Standard probabilistic sparse coding assumes a Laplace prior, a linear mapping from latents to observables, and Gaussian observable distributions. We here derive a solely entropy-based learning objective for the parameters of standard sparse coding. The novel variational objective has the following features: (A) unlike MAP approximations, it uses non-trivial posterior approximations for probabilistic inference; (B) the novel objective is fully analytic; and (C) the objective allows for a novel principled form of annealing. The objective is derived by first showing that the standard ELBO objective converges to a sum of entropies, which matches similar recent results for generative models with Gaussian priors. The conditions under which the ELBO becomes equal to entropies are then shown to have analytic solutions, which leads to the fully analytic objective. Numerical experiments are used to demonstrate the feasibility of learning with such entropy-based ELBOs. We investigate different posterior approximations including Gaussians with correlated latents and deep amortized approximations. Furthermore, we numerically investigate entropy-based annealing which results in improved learning. Our main contributions are theoretical, however, and they are twofold: (1) we provide the first demonstration on how a recently shown convergence of the ELBO to entropy sums can be used for learning; and (2) using the entropy objective, we derive a fully analytic ELBO objective for the standard sparse coding generative model.

Conventional and electron microscopy visualize structures in the micrometer to nanometer range, and such visualizations contribute decisively to our understanding of biological processes. Due to different factors in recording processes, microscopy images are subject to noise. Especially at their respective resolution limits, a high degree of noise can negatively effect both image interpretation by experts and further automated processing. However, the deteriorating effects of strong noise can be alleviated to a large extend by image enhancement algorithms. Because of the inherent high noise, a requirement for such algorithms is their applicability directly to noisy images or, in the extreme case, to just a single noisy image without a priori noise level information (referred to as blind zero-shot setting). This work investigates blind zero-shot algorithms for microscopy image denoising. The strategies for denoising applied by the investigated approaches include: filtering methods, recent feed-forward neural networks which were amended to be trainable on noisy images, and recent probabilistic generative models. As datasets we consider transmission electron microscopy images including images of SARS-CoV-2 viruses and fluorescence microscopy images. A natural goal of denoising algorithms is to simultaneously reduce noise while preserving the original image features, e.g., the sharpness of structures. However, in practice, a tradeoff between both aspects often has to be found. Our performance evaluations, therefore, focus not only on noise removal but set noise removal in relation to a metric which is instructive about sharpness. For all considered approaches, we numerically investigate their performance, report their denoising/sharpness tradeoff on different images, and discuss future developments. We observe that, depending on the data, the different algorithms can provide significant advantages or disadvantages in terms of their noise removal vs. sharpness preservation capabilities, which may be very relevant for different virological applications, e.g., virological analysis or image segmentation.

The central objective function of a variational autoencoder (VAE) is its variational lower bound (the ELBO). Here we show that for standard (i.e., Gaussian) VAEs the ELBO converges to a value given by the sum of three entropies: the (negative) entropy of the prior distribution, the expected (negative) entropy of the observable distribution, and the average entropy of the variational distributions (the latter is already part of the ELBO). Our derived analytical results are exact and apply for small as well as for intricate deep networks for encoder and decoder. Furthermore, they apply for finitely and infinitely many data points and at any stationary point (including local maxima and saddle points). The result implies that the ELBO can for standard VAEs often be computed in closed-form at stationary points while the original ELBO requires numerical approximations of integrals. As a main contribution, we provide the proof that the ELBO for VAEs is at stationary points equal to entropy sums. Numerical experiments then show that the obtained analytical results are sufficiently precise also in those vicinities of stationary points that are reached in practice. Furthermore, we discuss how the novel entropy form of the ELBO can be used to analyze and understand learning behavior. More generally, we believe that our contributions can be useful for future theoretical and practical studies on VAE learning as they provide novel information on those points in parameters space that optimization of VAEs converges to.

Latent variable models (LVMs) represent observed variables by parameterized functions of latent variables. Prominent examples of LVMs for unsupervised learning are probabilistic PCA or probabilistic sparse coding which both assume a weighted linear summation of the latents to determine the mean of a Gaussian distribution for the observables. In many cases, however, observables do not follow a Gaussian distribution. For unsupervised learning, LVMs which assume specific non-Gaussian observables (e.g., Bernoulli or Poisson) have therefore been considered. Already for specific choices of distributions, parameter optimization is challenging and only a few previous contributions considered LVMs with more generally defined observable distributions. In this contribution, we do consider LVMs that are defined for a range of different distributions, i.e., observables can follow any (regular) distribution of the exponential family. Furthermore, the novel class of LVMs presented here is defined for binary latents, and it uses maximization in place of summation to link the latents to observables. In order to derive an optimization procedure, we follow an expectation maximization approach for maximum likelihood parameter estimation. We then show, as our main result, that a set of very concise parameter update equations can be derived which feature the same functional form for all exponential family distributions. The derived generic optimization can consequently be applied (without further derivations) to different types of metric data (Gaussian and non-Gaussian) as well as to different types of discrete data. Moreover, the derived optimization equations can be combined with a recently suggested variational acceleration which is likewise generically applicable to the LVMs considered here. Thus, the combination maintains generic and direct applicability of the derived optimization procedure, but, crucially, enables efficient scalability. We numerically verify our analytical results using different observable distributions, and, furthermore, discuss some potential applications such as learning of variance structure, noise type estimation and denoising.

We combine two popular optimization approaches to derive learning algorithms for generative models: variational optimization and evolutionary algorithms. The combination is realized for generative models with discrete latents by using truncated posteriors as the family of variational distributions. The variational parameters of truncated posteriors are sets of latent states. By interpreting these states as genomes of individuals and by using the variational lower bound to define a fitness, we can apply evolutionary algorithms to realize the variational loop. The used variational distributions are very flexible and we show that evolutionary algorithms can effectively and efficiently optimize the variational bound. Furthermore, the variational loop is generally applicable (“black box”) with no analytical derivations required. To show general applicability, we apply the approach to three generative models (we use Noisy-OR Bayes Nets, Binary Sparse Coding, and Spike-and-Slab Sparse Coding). To demonstrate effectiveness and efficiency of the novel variational approach, we use the standard competitive benchmarks of image denoising and inpainting. The benchmarks allow quantitative comparisons to a wide range of methods including probabilistic approaches, deep deterministic and generative networks, and non-local image processing methods. In the category of “zero-shot” learning (when only the corrupted image is used for training), we observed the evolutionary variational algorithm to significantly improve the state-of-the-art in many benchmark settings. For one well-known inpainting benchmark, we also observed state-of-the-art performance across all categories of algorithms although we only train on the corrupted image. In general, our investigations highlight the importance of research on optimization methods for generative models to achieve performance improvements.

How can we efficiently find very large numbers of clustersCin very large datasetsNof potentially high dimensionalityD? Here we address the question by using a novel variational approach to optimize Gaussian mixture models (GMMs) with diagonal covariance matrices. The variational method approximates expectation maximization (EM) by applying truncated posteriors as variational distributions and partial E-steps in combination with coresets. Run time complexity to optimize the clustering objective then reduces fromO(NCD) per conventional EM iteration toO(N′G2D) for a variational EM iteration on coresets (with coreset sizeN ′ ≤ Nand truncation parameterG≪C). Based on the strongly reduced run time complexity per iteration, which scales sublinearly withNC, we then provide a concrete, practically applicable, parallelized and highly efficient clustering algorithm. In numerical experiments on standard large-scale benchmarks we (A) show that also overall clustering times scale sublinearly withNC, and (B) observe substantial wall-clock speedups compared to already highly efficient recently reported results. The algorithm’s sublinear scaling allows for applications at scales where alternative methods cease to be applicable. We demonstrate such very large-scale applicability using the YFCC100M benchmark, for which we realize with a GMM of up to 50.000 clusters an optimization of a data density model with up to 150 M parameters.

We investigate how the neural processing in auditory cortex is shaped by the statistics of natural sounds. Hypothesising that auditory cortex (A1) represents the structural primitives out of which sounds are composed, we employ a statistical model to extract such components. The input to the model are cochleagrams which approximate the non-linear transformations a sound undergoes from the outer ear, through the cochlea to the auditory nerve. Cochleagram components do not superimpose linearly, but rather according to a rule which can be approximated using the max function. This is a consequence of the compression inherent in the cochleagram and the sparsity of natural sounds. Furthermore, cochleagrams do not have negative values. Cochleagrams are therefore not matched well by the assumptions of standard linear approaches such as sparse coding or ICA. We therefore consider a new encoding approach for natural sounds, which combines a model of early auditory processing with maximal causes analysis (MCA), a sparse coding model which captures both the non-linear combination rule and non-negativity of the data. An efficient truncated EM algorithm is used to fit the MCA model to cochleagram data. We characterize the generative fields (GFs) inferred by MCA with respect to in vivo neural responses in A1 by applying reverse correlation to estimate spectro-temporal receptive fields (STRFs) implied by the learned GFs. Despite the GFs being non-negative, the STRF estimates are found to contain both positive and negative subfields, where the negative subfields can be attributed to explaining away effects as captured by the applied inference method. A direct comparison with ferret A1 shows many similar forms, and the spectral and temporal modulation tuning of both ferret and model STRFs show similar ranges over the population. In summary, our model represents an alternative to linear approaches for biological auditory encoding while it captures salient data properties and links inhibitory subfields to explaining away effects.

One iteration of standard k-means (i.e., Lloyd’s algorithm) or standard EM for Gaussian mixture models (GMMs) scales linearly with the number of clusters C, data points N, and data dimensionality D. In this study, we explore whether one iteration of k-means or EM for GMMs can scale sublinearly with C at run-time, while improving the clustering objective remains effective. The tool we apply for complexity reduction is variational EM, which is typically used to make training of generative models with exponentially many hidden states tractable. Here, we apply novel theoretical results on truncated variational EM to make tractable clustering algorithms more efficient. The basic idea is to use a partial variational E-step which reduces the linear complexity of O(NCD) required for a full E-step to a sublinear complexity. Our main observation is that the linear dependency on C can be reduced to a dependency on a much smaller parameter G which relates to cluster neighborhood relations. We focus on two versions of partial variational EM for clustering: variational GMM, scaling with O(NG²D), and variational k-means, scaling with O(NGD) per iteration. Empirical results show that these algorithms still require comparable numbers of iterations to improve the clustering objective to same values as k-means. For data with many clusters, we consequently observe reductions of net computational demands between two and three orders of magnitude. More generally, our results provide substantial empirical evidence in favor of clustering to scale sublinearly with C.

We establish a link between evolutionary algorithms (EAs) and learning of probabilistic generative models with binary hidden variables. Learning is formulated as approximate maximum likelihood optimization using variational expectation maximization. When choosing truncated posteriors as variational distributions, the variational parameters take the form of sets of latent states. By (A) identifying these sets with populations of genomes, and by (B) defining the fitness of individuals as the joint probability of the chosen generative model, the sets of latent states can be optimized using EAs. We obtain scalable learning algorithms that effectively improve the tractable free energy objective of truncated posteriors. While this novel approach is applicable to any model with binary latents and tractable joint probability, as a proof of concept, we here apply it to the optimization of parameters of noisy-OR Bayes Nets (modeling binary data) and Binary Sparse Coding (modelling continuous data). We find that the data likelihood is efficiently improved by employing genetic algorithms with point mutations and single-point cross-over as EAs. In general we believe that, with the novel link established here, standard as well as recent results in evolutionary optimization can be leveraged to address the difficult problem of parameter optimization in generative models.

We explore classifier training for data sets with very few labels. We investigate this task using a neural network for nonnegative data. The network is derived from a hierarchical normalized Poisson mixture model with one observed and two hidden layers. With the single objective of likelihood optimization, both labeled and unlabeled data are naturally incorporated into learning. The neural activation and learning equations resulting from our derivation are concise and local. As a consequence, the network can be scaled using standard deep learning tools for parallelized GPU implementation. Using standard benchmarks for nonnegative data, such as text document representations, MNIST, and NIST SD19, we study the classification performance when very few labels are used for training. In different settings, the network's performance is compared to standard and recently suggested semisupervised classifiers. While other recent approaches are more competitive for many labels or fully labeled data sets, we find that the network studied here can be applied to numbers of few labels where no other system has been reported to operate so far.

Inference and learning for probabilistic generative networks is often very challenging and typically prevents scalability to as large networks as used for deep discriminative approaches. To obtain efficiently trainable, large-scale and well performing generative networks for semi-supervised learning, we here combine two recent developments: a neural network reformulation of hierarchical Poisson mixtures (Neural Simpletrons), and a novel truncated variational EM approach (TV-EM). TV-EM provides theoretical guarantees for learning in generative networks, and its application to Neural Simpletrons results in particularly compact, yet approximately optimal, modifications of learning equations. If applied to standard benchmarks, we empirically find: that learning converges in fewer EM iterations, that the complexity per EM iteration is reduced, and that final likelihood values are higher on average. For the task of classification on data sets with few labels, learning improvements result in consistently lower error rates if compared to applications without truncation. Experiments on the MNIST data set herein allow for comparison to standard and state-of-the-art models in the semi-supervised setting. Further experiments on the NIST SD19 data set show the scalability of the approach when a very large amount of additional unlabeled data is available.

Sparse coding algorithms with continuous latent variables have been the subject of a large number of studies. However, discrete latent spaces for sparse coding have been largely ignored. In this work, we study sparse coding with latents described by discrete instead of continuous prior distributions. We consider the general case in which the latents (while being sparse) can take on any value of a finite set of possible values and in which we learn the prior probability of any value from data. This approach can be applied to any data generated by discrete causes, and it can be applied as an approximation of continuous causes. As the prior probabilities are learned, the approach then allows for estimating the prior shape without assuming specific functional forms. To efficiently train the parameters of our probabilistic generative model, we apply a truncated expectation-maximization approach (expectation truncation) that we modify to work with a general discrete prior. We evaluate the performance of the algorithm by applying it to a variety of tasks: (1) we use artificial data to verify that the algorithm can recover the generating parameters from a random initialization, (2) use image patches of natural images and discuss the role of the prior for the extraction of image components, (3) use extracellular recordings of neurons to present a novel method of analysis for spiking neurons that includes an intuitive discretization strategy, and (4) apply the algorithm on the task of encoding audio waveforms of human speech. The diverse set of numerical experiments presented in this letter suggests that discrete sparse coding algorithms can scale efficiently to work with realistic data sets and provide novel statistical quantities to describe the structure of the data.

We propose a nonparametric procedure to achieve fast inference in generative graphical models when the number of latent states is very large. The approach is based on iterative latent variable preselection, where we alternate between learning a selection function to reveal the relevant latent variables and using this to obtain a compact approximation of the posterior distribution for EM. This can make inference possible where the number of possible latent states is, for example, exponential in the number of latent variables, whereas an exact approach would be computationally infeasible. We learn the selection function entirely from the observed data and current expectation-maximization state via gaussian process regression. This is in contrast to earlier approaches, where selection functions were manually designed for each problem setting. We show that our approach performs as well as these bespoke selection functions on a wide variety of inference problems. In particular, for the challenging case of a hierarchical model for object localization with occlusion, we achieve results that match a customized state-of-the-art selection method at a far lower computational cost.

In this study, we propose an unsupervised method for dictionary learning in audio signals. The new method, called binary nonnegative matrix deconvolution (BNMD), is developed and used to discover patterns from magnitude-scale spectrograms. The BNMD models an audio spectrogram as a sum of delayed patterns having binary gains (activations). Only small subsets of patterns can be active for a given spectrogram excerpt. The proposed method was applied to speaker identification and separation tasks. The experimental results show that dictionaries obtained by the BNMD bring much higher speaker identification accuracies averaged over a range of SNRs from -6 dB to 9 dB (91.3%) than the NMD-based dictionaries (37.8-75.4%). The BNMD also gives a benefit over dictionaries obtained using vector quantization (87.8%). For bigger dictionaries the difference between the BNMD and the vector quantization (VQ) is getting smaller. For the speech separation task the BNMD dictionary gave a slight improvement over the VQ.

Experience constantly shapes neural circuits through a variety of plasticity mechanisms. While the functional roles of some plasticity mechanisms are well-understood, it remains unclear how changes in neural excitability contribute to learning. Here, we develop a normative interpretation of intrinsic plasticity (IP) as a key component of unsupervised learning. We introduce a novel generative mixture model that accounts for the class-specific statistics of stimulus intensities, and we derive a neural circuit that learns the input classes and their intensities. We will analytically show that inference and learning for our generative model can be achieved by a neural circuit with intensity-sensitive neurons equipped with a specific form of IP. Numerical experiments verify our analytical derivations and show robust behavior for artificial and natural stimuli. Our results link IP to non-trivial input statistics, in particular the statistics of stimulus intensities for classes to which a neuron is sensitive. More generally, our work paves the way toward new classification algorithms that are robust to intensity variations.

Probabilistic inference serves as a popular model for neural processing. It is still unclear, however, how approximate probabilistic inference can be accurate and scalable to very high-dimensional continuous latent spaces. Especially as typical posteriors for sensory data can be expected to exhibit complex latent dependencies including multiple modes. Here, we study an approach that can efficiently be scaled while maintaining a richly structured posterior approximation under these conditions. As example model we use spike-and-slab sparse coding for V1 processing, and combine latent subspace selection with Gibbs sampling (select-and-sample). Unlike factored variational approaches, the method can maintain large numbers of posterior modes and complex latent dependencies. Unlike pure sampling, the method is scalable to very high-dimensional latent spaces. Among all sparse coding approaches with non-trivial posterior approximations (MAP or ICA-like models), we report the largest-scale results. In applications we firstly verify the approach by showing competitiveness in standard denoising benchmarks. Secondly, we use its scalability to, for the first time, study highly-overcomplete settings for V1 encoding using sophisticated posterior representations. More generally, our study shows that very accurate probabilistic inference for multi-modal posteriors with complex dependencies is tractable, functionally desirable and consistent with models for neural inference.

We study the task of cleaning scanned text documents that are strongly corrupted by dirt such as manual line strokes, spilled ink, etc. We aim at autonomously removing such corruptions from a single letter-size page based only on the information the page contains. Our approach first learns character representations from document patches without supervision. For learning, we use a probabilistic generative model parameterizing pattern features, their planar arrangements and their variances. The model's latent variables describe pattern position and class, and feature occurrences. Model parameters are efficiently inferred using a truncated variational EM approach. Based on the learned representation, a clean document can be recovered by identifying, for each patch, pattern class and position while a quality measure allows for discrimination between character and non-character patterns. For a full Latin alphabet we found that a single page does not contain sufficiently many character examples. However, even if heavily corrupted by dirt, we show that a page containing a lower number of character types can efficiently and autonomously be cleaned solely based on the structural regularity of the characters it contains. In different example applications with different alphabets, we demonstrate and discuss the effectiveness, efficiency and generality of the approach.

We study optimal image encoding based on a generative approach with non-linear feature combinations and explicit position encoding. By far most approaches to unsupervised learning learning of visual features, such as sparse coding or ICA, account for translations by representing the same features at different positions. Some earlier models used a separate encoding of features and their positions to facilitate invariant data encoding and recognition. All probabilistic generative models with explicit position encoding have so far assumed a linear superposition of components to encode image patches. Here, we for the first time apply a model with non-linear feature superposition and explicit position encoding. By avoiding linear superpositions, the studied model represents a closer match to component occlusions which are ubiquitous in natural images. In order to account for occlusions, the non-linear model encodes patches qualitatively very different from linear models by using component representations separated into mask and feature parameters. We first investigated encodings learned by the model using artificial data with mutually occluding components. We find that the model extracts the components, and that it can correctly identify the occlusive components with the hidden variables of the model. On natural image patches, the model learns component masks and features for typical image components. By using reverse correlation, we estimate the receptive fields associated with the model's hidden units. We find many Gabor-like or globular receptive fields as well as fields sensitive to more complex structures. Our results show that probabilistic models that capture occlusions and invariances can be trained efficiently on image patches, and that the resulting encoding represents an alternative model for the neural encoding of images in the primary visual cortex.

Simple cells in primary visual cortex were famously found to respond to low-level image components such as edges. Sparse coding and independent component analysis (ICA) emerged as the standard computational models for simple cell coding because they linked their receptive fields to the statistics of visual stimuli. However, a salient feature of image statistics, occlusions of image components, is not considered by these models. Here we ask if occlusions have an effect on the predicted shapes of simple cell receptive fields. We use a comparative approach to answer this question and investigate two models for simple cells: a standard linear model and an occlusive model. For both models we simultaneously estimate optimal receptive fields, sparsity and stimulus noise. The two models are identical except for their component superposition assumption. We find the image encoding and receptive fields predicted by the models to differ significantly. While both models predict many Gabor-like fields, the occlusive model predicts a much sparser encoding and high percentages of 'globular' receptive fields. This relatively new center-surround type of simple cell response is observed since reverse correlation is used in experimental studies. While high percentages of 'globular' fields can be obtained using specific choices of sparsity and overcompleteness in linear sparse coding, no or only low proportions are reported in the vast majority of studies on linear models (including all ICA models). Likewise, for the here investigated linear model and optimal sparsity, only low proportions of 'globular' fields are observed. In comparison, the occlusive model robustly infers high proportions and can match the experimentally observed high proportions of 'globular' fields well. Our computational study, therefore, suggests that 'globular' fields may be evidence for an optimal encoding of visual occlusions in primary visual cortex.

We study unsupervised learning of occluding objects in images of visual scenes. The derived learning algorithm is based on a probabilistic generative model which parameterizes object shapes, object features and the background. No assumptions are made for the object orders in depth or the objects' planar positions. Parameter optimization is thus subject to the large combinatorics of depth orders and positions. Previous approaches constrained this combinatorics but were still only able to learn a very small number of objects. By applying a novel variational EM approach, we show that even without constraints on the object combinatorics, a relatively large number of objects can be learned. In different numerical experiments, our unsupervised approach extracts explicit object representations with object masks and object features closely aligned with the true objects in the scenes. We investigate the robustness of the approach and the use of the learned representations for inference. Furthermore, we demonstrate generality of the approach by applying it to grayscale images, color-vector images, and Gabor-vector images as well as to motion trajectory data for which the extracted components correspond to motion primitives.

We study the task of cleaning scanned text documents that are strongly corrupted by dirt such as manual line strokes, spilled ink etc. We aim at autonomously removing dirt from a single letter-size page based only on the information the page contains. Our approach, therefore, has to learn character representations without supervision and requires a mechanism to distinguish learned representations from irregular patterns. To learn character representations, we use a probabilistic generative model parameterizing pattern features, feature variances, the features' planar arrangements, and pattern frequencies. The latent variables of the model describe pattern class, pattern position, and the presence or absence of individual pattern features. The model parameters are optimized using a novel variational EM approximation. After learning, the parameters represent, independently of their absolute position, planar feature arrangements and their variances. A quality measure defined based on the learned representation then allows for an autonomous discrimination between regular character patterns and the irregular patterns making up the dirt. The irregular patterns can thus be removed to clean the document. For a full Latin alphabet we found that a single page does not contain sufficiently many character examples. However, even if heavily corrupted by dirt, we show that a page containing a lower number of character types can efficiently and autonomously be cleaned solely based on the structural regularity of the characters it contains. In different examples using characters from different alphabets, we demonstrate generality of the approach and discuss its implications for future developments.

Modelling natural images with sparse coding (SC) has faced two main challenges: flexibly representing varying pixel intensities and realistically representing low- level image components. 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 lin- ear 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 piece- wise Gibbs sampling 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. In contrast to standard SC, we find that the optimal prior favors asymmetric and bimodal ac- tivity of simple cells. Testing our model for consistency we find that the average posterior is approximately equal to the prior. Furthermore, we find that the model predicts a high percentage of globular receptive fields alongside Gabor-like fields. Similarly high percentages are observed in vivo. Our results thus argue in favor of improvements of the standard sparse coding model for simple cells by using flexible priors and nonlinear combinations.

Feedforward inhibition and synaptic scaling are important adaptive processes that control the total input a neuron can receive from its afferents. While often studied in isolation, the two have been reported to co-occur in various brain regions. The functional implications of their interactions remain unclear, however. Based on a probabilistic modeling approach, we show here that fast feedforward inhibition and synaptic scaling interact synergistically during unsupervised learning. In technical terms, we model the input to a neural circuit using a normalized mixture model with Poisson noise. We demonstrate analytically and numerically that, in the presence of lateral inhibition introducing competition between different neurons, Hebbian plasticity and synaptic scaling approximate the optimal maximum likelihood solutions for this model. Our results suggest that, beyond its conventional use as a mechanism to remove undesired pattern variations, input normalization can make typical neural interaction and learning rules optimal on the stimulus subspace defined through feedforward inhibition. Furthermore, learning within this subspace is more efficient in practice, as it helps avoid locally optimal solutions. Our results suggest a close connection between feedforward inhibition and synaptic scaling which may have important functional implications for general cortical processing.

An increasing number of experimental studies indicate that perception encodes a posterior probability distribution over possible causes of sensory stimuli, which is used to act close to optimally in the environment. One outstanding difficulty with this hypothesis is that the exact posterior will in general be too complex to be represented directly, and thus neurons will have to represent an approximation of this distribution. Two influential proposals of efficient posterior representation by neural populations are: 1) neural activity represents samples of the underlying distribution, or 2) they represent a parametric representation of a variational approximation of the posterior. We show that these approaches can be combined for an inference scheme that retains the advantages of both: it is able to represent multiple modes and arbitrary correlations, a feature of sampling methods, and it reduces the represented space to regions of high probability mass, a strength of variational approximations. Neurally, the combined method can be interpreted as a feed-forward preselection of the relevant state space, followed by a neural dynamics implementation of Markov Chain Monte Carlo (MCMC) to approximate the posterior over the relevant states. We demonstrate the effectiveness and efficiency of this approach on a sparse coding model. In numerical experiments on artificial data and image patches, we compare the performance of the algorithms to that of exact EM, variational state space selection alone, MCMC alone, and the combined select and sample approach. The select and sample approach integrates the advantages of the sampling and variational approximations, and forms a robust, neurally plausible, and very efficient model of processing and learning in cortical networks. For sparse coding we show applications easily exceeding a thousand observed and a thousand hidden dimensions.

We study the application of a strongly non-linear generative model to image patches. As in standard approaches such as Sparse Coding or Independent Component Analysis, the model assumes a sparse prior with independent hidden variables. However, in the place where standard approaches use the sum to combine basis functions we use the maximum. To derive tractable approximations for parameter estimation we apply a novel approach based on variational Expectation Maximization. The derived learning algorithm can be applied to large-scale problems with hundreds of observed and hidden variables. Furthermore, we can infer all model parameters including observation noise and the degree of sparseness. In applications to image patches we find that Gabor-like basis functions are obtained. Gabor-like functions are thus not a feature exclusive to approaches assuming linear superposition. Quantitatively, the inferred basis functions show a large diversity of shapes with many strongly elongated and many circular symmetric functions. The distribution of basis function shapes reflects properties of simple cell receptive fields that are not reproduced by standard linear approaches. In the study of natural image statistics, the implications of using different superposition assumptions have so far not been investigated systematically because models with strong non-linearities have been found analytically and computationally challenging. The presented algorithm represents the first large-scale application of such an approach.

We study unsupervised learning in a probabilistic generative model for occlusion. The model uses two types of latent variables: one indicates which objects are present in the image, and the other how they are ordered in depth. This depth order then determines how the positions and appearances of the objects present, specified in the model parameters, combine to form the image. We show that the object parameters can be learnt from an unlabelled set of images in which objects occlude one another. Exact maximum-likelihood learning is intractable. However, we show that tractable approximations to Expectation Maximization (EM) can be found if the training images each contain only a small number of objects on average. In numerical experiments it is shown that these approximations recover the correct set of object parameters. Experiments on a novel version of the bars test using colored bars, and experiments on more realistic data, show that the algorithm performs well in extracting the generating causes. Experiments based on the standard bars benchmark test for object learning show that the algorithm performs well in comparison to other recent component extraction approaches. The model and the learning algorithm thus connect research on occlusion with the research field of multiple-cause component extraction methods.

We study a dynamical model of processing and learning in the visual cortex, which reflects the anatomy of V1 cortical columns and properties of their neuronal receptive fields. Based on recent results on the fine-scale structure of columns in V1, we model the activity dynamics in subpopulations of excitatory neurons and their interaction with systems of inhibitory neurons. We find that a dynamical model based on these aspects of columnar anatomy can give rise to specific types of computations that result in self-organization of afferents to the column. For a given type of input, self-organization reliably extracts the basic input components represented by neuronal receptive fields. Self-organization is very noise tolerant and can robustly be applied to different types of input. To quantitatively analyze the system's component extraction capabilities, we use two standard benchmarks: the bars test and natural images. In the bars test, the system shows the highest noise robustness reported so far. If natural image patches are used as input, self-organization results in Gabor-like receptive fields. In quantitative comparison with in vivo measurements, we find that the obtained receptive fields capture statistical properties of V1 simple cells that algorithms such as independent component analysis or sparse coding do not reproduce.

We describe a neural network able to rapidly establish correspondence between neural feature layers. Each of the network's two layers consists of interconnected cortical columns, and each column consists of inhibitorily coupled subpopulations of excitatory neurons. The dynamics of the system builds on a dynamic model of a single column, which is consistent with recent experimental findings. The network realizes dynamic links between its layers with the help of specialized columns that evaluate similarities between the activity distributions of local feature cell populations, are subject to a topology constraint, and can gate the transfer of feature information between the neural layers. The system can robustly be applied to natural images, and correspondences are found in time intervals estimated to be smaller than 100 ms in physiological terms.

We study clustering, i.e., unsupervised data classification, by a model of the cortical macrocolumn. Continuous valued input vectors are encoded using a population place code. The macrocolumn model self-organizes its minicolumnar receptive fields (RFs) such that the input is hierarchically subdivided into increasingly finer classes. If input superpositions are used for training, the system is able to find an appropriate classification of the input and a suitable representation of input superpositions. Together with fast reaction times the model satisfies major requirements of biological information processing and distinguishes itself from other suggested models of continuous value processing in biological neural networks.

We study a model of the cortical macrocolumn consisting of a collection of inhibitorily coupled minicolumns. The proposed system overcomes several severe deficits of systems based on single neurons as cerebral functional units, notably limited robustness to damage and unrealistically large computation time. Motivated by neuroanatomical and neurophysiological findings, the utilized dynamics is based on a simple model of a spiking neuron with refractory period, fixed random excitatory interconnection within minicolumns, and instantaneous inhibition within one macrocolumn. A stability analysis of the system's dynamical equations shows that minicolumns can act as monolithic functional units for purposes of critical, fast decisions and learning. Oscillating inhibition (in the gamma frequency range) leads to a phase-coupled population rate code and high sensitivity to small imbalances in minicolumn inputs. Minicolumns are shown to be able to organize their collective inputs without supervision by Hebbian plasticity into selective receptive field shapes, thereby becoming classifiers for input patterns. Using the bars test, we critically compare our system's performance with that of others and demonstrate its ability for distributed neural coding.