The linear correlation coefficient is typically used to characterize and analyze dependencies of neural spike counts. Here, we show that the correlation coefficient is in general insufficient to characterize these dependencies. We construct two neuron spike count models with Poisson-like marginals and vary their dependence structure using copulas. To this end, we construct a copula that allows to keep the spike counts uncorrelated while varying their dependence strength. Moreover, we employ a network of leaky integrate-and-fire neurons to investigate whether weakly correlated spike counts with strong dependencies are likely to occur in real networks. We find that the entropy of uncorrelated but dependent spike count distributions can deviate from the corresponding distribution with independent components by more than 25% and that weakly correlated but strongly dependent spike counts are very likely to occur in biological networks. Finally, we introduce a test for deciding whether the dependence structure of distributions with Poisson-like marginals is well characterized by the linear correlation coefficient and verify it for different copula-based models.

This paper extends k-means algorithms from the Euclidean domain to the domain of graphs. To recompute the centroids, we apply subgradient methods for solving the optimization-based formulation of the sample mean of graphs. To accelerate the k-means algorithm for graphs without trading computational time against solution quality, we avoid unnecessary graph distance calculations by exploiting the triangle inequality of the underlying distance metric following Elkan's k-means algorithm proposed in \cite{Elkan03}. In experiments we show that the accelerated k-means algorithm are faster than the standard k-means algorithm for graphs provided there is a cluster structure in the data.

This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between graph matching and clique search reduces to showing that a given graph matching problem satisfies the proposed condition; and third, the result sets the scene for generic continuous solutions for a broad range of graph matching problems. To illustrate the mathematical framework, we apply it to a number of graph matching problems, including the problem of determining the graph edit distance.

In this analytical study we derive the optimal unbiased value estimator (MVU) and compare its statistical risk to three well known value estimators: Temporal Difference learning (TD), Monte Carlo estimation (MC) and Least-Squares Temporal Difference Learning (LSTD). We demonstrate that LSTD is equivalent to the MVU if the Markov Reward Process (MRP) is acyclic and show that both differ for most cyclic MRPs as LSTD is then typically biased. More generally, we show that estimators that fulfill the Bellman equation can only be unbiased for special cyclic MRPs. The main reason being the probability measures with which the expectations are taken. These measure vary from state to state and due to the strong coupling by the Bellman equation it is typically not possible for a set of value estimators to be unbiased with respect to each of these measures. Furthermore, we derive relations of the MVU to MC and TD. The most important one being the equivalence of MC to the MVU and to LSTD for undiscounted MRPs in which MC has the same amount of information. In the discounted case this equivalence does not hold anymore. For TD we show that it is essentially unbiased for acyclic MRPs and biased for cyclic MRPs. We also order estimators according to their risk and present counter-examples to show that no general ordering exists between the MVU and LSTD, between MC and LSTD and between TD and MC. Theoretical results are supported by examples and an empirical evaluation.

The standard approach to the classification of objects is to consider the examples as independent and identically distributed (iid). In many real world settings, however, this assumption is not valid, because a topographical relationshipexists between the objects. In this contribution we consider the special case of image segmentation, where the objects are pixels and where the underlying topography is a 2D regular rectangular grid. We introduce a classification method which not only uses measured vectorial feature information but also the label configuration within a topographic neighborhood.Due to the resulting dependence between the labels of neighboring pixels, a collective classification of a set of pixels becomes necessary. We propose a new method called'Topographic Support VectorMachine' (TSVM), which is based on a topographic kernel and a self-consistent solution to the label assignment shown to be equivalent toa recurrent neural network. The performance of the algorithm is compared to a conventional SVM on a cell image segmentation task.

Nonlinear filtering can solve very complex problems, but typically involve very time consuming calculations. Here we show that for filters that are constructed as a RBF network with Gaussian basis functions, a decomposition into linear filters exists, which can be computed efficiently in the frequency domain, yielding dramatic improvement in speed. We present an application of this idea to image processing. In electron micrograph images of photoreceptor terminals of the fruit fly, Drosophila, synaptic vesicles containing neurotransmitter should be detected and labeled automatically. We use hand labels, provided by human experts, to learn a RBF filter using Support Vector Regression with Gaussian kernels. We will show that the resulting nonlinear filter solves the task to a degree of accuracy, which is close to what can be achieved by human experts. This allows the very time consuming task of data evaluation to be done efficiently.

The most simple filter is linear mapping.

We introduce a family of classifiers based on a physical analogy to an electrostatic system of charged conductors. The family, called Coulomb classifiers, includes the two best-known support-vector machines (SVMs), the ν-SVM and the C-SVM. In the electrostatics analogy,a training example corresponds to a charged conductor at a given location in space, the classification function corresponds to the electrostatic potential function, and the training objective function corresponds to the Coulomb energy. The electrostatic framework provides not only a novel interpretation of existing algorithms andtheir interrelationships, but it suggests a variety of new methods for SVMs including kernels that bridge the gap between polynomial and radial-basis functions, objective functions that do not require positive-definite kernels, regularization techniques that allow for the construction of an optimal classifier in Minkowski space. Based on the framework, we propose novel SVMs and perform simulationstudies to show that they are comparable or superior tostandard SVMs. The experiments include classification tasks on data which are represented in terms of their pairwise proximities, wherea Coulomb Classifier outperformed standard SVMs.

We investigate the problem of learning a classification task for datasets which are described by matrices. Rows and columns of these matrices correspond to objects, where row and column objects may belong to different sets, and the entries in the matrix express the relationships between them. We interpret the matrix elements as being produced by an unknown kernel which operates on object pairs and we show that - under mild assumptions - these kernels correspond to dot products in some (unknown) feature space. Minimizing a bound for the generalization error of a linear classifier which has been obtained using covering numbers we derive an objective function for model selection according to the principle of structural risk minimization. The new objective function has the advantage that it allows the analysis of matrices which are not positive definite, and not even symmetric or square.

We introduce a family of classifiers based on a physical analogy to an electrostatic system of charged conductors. The family, called Coulomb classifiers, includes the two best-known support-vector machines (SVMs), the ν-SVM and the C-SVM. In the electrostatics analogy, a training example corresponds to a charged conductor at a given location in space, the classification function corresponds to the electrostatic potential function, and the training objective function corresponds to the Coulomb energy. The electrostatic framework provides not only a novel interpretation of existing algorithms and their interrelationships, but it suggests a variety of new methods for SVMs including kernels that bridge the gap between polynomial and radial-basis functions, objective functions that do not require positive-definite kernels, regularization techniques that allow for the construction of an optimal classifier in Minkowski space. Based on the framework, we propose novel SVMs and perform simulation studies to show that they are comparable or superior to standard SVMs. The experiments include classification tasks on data which are represented in terms of their pairwise proximities, where a Coulomb Classifier outperformed standard SVMs.