linear models for classification - chapter 4 in prml
TRANSCRIPT
LINEAR MODELS FOR CLASSIFICATIONChapter 4 in PRML
Yun Gu
Department of AutomationShanghai Jiao Tong University
Shanghai, CHINA
May 12, 2014
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Outline
1 Overview of Classification
2 Discriminant FunctionsFrom Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
3 Prob. Generative ModelsLinear Discriminant AnalysisQuadratic Discriminant Analysis
4 Prob. Discrimintive ModelsLogistic RegressionLDA or Logistic Regression?
5 BonusMulti-Class, Multi-Label, Struct-Output
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Outline of this section
1 Overview of Classification
2 Discriminant Functions
3 Prob. Generative Models
4 Prob. Discrimintive Models
5 Bonus
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
What is Classification?
Definition (Classification)
Classification is the problem of identifying to which of a set ofcategories (sub-populations) a new observation belongs, on the basis of atraining set of data containing observations (or instances) whose categorymembership is known.
Input: Feature vector x, continuous or discrete;
Output: Qualitative values (i.e. categorical or discrete values) y ,where y ∈ Y, Y = {y1, y2, ..., yn};Mapping: y = f (x) : x→ y ;
Supervised Learning/Semi-supervised Learning.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Example:Image Classification
Figure : An example of image classfication. Input: Visual Feaures(e.g. SIFT,Color Moment); Output: Scene Category (e.g. ”Bridge”, ”Castle”); Mapping:A Sparse Factor Representation (see ”Multi-Label Image Categorization WithSparse Factor Representation”, Sun, et.al. IEEE TIP 2014)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Outline of this section
1 Overview of Classification
2 Discriminant FunctionsFrom Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
3 Prob. Generative Models
4 Prob. Discrimintive Models
5 Bonus
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Two-Class Problem
Let’s start with the simplest case: Two-class Classification, where:
Input: Feature vector x, continuous or discrete;
Output: Binary values y , where y ∈ Y, Y = {−1,+1} orY = {0, 1};Model: y(X) = wTx + w0, x is assigned with −1, if y(x) ≥ 0;Otherwise, x is assigned with +1.
Examples: Male or Female; Toy Data.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Example of Two-Class Problem: On ”Toy” Dataset
Figure : A demonstration of Classification on ”toy” dataset via Semi-supervisedLearning. (see ”Graph Transduction via Alternating”, Wang, et.al, ICML, 2008)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
From Two-Class to Multiple-Class
A complex case: Multiple-class Classification, where:
Input: Feature vector x, continuous or discrete;
Output: For a n-class problem, Qualitive values C , where C ∈ C,C = {C1,C2, ...,Cn};Model:
Binary: One-vs-OneBinary: One-vs-RestBinary: ECOCDirectly comprising n Linear functions
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
From Two-Class to Multiple-Class (Cont.)
Scheme 1: Decompose a n-class problem into several two-classesproblems:
One-vs-Rest: Use n − 1 binary classifiers each of which solves atwo-class problem of separating points in a particular class Ck frompoints not in that class.
Pros: Easy and Efficient;Cons: Ambigious samples; Unbalanced Samples.
One-vs-One: Use (n − 1)n/2 binary classifiers, one for evertpossible pairs of classes. Each point is classified according to amajority vote amongst the discriminant functions.
Pros: ”Better” performance than ”One-vs-Rest”Cons: Ambigious samples; Too many classifiers;
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Problems from 1-vs-1 and 1-vs-R
Figure : Attempting to construct a n class discriminant from a set of two classdiscriminants leads to ambiguous regions, shown in green. On the left is anexample involving the use of two discriminants designed to distinguish points inclass Ck from points not in class Ck . On the right is an example involving threediscriminant functions each of which is used to separate a pair of classes Ck
and Cj . (PRML pp.183)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
From Two-Class to Multiple-Class (Cont.)
ECOC (Error-Correcting Output Codes:) Given a set of nclasses, the basis of the ECOC framework consists of designing acodeword for each of the classes. These codewords encode themembership information of each class for a given binary problem.
Pros: Robust;Cons: The construction of codebook.
4-Class One-vs-Rest Scheme
y1
y2
y3
y4
ECOC Extension
Figure : ECOC for a 4-Class ProblemY. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
From Two-Class to Multiple-Class (Cont.)
A comprehensive evaluation on different schemes for”TwoClass2MultipleClass” has been conducted (R. Rifkin and A.Klautau,”In Defense of One-Vs-All Classification”,JMLR,2004). Their main thesisis that a simple ”one-vs-all” scheme is as accurate as any other approach,assuming that the underlying binary classiers are well-tuned regularizedclassiers such as support vector machines. This thesis is interesting inthat it disagrees with a large body of recent published work on multiclassclassication. They support our position by means of a critical review ofthe existing literature, a substantial collection of carefully controlledexperimental work, and theoretical arguments.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
From Two-Class to Multiple-Class (Cont.)
Directly comprising n Linear functions:
yk(x) = wTx + w0, k = 1, 2, ..., n
x is assigned with Class Ck if yk≥yj(k ∈ {1, 2, ..., n}, k 6=j)
Pros: Likelihood Score, Convex;Cons: Easy to solve? Cannot use the binary mode;
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Methods for Classification
Discriminant Functions:
Data Fitting: Least Squares Estimation;Feature Reduction: Fisher’s Discriminant Analysis;Find a Separating Hyperplane: Perceptron and SVM.
Probalistic Generative Model: LDA/QDA.
Probalistic Discriminative Model: Logistic Regression.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Least Squares For Classification
Least Squares: Similar to linear regression, we can solve theclassification via least squares estimation.
Data: Given the training set {(xn, tn)}, n = 1, 2, ...,N, xn is thefeature vector; tn is a indicator vector. For a K-Class problem,tn,k = 1, tn,i = 0(i 6=k), if xn belongs to Ck .Decision Function: For Class Ck , yk((x)) = wk
Tx + wk,0. x isassigned with Ck , if yk(x)≥yi (x), k 6=iOpt Problem:
minW
L(W) = 12Tr{(WTX− T)(WTX− T)T}
s.t. X = {x1, x2, ..., xN};W = {w1
T ,w2T , ...,wK
T}T ;
T = {tT1 , tT2 , ..., tTK}T .
Cons: Sensitive to singular data; Poor performance with asymetrydistributed data.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Least Squares For Classification (Cont.)
Figure : Failure cases: Left:Singular data; Right: Asymetry data
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Fisher’s Linear Discriminant (Fisher, 1936)
Idea: Treat the classification problem as feature reduction(projected to 1-D);
Figure : The corresponding projection based on the Fisher linear discriminant.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Fisher’s Linear Discriminant (Cont.)
Criterion: Maximize the between-class variance and Minimize thewithin-class variance.
Data: For a 2-Class problem, training set {(xn, yn}.n = 1, 2, ...,N,yn∈{−1,+1}Decision Function: y = wTx
Opt Problem:
maxw
J(w) = (m2−m1)2
S21 +S2
2= wT SBw
wT SWw
s.t. mk = wTmk; mk = 1Nk
∑xn∈Ck
xn;
s2k =
∑xn∈Ck
(yn −mk)2; yk = wTxk;
SB = (m2 −m1)(m2 −m1)T ;
SW =∑
k
∑n∈Ck
(xn −mk)(xn −mk)T .
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Fisher’s Linear Discriminant (Cont.)
MultiClass Cases: (Reduced to K-Dimension)
Data: For a K-Class problem, training set {(xn, tn}.n = 1, 2, ...,N, tnis a K-dim indicator vector.Decision Function: y = WTxOpt Problem:
maxw
J(W) = Tr{s−1W sB}
s.t. sW =∑N
k=1
∑n∈Ck
(yn − µk)(yn − µk)T
sB =∑K
k=1 Nk(µk − µ)(µk − µ)T
µk = 1Nk
∑n∈Ck
yn, µ = 1N
∑Nk=1 µkNk .
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Find a Separating Hyperplane
This procedure tries to conduct linear decision boundaries thatexplicitly separate the data into different classes.
PerceptronSupport Vector Machine
Figure : The orange line is the least squares solution, which misclassifies one ofthe training points. Two blue separating hyperplanes are found by theperceptron with different random starts.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Preliminary
The distance between decision boundary and point:
Decision boundary: y(x) = wTx + w0 = 0;
Point: x = x⊥ + r w‖w‖ ;
Projection:r = y(x)‖w‖ (Pos/Neg).
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Perceptron (Rosenblatt,1962)
The Perceptron learning algorithms tries to find a separating hyperplaneby minimizing the distance of misclassified points to the decisionboundary. If a response yi = 1 is misclassified, then y = wTφ(x), andthe opposite for a misclassified response with yi = −1.
Problem: Binary Classification;
Decision Function: y = wTφ(x) + w0;
Opt Model:
minw,‖w‖=1
−∑i∈M
yi (wTφ(x))
where M indexes the set of missclassifield points. This problem canbe solved via Stochastic Gradient Descent (SGD) algorithm.
Cons: The solution is not unique; The number of iteration can bevery large.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
From Two Classes to Multiple ClassesLeast SquareFisher’s Linear DiscriminantFind a Separating Hyperplane
Optimal Separating Hyperplanes (Vapnik, 1996)
This approach tries to provides a unique solution to separatinghyperplane problem via maximizing the margin between the two classes.For a binary problem with N training data, we have:
maxw,w0
C
s.t. yi(wT xk+w0)‖w‖ ≥ C , i = 1, 2, ...,N
Set C‖w‖ = 1, the problem can be written as:
minw,w0
12‖w‖
s.t. yi (wTxk + w0) ≥ 1, i = 1, 2, ...,N
This is the basic form of Linear Support Vector Machine.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Linear Discriminant AnalysisQuadratic Discriminant Analysis
Outline of this section
1 Overview of Classification
2 Discriminant Functions
3 Prob. Generative ModelsLinear Discriminant AnalysisQuadratic Discriminant Analysis
4 Prob. Discrimintive Models
5 Bonus
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Linear Discriminant AnalysisQuadratic Discriminant Analysis
Generative Model Review
This part is based on Bayesian Posterior Inference as follows:
P(Ck |x) =P(x|Ck)P(Ck)
P(x)=
P(x|Ck)P(Ck)∑j P(x|Cj)P(Cj)
If P(x|Ck) is Gaussian, then the ”log-odds” c is:
logP(Ci |x)
P(Cj |x)= 1
2 log|Σj ||Σi | + 1
2 [(x− µj)TΣ−1
j (x− µj)− (x− µi)TΣ−1
i (x− µi)]
+ log P(Ci )P(Cj )
The task is to estimate the covariance and expectation of each class.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Linear Discriminant AnalysisQuadratic Discriminant Analysis
Linear Discriminant Analysis
Linear Discriminant Analysis (LDA) arises in the special case when weassume that the classes have a common covariance matrix Σk = Σ,∀k.So we have:
logP(Ci |x)
P(Cj |x)= log
P(Ci )
P(Cj)− 1
2(µi + µj)
TΣ−1(µi − µj) + xTΣ−1(µi − µj)
an equation linear in x. For each class, we have the linear discriminantfunctions:
δk(x) = xTΣ−1µk −1
2µk
TΣ−1µk + logP(Ck)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Linear Discriminant AnalysisQuadratic Discriminant Analysis
Linear Discriminant Analysis (Cont.)
How to estimate the parameters? MLE.ˆP(Ck) = Nk
N , where Nk is the number of Class-k observations;
µ̂k = 1Nk
∑n∈Ck
xn;
Σ̂ =∑K
k=1
∑n∈Ck
1N−K (x− µk)(x− µk)T
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Linear Discriminant AnalysisQuadratic Discriminant Analysis
Quadratic Discriminant Analysis
Getting back to the general discriminant problem, if the Σk are notassumed to be equal, then the pieces quadratic in x remain. We the getquadratic discriminant functions:
δk(x) = −1
2log |Σk | −
1
2(x− µk)TΣ−1
k (x− µk) + logP(Ck)
The decision boundary between each pair of classes K and L is describedby a quadratic equation {x : δk(x) = δl(x)}
Note: Compared with LDA, QDA performed better for unlinearproblems.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Logistic RegressionLDA or Logistic Regression?
Outline of this section
1 Overview of Classification
2 Discriminant Functions
3 Prob. Generative Models
4 Prob. Discrimintive ModelsLogistic RegressionLDA or Logistic Regression?
5 Bonus
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Logistic RegressionLDA or Logistic Regression?
Logistic Regression
The Logistic Regression Model arises from the desire to model theposterior probabilities of the K classes via linear functions in x , while atthe same time they sum to one and remain in [0, 1]. The model has theform:
logP(C1|x)
P(Ck |x)= w10 + w1
Tx
logP(C2|x)
P(Ck |x)= w20 + w2
Tx
...
logP(Ck−1|x)
P(Ck |x)= w(k−1)0 + wk−1
Tx
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Logistic RegressionLDA or Logistic Regression?
Logistic Regression (Cont.)
The model is specified in terms of K − 1 log-odds. We use the logisticfunction f (σ) = 1
1+exp(−σ) and we have:
P(Ck |x) =exp(wk0 + wk
Tx)
1 +∑K−1
l=1 exp(wl0 + wlTx)
, k = 1, 2, ...,K − 1
P(CK |x) =1
1 +∑K−1
l=1 exp(wl0 + wlTx)
Computation: Maximum Likelihood Estimation;
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Logistic RegressionLDA or Logistic Regression?
LDA or Logistic Regression?
For LDA, we find that the log-posterior odss between class k and K arelinear functions of x :
logP(Ci |x)
P(Cj |x)= log
P(Ci )
P(Cj)− 1
2(µi + µj)
TΣ−1(µi − µj) + xTΣ−1(µi − µj)
= αk0 + αkTx
For Logistic Regression, we have:
logP(Ck−1|x)
P(Ck |x)= w(k−1)0 + wk−1
Tx
It seems that the models are the same. The Logistic Regression model ismore general, in that it makes less assumptions.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Outline of this section
1 Overview of Classification
2 Discriminant Functions
3 Prob. Generative Models
4 Prob. Discrimintive Models
5 BonusMulti-Class, Multi-Label, Struct-Output
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Roadmap of Machine Learning in Computer Vision
Classification
Binary Classification
Multiple-Labeling
MutiClass Classification
Multi-Label Multi-Instance
Structure Output
Bounding-Box Attributive
Pixel-wise Recognition
Early 1990s 2000s Today
Event & Behaviour
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Multi-Label and Multi-Instance
Figure : Examples of Multi-label Learning (Li,CIKM’13)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Structure-Output: Bounding-Box
Figure : Examples of Bounding Box (Lan,ECCV’12)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Structure-Output: Sentence
Figure : Examples of ”Sentence” (Lan, ECCV’12)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Structure-Output: Semantic Description
Figure : Examples of ”Semantic Description” (Lan, CVPR’12)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Structure-Output: Pixel-wise Annotation
Figure : Examples of ”Pixel-wise Annotation” (Ladicky, CVPR’13)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Fine-grained Categorization
Figure : Examples of ”Fine-grained Categorization” (Yao, CVPR’11)
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Challenges & Future Work
Figure : Challenges of Visual Recognition
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Challenges & Future Work
Learning:
Transfer Learning;Weakly-supervised Learning;Deep Learning.
Data:
Visual Saliency, Objectness, Descriptors;Semantic Network.
Y. Gu LINEAR MODELS FOR CLASSIFICATION
Overview of ClassificationDiscriminant Functions
Prob. Generative ModelsProb. Discrimintive Models
Bonus
Multi-Class, Multi-Label, Struct-Output
Thank you.Q&A.
Y. Gu LINEAR MODELS FOR CLASSIFICATION