict


x
z
z1
zd
yq
y2
y1
encoding
decoding

yq
y2
y1
y
Yi Zhang and Jeff Schneider. Multi-label Output
Codes using Canonical Correlation Analysis.
AISTATS 2011
67
Multi-label ECOCs using CCA

• Encoding and learning
• Perform CCA
• Code includes both original labels and label
  projections
• Learn classifiers for original labels
• Learn regression for label projections

Learn to predict


x
z
z1
zd
yq
y2
y1
encoding
decoding

yq
y2
y1
y
Yi Zhang and Jeff Schneider. Multi-label Output
Codes using Canonical Correlation Analysis.
AISTATS 2011
68
Multi-label ECOCs using CCA

• Decoding
• Classifiers Bernoulli on q
  original labels
• Regression Gaussian on d
  label projections

Learn to predict


x
z
z1
zd
yq
y2
y1
encoding
decoding

yq
y2
y1
y
Yi Zhang and Jeff Schneider. Multi-label Output
Codes using Canonical Correlation Analysis.
AISTATS 2011
69
Multi-label ECOCs using CCA

• Decoding
• Classifiers Bernoulli on q
  original labels
• Regression Gaussian on d
  label projections
• Mean-field approximation

Learn to predict


x
z
z1
zd
yq
y2
y1
encoding
decoding

yq
y2
y1
y
Yi Zhang and Jeff Schneider. Multi-label Output
Codes using Canonical Correlation Analysis.
AISTATS 2011
70
Empirical results

• Similar results on other criteria (macro/micro
  F-1 scores)
• Similar results on other data (emotions)
• Similar results on other base learners (decision
  trees, SVMs)

1 Furnkranz et. al. Multi-label classification
via calibrated label ranking. MLJ, 2008
2 D. Hsu, et. al. Multi-label prediction via
compressed sensing. NIPS, 2009
3 Zhang and Schneider. A composite likelihood
view for multi-label classification. AISTATS 2012
71
Outline
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Encode a dimension reduction
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
problem-dependent coding and code predictability
Composite likelihood for pairwise coding
Discriminative and predictable codes
Multi-label output codes with CCA
Maximum-margin output coding
72
Recall coding with CCA

• CCA finds label projections z that are most
  predictable
• Low transmission errors in channel coding

Learn to predict


x
z
z1
zd
yq
y2
y1
encoding
decoding

yq
y2
y1
y
z
predict
x
73
A recent paper 1 coding with PCA

• Label projections z obtained by PCA
• z has maximum sample variance, i.e., far away
  from each other.
• Minimum code distance?

Learn to predict


x
z
z1
zd
yq
y2
y1
encoding
decoding

yq
y2
y1
y
z
1 Tai and Lin, 2010
74
Goal predictable and discriminative codes

• Predictable prediction is closed to the correct
  codeword
• Discriminative prediction is far away from
  incorrect codewords

Learn to predict


x
z
z1
zd
yq
y2
y1
encoding
decoding

yq
y2
y1
y
z
predict
x
75
Maximum margin output coding

• A max-margin formulation

z
predict
x
76
Maximum margin output coding

• A max-margin formulation
• Assume M is best linear predictor (in closed form
  of X, Y, V)
• Reformulate using metric learning
• Deal with the exponentially large number of
  constraints
• The cutting plane method
• Overgenerating

77
Maximum margin output coding

• A max-margin formulation
• Assume M is best linear predictor, and define

78
Maximum margin output coding

• A max-margin formulation
• Metric learning formulation define the
  Mahalanobis metric
• and the notation

79
Maximum margin output coding

• The metric learning problem
• An exponentially large number of constraints
• Cutting plane method? No polynomial-time
  separation oracle!

80
Maximum margin output coding

• The metric learning problem
• An exponentially large number of constraints
• Cutting plane method? No polynomial-time
  separation oracle!
• Cutting plane method with overgenerating
  (relaxation)
• Relax into
• Linearize for the relaxed domain
• New separation oracle box-constrained QP

81
Empirical results

• Similar results on other data (emotions and
  medical)

1 Furnkranz et. al. Multi-label classification
via calibrated label ranking. MLJ, 2008
2 Zhang et. al. Multi-label learning by
exploiting label dependency. KDD, 2010
3 D. Hsu, et. al. Multi-label prediction via
compressed sensing. NIPS, 2009
4 Tai and Lin. Multi-label Classification with
Principal Label Space Transformation. Neur. Comp.
5 Zhang and Schneider. Multi-label output codes
via canonical correlation analysis, AISTATS 2011
82
Conclusion

• Regularization to exploit input information
• Semi-supervised learning with word correlation
• Multi-task learning with a matrix-normal penalty
• Learning compressible models
• Projection penalties for dimension reduction
• Output coding to exploit output information
• Composite pairwise coding
• Coding via CCA
• Coding via max-margin formulation
• Future

83
Thank you! Questions?
Part I Encoding Input Information by
Regularization
Multi-task generalization
Learning with word correlation
A matrix-normal penalty for multi-task learning
Go beyond covariance and correlation structures
Encode a dimension reduction
Learn compressible models
Projection penalties
Part II Encoding Output Information by Output
Codes
problem-dependent coding and code predictability
Composite likelihood for pairwise coding
Discriminative and predictable codes
Multi-label output codes with CCA
Maximum-margin output coding
84
(No Transcript)
85
(No Transcript)
86
Local smoothness

• Smoothness of model coefficients
• Key property certain order of derivatives are
  sparse

Differentiation operator
87
Brain computer interaction

• Classify Electroencephalography (EEG) signals
• Sparse models vs. piecewise
  smooth models

88
Projection penalties linear cases

• Learn a linear model with a given linear
  reduction P

89
Projection penalties linear cases

• Learn a linear model with a given linear
  reduction P

90
Projection penalties linear cases

• Learn a linear model with projection penalties

projection distance
91
Projection penalties RKHS cases
M
w

• Learning in RKHS with projection penalties
• Primal
• Solve for in the dual (see the next page)
• Solve for v and b in the primal

MP
P
wP
F
F
X
92
Projection penalties RKHS cases
M
w

• Representer theorem for
• Dual

MP
P
wP
F
F
X
93
Projection penalties nonlinear cases

• Learning linear models
• Learning RKHS models

P
P
P
F
Rd
F
F
Rp
F
X
?
P(xi)
94
Empirical results

• Face recognition (Yale), using SVM (poly-2)
• Dimension reduction KPCA, KDA, OLaplacian

Classification Errors
95
Empirical results

• Face recognition (Yale), using SVM (poly-2)
• Dimension reduction KPCA, KDA, OLaplacian

Classification Errors
96
Empirical results

• Face recognition (Yale), SVM (poly-2)
• Dimension reduction KPCA, KDA, OLaplacian

Classification Errors
97
Empirical results

• Price forecasting (Boston house), ridge
  regression
• Dimension reduction partial least squares

1-R2
98
Binary relevance
Independently

• Binary relevance (a.k.a. 1-vs-all)
• Subproblems classify each label independently
• Decision making same
• Assume no label dependency

Learn to predict

x
yq
y2
y1
99
Binary relevance
Independently

• Binary relevance (a.k.a. 1-vs-all)
• Subproblems classify each label independently
• Decision making same
• Assume no label dependency
• The composite likelihood view

Learn to predict

x
yq
y2
y1
100
Empirical results

• Emotion data (classify music to different
  emotions)
• Evaluation measure subset accuracy

1 Furnkranz et. al. Multi-label classification
via calibrated label ranking. MLJ, 2008
2 D. Hsu, et. al. Multi-label prediction via
compressed sensing. NIPS, 2009