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# SVM Support Vectors Machines - PowerPoint PPT Presentation

SVM Support Vectors Machines. Based on Statistical Learning Theory of Vapnik, Chervonenkis, Burges, Scholkopf, Smola, Bartlett, Mendelson, Cristianini Presented By: Tamer Salman. The addressed Problems. SVM can deal with three kinds of problems: Pattern Recognition / Classification.

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### SVMSupport Vectors Machines

Based on Statistical Learning Theory of Vapnik, Chervonenkis, Burges, Scholkopf, Smola, Bartlett, Mendelson, Cristianini

Presented By: Tamer Salman

• SVM can deal with three kinds of problems:

• Pattern Recognition / Classification.

• Regression Estimation.

• Density Estimation.

• Given:

• A set of M labeled patterns:

• The patterns are drawn i.i.d from an unknown P(X,Y).

• A set of functions F.

• Chose a function f in F, such that an unseen pattern x will be correctly classified with high probability?

• Binary classification: Two classes, +1 and -1.

• What is the probability for error of a function f?

where c is some cost function on errors.

• The risk is not computable due to dP(x,y).

• A proper estimation must be found.

Linear SVM

Linear SVMLinearly Separable Case

• Linear SVM produces the maximal margin hyper plane, which is as far as possible from the closest training points.

• Given the training set, we seek w and b such that:

• In Addition, we seek the maximal margin hyperplane.

• What is the margin?

• How do we maximize it?

• The margin is the sum of distances of the two closest points from each side to the hyper plane.

• The distance of the hyper plane (w,b) from the origin is w/b.

• The margin is 2/||w||.

• Maximizing the margin is equivalent to minimizing ½||w||².

• The LaGrangian is:

• Requiring the derivatives with respect to w,b to vanish yields:

• KKT conditions yield:

• Where:

• The resulting separating function is:

• Notes:

• The points with α=0 do not affect the solution.

• The points with α≠0 are called support vectors.

• The equality conditions hold true only for the SVs.

• We introduce slack variables ξi and allow mistakes.

• We demand:

• And minimize:

• The modifications yield the following problem:

• Note that the training data appears in the solution only in inner products.

• If we pre-map the data into a higher and sparser space we can get more separability and a stronger separation family of functions.

• The pre-mapping might make the problem infeasible.

• We want to avoid pre-mapping and still have the same separation ability.

• Suppose we have a simple function that operates on two training points and implements an inner product of their pre-mappings, then we achieve better separation with no added cost.

• A Mercer kernel is a function:

for which there exists a function:

such that:

• A funtion k(.,.) is a Mercer kernel if

for any function g(.), such that:

the following holds true:

• Homogeneous Polynomial Kernels:

• Non-homogeneous Polynomial Kernels:

• Radial Basis Function (RBF) Kernels:

• The problem:

• The separating function:

• The solutions of non-linear SVM is linear in H (Feature Space).

• In non-linear SVM w exists in H.

• The complexity of computing the kernel values is not higher than the complexity of the solution and can be done a priory in a kernel matrix.

• SVM is suitable for large scale problems due to chunking ability.

• Due to the fact that the actual risk is not computable, we seek to estimate the error rate of a machine given a finite set of m patterns.

• Empirical Risk.

• Training and Testing.

• k-fold cross validation.

• Leave One out.

• We seek faster estimates of the solution.

• The bound should be tight and informative.

• Theoretical VC bound:

Risk < Empirical Risk + Complexity (VC-dimension / m)

Loose and not always informative.

Risk < R² / margin²

Where R is the radius of the smallest enclosing sphere of the data in feature space.

Tight and informative.

Error

Bound

LOO Error

Parameter

• One of the tightest sample-based bounds depend on the Rademacher Complexity term defined as follows:

where:

F is the class of functions mapping the domain of the input into R.

Ep(x) expectation with respect to the probability distribution of the input data.

Eσexpectation with respect to σi: independent uniform random variable of {±1}

• Rademacher complexity is a measure of the ability of the class of resulting functions to classify the input samples if associated with a random class.

• The following bound holds true with probability (1-δ):

Where:

Êm is the error on the input data measured through a loss function h(.) with Lipshitz constant L. That is:

And the loss function can be one of:

Vapnik’s: Bartlett & Mendelson’s: