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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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svm support vectors machines

SVMSupport Vectors Machines

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

Presented By: Tamer Salman

the addressed problems
The addressed Problems
  • SVM can deal with three kinds of problems:
    • Pattern Recognition / Classification.
    • Regression Estimation.
    • Density Estimation.
pattern recognition
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
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 svm linearly separable case

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
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
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
Linear SVM. Cont.
  • The LaGrangian is:
linear svm cont9
Linear SVM. Cont.
  • Requiring the derivatives with respect to w,b to vanish yields:
  • KKT conditions yield:
  • Where:
linear svm cont10
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
Linear SVM. Non-separable case.
  • We introduce slack variables ξi and allow mistakes.
  • We demand:
  • And minimize:
non separable case cont
Non-separable case. Cont.
  • The modifications yield the following problem:
non linear svm
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
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
Some Mercer Kernels
  • Homogeneous Polynomial Kernels:
  • Non-homogeneous Polynomial Kernels:
  • Radial Basis Function (RBF) Kernels:
solution of non linear svm
Solution of non-linear SVM
  • 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.
error estimates
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
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.

  • Margin Radius bound:

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 Bounds. Cont.



LOO Error


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


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.
rademacher risk bound
Rademacher Risk Bound
  • The following bound holds true with probability (1-δ):


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: