We present a hierarchical, control theory inspired method for variational inference (VI) for neural stochastic differential equations (SDEs).While VI for neural SDEs is a promising avenue for uncertaintyaware reasoning in time-series, it is computationally challenging due to the iterative nature of maximizing the ELBO.In this work, we propose to decompose the control term into linear and residual non-linear components and derive an optimal control term for linear SDEs, using stochastic optimal control.Modeling the non-linear component by a neural network, we show how to efficiently train neural SDEs without sacrificing their expressive power.Since the linear part of the control term is optimal and does not need to be learned, the training is initialized at a lower cost and we observe faster convergence.* MO acknowledges funding

We present *Fractional Diffusion Bridge Models* (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). As a remedy, leveraging a recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the existence of a coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared to the Brownian baselines, yielding lower root mean squared deviation (RMSD) of C$_\alpha$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.

We consider unsourced random access (uRA) in a cell-free (CF) user-centric wireless network, where a large number of potential users compete for a random access slot, while only a finite subset is active. The random access users transmit codewords of lengthLsymbols from a shared codebook, which are received byBgeographically distributed radio units (RUs), each equipped withMantennas. Our goal is to devise and analyze acentralizeddecoder to detect the transmitted messages (without prior knowledge of the active users) and estimate the corresponding channel state information. A specific challenge lies in the fact that, due to the geographically distributed nature of the CF network, there is no fixed correspondence between codewords and large-scale fading coefficients (LSFCs). This makes current activity detection approaches which make use of this fixed LSFC-codeword association not directly applicable. To overcome this problem, we propose a scheme where the access codebook is partitioned in location-based subcodes, such that users in a particular location make use of the corresponding subcode. The joint message detection and channel estimation is obtained via a novelApproximated Message Passing(AMP) algorithm for a linear superposition of matrix-valued sources corrupted by noise. The statistical asymmetry in the fading profile and message activity leads todifferent statisticsfor the matrix sources, which distinguishes the AMP formulation from previous cases. In the regime where the codebook size scales linearly withL, whileBandMare fixed, we present a rigorous high-dimensional (but finite-sample) analysis of the proposed AMP algorithm. Exploiting this, we then present a precise (and rigorous) large-system analysis of the message missed-detection and false-alarm rates, as well as the channel estimation mean-square error. The resulting system allows the seamless formation of user-centric clusters and very low latency beamformed uplink-downlink communication without explicit user-RU association, pilot allocation, and power control. This makes the proposed scheme highly appealing for low-latency random access communications in CF networks.

We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression for optimal approximation coefficients and propose to use neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction,—an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.

Generative modeling via stochastic processes has led to remarkable empirical results as well as to recent advances in their theoretical understanding. In principle, both space and time of the processes can be discrete or continuous. In this work, we study time-continuous Markov jump processes on discrete state spaces and investigate their correspondence to state-continuous diffusion processes given by SDEs. In particular, we revisit the $\textit{Ehrenfest process}$, which converges to an Ornstein-Uhlenbeck process in the infinite state space limit. Likewise, we can show that the time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process. This observation bridges discrete and continuous state spaces and allows to carry over methods from one to the respective other setting, such as for instance loss functions that lead to improved convergence. Additionally, we suggest an algorithm for training the time-reversal of Markov jump processes which relies on conditional expectations and can thus be directly related to denoising score matching. We demonstrate our methods in multiple convincing numerical experiments.

Motivated by the recent application of approximate message passing (AMP) to the analysis of convex optimizations in multi-class classifications [Loureiro, et. al., 2021], we present a convergence analysis of AMP dynamics with non-separable multivariate nonlinearities. As an application, we present a complete (and independent) analysis of the motivated convex optimization problem.

We introduce the first continuous-time score-based generative model that leverages fractional diffusion processes for its underlying dynamics. Although diffusion models have excelled at capturing data distributions, they still suffer from various limitations such as slow convergence, mode-collapse on imbalanced data, and lack of diversity. These issues are partially linked to the use of light-tailed Brownian motion (BM) with independent increments. In this paper, we replace BM with an approximation of its non-Markovian counterpart, fractional Brownian motion (fBM), characterized by correlated increments and Hurst index $H \in (0,1)$, where $H=0.5$ recovers the classical BM. To ensure tractable inference and learning, we employ a recently popularized Markov approximation of fBM (MA-fBM) and derive its reverse-time model, resulting in *generative fractional diffusion models* (GFDM). We characterize the forward dynamics using a continuous reparameterization trick and propose *augmented score matching* to efficiently learn the score function, which is partly known in closed form, at minimal added cost. The ability to drive our diffusion model via MA-fBM offers flexibility and control. $H \leq 0.5$ enters the regime of *rough paths* whereas $H>0.5$ regularizes diffusion paths and invokes long-term memory. The Markov approximation allows added control by varying the number of Markov processes linearly combined to approximate fBM. Our evaluations on real image datasets demonstrate that GFDM achieves greater pixel-wise diversity and enhanced image quality, as indicated by a lower FID, offering a promising alternative to traditional diffusion models

We propose automated augmented conjugate inference, a new inference method for non-conjugate Gaussian processes (GP) models.Our method automatically constructs an auxiliary variable augmentation that renders the GP model conditionally conjugate. Building on the conjugate structure of the augmented model, we develop two inference methods. First, a fast and scalable stochastic variational inference method that uses efficient block coordinate ascent updates, which are computed in closed form. Second, an asymptotically correct Gibbs sampler that is useful for small datasets.Our experiments show that our method is up two orders of magnitude faster and more robust than existing state-of-the-art black-box methods.

Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two-fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.

We propose a scalable stochastic variational approach to GP classification building on Pólya-Gamma data augmentation and inducing points. Unlike former approaches, we obtain closed-form updates based on natural gradients that lead to efficient optimization. We evaluate the algorithm on real-world datasets containing up to 11 million data points and demonstrate that it is up to two orders of magnitude faster than the state-of-the-art while being competitive in terms of prediction performance.

We propose an iterative algorithm for solving the Thouless-Anderson-Palmer (TAP) equations of Ising models with arbitrary rotation invariant (random) coupling matrices. In the (thermodynamic) limit of large-systems, we prove by means of the dynamical functional method that the proposed algorithm converges when the so-called de Almeida Thouless (AT) criterion is fulfilled. Moreover, we obtain an exact analytical expression for the rate of the convergence.

We study asymptotic properties of expectation propagation (EP) - a method for approximate inference originally developed in the field of machine learning. Applied to generalized linear models, EP iteratively computes a multivariate Gaussian approximation to the exact posterior distribution. The computational complexity of the repeated update of covariance matrices severely limits the application of EP to large problem sizes. In this study, we present a rigorous analysis by means of free probability theory that allows us to overcome this computational bottleneck if specific data matrices in the problem fulfill certain properties of asymptotic freeness. We demonstrate the relevance of our approach on the gene selection problem of a microarray dataset.

We present an approximate Bayesian inference approach for estimating the intensity of a inhomogeneous Poisson process, where the intensity function is modelled using a Gaussian process (GP) prior via a sigmoid link function. Augmenting the model using a latent marked Poisson process and Polya--Gamma random variables we obtain a representation of the likelihood which is conjugate to the GP prior. We estimate the posterior using a variational free--form mean field optimisation together with the framework of sparse GPs. Furthermore, as alternative approximation we suggest a sparse Laplace's method for the posterior, for which an efficient expectation--maximisation algorithm is derived to find the posterior's mode. Both algorithms compare well against exact inference obtained by a Markov Chain Monte Carlo sampler and standard variational Gauss approach solving the same model, while being one order of magnitude faster. Furthermore, the performance and speed of our method is competitive with that of another recently proposed Poisson process model based on a quadratic link function, while not being limited to GPs with squared exponential kernels and rectangular domains.

Neural decoding may be formulated as dynamic state estimation (filtering) based on point-process observations, a generally intractable problem. Numerical sampling techniques are often practically useful for the decoding of real neural data. However, they are less useful as theoretical tools for modeling and understanding sensory neural systems, since they lead to limited conceptual insight into optimal encoding and decoding strategies. We consider sensory neural populations characterized by a distribution over neuron parameters. We develop an analytically tractable Bayesian approximation to optimal filtering based on the observation of spiking activity that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. Continuous distributions are used to approximate large populations with few parameters, resulting in a filter whose complexity does not grow with population size and allowing optimization of population parameters rather than individual tuning functions. Numerical comparison with particle filtering demonstrates the quality of the approximation. The analytic framework leads to insights that are difficult to obtain from numerical algorithms and is consistent with biological observations about the distribution of sensory cells' preferred stimuli.

We introduce a theoretical approach for designing generalizations of the approximate message passing (AMP) algorithm for compressed sensing which are valid for large observation matrices that are drawn from an invariant random matrix ensemble. By design, the fixed points of the algorithm obey the Thouless-Anderson-Palmer (TAP) equations corresponding to the ensemble. Using a dynamical functional approach we are able to derive an effective stochastic process for the marginal statistics of a single component of the dynamics. This allows us to design memory terms in the algorithm in such a way that the resulting fields become Gaussian random variables allowing for an explicit analysis. The asymptotic statistics of these fields are consistent with the replica ansatz of the compressed sensing problem.

Black box variational inference (BBVI) with reparameterization gradients triggered the exploration of divergence measures other than the Kullback-Leibler (KL) divergence, such as alpha divergences. In this paper, we view BBVI with generalized divergences as a form of estimating the marginal likelihood via biased importance sampling. The choice of divergence determines a bias-variance trade-off between the tightness of a bound on the marginal likelihood (low bias) and the variance of its gradient estimators. Drawing on variational perturbation theory of statistical physics, we use these insights to construct a family of new variational bounds. Enumerated by an odd integer order $K$, this family captures the standard KL bound for $K=1$, and converges to the exact marginal likelihood as $K\to\infty$. Compared to alpha-divergences, our reparameterization gradients have a lower variance. We show in experiments on Gaussian Processes and Variational Autoencoders that the new bounds are more mass covering, and that the resulting posterior covariances are closer to the true posterior and lead to higher likelihoods on held-out data.

The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal encoding/decoding strategies, which are of significant relevance to Computational Neuroscience. We develop an analytically tractable Bayesian approximation to optimal filtering based on point process observations, which allows us to introduce distributional assumptions about sensory cell properties, that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with experiments about the distribution of tuning curve centers. Interestingly, we find that the information gained from the absence of spikes may be crucial to performance.

Data rate adaptation (RA) schemes are the key means by which WLAN adapters adjust their operation to the variable quality of wireless channels. The IEEE 802.11 standard does not specify any RA preferences allowing for a competition in performance among vendors, thus numerous proprietary solutions coexist. While the RA schemes implemented in individual user terminals are unknown to the AP of a hotspot, it is well known that the way how individual stations adapt their rates strongly influences the performance of the whole WLAN cell. Therefore, the knowledge of the scheme applied by each station may be useful for the radio resource management in complex networks (e.g., HetNets or dense WLAN deployments in enterprise networks). In this paper, we present a novel approach to estimate the features of the RA schemes implemented in individual stations and demonstrate its efficiency using both simulated WLAN configurations as well as measurements.

We propose an approximate inference algorithm for continuous time Gaussian-Markov process models with both discrete and continuous time likelihoods. We show that the continuous time limit of the expectation propagation algorithm exists and results in a hybrid fixed point iteration consisting of (1) expectation propagation updates for the discrete time terms and (2) variational updates for the continuous time term. We introduce corrections methods that improve on the marginals of the approximation. This approach extends the classical Kalman-Bucy smoothing procedure to non-Gaussian observations, enabling continuous-time inference in a variety of models, including spiking neuronal models (state-space models with point process observations) and box likelihood models. Experimental results on real and simulated data demonstrate high distributional accuracy and significant computational savings compared to discrete-time approaches in a neural application.

We introduce a nonparametric approach for estimating drift functions in systems of stochastic differential equations from incomplete observations of the state vector. Using a Gaussian process prior over the drift as a function of the state vector, we develop an approximate EM algorithm to deal with the unobserved, latent dynamics between observations. The posterior over states is approximated by a piecewise linearized process and the MAP estimation of the drift is facilitated by a sparse Gaussian process regression.

We reformulate a class of non-linear stochastic optimal control problems introduced by Todorov (in Advances in Neural Information Processing Systems, vol. 19, pp. 1369–1376, 2007 ) as a Kullback-Leibler (KL) minimization problem. As a result, the optimal control computation reduces to an inference computation and approximate inference methods can be applied to efficiently compute approximate optimal controls. We show how this KL control theory contains the path integral control method as a special case. We provide an example of a block stacking task and a multi-agent cooperative game where we demonstrate how approximate inference can be successfully applied to instances that are too complex for exact computation. We discuss the relation of the KL control approach to other inference approaches to control.

We consider the problem of Bayesian inference for continuous time multi-stable stochastic systems which can change both their diffusion and drift parameters at discrete times. We propose exact inference and sampling methodologies for two specific cases where the discontinuous dynamics is given by a Poisson process and a two-state Markovian switch. We test the methodology on simulated data, and apply it to two real data sets in finance and systems biology. Our experimental results show that the approach leads to valid inferences and non-trivial insights.

Bayesian filtering of stochastic stimuli has received a great deal of attention re- cently. It has been applied to describe the way in which biological systems dy- namically represent and make decisions about the environment. There have been no exact results for the error in the biologically plausible setting of inference on point process, however. We present an exact analysis of the evolution of the mean- squared error in a state estimation task using Gaussian-tuned point processes as sensors. This allows us to study the dynamics of the error of an optimal Bayesian decoder, providing insights into the limits obtainable in this task. This is done for Markovian and a class of non-Markovian Gaussian processes. We find that there is an optimal tuning width for which the error is minimized. This leads to a char- acterization of the optimal encoding for the setting as a function of the statistics of the stimulus, providing a mathematically sound primer for an ecological theory of sensory processing.

We discuss the expectation propagation (EP) algorithm for approximate Bayesian inference using a factorizing posterior approximation. For neural network models, we use a central limit theorem argument to make EP tractable when the number of parameters is large. For two types of models, we show that EP can achieve optimal generalization performance when data are drawn from a simple distribution.

We present a novel approach to inference in conditionally Gaussian continuous time stochastic processes, where the latent process is a Markovian jump process. We first consider the case of jump-diffusion processes, where the drift of a linear stochastic differential equation can jump at arbitrary time points. We derive partial differential equations for exact inference and present a very efficient mean field approximation. By introducing a novel lower bound on the free energy, we then generalise our approach to Gaussian processes with arbitrary covariance, such as the non-Markovian RBF covariance. We present results on both simulated and real data, showing that the approach is very accurate in capturing latent dynamics and can be useful in a number of real data modelling tasks.

MOTIVATION: mRNA transcriptional dynamics is governed by a complex network of transcription factor (TF) proteins. Experimental and theoretical analysis of this process is hindered by the fact that measurements of TF activity in vivo is very challenging. Current models that jointly infer TF activities and model parameters rely on either of the two main simplifying assumptions: either the dynamics is simplified (e.g. assuming quasi-steady state) or the interactions between TFs are ignored, resulting in models accounting for a single TF. RESULTS: We present a novel approach to reverse engineer the dynamics of multiple TFs jointly regulating the expression of a set of genes. The model relies on a continuous time, differential equation description of transcriptional dynamics where TFs are treated as latent on/off variables and are modelled using a switching stochastic process (telegraph process). The model can not only incorporate both activation and repression, but allows any non-trivial interaction between TFs, including AND and OR gates. By using a factorization assumption within a variational Bayesian treatment we formulate a framework that can reconstruct both the activity profiles of the TFs and the type of regulation from time series gene expression data. We demonstrate the identifiability of the model on a simple but non-trivial synthetic example, and then use it to formulate non-trivial predictions about transcriptional control during yeast metabolism. AVAILABILITY: http://homepages.inf.ed.ac.uk/gsanguin/ CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Stress response in cells is often mediated by quick activation of transcription factors (TFs). Given the difficulty in experimentally assaying TF activities, several statistical approaches have been proposed to infer them from microarray time courses. However, these approaches often rely on prior assumptions which rule out the rapid responses observed during stress response. RESULTS: We present a novel statistical model to infer how TFs mediate stress response in cells. The model is based on the assumption that sensory TFs quickly transit between active and inactive states. We therefore model mRNA production using a bistable dynamical systems whose behaviour is described by a system of differential equations driven by a latent stochastic process. We assume the stochastic process to be a two-state continuous time jump process, and devise both an exact solution for the inference problem as well as an efficient approximate algorithm. We evaluate the method on both simulated data and real data describing Escherichia coli's response to sudden oxygen starvation. This highlights both the accuracy of the proposed method and its potential for generating novel hypotheses and testable predictions. AVAILABILITY: MATLAB and C++ code used in the article can be downloaded from http://www.dcs.shef.ac.uk/~guido/.

The variational approximation of posterior distributions by multivariate gaussians has been much less popular in the machine learning community compared to the corresponding approximation by factorizing distributions. This is for a good reason: the gaussian approximation is in general plagued by an Omicron(N)(2) number of variational parameters to be optimized, N being the number of random variables. In this letter, we discuss the relationship between the Laplace and the variational approximation, and we show that for models with gaussian priors and factorizing likelihoods, the number of variational parameters is actually Omicron(N). The approach is applied to gaussian process regression with nongaussian likelihoods.

We develop as series of corrections to Expectation Propagation (EP), which is one of the most popular methods for approximate probabilistic inference. These corrections can lead to improvements of the inference approximation or serve as a sanity check, indicating when EP yields unrealiable results.

Diffusion processes are a family of continuous-time continuous-state stochastic processes that are in general only partially observed. The joint estimation of the forcing parameters and the system noise (volatility) in these dynamical systems is a crucial, but non-trivial task, especially when the system is nonlinear and multi-modal. We propose a variational treatment of diffusion processes, which allows us to estimate these parameters by simple gradient techniques and which is computationally less demanding than most MCMC approaches. Furthermore, our parameter inference scheme does not break down when the time step gets smaller, unlike most current approaches. Finally, we show how a cheap estimate of the posterior over the parameters can be constructed based on the variational free energy.

Markov jump processes play an important role in a large number of application domains. However, realistic systems are analytically intractable and they have traditionally been analysed using simulation based techniques, which do not provide a framework for statistical inference. We propose a mean field approximation to perform posterior inference and parameter estimation. The approximation allows a practical solution to the inference problem, {while still retaining a good degree of accuracy.} We illustrate our approach on two biologically motivated systems.

The problem of computing a resample estimate for the reconstruction error in PCA is reformulated as an inference problem with the help of the replica method. Using the expectation consistent (EC) approximation, the intractable inference problem can be solved efficiently using only two variational parameters. A perturbative correction to the result is computed and an alternative simplified derivation is also presented.

We propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation. The approximation can be understood as replacing an average over the original intractable distribution with a tractable one. It requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution. In this way we are able to use Gaussian approximations for models with discrete or bounded variables which allow us to include non-trivial correlations. These are neglected in many other methods. We test the framework on toy benchmark problems for binary variables on fully connected graphs and 2D grids and compare with other methods, such as loopy belief propagation. Good performance is already achieved by using single nodes as tractable substructures. Significant improvements are obtained when a spanning tree is used instead.

We propose a novel a framework for deriving approximations for in- tractable probabilistic models. This framework is based on a free energy (negative log marginal likelihood) and can be seen as a generalization of adaptive TAP [1, 2, 3] and expectation propagation (EP) [4, 5]. The free energy is constructed from two approximating distributions which encode different aspects of the intractable model such a single node con- straints and couplings and are by construction consistent on a chosen set of moments. We test the framework on a difficult benchmark problem with binary variables on fully connected graphs and 2D grid graphs. We find good performance using sets of moments which either specify fac- torized nodes or a spanning tree on the nodes (structured approximation). Surprisingly, the Bethe approximation gives very inferior results even on grids.

We compute approximate analytical bootstrap averages for support vec- tor classification using a combination of the replica method of statistical physics and the TAP approach for approximate inference. We test our method on a few datasets and compare it with exact averages obtained by extensive Monte-Carlo sampling.

A general linear response method for deriving improved estimates of cor- relations in the variational Bayes framework is presented. Three applica- tions are given and it is discussed how to use linear response as a general principle for improving mean field approximations.

We apply the replica method of Statistical Physics combined with a vari- ational method to the approximate analytical computation of bootstrap averages for estimating the generalization error. We demonstrate our ap- proach on regression with Gaussian processes and compare our results with averages obtained by Monte-Carlo sampling.

We develop an approach for sparse representations of gaussian process (GP) models (which are Bayesian types of kernel machines) in order to overcome their limitations for large data sets. The method is based on a combination of a Bayesian on-line algorithm, together with a sequential construction of a relevant subsample of the data that fully specifies the prediction of the GP model. By using an appealing parameterization and projection techniques in a reproducing kernel Hilbert space, recursions for the effective parameters and a sparse gaussian approximation of the posterior process are obtained. This allows for both a propagation of predictions and Bayesian error measures. The significance and robustness of our approach are demonstrated on a variety of experiments.

The seeded region growing (SRG) algorithm is a fast robust parameter-free method for segmenting intensity images given initial seed locations for each region. The requirement of predetermined seeds means that the model cannot operate fully autonomously. In this paper, we demonstrate a novel region growing variant of the pulse-coupled neural network (PCNN), which offers comparable performance to the SRG and is able to generate seed locations internally, opening the way to fully autonomous operation.

The adaptive TAP Gibbs free energy for a general densely connected probabilistic model with quadratic interactions and arbritary single site constraints is derived. We show how a specific sequential minimization of the free energy leads to a generalization of Minka’s expectation propa- gation. Lastly, we derive a sparse representation version of the sequential algorithm. The usefulness of the approach is demonstrated on classifica- tion and density estimation with Gaussian processes and on an indepen- dent component analysis problem.

We combine the replica approach from statistical physics with a varia- tional approach to analyze learning curves analytically. We apply the method to Gaussian process regression. As a main result we derive ap- proximative relations between empirical error measures, the generaliza- tion error and the posterior variance.

Using methods of Statistical Physics, we investigate the rOle of model complexity in learning with support vector machines (SVMs). We show the advantages of using SVMs with kernels of infinite complexity on noisy target rules, which, in contrast to common theoretical beliefs, are found to achieve optimal general(cid:173) ization error although the training error does not converge to the generalization error. Moreover, we find a universal asymptotics of the learning curves which only depend on the target rule but not on the SVM kernel.

We develop an approach for a sparse representation for Gaussian Process (GP) models in order to overcome the limitations of GPs caused by large data sets. The method is based on a combination of a Bayesian online al(cid:173) gorithm together with a sequential construction of a relevant subsample of the data which fully specifies the prediction of the model. Experi(cid:173) mental results on toy examples and large real-world data sets indicate the efficiency of the approach.

Based on a statistical mechanics approach, we develop a method for approximately computing average case learning curves for Gaus(cid:173) sian process regression models. The approximation works well in the large sample size limit and for arbitrary dimensionality of the input space. We explain how the approximation can be systemati(cid:173) cally improved and argue that similar techniques can be applied to general likelihood models.

We derive a mean-field algorithm for binary classification with gaussian processes that is based on the TAP approach originally proposed in statistical physics of disordered systems. The theory also yields an approximate leave-one-out estimator for the generalization error, which is computed with no extra computational cost. We show that from the TAP approach, it is possible to derive both a simpler "naive" mean-field theory and support vector machines (SVMs) as limiting cases. For both mean-field algorithms and support vector machines, simulation results for three small benchmark data sets are presented. They show that one may get state-of-the-art performance by using the leave-one-out estimator for model selection and the built-in leave-one-out estimators are extremely precise when compared to the exact leave-one-out estimate. The second result is taken as strong support for the internal consistency of the mean-field approach.

We propose an algorithm called query by commitee, in which a committee of students is trained on the same data set. The next query is chosen according to the principle of maximal disagreement. The algorithm is studied for two toy models: the high-low game and perceptron learning of another perceptron. As the number of queries goes to infinity, the committee algorithm yields asymptotically finite information gain. This leads to generalization error that decreases exponentially with the number of examples. This in marked contrast to learning from randomly chosen inputs, for which the information gain approaches zero and the generalization error decreases with a relatively slow inverse power law. We suggest that asymptotically finite information gain may be an important characteristic of good query algorithms.