Transcript exin6
unit #6 Giansalvo EXIN Cirrincione 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 N basis functions Fnn’ = F(xn- xn’ ) (wn) (t n) generalized linear discriminant function usually Euclidean Radial basis functions Exact interpolation localized x e x2 2 2 x x 2 2 0 Many properties of the interpolating function are relatively insensitive to the precise form of the non-linear kernel. thin-plate spline function x x 2 lnx multi-quadric function for = 1/2 x x 2 0 1 2 x x 3 x x non-linear in the components of x Radial basis functions Exact interpolation Gaussian noise, zero mean, = 0.05 hx 0.5 0.4 sin2 x 30 points Gaussian basis functions with = 0.067 (roughly twice the spacing of the data points) highly oscillatory Radial basis functions Exact interpolation more than one output variable 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 j 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. RBF universal approximator fixed at 1 Gaussian mMd Elements mji kernel best approximator 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 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 homework 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 2 = 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. homework RBF training first-layer weights decoupling second-layer weights w kj j fixed first-layer weights generalized linear discriminant fast learning RBF training normal equations = 10.0 RBF training = 0.4 = 0.08 •M=5 • centres = random subset of the TS `Regularization theory differential Set the functional derivative w.r.t. y(x) to zero: operator Euler-Lagrange equations adjoint differential operator to Pof the The solution can be written down in terms ^ Green’s functions G(x,x’) of the operator PP: If P is translationally and rotationally invariant, G depends only on the distance between x and x’ (radial). The solution is given by: RBF Regularization theory 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 if the operator P is chosen as: n = 0 implies RB exact interpolation Regularization theory • = 0.067 • n = 40 • RB’s centred on each data n = 0 implies RB exact interpolation homework 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 Pˆ P 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: 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: 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: 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. Parzen Gaussian kernel estimator Relation to kernel regression The optimal mapping is given by forming the regression, or conditional average tx, of the target data, conditioned on the input variables. Relation to kernel regression The optimal mapping is given by forming the regression, or conditional average tx, of the target data, conditioned on the input variables. Nadaraya-Watson estimator Relation to kernel regression Extension: replace the kernel estimator with an adaptive mixture model second-layer weights normalized Gaussians Nadaraya-Watson estimator RBF’s for classification px Ck Model the class distributions by local kernel functions Multilayer perceptron RBF’s for classification The outputs represent approximations to the posterior probabilities RBF Hidden-to-output weight 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 where RBF’s for classification PCk x wkj j x M j 1 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. 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). homework 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- m||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 threedimensional input space. 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). 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). basis function optimization The basis function parameters should be chosen to form a representation of the pdf of the input data. ignore any target information mj’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). 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). 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). basis function optimization The optimal choice of basis function parameters for input density estimation needs not be optimal for representing the mapping of the output variables. input input pdf target pdf basis function optimization subsets of data points basis function centres basis function widths set them equal to a start with all data points random subset of TS data as kernel centres and then all equal and set to some determined from the (use as initial conditions) selectively remove centres multiple of the average in such a way as to haveaverage distance of each distance between the minimum disruption on kernel to its L nearest kernel centres (overlap the system performance neighbours, where L is for smoothing) typically small • very fast • significantly sub-optimal basis function optimization K-means (basic ISODATA) clustering algorithm Suppose there are N data points xn in total; it tries to find decidedainsetadvance of K representatives vectors mj 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 mj is the mean of the data points in set Sj and is given by: 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. basis function optimization K-means (basic ISODATA) clustering algorithm batch (Lloyd, 1982) K=2 basis function optimization K-means (basic ISODATA) clustering algorithm Robbins-Monro for finding the root of a regression function given sequential (MacQueen, 1967) by the derivative of J w.r.t. mj The initial centres are randomly chosen from the data points, and as each data point xn is presented, the nearest mj 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. homework Write a numerical implementation of the K-means clustering algorithm using both the batch and online 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. 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. max • P(j) • kernel parameters 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. K-means as a particular limit of the EM optimization of a Gaussian mixture model 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. a sum-of-squares error a spherical Gaussian basis functions non-linear computationally intensive optimization problem basis function optimization (supervised training) RBF training well localized kernels input basis function optimization (supervised training) RBF training basis function optimization (supervised training) RBF training activation only for a small fraction of kernels training procedures can be speeded up significantly by identifying the relevant kernels and therefore avoiding unnecessary computation basis function optimization (supervised training) RBF training coarse (unsupervised) to fine (supervised) no guarantee basis function will remain localized !