Giansalvo EXIN Cirrincione

1. unit #6 Neural Networks and Pattern Recognition Giansalvo EXIN Cirrincione

2. N basis functions usually Euclidean Fnn’ = F(xn- xn’ ) (wn) (t n) Radial basis functions Exact interpolation The exact interpolation problem requires every input vector to be mapped exactly onto the corresponding target vector. Problem: given a mapping: xd  t and a TS of N points, find a function h(x) such that: h(x n) = t n n = 1, … , N generalized linear discriminant function

3. localized Radial basis functions Exact interpolation Many properties of the interpolating function are relatively insensitive to the precise form of the non-linear kernel. thin-plate spline function multi-quadric function for b = 1/2 non-linear in the components of x

4. Gaussian noise, zero mean, s = 0.05 30 points Radial basis functions Exact interpolation Gaussian basis functions with s = 0.067 (roughly twice the spacing of the data points) highly oscillatory

5. more than one output variable Radial basis functions Exact interpolation

6. RBF RBF’s provide a smooth interpolating function in which the number of basis functions is determined by the complexity of the mapping to be represented rather than by the size of the data set. The number M of basis functions is less than N. The centres of the basis functions are determined by the training process. Each basis function has its own width sj which is adaptive. Bias parameters are included in the linear sum and compensate for the difference between the average value over the TS of the basis function activations and the corresponding average value of the targets.

7. universal approximator best approximator fixed at 1 Elements mji Gaussian kernel RBF mMd

8. RBF There is a trade-off between using a smaller number of basis with many adjustable parameters and a larger number of less flexible functions

9. homework homework RBF There is a trade-off between using a smaller number of basis with many adjustable parameters and a larger number of less flexible functions Suppose that all of the Gaussian basis functions in the network share a common covariance matrix S. Show that the mapping represented by such a network is equivalent to that of a network of spherical Gaussian basis functions with common variance s2 = 1, provided the input vector x is first transformed by an appropriate linear transformation. Find expressions relating the transformed input vector and kernel centres to the corresponding original vectors.

10. first-layer weights second-layer weights fixed first-layer weights generalized linear discriminant RBF training decoupling fast learning

11. normal equations RBF training

12. s = 10.0 s = 0.4 s = 0.08 RBF training • M = 5 • centres = random subset of the TS

13. differential operator adjoint differential operator to P The solution can be written down in terms of the Green’s functionsG(x,x’)of the operator PP: ^ `Regularization theory Set the functional derivative w.r.t. y(x) to zero: Euler-Lagrange equations If P is translationally and rotationally invariant, G depends onlyon the distance between x and x’ (radial). The solution is given by: RBF

14. Regularization theory

15. Regularization theory Integrate over a small region around xn Using the solution in terms of the Green’s functions: The Green’s function is Gaussian with width parameter s if the operator P is chosen as: n = 0 implies RB exact interpolation

16. Regularization theory • s = 0.067 • n = 40 • RB’s centred on each data n = 0 implies RB exact interpolation

17. homework

18. Consider the functional derivative of the regularization functional (w.r.t. y(x)) given by: By using successive integration by parts, and making use of identities: show that the operator is given by: It should be assumed that boundary terms arising from the integration by parts can be neglected. Now find the radial Green’s function of this operator as follows. First introduce the multidimensional Fourier transform of G in the form:

19. By using the last two formulae and using the following formula for the Fourier transform of the Dirac function: where d is the dimensionality of x and s, show that the Fourier transform of the Green’s function is given by: Now substitute this result into the inverse Fourier transform of G and then show that the Green’s function is given by:

20. Regularization theory Regularization can also be applied to general RBF’s. Also, regularization terms can be considered for which the kernels are not necessarily the Green’s functions. For example: penalizes mappings which have large curvatures. This regularizer leads to second-layer weights which are found by the solution of:

21. Parzen Gaussian kernel estimator Relation to kernel regression Technique for estimating regression functions from noisy data, based on methods of kernel density estimation Consider a mapping : x  y and a corresponding TS; a complete description of the statistical properties of the generator of the data is given by the probability density p(x,t) in the joint input-target space.

22. Relation to kernel regression The optimal mapping is given by forming the regression, or conditional average tx, of the target data, conditioned on the input variables.

23. Relation to kernel regression The optimal mapping is given by forming the regression, or conditional average tx, of the target data, conditioned on the input variables. Nadaraya-Watson estimator

24. second-layer weights normalized Gaussians Relation to kernel regression Extension: replace the kernel estimator with an adaptive mixture model Nadaraya-Watson estimator

25. Multilayer perceptron RBF’s for classification Model the class distributions by local kernel functions

26. Hidden-to-output weight RBF’s for classification The outputs represent approximations to the posterior probabilities RBF

27. where RBF’s for classification Use a common pool of M basis functions, labelled by an index j, to represent all of the class-conditional densities

28. RBF’s for classification The activations of the basis functions can be interpreted as the posterior probabilities of the presence of corresponding features in the input space, and the weights can similarly be interpreted as the posterior probabilities of class membership, given the presence of the features. The outputs represent the posterior probabilities of class membership.

29. homework homework Comparison with the multilayer perceptron The hidden unit activation in a MLP is constant on parallel(d-1)-dimensional hyperplanes. The hidden unit (RB) activation in a RBF is constant on concentric(d-1)-dimensional hyperspheres (more generally hyperellipsoids). In a MLP a hidden unit has a constant activation function for input vectors which lie on a hyperplanar surface in input space given by wTx+w0=const., while for a spherical RBF a hidden unit has constant activation on a hyperspherical surface defined by ||x- ||2 =const.. Show that, for suitable choices of the parameters, these surfaces coincide if the input vectors are normalized to unit length. Illustrate this equivalence geometrically for vectors in a three-dimensional input space.

30. Comparison with the multilayer perceptron The hidden unit activation in a MLP is constant on parallel(d-1)-dimensional hyperplanes. The hidden unit (RB) activation in a RBF is constant on concentric(d-1)-dimensional hyperspheres (more generally hyperellipsoids). The MLP forms a distributed representation in the space of activation values for the hidden units (problems: local minima, flat regions). The RBF forms a representation in the space of activation values for the hidden units which is local w.r.t. the input space. All of the parameters in a MLP are usually determined at the same time as part of a single global supervised training strategy. RBF is trained in two steps (kernels determined first by unsupervised methods, weights then found by fast linear supervised methods).

31. basis function optimization ignore any target information The basis function parameters should be chosen to form a representation of the pdf of the input data. j’s as prototypes of the inputs Problem: if the basis function centres are used to fill out a compact d-dimensional region of the input space, then the number of kernel centres will be an exponential function of d. Problem: input variables which have significant variance but play little role in determining the appropriate output variables (irrelevant inputs).

32. basis function optimization y independent of x2 Problem: input variables which have significant variance but play little role in determining the appropriate output variables (irrelevant inputs).

33. basis function optimization A MLP can learn to ignore irrelevant inputs and obtain accurate results with a small number of hidden units. y independent of x2 Problem: input variables which have significant variance but play little role in determining the appropriate output variables (irrelevant inputs).

34. basis function optimization input target pdf input pdf The optimal choice of basis function parameters for input density estimation needs not be optimal for representing the mapping of the output variables.

35. basis function optimization basis function widths all equal and set to some multiple of the average distance between the kernel centres (overlap for smoothing) determined from the average distance of each kernel to its L nearest neighbours, where L is typically small subsets of data points basis function centres set them equal to a random subset of TS data (use as initial conditions) start with all data points as kernel centres and then selectively remove centres in such a way as to have minimum disruption on the system performance • very fast • significantly sub-optimal

36. basis function optimization decided in advance K-means (basic ISODATA) clustering algorithm Suppose there are N data points xn in total; it tries to find a set of K representatives vectors j where j = 1, ... ,k. It seeks to partition the data points into K disjoint subsets Sj containing Nj data points in such a way as to minimize the sum-of-squares clustering function given by: wherej is the mean of the data points in set Sj and is given by:

37. basis function optimization K-means (basic ISODATA) clustering algorithm batch (Lloyd, 1982) It begins by assigning the points at random to K sets and then computing the mean vectors of the points in each set. Next, each point is re-assigned to a new set according to which is the nearest mean vector. The means of the sets are then recomputed. This procedure is repeated until there is no further change in the grouping of the data points. At each such iteration, the value of J will not increase.

38. basis function optimization K = 2 K-means (basic ISODATA) clustering algorithm batch (Lloyd, 1982)

39. basis function optimization Robbins-Monro for finding the root of a regression function given by the derivative of J w.r.t. j K-means (basic ISODATA) clustering algorithm sequential (MacQueen, 1967) The initial centres are randomly chosen from the data points, and as each data point xn is presented, the nearest j is updated using: learning rate Once the centres of the kernels have been found, the covariance matrices of the kernels can be set to the covariances of the points assigned to the corresponding clusters.

40. homework

41. Write a numerical implementation of the K-means clustering algorithm using both the batch and on-line versions. Illustrate the operation of the algorithm by generating data sets in two dimensions from a mixture of Gaussian distributions, and plotting the data points together with the trajectories of the estimated means during the course of the algorithm. Investigate how the results depend on the value of K in relation to the number of Gaussian distributions, and how they depend on the variances of the distributions in relation to their separation. Study the performance of the on-line version of the algorithm for different values of the learning rate parameter  and compare the algorithm with the batch version.

42. basis functions • P(j) • kernel parameters input data pdf max Gaussian mixture models The basis functions of the RBF can be regarded as the components of a mixture density model whose parameters are to be optimized by ML. Once the mixture model has been optimized, the mixing coefficients P(j) can be discarded, and the basis functions then used in the RBF in which the second-layer weights are found by supervised training.

43. K-means as a particular limit of the EM optimization of a Gaussian mixture model

44. basis function optimization (supervised training) The basis function parameters for regression can be found by treating the kernel centres and widths, along with the second-layer weights, as adaptive parameters to be determined by minimization of an error function. • sum-of-squares error • spherical Gaussian basis functions non-linear computationally intensive optimization problem

45. basis function optimization (supervised training) input RBF training well localized kernels

46. basis function optimization (supervised training) RBF training

47. basis function optimization (supervised training) activation only for a small fraction of kernels RBF training training procedures can be speeded up significantly by identifying the relevant kernels and therefore avoiding unnecessary computation

48. basis function optimization (supervised training) RBF training coarse (unsupervised) to fine (supervised) no guarantee basis function will remain localized !

49. FINE