Adaptive boosting methods are simple modular algorithms that operate as follows. Let 9: X -t Y be the function to be learned, where the label set Y is finite, typically binary-valued. The algorithm uses a learning procedure, which has access to n training examples, {(Xl, Y1),..., (xn, Yn)}, drawn randomly from X x Yaccording to distribution D; it outputs a hypothesis I:

So which of the two variances is "correct"? From a modelling point of view, the variance from the test example tells us the true story, so the training set variance should be regarded as biased.

Using statistical mechanics results, I calculate learning curves (average generalization error) for Gaussian processes (GPs) and Bayesian neural networks (NNs) used for regression. Applying the results to learning a teacher defined by a two-layer network, I can directly compare GP and Bayesian NN learning.

Becausethe training examples in an ill-posed data set do not fully span the signal space the observed training set variances in each basis vector will be too high compared to the average variance ofthe test set projections onto the same basis vectors. On basis of this understanding we introduce the Generalizable Singular ValueDecomposition (GenSVD) as a means to reduce this bias by re-estimation of the singular values obtained in a conventional Singular Value Decomposition, allowing for a generalization performance increaseof a subsequent statistical model. We demonstrate that the algorithm succesfully corrects bias in a data set from a functional PET activation study of the human brain. 1 Ill-posed Data Sets An ill-posed data set has more dimensions in each example than there are examples. Such data sets occur in many fields of research typically in connection with image measurements. The associated statistical problem is that of extracting structure from the observed high-dimensional vectors in the presence of noise. The statistical analysis can be done either supervised (Le.

Adaptive boosting methods are simple modular algorithms that operate as follows. Let 9: X -t Y be the function to be learned, where the label set Y is finite, typically binary-valued.The algorithm uses a learning procedure, which has access to n training examples, {(Xl, Y1), ..., (xn, Yn)}, drawn randomly from X x Yaccording todistribution D; it outputs a hypothesis I:

Using statistical mechanics results, I calculate learning curves (average generalization error) for Gaussian processes (GPs) and Bayesian neural networks (NNs) used for regression. Applying the results to learning a teacher defined by a two-layer network, I can directly compare GP and Bayesian NN learning.

In this paper, we consider the problem of active learning in trigonometric polynomial networks and give a necessary and sufficient condition of sample points to provide the optimal generalization capability. By analyzing the condition from the functional analytic point of view, we clarify the mechanism of achieving the optimal generalization capability. We also show that a set of training examples satisfying the condition does not only provide the optimal generalization but also reduces the computational complexity and memory required for the calculation of learning results. Finally, examples of sample points satisfying the condition are given and computer simulations are performed to demonstrate the effectiveness of the proposed active learning method.

We generalize a recent formalism to describe the dynamics of supervised learning in layered neural networks, in the regime where data recycling is inevitable, to the case of noisy teachers. Our theory generates reliable predictions for the evolution in time of training-and generalization errors, and extends the class of mathematically solvable learning processes in large neural networks to those situations where overfitting can occur.

In this paper, we consider the problem of active learning in trigonometric polynomial networks and give a necessary and sufficient condition of sample points to provide the optimal generalization capability. By analyzing the condition from the functional analytic point of view, we clarify the mechanism of achieving the optimal generalization capability. We also show that a set of training examples satisfying the condition does not only provide the optimal generalization but also reduces the computational complexity and memory required for the calculation of learning results. Finally, examples of sample points satisfying the condition are given and computer simulations are performed to demonstrate the effectiveness of the proposed active learning method.

We generalize a recent formalism to describe the dynamics of supervised learning in layered neural networks, in the regime where data recycling is inevitable, to the case of noisy teachers. Our theory generates reliable predictions for the evolution in time of training-and generalization errors, and extends the class of mathematically solvable learning processes in large neural networks to those situations where overfitting can occur.