We consider the estimation of an i.i.d.\ vector $\xbf \in \R^n$ from measurements $\ybf \in \R^m$ obtained by a general cascade model consisting of a known linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. We present a method, called adaptive generalized approximate message passing (Adaptive GAMP), that enables joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector $\xbf$. The proposed algorithm is a generalization of a recently-developed method by Vila and Schniter that uses expectation-maximization (EM) iterations where the posteriors in the E-steps are computed via approximate message passing. The techniques can be applied to a large class of learning problems including the learning of sparse priors in compressed sensing or identification of linear-nonlinear cascade models in dynamical systems and neural spiking processes. We prove that for large i.i.d.\ Gaussian transform matrices the asymptotic componentwise behavior of the adaptive GAMP algorithm is predicted by a simple set of scalar state evolution equations. This analysis shows that the adaptive GAMP method can yield asymptotically consistent parameter estimates, which implies that the algorithm achieves a reconstruction quality equivalent to the oracle algorithm that knows the correct parameter values. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of complex linear-nonlinear models with provable guarantees.

Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending functional data analysis methods to nonlinear contexts. We study this problem in the case of kernel ridge regression for functional responses with an lr-norm constraint on the combination coefficients. The resulting optimization problem is more involved than those of multiple scalar-valued kernel learning since operator-valued kernels pose more technical and theoretical issues. We propose a multiple operator-valued kernel learning algorithm based on solving a system of linear operator equations by using a block coordinate-descent procedure. We experimentally validate our approach on a functional regression task in the context of finger movement prediction in brain-computer interfaces.

Modelling natural images with sparse coding (SC) has faced two main challenges: flexibly representing varying pixel intensities and realistically representing lowlevel imagecomponents. This paper proposes a novel multiple-cause generative model of low-level image statistics that generalizes the standard SC model in two crucial points: (1) it uses a spike-and-slab prior distribution for a more realistic representation of component absence/intensity, and (2) the model uses the highly nonlinear combination rule of maximal causes analysis (MCA) instead of a linear combination.The major challenge is parameter optimization because a model with either (1) or (2) results in strongly multimodal posteriors. We show for the first time that a model combining both improvements can be trained efficiently while retaining the rich structure of the posteriors. We design an exact piecewise Gibbssampling method and combine this with a variational method based on preselection of latent dimensions. This combined training scheme tackles both analytical and computational intractability and enables application of the model to a large number of observed and hidden dimensions. Applying the model to image patches we study the optimal encoding of images by simple cells in V1 and compare the model's predictions with in vivo neural recordings.

We develop a novel Bayesian nonparametric model for random bipartite graphs. The model is based on the theory of completely random measures and is able to handle a potentially infinite number of nodes. We show that the model has appealing properties and in particular it may exhibit a power-law behavior. We derive a posterior characterization, an Indian Buffet-like generative process for network growth, and a simple and efficient Gibbs sampler for posterior simulation. Our model is shown to be well fitted to several real-world social networks.

Multi-metric learning techniques learn local metric tensors in different parts of a feature space. With such an approach, even simple classifiers can be competitive with the state-of-the-art because the distance measure locally adapts to the structure of the data. The learned distance measure is, however, non-metric, which has prevented multi-metric learning from generalizing to tasks such as dimensionality reduction and regression in a principled way. We prove that, with appropriate changes, multi-metric learning corresponds to learning the structure of a Riemannian manifold. We then show that this structure gives us a principled way to perform dimensionality reduction and regression according to the learned metrics. Algorithmically, we provide the first practical algorithm for computing geodesics according to the learned metrics, as well as algorithms for computing exponential and logarithmic maps on the Riemannian manifold. Together, these tools let many Euclidean algorithms take advantage of multi-metric learning. We illustrate the approach on regression and dimensionality reduction tasks that involve predicting measurements of the human body from shape data.

A key function of brains is undoubtedly the abstraction and maintenance of information from the environment for later use. Neurons in association cortex play an important role in this process: during learning these neurons become tuned to relevant features and represent the information that is required later as a persistent elevation of their activity. It is however not well known how these neurons acquire their task-relevant tuning. Here we introduce a biologically plausible learning scheme that explains how neurons become selective for relevant information when animals learn by trial and error. We propose that the action selection stage feeds back attentional signals to earlier processing levels. These feedback signals interact with feedforward signals to form synaptic tags at those connections that are responsible for the stimulus-response mapping. A globally released neuromodulatory signal interacts with these tagged synapses to determine the sign and strength of plasticity. The learning scheme is generic because it can train networks in different tasks, simply by varying inputs and rewards. It explains how neurons in association cortex learn to (1) temporarily store task-relevant information in non-linear stimulus-response mapping tasks and (2) learn to optimally integrate probabilistic evidence for perceptual decision making.

Statistical models for networks have been typically committed to strong prior assumptions concerning the form of the modeled distributions. Moreover, the vast majority of currently available models are explicitly designed for capturing some specific graph properties (such as power-law degree distributions), which makes them unsuitable for application to domains where the behavior of the target quantities is not known a priori. The key contribution of this paper is twofold. First, we introduce the Fiedler delta statistic, based on the Laplacian spectrum of graphs, which allows to dispense with any parametric assumption concerning the modeled network properties. Second, we use the defined statistic to develop the Fiedler random field model, which allows for efficient estimation of edge distributions over large-scale random networks. After analyzing the dependence structure involved in Fiedler random fields, we estimate them over several real-world networks, showing that they achieve a much higher modeling accuracy than other well-known statistical approaches.

Neural adaptation underlies the ability of neurons to maximize encoded information over a wide dynamic range of input stimuli. While adaptation is an intrinsic feature of neuronal models like the Hodgkin-Huxley model, the challenge is to integrate adaptation in models of neural computation. Recent computational models like the Adaptive Spike Response Model implement adaptation as spike-based addition of fixed-size fast spike-triggered threshold dynamics and slow spike-triggered currents. Such adaptation has been shown to accurately model neural spiking behavior over a limited dynamic range. Taking a cue from kinetic models of adaptation, we propose a multiplicative Adaptive Spike Response Model where the spike-triggered adaptation dynamics are scaled multiplicatively by the adaptation state at the time of spiking. We show that unlike the additive adaptation model, the firing rate in the multiplicative adaptation model saturates to a maximum spike-rate. When simulating variance switching experiments, the model also quantitatively fits the experimental data over a wide dynamic range. Furthermore, dynamic threshold models of adaptation suggest a straightforward interpretation of neural activity in terms of dynamic signal encoding with shifted and weighted exponential kernels. We show that when thus encoding rectified filtered stimulus signals, the multiplicative Adaptive Spike Response Model achieves a high coding efficiency and maintains this efficiency over changes in the dynamic signal range of several orders of magnitude, without changing model parameters.

We present a novel method for learning densities with bounded support which enables us to incorporate `hard' topological constraints. In particular, we show how emerging techniques from computational algebraic topology and the notion of Persistent Homology can be combined with kernel based methods from Machine Learning for the purpose of density estimation. The proposed formalism facilitates learning of models with bounded support in a principled way, and -- by incorporating Persistent Homology techniques in our approach -- we are able to encode algebraic-topological constraints which are not addressed in current state-of the art probabilistic models. We study the behaviour of our method on two synthetic examples for various sample sizes and exemplify the benefits of the proposed approach on a real-world data-set by learning a motion model for a racecar. We show how to learn a model which respects the underlying topological structure of the racetrack, constraining the trajectories of the car.

Conditional Markov Chains (also known as Linear-Chain Conditional Random Fields in the literature) are a versatile class of discriminative models for the distribution of a sequence of hidden states conditional on a sequence of observable variables. Large-sample properties of Conditional Markov Chains have been first studied by Sinn and Poupart [1]. The paper extends this work in two directions: first, mixing properties of models with unbounded feature functions are being established; second, necessary conditions for model identifiability and the uniqueness of maximum likelihood estimates are being given.