Feature Selection.pdf

Feature Selection
IFT6758 - Data Science
http://www.public.asu.edu/~jundongl/tutorial/KDD17/KDD17.pdf
Resources:
http://dongguo.me

What is feature selection?
• A procedure in machine learning to find a subset of features that produces
‘better’ model for given dataset
• Avoid overfitting and achieve better generalization ability
• Reduce the storage requirement and training time
• Interpretability
!2

Relevant vs. Redundant features
• Feature selection keeps relevant features for learning and removes redundant
and irrelevant features
• For example, for a binary classification task (f1 is relevant; f2 is redundant
given f1; f3 is irrelevant)
!3

Feature Extraction vs. Feature Selection
• Commonalities
– Speed up the learning process
– Reduce the storage requirements
– Improve the learning performance
– Build more generalized models
• Diﬀerences
– Feature extraction obtains new features while feature selection selects a
subset of original ones
– Feature selection maintains physical meanings and gives models better
readability and interpretability
!4

Interpretability of Learning
Algorithms
!5
More about this topic (Week 14)

When feature selection is important?
• Noisy data
• Lots of low frequent features
• Use multi-type features
• Too many features comparing to samples
• Complex model
• Samples in real scenario is inhomogeneous with training & test samples
!6

Feature Selection Algorithms
• From the label perspective (whether label information is involved during the
selection phase):
– Supervised
– Unsupervised
– Semi-Supervised
• From the selection strategy perspective (how the features are selected):
– Wrapper methods
– Filter methods
– Embedded methods
!7

Supervised Feature Selection
• Supervised feature selection is often for classification or regression problems
• Find discriminative features that separate samples from diﬀerent classes
(classification) or approximate target variables (regression)
!8

Unsupervised Feature Selection
• It is often for clustering problems
• Label information is expensive to obtain which requires both time and eﬀorts
• Unsupervised methods seek alternative criteria to define feature relevance
!9

Semi-Supervised Feature Selection
• We often have a small amount of labeled data and a large amount of
unlabeled data
• Semi-supervised methods exploit both labeled and unlabeled data to find
relevant features
!10

!11
• Step 1: search for a subset of features
• Step 2: evaluate the selected features
• Repeat Step 1 and Step 2 until stopped
Wrapper Methods

!12
• Subset selection method : Two types: Forward Search and Backward Search
• Forward Search
• Start with no features
• Greedily include the most relevant feature
• Stop when selected the desired number of features
• Backward Search
• Start with all the features
• Greedily remove the least relevant feature
• Stop when selected the desired number of features
• Inclusion/Removal criteria uses cross-validation
Feature Selection Techniques

Wrapper Methods
• Can be applied for ANY model!
• Rely on the predictive performance of a predefined learning algorithm to assess features
• Shrink / grow feature set by greedy search
• Repeat until some stopping criteria are satisfied
• Achieve high accuracy for a particular learning method
• Run CV / train-val split per feature
• Computational expensive (worst case search space is 2 ) , some typical search strategies are
– Sequential search
– Best-first search
– Branch-and-bound search
!13
d

Filter Methods
• Independent of any learning algorithms
• Relying on certain characteristics of data to assess feature importance (e.g.,
feature correlation, mutual information…)
• More eﬃcient than wrapper methods
• The selected features may not be optimal for a particular learning algorithm
!14

!15
• Single feature evaluation: Measure quality of features by all
kinds of metrics
• Frequency based
• Dependence of feature and label (Co-occurrence), e.g., Mutual
information, Chi square statistic
• Information theory, KL divergence, Information gain
• Gini indexing

Embedded Methods
• A trade-oﬀ between wrapper and filter methods by embedding feature
selection into the model learning, e.g., ID3
• Inherit the merits of wrapper and filter methods
– Include the interactions with the learning algorithm
– More eﬃcient than wrapper methods
• Like wrapper methods, they are biased to the underlying learning algorithms
!16

Traditional Feature Selection
!18
Similarity based
methods
Information
Theoretical based
methods
Sparse Learning
based methods
Statistical based
methods

Similarity Technique
• Pairwise data similarity is often encoded in the data similarity matrix
• E.g., without class label information, it can be defined by the RBF kernel
• E.g., using the class labels, the similarity can be obtained as
!19

Similarity based Feature Selection
• Similarity based methods assess the importance of features by their ability to
preserve data similarity
• A good feature should not randomly assign values to data instances
• A good feature should assign similar values to instances that are close to
each other – (the “closeness” is obtained from data similarity matrix)
!20
Different shapes denote different
values assigned by a feature

• Suppose data similarity matrix is to find the most relevant
features , we need to maximize:
• It is often solved by greedily selecting the top features that maximize their
individual utility
• Diﬀerent methods vary in the way how the vector f and similarity matrix S are
transformed to and
!21
Similarity based Methods – A
General Framework

Laplacian Score [He et al., 2005]
• First, it builds the data similarity matrix S, diagonal matrix D and Laplacian
matrix L without using class labels
• Motivation: a good feature should (1) preserve data similarity structure; and (2)
have high feature variance
• Laplacian score is also equivalent to:
• A special case of the similarity-based FS framework
!22
• Then the Laplacian Score of feature
• Then the Laplacian Score of feature is :

Spectral Feature Selection [Zhao and
Liu, 2007]
• Eigenvectors of similarity matrix S carry the data distribution
• Observation: eigenvectors assign similar values to the samples that are of the
same aﬃliations
!23

• Measure features’ consistency by comparing it with the eigenvectors (e.g. , )
using inner product ′
• By considering all eigenvectors, the feature score is:
•
!24
Spectral Feature Selection [Zhao and
Liu, 2007]

Fisher Score [Duda et al., 2001]
• Given class labels, within class and between class data similarity matrix
(local affinity) and (global affinity) are defined as
• is larger if and belong to the same class, smaller otherwise
• is larger if and belong to the different classes, smaller otherwise
• A good feature should make instances from different classes far away and
make instances from the same class close to each other
!25

• The score of the i-th feature is
• Fisher Score can be calculated from Laplacian Score
!26
Fisher Score [Duda et al., 2001]

Trace Ratio Criteria [Nie et al., 2008]
• Fisher score evaluates the importance of features individually, which may
lead to suboptimal solution
• Trace Ratio attempts to assess the importance of a subset of features ℱ
simultaneously
• Maximizing the above score is equivalent to maximize the following, which is
a special case of the general framework
!27

Similarity based Methods
Summary
• Many others can also be reduced to the general similarity based feature
selection framework
– Batch-mode Laplacian score [Nie et al. 2008]
– RelieF [Robnik-Sikonja and Kononenko, 2003]
– HSIC Criterion [Song et al. 2007] …
• Pros
– Simple and easy to calculate the feature scores
– Selected features can be generalized to subsequent learning tasks
• Cons
– Most methods cannot handle feature redundancy
!28

!29
Similarity based
methods
Information
Theoretical based
methods
Sparse Learning
based methods
Statistical based
methods

Information Theoretical based
Methods
• Exploit diﬀerent heuristic filter criteria to measure the importance of features
• Our target is to find these “optimal” features
!30

Preliminary - Information Theoretical
Measures
• Entropy of a discrete variable X
• Conditional entropy of X given Y
!31

Preliminary - Information Theoretical
Measures
• Information gain between X and Y
• Conditional information gain
!32

Information Theoretic based
Methods - A General Framework
• Searching for the best feature subset is NP-hard, most methods employ
forward/backward sequential search heuristics
• E.g., for forward search, given selected features S, we should do the
following for the next selected feature
– Maximize its correlation with class labels Y:
– Minimize the redundancy w.r.t. selected features in S:
– Maximize its complementary info w.r.t. selected features in S:
!33

• Given selected features S, the feature score for the next selected feature
can be determined by
• If g(*) is a linear function, then it can be represented as
• But also, g(*) can be a nonlinear function
!34
Information Theoretic based
Methods - A General Framework

Information Gain [Lewis, 1992]
• Information gain only measures the feature importance by its correlation with
class labels
• The information gain of a new unselected feature
• Selecting features independently
• It is a special case of the linear function by setting
!35

Mutual Information Feature Selection
[Battiti, 1994]
• Information gain only considers feature relevance
• Features also should not be redundant to each other
• The score of a new unselected feature
• It is also a special case of the linear function by setting
!36

Minimum Redundancy Maximum
Relevance [Peng et al., 2005]
• Intuitively, with more selected features, the eﬀect of feature redundancy
should gradually decrease
• Meanwhile, pairwise feature independence becomes stronger
• The score of a new unselected feature x is
• MRMR is also a special case of the linear function by setting and
adjusting adaptively
!37

Conditional Infomax Feature
Extraction [Lin and Tang, 2006]
• Correlated feature is useful if the correlation within classes is stronger than
the overall correlation
• Correlation does not imply redundancy! [Guyon et al. 2006]
• It is also a special case of the linear function by
!38

Function g(∗) Can Also Be Nonlinear
• Conditional Mutual Information Maximization [Fleuret, 2004]
• Information Fragments [Vidal-Naquet and Ullman, 2003]
!39

Function g(∗) Can Also Be Nonlinear
• Interaction Capping [Jakulin, 2005]
• Double Input Sym Relevance [Meyer and Bontempi, 2006]
!40

Information Theoretical based
Methods - Summary
• Other information theoretical based methods
– Fast Correlation Based Filter [Yu and Liu, 2004]
– Interaction Gain Feature Selection [El Akadi et al. 2008]
– Conditional MIFS [Cheng et al. 2011]…
• Pros
– Can handle both feature relevance and redundancy
– Selected features can be generalized for subsequent learning tasks
• Cons
– Most algorithms can only work in a supervised scenario
– Can only handle discretized data
!41

!42
Similarity based
methods
Information
Theoretical based
methods
Sparse Learning
based methods
Statistical based
methods

What is Feature Sparsity?
• The model parameters in many data mining tasks can be represented as a
vector w or a matrix W
• Sparsity indicates that many elements in w and W are small or exactly zero
!43

Sparse Learning Methods - A
General Framework
• Let us start from the binary classification or the univariate regression problem
• Let w denote the model parameter (a.k.a. feature coeﬃcient), it can be
obtained by solving
!44

Lasso [Tibshirani, 1996]
• Based on ℓ=-norm regularization on weight
• In the case of least square loss with oﬀset value, it looks like this …
• It is also equivalent to the following model
!45

Why ℓ=-norm Induces Sparsity?
!46

Extension to Multi-Class or Multi-
Variate Problems
• Require feature selection results to be consistent across multiple targets in
multi-class classification or multi-variate regression
• ||W||2 achieves joint feature sparsity across multiple targets • In the case of
least square loss with oﬀset, it looks like this
!47

Unsupervised Sparse Learning
based Feature Selection
• Without class labels, we attempt to find discriminative features that can
preserve data clustering structure
• There are two options
– Obtain clusters and then perform FS (e.g., MCFS)
– Embed FS into clustering (e.g., NDFS)
• The 2nd option is preferred as not all features are useful to find clustering
structure
!48

Multi-Cluster Feature Selection
(MCFS) [Cai et al., 2011]
• Basic idea: the selected features should preserve cluster structure
• Step 1: spectral clustering to find intrinsic cluster structure
• Step 2: perform Lasso on each cluster
• Step 3: combine multiple feature coeﬃcient
together and get feature score
!49

Nonnegative Unsupervised Feature
Selection (NDFS) [Li et al., 2012]
• Perform spectral clustering and feature selection jointly
• The weighted cluster indicator matrix G can be obtained by using nonnegative
spectral analysis
• Embed cluster matrix into feature selection
• Feature score obtained from W (higher the value, the better)
!50

Sparse Learning based Methods -
Summary
• Other sparse learning based methods
– Multi-label informed feature selection [Jian et al. 2016]
– Embedded unsupervised feature selection [Wang et al. 2015]
– Adaptive structure learning feature selection [Du et al. 2015]
• Pros
– Obtain good performance for the underlying learning method
– With good model interpretability
• Cons
– The selected features may not be suitable for other tasks
– Require solving non-smooth optimization problems, which is computational
expensive
!51

!52
Similarity based
methods
Information
Theoretical based
methods
Sparse Learning
based methods
Statistical based
methods

Statistical based Methods
• This family of algorithms are based on diﬀerent statistical measures to
measure feature importance
• Most of them are filter feature selection methods
• Most algorithms evaluate features individually, so the feature redundancy is
inevitably ignored
• Most algorithms can only handle discrete data, the numerical features have
to be discretized first
!53

T-Score [Davis and Sampson, 1986]
• It is used for binary classification problems
• Assess whether the feature makes the means of samples from two classes
statistically significant
• The t-score of each feature is
• The higher the T-score, the more important the feature is
!54

Chi-Square Score [Liu and Setiono,
1995]
• Utilize independence test to assess whether the feature is independent of
class label
• Given a feature with r values, its feature score is
• Higher chi-square indicates that the feature is more important
!55

Statistical based Methods -
Summary
• Other statistical based methods
– Low variance – CFS [Hall and Smith, 1999]
– Kruskal Wallis [McKnight, 2010]…
• Pros
– Computational eﬃcient
– The selected features can be generalized to subsequent learning tasks
• Cons
– Cannot handle feature redundancy
– Require data discretization techniques
– Many statistical measures are not that eﬀective in high-dim space
!56

!57
Similarity based
methods
Information
Theoretical based
methods
Sparse Learning
based methods
Statistical based
methods
• Reconstruction based Feature Selection
– Minimize reconstruction error of data with selected features
– Reconstruction function can be both linear and nonlinear
• Hybrid Feature Selection
– Construct a set of diﬀerent feature selection results
– Aggregate diﬀerent outputs into a consensus result
Other Types of Methods

Feature Selection Issues
• Recent popularity of big data presents challenges to conventional FS
– Streaming data and features
– Heterogeneous data
– Structures between features
– Volume of collected data
!58

Feature Selection with Heterogeneous Data
!59
Feature Selection with Structured Features
Multi-Source Feature Selection

Feature Selection with Structured
Features - A Framework
• A popular and successful approach is to minimize the fitting error penalized
with structural regularization
• The above formulation is flexible in incorporating various types of structures
among features
!60

Group Structure – Group Lasso [Yuan
and Lin, 2006]
• Features form group structure in many applications
• Group lasso selects or does not select a group as a whole
!61

Sparse Group Lasso [Friedman et al.,
2010]
• For certain applications, it is desirable to select representative features from
selected groups
• Sparse group lasso performs group selection and feature selection
simultaneously
!62

Group Structure - Summary
• Comparison between Lasso, Group Lasso and Sparse Group Lasso
!63

Tree Structure Among Features
• Features can also exhibit tree (hierarchical) structure
– Pixels of face images
– Gene expression
– Words of documents
!64

Tree-Guided Group Lasso [Liu and
Ye, 2010]
• Leaf nodes are individual features
• Internal nodes are a group of features
• Each internal node has a weight indicates how tight the features in its subtree
are correlated
!65

Graph Structure Among Features
• Features can also exhibit graph structure
– Synonyms and antonyms between diﬀerent words
–Regulatory relationships between genes
!66

Graph Lasso [Ye and Liu, 2012]
• Two nodes are connected if two features and tend to be selected
together
• Their feature weights are similar
• Impose a regularization on the feature graph
!67

Graph-Guided Fused Lasso
(GFLasso) [Kim and Xing, 2009]
• Graph Lasso assume features connected together have similar feature
coeﬃcients
• However, features can also be negatively correlated
• GFLasso explicitly considers both positive and negative feature correlations
!68

Feature Selection with Structured
Features - Summary
• Incorporate feature structures as prior knowledge
• Pros
– Improve the learning performance in many cases
– Make the learning process more interpretable
• Cons
–Require label information to guide feature selection
–Require to solve complex non-smooth optimization problems
!69

!70

Feature Selection with
Heterogeneous Data
!71

Feature Selection with
Heterogeneous Data
• Traditional feature selection algorithms are for a single source and are heavily
based on the data i.i.d. assumption
• Heterogeneous data is prevalent and is often not i.i.d.
– Networked data
– Data from multiple sources
• It is necessary to leverage feature selection to fuse multiple data sources
synergistically
!72

Why Performing Feature Selection
with Networks?
• Social Influence & Homophily: node features and network are inherently
correlated
• Many learning tasks are enhanced by modeling the correlation collective
classification
– Community detection
– Anomaly detection
– Collective classification
• But not all features are hinged with the network structure!
!73

Challenges of Feature Selection with
Networked Data
• Feature selection on networked data faces unique challenges
– How to model link information
– How to fuse heterogeneous information sources
– Label information is costly to obtain
• Unique properties from network and features of instances bring more
challenges
• Unique properties from network and features of instances bring more
challenges
!74

Feature Selection on Networks
(FSNet) [Gu and Han, 2011]
• Use a linear classifier to capture the relationship between content information
X and class labels Y
• Employ graph regularization to model links
• Objective function of FSNet
!75

Linked Feature Selection (LinkedFS)
[Tang and Liu, 2012]
• Investigate feature selection on social media data with various types of social
relations: four basic types
• These social relations are supported by social theories (Homophily and Social
Influence)
!76

Linked Feature Selection (LinkedFS)
[Tang and Liu, 2012]
• For CoPost hypothesis
– Posts by the same user are likely to be of similar topics
• Feature selection with the CoPost hypothesis
!77

Personalized Feature Selection [Li et
al., 2017]
• Content information of nodes are highly idiosyncratic
– E.g., blogs, posts and images of different users could be diverse and with
different social foci
– E.g., the same content could convey different meanings: “The price comes
down! #apple” [Wu and Huang 2016]
• But, nodes share some commonality to some extent
• How to tackle the idiosyncrasy and commonality of node features for learning
such as node classification?
!78

al., 2017]
• To find personalized features, we attempt to achieve feature sparsity within
each local feature weight
• The exclusive group lasso encourages intra-group competition but
discourages inter-group competition
!79

al., 2017]
• We cluster local weights into groups to reduce overfitting
• The objective function
!80

Linked Unsupervised Feature
Selection (LUFS) [Tang and Liu, 2012]
• Data is often unlabeled in networked data
• No explicit definition of feature relevance
• Fortunately, links provide additional constraints
!81

• Obtain within, between and total social dimension scatter matrix
• Instances are similar within social dimensions while dissimilar between social
dimensions
• Similar instances in terms of their contents are more likely to share similar
topics
!82
, and

• Optimization framework of LUFS
• Spectral relaxation on and impose. -norm regularization
!83

Robust Unsupervised on Networks
(NetFS) [Li et al., 2016]
• LUFS performs network structure modeling and feature selection separately
• NetFS embeds latent representation modeling into feature selection and is
more robust to noise links
!84

Robust Unsupervised on Networks
(NetFS) [Li et al., 2016]
!85

!86

Multi-Source Feature Selection [Zhao
and Liu, 2008]
• Given multiple local geometric patterns in aﬃnity matrix , the global
pattern is
• Geometry-dependent sample covariance matrix for the target source is
• Two ways to obtain relevant features from
– Sort the diagonal of and return the features with the highest variances
(consistent with global pattern)
– Apply sparse PCA to select features that are able to retain the total variance
maximally
!87

Evaluation of Feature Selection
• Feature weighting: given a desired feature number k, rank features according
to the feature scores, and then return the top-k
• Feature subset selection: directly return the obtained feature subset (cannot
specify beforehand)
!88

Evaluation of Feature Selection -
Supervised
Supervised feature selection
1. Divide data into training and testing set
2. Perform feature selection to obtain selected features
3. Obtain the training and testing data on the selected features
4. Feed into a classifier (e.g., SVM)
5. Obtain the classification performance on (e.g., F1, AUC)
!89
The higher the classification performance,
the better the selected features are

Evaluation of Feature Selection -
Unsupervised
Unsupervised feature selection
1. Perform feature selection on data to obtain selected features
2. Obtain new data on the selected features
3. Perform clustering (given #m clusters)
4. Compare the obtained clustering with the ground truth
5. Obtain clustering evaluation results (e.g., NMI)
!90
The higher the clustering performance,
the better the selected features are

Challenges of Feature Selection
• Scalability
• Stability Challenge
•
!91

Scalability Challange
Data size
• With the growth of data size, the scalability of most feature selection
algorithms is jeopardized
• Data of TB scale cannot be easily loaded into memory and limits the usage of
FS algorithms
• In many cases, one pass of data is desired, the second or more pass can be
impractical
!92
Potential Solution: use distributed programming framework
to perform parallel feature selection

Scalability Challange
Feature size
• Most existing feature selection algorithms have a time complexity
proportional to or even ?
• Data of ultrahigh-dimensionality emerges
– Text mining
– Information retrieval
– Brain image
• For many feature selection algorithms, eﬃciency deteriorates quickly as
could be very large
• Well-designed feature selection algorithms work in linear or sub-linear time
are preferred
!93

Stability Challange
• Stability of FS algorithms is also an important measure
• Definition: the sensitivity of a feature selection algorithm to the perturbation
of training data
!94

Achieving Stability
• Perturbation of training data in various formats
– Addition/deletion of training samples
– Inclusion of noisy/outlier samples
• Stability of feature selection helps domain experts be more confident with the
selected features
– Biologists would like to see the same set of genes selected each time when
they obtain new data; otherwise they will not trust the algorithm
• Many feature selection algorithms suﬀer from low stability with small
perturbation!
!95

Model Selection
Which Set of Features to Use?
• We usually need to specify the number of selected features in feature
weighting methods
• Finding the “optimal” number is diﬃcult
– A large number will increase the risk in including, irrelevant and redundant
features, jeopardizing learning performance
– A small number will miss some relevant features
• Solution: apply heuristics such as “grid search” strategy, but performing “grid
search” is very time-consuming
• Choosing the # of selected features is still an open problem!
!96

Model Selection for Unsupervised
Learning
• In unsupervised feature selection, we often need to specify the number of
cluster or pseudo class labels
• However, we often have limited knowledge about the intrinsic cluster
structure of data
• Different cluster number may lead to different cluster structures
– May merge smaller clusters into a big cluster
– May split one big cluster into multiple small clusters
• Lead to different feature selection results
• Without label information, we cannot perform cross validation
!97

Privacy and Security Issues in
Feature Selection
• Many collected data for learning are highly sensitive, e.g., medical details,
census records, …
• Most feature selection algorithms cannot address the privacy issues
– require privacy-preserving FS
• Feature privacy
– Find optimal feature subset with the total privacy degree less than a given
threshold
• Sample privacy (diﬀerential privacy)
– Know all but one entry of the data, and cannot gain additional info about
the entry with the output of the algorithm
!98
We will cover more in Week 14

Summary
• Feature selection is eﬀective to tackle the curse of dimensionality and is
essential to many data mining and machine learning problems
• The objectives of feature selection include
– Building simpler and more comprehensive models
– Improving learning performance
– Preparing clean and understandable data
• Feature selection is equally important in the age of deep learning and big data
• We provide a structured overview of feature selection from a data perspective
– Feature selection for conventional data (four main categories)
– Feature selection with structured features
– Feature selection with heterogeneous data
– Feature selection with streaming data
!99

References
• Check all the references at the end of:
http://www.public.asu.edu/~jundongl/tutorial/KDD17/KDD17.pdf
!100

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