Radial-Basis Function Networks

Radial-Basis Function Networks

CS/CMPE 537 – Neural Networks

CS/CMPE 537 - Neural Networks (Sp 2006-2007) - Asim Karim @ LUMS 2

Introduction

Typical tasks performed by neural networks are association, classification, and filtering. This categorization has historical significance as well.

These tasks involve input-output mappings, and the network is designed to learn the mapping from knowledge of the problem environment

Thus, the design of a neural network can be viewed as a curve-fitting or function approximation problem This viewpoint is the motivation for radial-basis function

networks


Radial-Basis Function Networks

RBF are 2 layer networks; input source nodes, hidden neurons with basis functions (nonlinear), and output neurons with linear/nonlinear activation functions

The theory of radial-basis function networks is built upon function approximation theory in mathematics

RBF networks were first used in 1988. Major work was done by Moody and Darken (1989) and Poggio and Girosi (1990)

In RBF networks, the mapping from input to high-dimension hidden space is nonlinear, while that from hidden to output space is linear

What is the basis for this ?


Radial-Basis Function Network

x1

xp

y1

y2

Φ1(.)

ΦM(.)

Sourcenodes

Hidden neurons with RBF activation functions

Outputneurons

w


Cover’s Theorem (1)

Cover’s theorem (1965) gives the motivation for RBF networks

Cover’s theorem on the separability of patterns Complex pattern-classification problems cast in high-

dimensional space nonlinearly is more likely to be linearly separable than low-dimensional space



Consider set X of N p-dimensional vectors (input patterns) x1 to xN. Let X+ and X- be a binary partition of X, and φ(x) = [φ1(x), φ2(x),…, φM(x)]T.

Cover’s theorem A binary partition (dichotomy) [X+, X-] of X is said to be φ-

separable if there exist an m-dimensional vector w such that

wTφ(x) ≥ 0 when x belong to X+

wTφ(x) < 0 when x belong to X- Decision boundary or surface

wTφ(x) = 0


Example (1)

Consider the XOR problem to illustrate the significance of φ-separability and Cover’s theorem.

Define a pair of Gaussian hidden functions

φ1(x) = exp(-||x – t1||2) t1 = [1, 1]T

φ2(x) = exp(-||x – t2|2|) t1 = [0, 0]T

Output of these function for each pattern


Example (2)


Function Approximation (1)

Function approximation seeks to describe the behavior of complex functions by ensembles of simpler functions Describe f(x) by F(x)

F(x) can be described in a compact region of input space by

F(x) = Σi=1 N wiφi(x)

Such that

|f(x) – F(x)| < ε ε can be made arbitrarily small Choice of function φ(.) ?


Function Approximation (2)

Find F(x) that “best” approximates the map/function f. The best approximation is problem dependent, and it can be strict interpolation or good generalization (regularized interpolation).

Design decisions Choice of elementary functions φ(.) How to compute the weights w ? How many elementary functions to use (i.e. what should be

N)? How to obtain a good generalization ?


Choice of Elementary Functions φ

Let f(x) belongs to the function space L2(Rp) (true for almost all physical systems)

We want φ to be a basis of L2 What is meant by a basis? A set of functions φi (i = 1, M) are a basis of L2 if linear

superposition of φi can generate any function in L2 . Moreover, they must be linearly independent:

w1 φ1 + w2 φ2 +…+ wM φM = 0 iff wi = 0 for all i


Interpolation Problem (1)

In general, the map from an input space to an output space is given by

f: Rp -> Rq p and q = input and out space dimensions; f = map or

hypersurface

Strict interpolation problem Given a set of N different points xi (i = 1, N) and a

corresponding set of N real numbers di (i = 1, N) find a function F: Rp -> R1 that satisfies the interpolation condition

F(xi) = di i = 1, N

The function F passes through all the points



A common type of φ(.) is radial-symmetric basis functions

F(x) = Σi=1 N wiφ||x – xi||

Substituting and writing in matrix format

Ф w = d Ф = φji (i, j = 1, N) = interpolation matrix; φji = φ||xj – xi||

w = linear weight vector; d = desired response vector



Ф is known to be positive definite for a certain class of radial-basis functions. Thus,

w = Ф-1 d In theory, w can be computed. In practice, however, Ф

is close to singular Then what ?

Regularization theory to perturb Ф to make it non-singular

But, there is another problem… poor generalization or overfitting


Ill-Posed Problems

Supervised learning is a an ill-posed problem There is not enough information in the training data to

reconstruct the input-output mapping uniquely The presence of noise or imprecision in the input data adds

uncertainty to the reconstruction of the input-outut mapping

To achieve good generalization additional information of the domain is needed In other words, the input-output patterns should exhibit

redundancy Redundancy is achieved when the physical generator of data

is smooth, and thus can be used to generate redundant input-output examples


Regularization Theory (1)

How to make an ill-posed problem well-posed ? By constraining the mapping with additional

information (e.g. smoothness) in the form of a nonnegative functional

Proposed by Tikhonov in 1963 in the context of function approximation in mathematics



Input-output examples: xi, di (i = 1, N)

Find the mapping F(x): Rp -> R1 for the input-output examples

In regularization theory, F is found by minimizing the cost functional ξ(F)

ξ(F) = ξs(F) + λξc(F)

Standard error term

ξs(F) = 0.5Σi=1 N (di – yi)2 = 0.5Σi=1 N (di – F(xi))2

Regularization term

ξc(F) = 0.5||PF(x)||2 P = linear differential operator



Regularization term depends on the geometric properties of the approximating function The selection of the operator P is therefore problem

dependent based on prior knowledge of the geometric properties of the actual function f(x) (e.g. the smoothness of f(x))

Regularization parameter λ: a positive real number This parameter indicates the sufficiency of the given input-

output examples in capturing the underlying function f(x) The solution of the regularization problem is a function

type F(x) We won’t go into the details of how to find F as it requires

good understanding of functional analysis



Solution of the regularization problem yields

F(x) = 1/λΣi=1 N [di - F(xi)]G(x, xi) = Σi=1 N wiG(x, xi) G(x,xi) = Green’s function centered on xi

In matrix form

F = Gw

or

(G + λI)w = d

and

w = (G + λI)-1d G depends only on the operator P


Type of Function G(x; xi)

If P is translationally invariant then G(x; xi) depends only on the difference of x and xi, i.e.

G(x; xi) = G(x - xi) If P is both translationally and rotationally invariant

then G(x; xi) depends only on Euclidean norm of the difference vector x - xi, i.e.

G(x; xi) = G(||x – xi||) This is a radial-basis function

If P is further constrained, and G(x; xi) is positive definite, then we have the Gaussian radial-basis function, i.e.

G(x; xi) = exp(- (1/2σ2) ||x – xi||2)


Regularization Network (1)


Regularization Network (2)

The regularization network is based on the regularized interpolation problem

F(x) = Σi=1 N wiG(x, xi)

It has 3 layers Input layer of p source nodes, where p is the dimension of

the input vector x (or number of independent variables) Hidden layer with N neurons, where N is the number of

input-output examples. Each neuron uses the activation function G(x; xi)

Output layer with q neurons, where q is the output dimension

The unknowns are the weights w (only) from the hidden layer to the output layer


RBF Networks (in Practice) (1)

The regularization network requires N hidden neurons, which becomes computationally expensive for large N The complexity of the network is reduced to obtain an

approximate solution to the regularization problem

The approximate solution F*(x) is then given by

F*(x) = Σi=1 M wiφi(x) φi(x) (i = 1,M) = new set of basis functions; M is typically

less than N

Using radial basis functions

F*(x) = Σi=1 M wiφi(||x – ti||)


RBF Networks (2)


RBF Networks (3)

Unknowns in the RBF network M, the number of hidden neurons (M < N) The centers ti of the radial-basis functions

And the weights w


How to Train RBF Networks - Learning

Normally the training of the hidden layer parameters (number of hidden neurons, centers and variance of Gaussian) is done prior to the training of the weights (i.e. on a different ‘time scale’) This is justified based on the fact that the hidden layer

performs a different task (nonlinear) than the output layer weights (linear)

The weights are learned by supervised learning using an appropriate algorithm (LMS or BP)

The hidden layer parameters are learned by (in general, but not always) unsupervised learning


Fixed Centers Selected at Random

Randomly select M inputs as centers for the activation functions

Fix the variance of the Gaussian based on the distance between the selected centers. A radial-basis function centered at ti is then given by

φ(||x – ti||) = exp(- M/d2 ||x – ti||2) d = max. distance between the chosen centers The width’ or standard deviation of the functions is fixed,

given by σ = d/√2M The linear weights are then computed by solving the

regularization problem or by using supervised learning


Self-Organized Selection of Centers

Use a self-organizing or clustering technique to determine the number and centers of the Gaussian functions A common algorithm is the k-means algorithm. This

algorithms assigns a label to a vector x by the majority label on the k-nearest neighbors

Then compute the weights using a supervised error-correction learning such as LMS


Supervised Selection of Centers

All unknown parameters are trained using error-correcting supervised learning

A gradient descent approach is used to find the minimum of the cost function wrt the weights wi and activation function centers ti and spread of centers σ


Example (1)

Classify between two ‘overlapping’ two-dimensional, Gaussian-distributed patterns

Conditional probability density function for the two classes

f(x | C1) = 1/2πσ12 exp[-1/2σ1

2 ||x – μ1||2]

μ1 = mean = [0 0]T and σ12 = variance = 1

f(x | C2) = 1/2πσ22 exp[-1/2σ2

2 ||x – μ2||2]

μ2 = mean = [2 0]T and σ22 = variance = 4

x = [x1 x2]T = two dimensional input

C1 and C2 = class labels


Example (2)


Example (3)


Example (4)

Consider a two-input, M hidden neurons, and two-output RBF Decision rule: an input x is classified to C1 if y1 >= 0

The training set is generated from the probability distribution functions

Using the perceptron algorithm, the network is trained for minimum mean-square-error

The testing set is generated from the probability distribution functions The trained network is tested for correct classification

For other implementation details, see the Matlab code


Example: Function Approximation (1)

Approximate relationship between a car’s fuel economy (in miles per gallon) and its characteristics

Input data description: 9 independent discrete valued, boolean, and continuous variables X1: number of cylinders

X2: displacement

X3: horsepower

X4: weight

X5: acceleration

X6: model year

X7: Made in US? (0,1)

X8: Made in Europe? (0,1)

X9: Made in Japan? (0,1)

Output f(X) is fuel economy in miles per gallon


Example: Function Approximation (2)

Using the NNET toolbox, create and train a RBF network with function newrb

The function parameters allows you to set the mean-squared-error goal of the training, the spread of the radial-basis functions, and the maximum number of hidden layer neurons.

Newrb uses the following approach to find the unknowns (it is a self-organizing approach) Start with one hidden neuron; compute network error Add another neuron with center equal to input vector that

produced the maximum error; compute network error If network error does not improve significantly, stop; other

go to previous step and add another neuron


Comparison of RBF Network and MLP (1)

Both are universal approximators. Thus, a RBF network exists for every MLP, and vice versa

An RBF has a single hidden layer, while an MLP can have multiple hidden layers

The model of the computational neurons of an MLP are all identical, while the neurons in the hidden and output layers of an RBF network have different models

The activation functions of the hidden nodes of an RBF network is based on the Euclidean norm of the input wrt to a center, while that of an MLP is based on the inner product of input and weights


Comparison of RBF Network and MLP (2)

MLPs construct global approximations to nonlinear input-output mapping. This is a consequence of the global activation function (sigmoidal) used in MLPs As a result, MLP can perform generalization in regions

where input data is not available (i.e. extrapolation)

RBF networks construct local approximations to input-output data. This is a consequence of the local Gaussian functions As a result, RBF networks are capable of fast learning from

the training data

Radial-Basis Function Networks

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