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**Linear Discriminant Functions Chapter 5 (Duda et al.)**

CS479/679 Pattern Recognition Dr. George Bebis

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**Generative vs Discriminant Approach**

Generative approaches find the discriminant function by first estimating the probability distribution of the patterns belonging to each class. Discriminant approaches find the discriminant function explicitly, without assuming a probability distribution.

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**Generative Approach – Example (two categories)**

More common to use a single discriminant function (dichotomizer) instead of two: Examples:

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**Discriminant Approach**

Specify parametric form of the discriminant function, for example, a linear discriminant: Decide w1 if g(x) > 0 and w2 if g(x) < 0 If g(x)=0, then x lies on the decision boundary and can be assigned to either class.

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**Discriminant Approach (cont’d)**

Find the “best” decision boundary (i.e., estimate w and w0) using a set of training examples xk.

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**Discriminant Approach (cont’d)**

The solution is found by minimizing a criterion function (e.g., “training error” or “empirical risk”): Learning algorithms can be applied to find the solution. correct class predicted class

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**Linear Discriminant Functions: two-categories case**

A linear discriminant function has the following form: The decision boundary, is a hyperplane where the orientation of the hyperplane is determined by w and its location by w0. w is the normal to the hyperplane If w0=0, the hyperplane passes through the origin

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**Geometric Interpretation of g(x)**

g(x) provides an algebraic measure of the distance of x from the hyperplane. x can be expressed as follows: direction of r

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**Geometric Interpretation of g(x) (cont’d)**

Substitute x in g(x): since and

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**Geometric Interpretation of g(x) (cont’d)**

Therefore, the distance of x from the hyperplane is given by: setting x=0:

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**Linear Discriminant Functions: multi-category case**

There are several ways to devise multi-category classifiers using linear discriminant functions: (1) One against the rest problem: ambiguous regions

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**Linear Discriminant Functions: multi-category case (cont’d)**

(2) One against another (i.e., c(c-1)/2 pairs of classes) problem: ambiguous regions

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**Linear Discriminant Functions: multi-category case (cont’d)**

To avoid the problem of ambiguous regions: Define c linear discriminant functions Assign x to wi if gi(x) > gj(x) for all j i. The resulting classifier is called a linear machine (see Chapter 2)

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**Linear Discriminant Functions: multi-category case (cont’d)**

A linear machine divides the feature space in c convex decisions regions. If x is in region Ri, the gi(x) is the largest. Note: although there are c(c-1)/2 pairs of regions, there typically less decision boundaries

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**Linear Discriminant Functions: multi-category case (cont’d)**

The decision boundary between adjacent regions Ri and Rj is a portion of the hyperplane Hij given by: (wi-wj) is normal to Hij and the signed distance from x to Hij is

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**Higher Order Discriminant Functions**

Can produce more complicated decision boundaries than linear discriminant functions.

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**Generalized discriminants**

- defined through special functions yi(x) called φ functions - α is a dimensional weight vector the φ functions yi(x) map a point from the d-dimensional x-space to a point in the -dimensional y-space (usually >> d ) φ

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**Generalized discriminants (cont’d)**

The resulting discriminant function is linear in y-space. Separates points in the transformed space by a hyperplane passing through the origin.

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Example The corresponding decision regions R1,R2 in the x-space are not simply connected! φ functions d=1,

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Example (cont’d) g(x) maps a line in x- space to a parabola in y- space. The plane αty=0 divides the y-space in two decision regions

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**Learning: two-category, linearly separable case**

Given a linear discriminant function the goal is to “learn” the parameters w and w0 from a set of n labeled samples xi where each xi has a class label ω1 or ω2.

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**Augmented feature/parameter space**

Simplify notation: dimensionality: d (d+1)

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**Classification in augmented space**

Classification rule: If αtyi>0 assign yi to ω1 else if αtyi<0 assign yi to ω2 g(x)=αty Discriminant:

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**Learning in augmented space: two-category, linearly separable case**

Given a linear discriminant function the goal is to learn the weights (parameters) α from a set of n labeled samples yi where each yi has a class label ω1 or ω2. g(x)=αty

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**Learning in augmented space: effect of training examples**

Every training sample yi places a constraint on the weight vector α. αty=0 defines a hyperplane in parameter space having y as a normal vector. Given n examples, the solution α must lie on the intersection of n half-spaces. a1 a2 parameter space (ɑ1, ɑ2)

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**Learning in augmented space: effect of training examples (cont’d)**

Visualize solution in the parameter or feature space. parameter space (ɑ1, ɑ2) feature space (y1, y2) a1 a2

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**Uniqueness of Solution**

Solution vector α is usually not unique; we can impose certain constraints to enforce uniqueness: “Find unit-length weight vector that maximizes the minimum distance from the training examples to the separating plane”

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**Iterative Optimization**

Define an error function J(α) (i.e., missclassifications) that is minimized if α is a solution vector. Minimize J(α) iteratively: α(k) α(k+1) search direction learning rate How should we define pk?

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**Choosing pk using Gradient Descent**

learning rate (note: replace a with α)

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**Gradient Descent (cont’d)**

solution space - J(α)

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**Gradient Descent (cont’d)**

What is the effect of the learning rate? η J(α) slow but converges to solution fast by overshoots solution

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**Gradient Descent (cont’d)**

How to choose the learning rate h(k)? If J(α) is quadratic, then H is constant which implies that the learning rate is constant. Taylor series approximation Hessian (2nd derivatives) (note:replace a with α) optimum learning rate

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**Choosing pk using Newton’s Method**

requires inverting H (note: replace a with α)

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**Newton’s method (cont’d)**

If J(α) is quadratic, Newton’s method converges in one step! J(α)

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**Gradient descent vs Newton’s method**

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**“Normalized” Problem If yi in ω2, replace yi by -yi**

Find α such that: αtyi>0 replace yi by -yi Seek a hyperplane that separates patterns from different categories Seek a hyperplane that puts normalized patterns on the same (positive) side

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**Perceptron rule Use Gradient Descent assuming:**

where Y(α) is the set of samples misclassified by α. If Y(α) is empty, Jp(α)=0; otherwise, Jp(α)>0 Find α such that: αtyi>0

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**Perceptron rule (cont’d)**

The gradient of Jp(α) is: The perceptron update rule is obtained using gradient descent: (note: replace a with α) or

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**Perceptron rule (cont’d)**

(note: replace a with α and Yk with Y(α)) missclassified examples

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**Perceptron rule (cont’d)**

Move the hyperplane so that training samples are on its positive side. a2 a1 Example:

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**Perceptron rule (cont’d)**

η(k)=1 one example at a time Perceptron Convergence Theorem: If training samples are linearly separable, then the sequence of weight vectors by the above algorithm will terminate at a solution vector in a finite number of steps.

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**Perceptron rule (cont’d)**

order of examples: y2 y3 y1 y3 “Batch” algorithm leads to a smoother trajectory in solution space.

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