Pattern Recognition and Machine Learning

Exercises 221 To do so, assume that one of the basis functionsφ 0 (x)=1so that the corresponding parameterw 0 plays the role of ...

222 4. LINEAR MODELS FOR CLASSIFICATION which represents the mean of those feature vectors assigned to classCk. Similarly, show ...

Exercises 223 4.17 ( ) www Show that the derivatives of the softmax activation function (4.104), where theakare defined by (4.10 ...

224 4. LINEAR MODELS FOR CLASSIFICATION 4.26 ( ) In this exercise, we prove the relation (4.152) for the convolution of a probit ...

5 Neural Networks In Chapters 3 and 4 we considered models for regression and classification that com- prised linear combination ...

226 5. NEURAL NETWORKS sparser models. Unlike the SVM it also produces probabilistic outputs, although this is at the expense of ...

5.1. Feed-forward Network Functions 227 5.1 Feed-forward Network Functions The linear models for regression and classification d ...

228 5. NEURAL NETWORKS Figure 5.1 Network diagram for the two- layer neural network corre- sponding to (5.7). The input, hidden, ...

5.1. Feed-forward Network Functions 229 notation for the two kinds of model. We shall see later how to give a probabilistic inte ...

230 5. NEURAL NETWORKS Figure 5.2 Example of a neural network having a general feed-forward topology. Note that each hidden and ...

5.1. Feed-forward Network Functions 231 Figure 5.3 Illustration of the ca- pability of a multilayer perceptron to approximate fo ...

232 5. NEURAL NETWORKS Figure 5.4 Example of the solution of a simple two- class classification problem involving synthetic data ...

5.2. Network Training 233 target vectors{tn}, we minimize the error function E(w)= 1 2 ∑N n=1 ‖y(xn,w)−tn‖^2. (5.11) However, we ...

234 5. NEURAL NETWORKS where we have discarded additive and multiplicative constants. The value ofwfound by minimizingE(w)will b ...

5.2. Network Training 235 If we consider a training set of independent observations, then the error function, which is given by ...

236 5. NEURAL NETWORKS Figure 5.5 Geometrical view of the error functionE(w)as a surface sitting over weight space. PointwAis a ...

5.2. Network Training 237 point in weight space such that the gradient of the error function vanishes, so that ∇E(w)=0 (5.26) as ...

238 5. NEURAL NETWORKS where cubic and higher terms have been omitted. Herebis defined to be the gradient ofEevaluated atŵ b≡∇E ...

5.2. Network Training 239 Figure 5.6 In the neighbourhood of a min- imumw
, the error function can be approximated by a quadrati ...

240 5. NEURAL NETWORKS evaluations, each of which would requireO(W)steps. Thus, the computational effort needed to find the mini ...