SVM Support Vectors Machines

1 / 22

# 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'SVM Support Vectors Machines' - davida

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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.
Pattern Recognition
• 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.
The Actual Risk
• 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 Neural Network

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.
Linearly Separable Case. Cont.
• 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?
Margin Maximization
• 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.
Linear SVM. Cont.
• The LaGrangian is:
Linear SVM. Cont.
• Requiring the derivatives with respect to w,b to vanish yields:
• KKT conditions yield:
• Where:
Linear SVM. Cont.
• 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.
Linear SVM. Non-separable case.
• We introduce slack variables ξi and allow mistakes.
• We demand:
• And minimize:
Non-separable case. Cont.
• The modifications yield the following problem:
Non Linear SVM
• 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.
Mercer Kernels
• 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:

Some Mercer Kernels
• Homogeneous Polynomial Kernels:
• Non-homogeneous Polynomial Kernels:
• Radial Basis Function (RBF) Kernels:
Solution of non-linear SVM
• The problem:
• The separating function:
Notes
• 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.
Error Estimates
• 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.
Error Bounds
• 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 Bounds. Cont.

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.