Search

We describe a method for removing the effect of confounders in order to reconstruct a latent quantity of interest. The method, referred to as half-sibling regression, is inspired by recent work in causal inference using additive noise models. We provide a theoretical justification and illustrate the potential of the method in a challenging astronomy application., Comment: Extended version of a paper appearing in the Proceedings of the 32nd International Conference on Machine Learning, Lille, France, 2015

We show how, and under which conditions, the equilibrium states of a first-order Ordinary Differential Equation (ODE) system can be described with a deterministic Structural Causal Model (SCM). Our exposition sheds more light on the concept of causality as expressed within the framework of Structural Causal Models, especially for cyclic models., Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI2013)

We propose one-class support measure machines (OCSMMs) for group anomaly detection which aims at recognizing anomalous aggregate behaviors of data points. The OCSMMs generalize well-known one-class support vector machines (OCSVMs) to a space of probability measures. By formulating the problem as quantile estimation on distributions, we can establish an interesting connection to the OCSVMs and variable kernel density estimators (VKDEs) over the input space on which the distributions are defined, bridging the gap between large-margin methods and kernel density estimators. In particular, we show that various types of VKDEs can be considered as solutions to a class of regularization problems studied in this paper. Experiments on Sloan Digital Sky Survey dataset and High Energy Particle Physics dataset demonstrate the benefits of the proposed framework in real-world applications., Comment: Conference on Uncertainty in Artificial Intelligence (UAI2013)

Information spreads across social and technological networks, but often the network structures are hidden from us and we only observe the traces left by the diffusion processes, called cascades. Can we recover the hidden network structures from these observed cascades? What kind of cascades and how many cascades do we need? Are there some network structures which are more difficult than others to recover? Can we design efficient inference algorithms with provable guarantees? Despite the increasing availability of cascade data and methods for inferring networks from these data, a thorough theoretical understanding of the above questions remains largely unexplored in the literature. In this paper, we investigate the network structure inference problem for a general family of continuous-time diffusion models using an $l_1$-regularized likelihood maximization framework. We show that, as long as the cascade sampling process satisfies a natural incoherence condition, our framework can recover the correct network structure with high probability if we observe $O(d^3 \log N)$ cascades, where $d$ is the maximum number of parents of a node and $N$ is the total number of nodes. Moreover, we develop a simple and efficient soft-thresholding inference algorithm, which we use to illustrate the consequences of our theoretical results, and show that our framework outperforms other alternatives in practice., To appear in the 31st International Conference on Machine Learning (ICML), 2014

Many applications require the analysis of complex interactions between time series. These interactions can be non-linear and involve vector valued as well as complex data structures such as graphs or strings. Here we provide a general framework for the statistical analysis of these interactions when random variables are sampled from stationary time-series of arbitrary objects. To achieve this goal we analyze the properties of the kernel cross-spectral density operator induced by positive definite kernels on arbitrary input domains. This framework enables us to develop an independence test between time series as well as a similarity measure to compare different types of coupling. The performance of our test is compared to the HSIC test using i.i.d. assumptions, showing improvement in terms of detection errors as well as the suitability of this approach for testing dependency in complex dynamical systems. Finally, we use this approach to characterize complex interactions in electrophysiological neural time series.

A mean function in a reproducing kernel Hilbert space (RKHS), or a kernel mean, is central to kernel methods in that it is used by many classical algorithms such as kernel principal component analysis, and it also forms the core inference step of modern kernel methods that rely on embedding probability distributions in RKHSs. Given a finite sample, an empirical average has been used commonly as a standard estimator of the true kernel mean. Despite a widespread use of this estimator, we show that it can be improved thanks to the well-known Stein phenomenon. We propose a new family of estimators called kernel mean shrinkage estimators (KMSEs), which benefit from both theoretical justifications and good empirical performance. The results demonstrate that the proposed estimators outperform the standard one, especially in a "large d, small n" paradigm., Comment: 41 pages

Purpose: Subject motion can severely degrade MR images. A retrospective motion correction algorithm, Gradient-based motion correction, which significantly reduces ghosting and blurring artifacts due to subject motion was proposed. The technique uses the raw data of standard imaging sequences; no sequence modifications or additional equipment such as tracking devices are required. Rigid motion is assumed. Methods: The approach iteratively searches for the motion trajectory yielding the sharpest image as measured by the entropy of spatial gradients. The vast space of motion parameters is efficiently explored by gradient-based optimization with a convergence guarantee. Results: The method has been evaluated on both synthetic and real data in two and three dimentions using standard imaging techniques. MR images are consistently improved over different kinds of motion trajectories. Using a graphics processing unit implementation, computation times are in the order of a few minutes for a full three-dimentional volume. Conclusion: The presented technique can be an alternative or a complement to prospective motion correction methods and is able to improve images with strong motion artifacts from standard imaging sequences without requiring additional data.

We show how, and under which conditions, the equilibrium states of a first-order Ordinary Differential Equation (ODE) system can be described with a deterministic Structural Causal Model (SCM). Our exposition sheds more light on the concept of causality as expressed within the framework of Structural Causal Models, especially for cyclic models.

Networks provide a skeleton for the spread of contagions, like, information, ideas, behaviors and diseases. Many times networks over which contagions diffuse are unobserved and need to be inferred. Here we apply survival theory to develop general additive and multiplicative risk models under which the network inference problems can be solved efficiently by exploiting their convexity. Our additive risk model generalizes several existing network inference models. We show all these models are particular cases of our more general model. Our multiplicative model allows for modeling scenarios in which a node can either increase or decrease the risk of activation of another node, in contrast with previous approaches, which consider only positive risk increments. We evaluate the performance of our network inference algorithms on large synthetic and real cascade datasets, and show that our models are able to predict the length and duration of cascades in real data., Comment: To appear at ICML '13

© 2013 Massachusetts Institute of Technology.Inference of human intention may be an essential step towards understanding human actions and is hence important for realizing efficient human-robot interaction. In this paper, we propose the Intention-Driven Dynamics Model (IDDM), a latent variable model for inferring unknown human intentions. We train the model based on observed human movements/actions. We introduce an efficient approximate inference algorithm to infer the humans intention from an ongoing movement. We verify the feasibility of the IDDM in two scenarios, i.e., target inference in robot table tennis and action recognition for interactive humanoid robots. In both tasks, the IDDM achieves substantial improvements over state-of-The-Art regression and classification.

Subjects operating a brain-computer interface (BCI) based on sensorimotor rhythms exhibit large variations in performance over the course of an experimental session. Here, we show that high-frequency gamma-oscillations, originating in fronto-parietal networks, predict such variations on a trial-to-trial basis. We interpret this finding as empirical support for an in uence of attentional networks on BCI-performance via modulation of the sensorimotor rhythm.

The problem of finding the optimal set of source nodes in a diffusion network that maximizes the spread of information, influence, and diseases in a limited amount of time depends dramatically on the underlying temporal dynamics of the network. However, this still remains largely unexplored to date. To this end, given a network and its temporal dynamics, we first describe how continuous time Markov chains allow us to analytically compute the average total number of nodes reached by a diffusion process starting in a set of source nodes. We then show that selecting the set of most influential source nodes in the continuous time influence maximization problem is NP-hard and develop an efficient approximation algorithm with provable near-optimal performance. Experiments on synthetic and real diffusion networks show that our algorithm outperforms other state of the art algorithms by at least ~20% and is robust across different network topologies., Comment: To appear in the 29th International Conference on Machine Learning (ICML), 2012. Website: http://www.stanford.edu/~manuelgr/influmax/

We address the challenging task of decoupling material properties from lighting properties given a single image. In the last two decades virtually all works have concentrated on exploiting edge information to address this problem. We take a different route by introducing a new prior on reflectance, that models reflectance values as being drawn from a sparse set of basis colors. This results in a Random Field model with global, latent variables (basis colors) and pixel-accurate output reflectance values. We show that without edge information high-quality results can be achieved, that are on par with methods exploiting this source of information. Finally, we are able to improve on state-of-the-art results by integrating edge information into our model. We believe that our new approach is an excellent starting point for future developments in this field.

We consider the problem of function estimation in the case where an underlying causal model can be inferred. This has implications for popular scenarios such as covariate shift, concept drift, transfer learning and semi-supervised learning. We argue that causal knowledge may facilitate some approaches for a given problem, and rule out others. In particular, we formulate a hypothesis for when semi-supervised learning can help, and corroborate it with empirical results., Comment: Appears in Proceedings of the 29th International Conference on Machine Learning (ICML 2012). arXiv admin note: substantial text overlap with arXiv:1112.2738

Diffusion and propagation of information, influence and diseases take place over increasingly larger networks. We observe when a node copies information, makes a decision or becomes infected but networks are often hidden or unobserved. Since networks are highly dynamic, changing and growing rapidly, we only observe a relatively small set of cascades before a network changes significantly. Scalable network inference based on a small cascade set is then necessary for understanding the rapidly evolving dynamics that govern diffusion. In this article, we develop a scalable approximation algorithm with provable near-optimal performance based on submodular maximization which achieves a high accuracy in such scenario, solving an open problem first introduced by Gomez-Rodriguez et al (2010). Experiments on synthetic and real diffusion data show that our algorithm in practice achieves an optimal trade-off between accuracy and running time., Comment: To appear in the 29th International Conference on Machine Learning (ICML), 2012. Website: http://www.stanford.edu/~manuelgr/network-inference-multitree/

We study a particular class of cyclic causal models, where each variable is a (possibly nonlinear) function of its parents and additive noise. We prove that the causal graph of such models is generically identifiable in the bivariate, Gaussian-noise case. We also propose a method to learn such models from observational data. In the acyclic case, the method reduces to ordinary regression, but in the more challenging cyclic case, an additional term arises in the loss function, which makes it a special case of nonlinear independent component analysis. We illustrate the proposed method on synthetic data.

We propose a method for inferring the existence of a latent common cause ('confounder') of two observed random variables. The method assumes that the two effects of the confounder are (possibly nonlinear) functions of the confounder plus independent, additive noise. We discuss under which conditions the model is identifiable (up to an arbitrary reparameterization of the confounder) from the joint distribution of the effects. We state and prove a theoretical result that provides evidence for the conjecture that the model is generically identifiable under suitable technical conditions. In addition, we propose a practical method to estimate the confounder from a finite i.i.d. sample of the effects and illustrate that the method works well on both simulated and real-world data., Comment: Appears in Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI2009)

An automatic particle picking algorithm for processing electron micrographs of a large molecular complex, the 26S proteasome, is described. The algorithm makes use of a coherence enhancing diffusion filter to denoise the data, and a random forest classifier for removing false positives. It does not make use of a 3D reference model, but uses a training set of manually picked particles instead. False positive and false negative rates of around 25 to 30 are achieved on a testing set. The algorithm was developed for a specific particle, but contains steps that should be useful for developing automatic picking algorithms for other particles.

We propose a novel method for inferring whether X causes Y or vice versa from joint observations of X and Y. The basic idea is to model the observed data using probabilistic latent variable models, which incorporate the effects of unobserved noise. To this end, we consider the hypothetical effect variable to be a function of the hypothetical cause variable and an independent noise term (not necessarily additive). An important novel aspect of our work is that we do not restrict the model class, but instead put general non-parametric priors on this function and on the distribution of the cause. The causal direction can then be inferred by using standard Bayesian model selection. We evaluate our approach on synthetic data and real-world data and report encouraging results.

Modelling camera shake as a space-invariant convolution simplifies the problem of removing camera shake, but often insufficiently models actual motion blur such as those due to camera rotation and movements outside the sensor plane or when objects in the scene have different distances to the camera. In an effort to address these limitations, (i) we introduce a taxonomy of camera shakes, (ii) we build on a recently introduced framework for space-variant filtering by Hirsch et al. and a fast algorithm for single image blind deconvolution for space-invariant filters by Cho and Lee to construct a method for blind deconvolution in the case of space-variant blur, and (iii), we present an experimental setup for evaluation that allows us to take images with real camera shake while at the same time recording the spacevariant point spread function corresponding to that blur. Finally, we demonstrate that our method is able to deblur images degraded by spatially-varying blur originating from real camera shake, even without using additionally motion sensor information.

Time plays an essential role in the diffusion of information, influence and disease over networks. In many cases we only observe when a node copies information, makes a decision or becomes infected -- but the connectivity, transmission rates between nodes and transmission sources are unknown. Inferring the underlying dynamics is of outstanding interest since it enables forecasting, influencing and retarding infections, broadly construed. To this end, we model diffusion processes as discrete networks of continuous temporal processes occurring at different rates. Given cascade data -- observed infection times of nodes -- we infer the edges of the global diffusion network and estimate the transmission rates of each edge that best explain the observed data. The optimization problem is convex. The model naturally (without heuristics) imposes sparse solutions and requires no parameter tuning. The problem decouples into a collection of independent smaller problems, thus scaling easily to networks on the order of hundreds of thousands of nodes. Experiments on real and synthetic data show that our algorithm both recovers the edges of diffusion networks and accurately estimates their transmission rates from cascade data., To appear in the 28th International Conference on Machine Learning (ICML), 2011. Website: http://www.stanford.edu/~manuelgr/netrate/

Brain-Computer Interfaces based on electrocorticography (ECoG) or electroencephalography (EEG), in combination with robot-assisted active physical therapy, may support traditional rehabilitation procedures for patients with severe motor impairment due to cerebrovascular brain damage caused by stroke. In this short report, we briefly review the state-of-the art in this exciting new field, give an overview of the work carried out at the Max Planck Institute for Biological Cybernetics and the University of Tübingen, and discuss challenges that need to be addressed in order to move from basic research to clinical studies.

This letter introduces a nonlinear measure of independence between random variables for remote sensing supervised feature selection. The so-called Hilbert–Schmidt independence criterion (HSIC) is a kernel method for evaluating statistical dependence and it is based on computing the Hilbert–Schmidt norm of the cross-covariance operator of mapped samples in the corresponding Hilbert spaces. The HSIC empirical estimator is easy to compute and has good theoretical and practical properties. Rather than using this estimate for maximizing the dependence between the selected features and the class labels, we propose the more sensitive criterion of minimizing the associated HSIC p-value. Results in multispectral, hyperspectral, and SAR data feature selection for classification show the good performance of the proposed approach.

Inferring the causal structure of a set of random variables from a finite sample of the joint distribution is an important problem in science. Recently, methods using additive noise models have been suggested to approach the case of continuous variables. In many situations, however, the variables of interest are discrete or even have only finitely many states. In this work we extend the notion of additive noise models to these cases. Whenever the joint distribution P(X;Y ) admits such a model in one direction, e.g. Y = f(X) + N; N ? X, it does not admit the reversed model X = g(Y ) + ~N ; ~N ? Y as long as the model is chosen in a generic way. Based on these deliberations we propose an efficient new algorithm that is able to distinguish between cause and effect for a finite sample of discrete variables. We show that this algorithm works both on synthetic and real data sets.

Embeddings of probability measures into reproducing kernel Hilbert spaces have been proposed as a straightforward and practical means of representing and comparing probabilities. In particular, the distance between embeddings (the maximum mean discrepancy, or MMD) has several key advantages over many classical metrics on distributions, namely easy computability, fast convergence and low bias of finite sample estimates. An important requirement of the embedding RKHS is that it be characteristic: in this case, the MMD between two distributions is zero if and only if the distributions coincide. Three new results on the MMD are introduced in the present study. First, it is established that MMD corresponds to the optimal risk of a kernel classifier, thus forming a natural link between the distance between distributions and their ease of classification. An important consequence is that a kernel must be characteristic to guarantee classifiability between distributions in the RKHS. Second, the class of characteristic kernels is broadened to incorporate all strictly positive definite kernels: these include non-translation invariant kernels and kernels on non-compact domains. Third, a generalization of the MMD is proposed for families of kernels, as the supremum over MMDs on a class of kernels (for instance the Gaussian kernels with different bandwidths). This extension is necessary to obtain a single distance measure if a large selection or class of characteristic kernels is potentially appropriate. This generalization is reasonable, given that it corresponds to the problem of learning the kernel by minimizing the risk of the corresponding kernel classifier. The generalized MMD is shown to have consistent finite sample estimates, and its performance is demonstrated on a homogeneity testing example.

The NIPS 2008 workshop on causality provided a forum for researchers from different horizons to share their view on causal modeling and address the difficult question of assessing causal models. There has been a vivid debate on properly separating the notion of causality from particular models such as graphical models, which have been dominating the field in the past few years. Part of the workshop was dedicated to discussing the results of a challenge, which offered a wide variety of applications of causal modeling. We have regrouped in these proceedings the best papers presented. Most lectures were videotaped or recorded. All information regarding the challenge and the lectures are found at http://www.clopinet.com/isabelle/Projects/NIPS2008/. This introduction provides a synthesis of the findings and a gentle introduction to causality topics, which are the object of active research.

The causal Markov condition (CMC) is a postulate that links observations to causality. It describes the conditional independences among the observations that are entailed by a causal hypothesis in terms of a directed acyclic graph. In the conventional setting, the observations are random variables and the independence is a statistical one, i.e., the information content of observations is measured in terms of Shannon entropy. We formulate a generalized CMC for any kind of observations on which independence is defined via an arbitrary submodular information measure. Recently, this has been discussed for observations in terms of binary strings where information is understood in the sense of Kolmogorov complexity. Our approach enables us to find computable alternatives to Kolmogorov complexity, e.g., the length of a text after applying existing data compression schemes. We show that our CMC is justified if one restricts the attention to a class of causal mechanisms that is adapted to the respective information measure. Our justification is similar to deriving the statistical CMC from functional models of causality, where every variable is a deterministic function of its observed causes and an unobserved noise term. Our experiments on real data demonstrate the performance of compression based causal inference., Comment: 21 pages, 4 figures

We develop an incremental generalized expectation maximization (GEM) framework to model the multiframe blind deconvolution problem. A simplistic version of this problem was recently studied by Harmeling etal~citeharmeling09}. We solve a more realistic version of this problem which includes the following major features: (i) super-resolution ability emph{despite noise and unknown blurring; (ii) saturation-correction, i.e., handling of overexposed pixels that can otherwise confound the image processing; and (iii) simultaneous handling of color channels. These features are seamlessly integrated into our incremental GEM framework to yield simple but efficient multiframe blind deconvolution algorithms. We present technical details concerning critical steps of our algorithms, especially to highlight how all operations can be written using matrix-vector multiplications. We apply our algorithm to real-world images from astronomy and super resolution tasks. Our experimental results show that our methods yield improve d resolution and deconvolution at the same time.

From 27.09.2009 to 02.10.2009, the Dagstuhl Seminar 09401 ``Machine learning approaches to statistical dependences and causality'' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general.

We present a methodology for incorporating prior knowledge on class probabilities into the registration process. By using knowledge from the imaging modality, pre-segmentations, and/or probabilistic atlases, we construct vectors of class probabilities for each image voxel. By defining new image similarity measures for distribution-valued images, we show how the class probability images can be nonrigidly registered in a variational framework. An experiment on nonrigid registration of MR and CT full-body scans illustrates that the proposed technique outperforms standard mutual information (MI) and normalized mutual information (NMI) based registration techniques when measured in terms of target registration error (TRE) of manually labeled fiducials.

From an information-theoretic perspective, a noisy transmission system such as a visual Brain-Computer Interface (BCI) speller could benefit from the use of errorcorrecting codes. However, optimizing the code solely according to the maximal minimum-Hamming-distance criterion tends to lead to an overall increase in target frequency of target stimuli, and hence a significantly reduced average target-to-target interval (TTI), leading to difficulties in classifying the individual event-related potentials (ERPs) due to overlap and refractory effects. Clearly any change to the stimulus setup must also respect the possible psychophysiological consequences. Here we report new EEG data from experiments in which we explore stimulus types and codebooks in a within-subject design, finding an interaction between the two factors. Our data demonstrate that the traditional, rowcolumn code has particular spatial properties that lead to better performance than one would expect from its TTIs and Hamming-distances alone, but nonetheless error-correcting codes can improve performance provided the right stimulus type is used.

We study the problem of domain transfer for a supervised classification task in mRNA splicing. We consider a number of recent domain transfer methods from machine learning, including some that are novel, and evaluate them on genomicsequence data from model organisms of varying evolutionary distance. We find that in cases where the organisms are not closely related, the use of domain adaptation methods can help improve classification performance.

Embeddings of random variables in reproducing kernel Hilbert spaces (RKHSs) may be used to conduct statistical inference based on higher order moments. For sufficiently rich (characteristic) RKHSs, each probability distribution has a unique embedding, allowing all statistical properties of the distribution to be taken into consideration. Necessary and sufficient conditions for an RKHS to be characteristic exist for Rn. In the present work, conditions are established for an RKHS to be characteristic on groups and semigroups. Illustrative examples are provided, including characteristic kernels on periodic domains, rotation matrices, and Rn+.

Multi-modal image registration is a challenging problem in medical imaging. The goal is to align anatomically identical structures; however, their appearance in images acquired with different imaging devices, such as CT or MR, may be very different. Registration algorithms generally deform one image, the floating image, such that it matches with a second, the reference image, by maximizing some similarity score between the deformed and the reference image. Instead of using a universal, but a priori fixed similarity criterion such as mutual information, we propose learning a similarity measure in a discriminative manner such that the reference and correctly deformed floating images receive high similarity scores. To this end, we develop an algorithm derived from max-margin structured output learning, and employ the learned similarity measure within a standard rigid registration algorithm. Compared to other approaches, our method adapts to the specific registration problem at hand and exploits correlations between neighboring pixels in the reference and the floating image. Empirical evaluation on CT-MR/PET-MR rigid registration tasks demonstrates that our approach yields robust performance and outperforms the state of the art methods for multi-modal medical image registration.

This paper introduces a new approach to constructing meaningful lower dimensional representations of sets of data points. We argue that constraining the mapping between the high and low dimensional spaces to be a diffeomorphism is a natural way of ensuring that pairwise distances are approximately preserved. Accordingly we develop an algorithm which diffeomorphically maps the data near to a lower dimensional subspace and then projects onto that subspace. The problem of solving for the mapping is transformed into one of solving for an Eulerian flow field which we compute using ideas from kernel methods. We demonstrate the efficacy of our approach on various real world data sets.

The discovery of causal relationships between a set of observed variables is a fundamental problem in science. For continuous-valued data linear acyclic causal models are often used because these models are well understood and there are well-known methods to fit them to data. In reality, of course, many causal relationships are more or less nonlinear, raising some doubts as to the applicability and usefulness of purely linear methods. In this contribution we show that in fact the basic linear framework can be generalized to nonlinear models with additive noise. In this extended framework, nonlinearities in the data-generating process are in fact a blessing rather than a curse, as they typically provide information on the underlying causal system and allow more aspects of the true data-generating mechanisms to be identified. In addition to theoretical results we show simulations and some simple real data experiments illustrating the identification power provided by nonlinearities.

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as $\gamma_k$, indexed by the kernel function $k$ that defines the inner product in the RKHS. We present three theoretical properties of $\gamma_k$. First, we consider the question of determining the conditions on the kernel $k$ for which $\gamma_k$ is a metric: such $k$ are denoted {\em characteristic kernels}. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g. on compact domains), and are difficult to check, our conditions are straightforward and intuitive: bounded continuous strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on $\bb{R}^d$, then it is characteristic if and only if the support of its Fourier transform is the entire $\bb{R}^d$. Second, we show that there exist distinct distributions that are arbitrarily close in $\gamma_k$. Third, to understand the nature of the topology induced by $\gamma_k$, we relate $\gamma_k$ to other popular metrics on probability measures, and present conditions on the kernel $k$ under which $\gamma_k$ metrizes the weak topology., Comment: 48 pages

We investigate an implicit method to compute a piecewise linear representation of a surface from a set of sample points. As implicit surface functions we use the weighted sum of piecewise linear kernel functions. For such a function we can partition Rd in such a way that these functions are linear on the subsets of the partition. For each subset in the partition we can then compute the zero level set of the function exactly as the intersection of a hyperplane with the subset.

We study a pattern classification algorithm which has recently been proposed by Vapnik and coworkers. It builds on a new inductive principle which assumes that in addition to positive and negative data, a third class of data is available, termed the Universum. We assay the behavior of the algorithm by establishing links with Fisher discriminant analysis and oriented PCA, as well as with an SVM in a projected subspace (or, equivalently, with a data-dependent reduced kernel). We also provide experimental results.

We propose a new measure of conditional dependence of random variables, based on normalized cross-covariance operators on reproducing kernel Hilbert spaces. Unlike previous kernel dependence measures, the proposed criterion does not depend on the choice of kernel in the limit of infinite data, for a wide class of kernels. At the same time, it has a straightforward empirical estimate with good convergence behaviour. We discuss the theoretical properties of the measure, and demonstrate its application in experiments.

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). The embedding function has been proven to be injective when the reproducing kernel is universal. In this case, the embedding induces a metric on the space of probability distributions defined on compact metric spaces. In the present work, we consider more broadly the problem of specifying characteristic kernels, defined as kernels for which the RKHS embedding of probability measures is injective. In particular, characteristic kernels can include non-universal kernels. We restrict ourselves to translation-invariant kernels on Euclidean space, and define the associated metric on probability measures in terms of the Fourier spectrum of the kernel and characteristic functions of these measures. The support of the kernel spectrum is important in finding whether a kernel is characteristic: in particular, the embedding is injective if and only if the kernel spectrum has the entire domain as its support. Characteristic kernels may nonetheless have difficulty in distinguishing certain distributions on the basis of finite samples, again due to the interaction of the kernel spectrum and the characteristic functions of the measures.

This report summarizes the theory and some main applications of a new non-monotonic algorithm for maximizing a Poisson Likelihood, which for Positron Emission Tomography (PET) is equivalent to minimizing the associated Kullback-Leibler Divergence, and for Transmission Tomography is similar to maximizing the dual of a maximum entropy problem. We call our method non-monotonic maximum likelihood (NMML) and show its application to different problems such as tomography and image restoration. We discuss some theoretical properties such as convergence for our algorithm. Our experimental results indicate that speedups obtained via our non-monotonic methods are substantial.

We propose a framework for analyzing and comparing distributions, allowing us to design statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS). We present two tests based on large deviation bounds for the test statistic, while a third is based on the asymptotic distribution of this statistic. The test statistic can be computed in quadratic time, although efficient linear time approximations are available. Several classical metrics on distributions are recovered when the function space used to compute the difference in expectations is allowed to be more general (eg.~a Banach space). We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

A system and method (12) for determining a property map (82) for an object, in particular for a living being based on at least one first image (84) in particular a nuclear magnetic resonance image, an MR image, of the object. According to the method (12), a structure for reference pairs is defined in a first step (96), each reference pair (16 - 26), having at least two entries (62). The first entry represents a property value, in particular, an attenuation value. The second entry (62) preferably represents a group of related image points (67), in particular, extracted from MR images (28) comprising an image point of interest corresponding to the property value. A number of training pairs (16 - 26) are prepared in a further step (98) of the method (12). A structure for the training pairs (16 - 26) corresponds to the structure of the reference pairs and the entries for the training pairs (16 - 26) are known. A relationship between the first entries and the further entries (62 - 66) of the training pairs (16 - 26) is established by machine learning in a further step (100) of the method (12) in order to be able to predict a corresponding property value (88) for any point (90) of the first image (84).

This technical report is merely an extended version of the appendix of Steinke et.al. "Manifold-valued Thin-Plate Splines with Applications in Computer Graphics" (2008) with complete proofs, which had to be omitted due to space restrictions. This technical report requires a basic knowledge of differential geometry. However, apart from that requirement the technical report is self-contained.

We consider the problem of constructing a function whose zero set is to represent a surface, given sample points with surface normal vectors. The contributions include a novel means of regularising multi-scale compactly supported basis functions that leads to the desirable properties previously only associated with fully supported bases, and show equivalence to a Gaussian process with modified covariance function. We also provide a regularisation framework for simpler and more direct treatment of surface normals, along with a corresponding generalisation of the representer theorem. We demonstrate the techniques on 3D problems of up to 14 million data points, as well as 4D time series data.

An overview of different approaches to brain-computer interfaces (BCIs) developed in our laboratory is given. An important clinical application of BCIs is to enable communication or environmental control in severely paralyzed patients. The BCI “Thought-Translation Device (TTD)” allows verbal communication through the voluntary self-regulation of brain signals (e.g., slow cortical potentials (SCPs)), which is achieved by operant feedback training. Humans' ability to self-regulate their SCPs is used to move a cursor toward a target that contains a selectable letter set. Two different approaches were followed to developWeb browsers that could be controlled with binary brain responses. Implementing more powerful classification methods including different signal parameters such as oscillatory features improved our BCI considerably. It was also tested on signals with implanted electrodes. Most BCIs provide the user with a visual feedback interface. Visually impaired patients require an auditory feedback mode. A procedure using auditory (sonified) feedback of multiple EEG parameters was evaluated. Properties of the auditory systems are reported and the results of two experiments with auditory feedback are presented. Clinical data of eight ALS patients demonstrated that all patients were able to acquire efficient brain control of one of the three available BCI systems (SCP, µ-rhythm, and P300), most of them used the SCP-BCI. A controlled comparison of the three systems in a group of ALS patients, however, showed that P300-BCI and the µ-BCI are faster and more easily acquired than SCP-BCI, at least in patients with some rudimentary motor control left. Six patients who started BCI training after entering the completely locked-in state did not achieve reliable communication skills with any BCI system. One completely locked-in patient was able to communicate shortly with a ph-meter, but lost control afterward.

We consider the scenario where training and test data are drawn from different distributions, commonly referred to as sample selection bias. Most algorithms for this setting try to first recover sampling distributions and then make appro- priate corrections based on the distribution estimate. We present a nonparametric method which directly produces resampling weights without distribution estima- tion. Our method works by matching distributions between training and testing sets in feature space. Experimental results demonstrate that our method works well in practice.