Vector-quantized variational autoencoders (VQ-VAEs) are central to models that rely on high reconstruction fidelity, from neural compression to generative pipelines. Hierarchical extensions, such as VQ-VAE2, are often credited with superior reconstruction performance because they split global and local features across multiple levels. However, since higher levels derive all their information from lower levels, they should not carry additional reconstructive content beyond what the lower-level already encodes. Combined with recent advances in training objectives and quantization mechanisms, this leads us to ask whether a single-level VQ-VAE, with matched representational budget and no codebook collapse, can equal the reconstruction fidelity of its hierarchical counterpart. Although the multi-scale structure of hierarchical models may improve perceptual quality in downstream tasks, the effect of hierarchy on reconstruction accuracy, isolated from codebook utilization and overall representational capacity, remains empirically underexamined. We revisit this question by comparing a two-level VQ-VAE and a capacity-matched single-level model on high-resolution ImageNet images. Consistent with prior observations, we confirm that inadequate codebook utilization limits single-level VQ-VAEs and that overly high-dimensional embeddings destabilize quantization and increase codebook collapse. We show that lightweight interventions such as initialization from data, periodic reset of inactive codebook vectors, and systematic tuning of codebook hyperparameters significantly reduce collapse. Our results demonstrate that when representational budgets are matched, and codebook collapse is mitigated, single-level VQ-VAEs can match the reconstruction fidelity of hierarchical variants, challenging the assumption that hierarchical quantization is inherently superior for high-quality reconstructions.

Predictive coding-inspired deep networks for visual computing integrate classification and reconstruction processes in shared intermediate layers.Although synergy between these processes is commonly assumed, it has yet to be convincingly demonstrated.In this study, we utilize a purposefully designed family of autoencoder-like architectures with an added classification head to examine the consequences of combining classificationand reconstruction-driven information within the models' latent layers.Our findings underscore a significant challenge: Classification-driven information diminishes reconstruction-driven information in shared representations and vice versa.Our results challenge prevailing assumptions in predictive coding and offer guidance for future iterations of predictive coding concepts in deep networks.

Curriculum analytics (CA) studies curriculum structure and student data to ensure the quality of educational programs. To gain statistical robustness, most existing CA techniques rely on the assumption of time-invariant course difficulty, preventing them from capturing variations that might occur over time. However, ensuring low temporal variation in course difficulty is crucial to warrant fairness in treating individual student cohorts and consistency in degree outcomes. We introduce item response theory (IRT) as a CA methodology that enables us to address the open problem of monitoring course difficulty variations over time. We use statistical criteria to quantify the degree to which course performance data meets IRT’s theoretical assumptions and verify validity and reliability of IRT-based course difficulty estimates. Using data from 664 Computer Science and 1,355 Mechanical Engineering undergraduate students, we show how IRT can yield valuable CA insights: First, by revealing temporal variations in course difficulty over several years, we find that course difficulty has systematically shifted downward during the COVID-19 pandemic. Second, time-dependent course difficulty and cohort performance variations confound conventional course pass rate measures. We introduce IRT-adjusted pass rates as an alternative to account for these factors. Our findings affect policymakers, student advisors, accreditation, and course articulation.

Curriculum Analytics (CA) studies curriculum structure and student data to ensure the quality of educational programs. One desirable property of courses within curricula is that they are not unexpectedly more difficult for students of different backgrounds. While prior work points to likely variations in course difficulty across student groups, robust methodologies for capturing such variations are scarce, and existing approaches do not adequately decouple course-specific difficulty from students' general performance levels. The present study introduces Differential Course Functioning (DCF) as an Item Response Theory (IRT)-based CA methodology. DCF controls for student performance levels and examines whether significant differences exist in how distinct student groups succeed in a given course. Leveraging data from over 20,000 students at a large public university, we demonstrate DCF's ability to detect inequities in undergraduate course difficulty across student groups described by grade achievement. We compare major pairs with high co-enrollment and transfer students to their non-transfer peers. For the former, our findings suggest a link between DCF effect sizes and the alignment of course content to student home department motivating interventions targeted towards improving course preparedness. For the latter, results suggest minor variations in course-specific difficulty between transfer and non-transfer students. While this is desirable, it also suggests that interventions targeted toward mitigating grade achievement gaps in transfer students should encompass comprehensive support beyond enhancing preparedness for individual courses. By providing more nuanced and equitable assessments of academic performance and difficulties experienced by diverse student populations, DCF could support policymakers, course articulation officers, and student advisors.

This study discusses the negative impact of the derivative of the activation functions in the output layer of artificial neural networks, in particular in continual learning. We propose Hebbian descent as a theoretical framework to overcome this limitation, which is implemented through an alternative loss function for gradient descent we refer to as Hebbian descent loss. This loss is effectively the generalized log-likelihood loss and corresponds to an alternative weight update rule for the output layer wherein the derivative of the activation function is disregarded. We show how this update avoids vanishing error signals during backpropagation in saturated regions of the activation functions, which is particularly helpful in training shallow neural networks and deep neural networks where saturating activation functions are only used in the output layer. In combination with centering, Hebbian descent leads to better continual learning capabilities. It provides a unifying perspective on Hebbian learning, gradient descent, and generalized linear models, for all of which we discuss the advantages and disadvantages. Given activation functions with strictly positive derivative (as often the case in practice), Hebbian descent inherits the convergence properties of regular gradient descent. While established pairings of loss and output layer activation function (e.g., mean squared error with linear or cross-entropy with sigmoid/softmax) are subsumed by Hebbian descent, we provide general insights for designing arbitrary loss activation function combinations that benefit from Hebbian descent. For shallow networks, we show that Hebbian descent outperforms Hebbian learning, has a performance similar to regular gradient descent, and has a much better performance than all other tested update rules in continual learning. In combination with centering, Hebbian descent implements a forgetting mechanism that prevents catastrophic interference notably better than the other tested update rules. When training deep neural networks, our experimental results suggest that Hebbian descent has better or similar performance as gradient descent.

Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically. This tutorial provides an in-depth introduction to Markov chains and explores their connection to graphs and random walks. We use tools from linear algebra and graph theory to describe the transition matrices of different types of Markov chains, with a particular focus on exploring properties of the eigenvalues and eigenvectors corresponding to these matrices. The results presented are relevant to a number of methods in machine learning and data mining, which we describe at various stages. Rather than being a novel academic study in its own right, this text presents a collection of known results, together with some new concepts. Moreover, the tutorial focuses on offering intuition to readers rather than formal understanding and only assumes basic exposure to concepts from linear algebra and probability theory. It is therefore accessible to students and researchers from a wide variety of disciplines.

Modern model-based reinforcement learning methods for high-dimensional inputs often incorporate an unsupervised learning step for dimensionality reduction. The training objective of these unsupervised learning methods often leverages only static inputs such as reconstructing observations. These representations are combined with predictor functions for simulating rollouts to navigate the environment. We advance this idea by taking advantage of the fact that we navigate dynamic environments with visual stimulus and create a representation that is specifically designed with control and actions in mind. We propose to learn a feature map that is maximally predictable for a predictor function. This results in representations that are well suited for the task of planning, where the predictor is used as a forward model. To this end, we introduce a new way of learning this representation along with the prediction function, a system we dub Latent Representation Prediction Network (LARP). The prediction function is used as a forward model for a search on a graph in a viewpoint-matching task, and the representation learned to maximize predictability is found to outperform other representations. The sample efficiency and overall performance of our approach are shown to rival standard reinforcement learning methods, and our learned representation transfers successfully to unseen environments.

Many studies have suggested that episodic memory is a generative process, but most computational models adopt a storage view. In this article, we present a model of the generative aspects of episodic memory. It is based on the central hypothesis that the hippocampus stores and retrieves selected aspects of an episode as a memory trace, which is necessarily incomplete. At recall, the neocortex reasonably fills in the missing parts based on general semantic information in a process we call semantic completion. The model combines two neural network architectures known from machine learning, the vector-quantized variational autoencoder (VQ-VAE) and the pixel convolutional neural network (PixelCNN). As episodes, we use images of digits and fashion items (MNIST) augmented by different backgrounds representing context. The model is able to complete missing parts of a memory trace in a semantically plausible way up to the point where it can generate plausible images from scratch, and it generalizes well to images not trained on. Compression as well as semantic completion contribute to a strong reduction in memory requirements and robustness to noise. Finally, we also model an episodic memory experiment and can reproduce that semantically congruent contexts are always recalled better than incongruent ones, high attention levels improve memory accuracy in both cases, and contexts that are not remembered correctly are more often remembered semantically congruently than completely wrong. This model contributes to a deeper understanding of the interplay between episodic memory and semantic information in the generative process of recalling the past.

One of the fundamental challenges in reinforcement learning (RL) is the one of data efficiency: modern algorithms require a very large number of training samples, especially compared to humans, for solving environments with high-dimensional observations. The severity of this problem is increased when the reward signal is sparse. In this work, we propose learning a state representation in a self-supervised manner for reward prediction. The reward predictor learns to estimate either a raw or a smoothed version of the true reward signal in environment with a single, terminating, goal state. We augment the training of out-of-the-box RL agents by shaping the reward using our reward predictor during policy learning. Using our representation for preprocessing high-dimensional observations, as well as using the predictor for reward shaping, is shown to significantly enhance Actor Critic using Kronecker-factored Trust Region and Proximal Policy Optimization in single-goal environments with visual inputs.

Abstract Slow feature analysis (SFA) is an unsupervised learning algorithm that extracts slowly varying features from a multi-dimensional time series. SFA has been extended to supervised learning (classification and regression) by an algorithm called graph-based SFA (GSFA). GSFA relies on a particular graph structure to extract features that preserve label similarities. Processing of high dimensional input data (e.g., images) is feasible via hierarchical GSFA (HGSFA), resulting in a multi-layer neural network. Although HGSFA has useful properties, in this work we identify a shortcoming, namely, that HGSFA networks prematurely discard quickly varying but useful features before they reach higher layers, resulting in suboptimal global slowness and an under-exploited feature space. To counteract this shortcoming, which we call unnecessary information loss, we propose an extension called hierarchical information-preserving GSFA (HiGSFA), where some features fulfill a slowness objective and other features fulfill an information preservation objective. The efficacy of the extension is verified in three experiments: (1) an unsupervised setup where the input data is the visual stimuli of a simulated rat, (2) the localization of faces in image patches, and (3) the estimation of human age from facial photographs of the MORPH-II database. Both HiGSFA and HGSFA can learn multiple labels and offer a rich feature space, feed-forward training, and linear complexity in the number of samples and dimensions. However, the proposed algorithm, HiGSFA, outperforms HGSFA in terms of feature slowness, estimation accuracy, and input reconstruction, giving rise to a promising hierarchical supervised-learning approach. Moreover, for age estimation, HiGSFA achieves a mean absolute error of 3.41 years, which is a competitive performance for this challenging problem.

We propose Power Slow Feature Analysis, a gradient-based method to extract temporally slow features from a high-dimensional input stream that varies on a faster time-scale, as a variant of Slow Feature Analysis (SFA) that allows end-to-end training of arbitrary differentiable architectures and thereby significantly extends the class of models that can effectively be used for slow feature extraction. We provide experimental evidence that PowerSFA is able to extract meaningful and informative low-dimensional features in the case of (a) synthetic low-dimensional data, (b) ego-visual data, and also for (c) a general dataset for which symmetric non-temporal similarities between points can be defined.

In this paper, we propose a new experimental protocol and use it to benchmark the data efficiency --- performance as a function of training set size --- of two deep learning algorithms, convolutional neural networks (CNNs) and hierarchical information-preserving graph-based slow feature analysis (HiGSFA), for tasks in classification and transfer learning scenarios. The algorithms are trained on different-sized subsets of the MNIST and Omniglot data sets. HiGSFA outperforms standard CNN networks when the models are trained on 50 and 200 samples per class for MNIST classification. In other cases, the CNNs perform better. The results suggest that there are cases where greedy, locally optimal bottom-up learning is equally or more powerful than global gradient-based learning.

The experimental evidence on the interrelation between episodic memory and semantic memory is inconclusive. Are they independent systems, different aspects of a single system, or separate but strongly interacting systems? Here, we propose a computational role for the interaction between the semantic and episodic systems that might help resolve this debate. We hypothesize that episodic memories are represented as sequences of activation patterns. These patterns are the output of a semantic representational network that compresses the high-dimensional sensory input. We show quantitatively that the accuracy of episodic memory crucially depends on the quality of the semantic representation. We compare two types of semantic representations: appropriate representations, which means that the representation is used to store input sequences that are of the same type as those that it was trained on, and inappropriate representations, which means that stored inputs differ from the training data. Retrieval accuracy is higher for appropriate representations because the encoded sequences are less divergent than those encoded with inappropriate representations. Consistent with our model prediction, we found that human subjects remember some aspects of episodes significantly more accurately if they had previously been familiarized with the objects occurring in the episode, as compared to episodes involving unfamiliar objects. We thus conclude that the interaction with the semantic system plays an important role for episodic memory.

The computational principles of slowness and predictability have been proposed to describe aspects of information processing in the visual system. From the perspective of slowness being a limited special case of predictability we investigate the relationship between these two principles empirically. On a collection of real-world data sets we compare the features extracted by slow feature analysis (SFA) to the features of three recently proposed methods for predictable feature extraction: forecastable component analysis, predictable feature analysis, and graph-based predictable feature analysis. Our experiments show that the predictability of the learned features is highly correlated, and, thus, SFA appears to effectively implement a method for extracting predictable features according to different measures of predictability.

We propose graph-based predictable feature analysis (GPFA), a new method for unsupervised learning of predictable features from high-dimensional time series, where high predictability is understood very generically as low variance in the distribution of the next data point given the previous ones. We show how this measure of predictability can be understood in terms of graph embedding as well as how it relates to the information-theoretic measure of predictive information in special cases. We confirm the effectiveness of GPFA on different datasets, comparing it to three existing algorithms with similar objectives—namely slow feature analysis, forecastable component analysis, and predictable feature analysis—to which GPFA shows very competitive results.

Slow feature analysis (SFA) is an unsupervised learning algorithm that extracts slowly varying features from a multi- dimensional time series. Graph-based SFA (GSFA) is an extension to SFA for supervised learning that can be used to successfully solve regression problems if combined with a simple supervised post-processing step on a small number of slow features. The objective function of GSFA minimizes the squared output differences between pairs of samples specified by the edges of a structure called training graph. The edges of current training graphs, however, are derived only from the relative order of the labels. Exploiting the exact numerical value of the labels enables further improvements in label estimation accuracy. In this article, we propose the exact label learning (ELL) method to create a more precise training graph that encodes the desired labels explicitly and allows GSFA to extract a normalized version of them directly (i.e., without supervised post- processing). The ELL method is used for three tasks: (1) We estimate gender from artificial images of human faces (regression) and show the advantage of coding additional labels, particularly skin color. (2) We analyze two existing graphs for regression. (3) We extract compact discriminative features to classify traffic sign images. When the number of output features is limited, such compact features provide a higher classification rate compared to a graph that generates features equivalent to those of nonlinear Fisher discriminant analysis. The method is versatile, directly supports multiple labels, and provides higher accuracy compared to current graphs for the problems considered.

This work analyzes centered Restricted Boltzmann Machines (RBMs) and centered Deep Boltzmann Machines (DBMs), where centering is done by subtracting offset values from visible and hidden variables. We show analytically that (i) centered and normal Boltzmann Machines (BMs) and thus RBMs and DBMs are different parameterizations of the same model class, such that any normal BM/RBM/DBM can be transformed to an equivalent centered BM/RBM/DBM and vice versa, and that this equivalence generalizes to artificial neural networks in general, (ii) the expected performance of centered binary BMs/RBMs/DBMs is invariant under simultaneous flip of data and offsets, for any offset value in the range of zero to one, (iii) centering can be reformulated as a different update rule for normal BMs/RBMs/DBMs, and (iv) using the enhanced gradient is equivalent to setting the offset values to the average over model and data mean. Furthermore, we present numerical simulations suggesting that (i) optimal generative performance is achieved by subtracting mean values from visible as well as hidden variables, (ii) centered binary RBMs/DBMs reach significantly higher log-likelihood values than normal binary RBMs/DBMs, (iii) centering variants whose offsets depend on the model mean, like the enhanced gradient, suffer from severe divergence problems, (iv) learning is stabilized if an exponentially moving average over the batch means is used for the offset values instead of the current batch mean, which also prevents the enhanced gradient from severe divergence, (v) on a similar level of log-likelihood values centered binary RBMs/DBMs have smaller weights and bigger bias parameters than normal binary RBMs/DBMs, (vi) centering leads to an update direction that is closer to the natural gradient, which is extremely efficient for training as we show for small binary RBMs, (vii) centering eliminates the need for greedy layer-wise pre-training of DBMs, which often even deteriorates the results independently of whether centering is used or not, and (ix) centering is also beneficial for auto encoders.

Every organism in an environment, whether biological, robotic or virtual, must be able to predict certain aspects of its environment in order to survive or perform whatever task is intended. It needs a model that is capable of estimating the consequences of possible actions, so that planning, control, and decision-making become feasible. For scientific purposes, such models are usually created in a problem specific manner using differential equations and other techniques from control-and system-theory. In contrast to that, we aim for an unsupervised approach that builds up the desired model in a self-organized fashion. Inspired by Slow Feature Analysis (SFA), our approach is to extract subsignals from the input, that behave as predictable as possible. These "predictable features" are highly relevant for modeling, because predictability is a desired property of the needed consequence-estimating model by definition. In our approach, we measure predictability with respect to a certain prediction model. We focus here on the solution of the arising optimization problem and present a tractable algorithm based on algebraic methods which we call Predictable Feature Analysis (PFA). We prove that the algorithm finds the globally optimal signal if this signal can be predicted with low error. To deal with cases where the optimal signal has a significant prediction error, we provide a robust, heuristically motivated variant of the algorithm and verify it empirically. Additionally, we give formal criteria a prediction model must meet to be suitable for measuring predictability in the PFA setting and also provide a suitable default model along with a formal proof that it meets these criteria.

In the last decades a standard model regarding the function of the hippocampus in memory formation has been established and tested computationally. It has been argued that the CA3 region works as an auto-associative memory and that its recurrent fibers are the actual storing place of the memories. Furthermore, to work properly CA3 requires memory patterns that are mutually uncorrelated. It has been suggested that the dentate gyrus orthogonalizes the patterns before storage, a process known as pattern separation. In this study we review the model when random input patterns are presented for storage and investigate whether it is capable of storing patterns of more realistic entorhinal grid cell input. Surprisingly, we find that an auto-associative CA3 net is redundant for random inputs up to moderate noise levels and is only beneficial at high noise levels. When grid cell input is presented, auto-association is even harmful for memory performance at all levels. Furthermore, we find that Hebbian learning in the dentate gyrus does not support its function as a pattern separator. These findings challenge the standard framework and support an alternative view where the simpler EC-CA1-EC network is sufficient for memory storage.

The developing visual system of many mammalian species is partially structured and organized even before the onset of vision. Spontaneous neural activity, which spreads in waves across the retina, has been suggested to play a major role in these prenatal structuring processes. Recently, it has been shown that when employing an efficient coding strategy, such as sparse coding, these retinal activity patterns lead to basis functions that resemble optimal stimuli of simple cells in primary visual cortex (V1). Here we present the results of applying a coding strategy that optimizes for temporal slowness, namely Slow Feature Analysis (SFA), to a biologically plausible model of retinal waves. Previously, SFA has been successfully applied to model parts of the visual system, most notably in reproducing a rich set of complex-cell features by training SFA with quasi-natural image sequences. In the present work, we obtain SFA units that share a number of properties with cortical complex-cells by training on simulated retinal waves. The emergence of two distinct properties of the SFA units (phase invariance and orientation tuning) is thoroughly investigated via control experiments and mathematical analysis of the input-output functions found by SFA. The results support the idea that retinal waves share relevant temporal and spatial properties with natural visual input. Hence, retinal waves seem suitable training stimuli to learn invariances and thereby shape the developing early visual system such that it is best prepared for coding input from the natural world.

Computational modeling of the primate visual system yields insights of potential relevance to some of the challenges that computer vision is facing, such as object recognition and categorization, motion detection and activity recognition, or vision-based navigation and manipulation. This paper reviews some functional principles and structures that are generally thought to underlie the primate visual cortex, and attempts to extract biological principles that could further advance computer vision research. Organized for a computer vision audience, we present functional principles of the processing hierarchies present in the primate visual system considering recent discoveries in neurophysiology. The hierarchical processing in the primate visual system is characterized by a sequence of different levels of processing (on the order of 10) that constitute a deep hierarchy in contrast to the flat vision architectures predominantly used in today's mainstream computer vision. We hope that the functional description of the deep hierarchies realized in the primate visual system provides valuable insights for the design of computer vision algorithms, fostering increasingly productive interaction between biological and computer vision research.

We generalize the method of Slow Feature Analysis (SFA) for vector-valued functions of several variables and apply it to the problem of blind source separation, in particular to image separation. It is generally necessary to use multivariate SFA instead of univariate SFA for separating multi-dimensional signals. For the linear case, an exact mathematical analysis is given, which shows in particular that the sources are perfectly separated by SFA if and only if they and their first-order derivatives are uncorrelated. When the sources are correlated, we apply the following technique called Decorrelation Filtering: use a linear filter to decorrelate the sources and their derivatives in the given mixture, then apply the unmixing matrix obtained on the filtered mixtures to the original mixtures. If the filtered sources are perfectly separated by this matrix, so are the original sources. A decorrelation filter can be numerically obtained by solving a nonlinear optimization problem. This technique can also be applied to other linear separation methods, whose output signals are decorrelated, such as ICA. When there are more mixtures than sources, one can determine the actual number of sources by using a regularized version of SFA with decorrelation filtering. Extensive numerical experiments using SFA and ICA with decorrelation filtering, supported by mathematical analysis, demonstrate the potential of our methods for solving problems involving blind source separation.

We generalize the method of Slow Feature Analysis for vector-valued functions of multivariables and apply it to the problem of blind source separation, in particular image separation. For the linear case, exact mathematical analysis is given, which shows in particular that the sources are perfectly separated by SFA if and only if they and their first order derivatives are uncorrelated. When the sources are correlated, we apply the following technique called decorrelation filtering: use a linear filter to decorrelate the sources and their derivatives, then apply the separating matrix obtained on the filtered sources to the original sources. We show that if the filtered sources are perfectly separated by this matrix, then so are the original sources. We show how to numerically obtain such a decorrelation filter by solving a nonlinear optimization problem. This technique can also be applied to other linear separation methods, whose output signals are decorrelated, such as ICA.

Primates are very good at recognizing objects independent of viewing angle or retinal position, and they outperform existing computer vision systems by far. But invariant object recognition is only one prerequisite for successful interaction with the environment. An animal also needs to assess an object's position and relative rotational angle. We propose here a model that is able to extract object identity, position, and rotation angles. We demonstrate the model behavior on complex three-dimensional objects under translation and rotation in depth on a homogeneous background. A similar model has previously been shown to extract hippocampal spatial codes from quasi-natural videos. The framework for mathematical analysis of this earlier application carries over to the scenario of invariant object recognition. Thus, the simulation results can be explained analytically even for the complex high-dimensional data we employed.

We develop a group-theoretical analysis of slow feature analysis for the case where the input data are generated by applying a set of continuous transformations to static templates. As an application of the theory, we analytically derive nonlinear visual receptive fields and show that their optimal stimuli, as well as the orientation and frequency tuning, are in good agreement with previous simulations of complex cells in primary visual cortex (Berkes and Wiskott, 2005). The theory suggests that side and end stopping can be interpreted as a weak breaking of translation invariance. Direction selectivity is also discussed.

Recently, we presented a study of adult neurogenesis in a simplified hippocampal memory model. The network was required to encode and decode memory patterns despite changing input statistics. We showed that additive neurogenesis was a more effective adaptation strategy compared to neuronal turnover and conventional synaptic plasticity as it allowed the network to respond to changes in the input statistics while preserving representations of earlier environments. Here we extend our model to include realistic, spatially driven input firing patterns in the form of grid cells in the entorhinal cortex. We compare network performance across a sequence of spatial environments using three distinct adaptation strategies: conventional synaptic plasticity, where the network is of fixed size but the connectivity is plastic; neuronal turnover, where the network is of fixed size but units in the network may die and be replaced; and additive neurogenesis, where the network starts out with fewer initial units but grows over time. We confirm that additive neurogenesis is a superior adaptation strategy when using realistic, spatially structured input patterns. We then show that a more biologically plausible neurogenesis rule that incorporates cell death and enhanced plasticity of new granule cells has an overall performance significantly better than any one of the three individual strategies operating alone. This adaptation rule can be tailored to maximise performance of the network when operating as either a short- or long-term memory store. We also examine the time course of adult neurogenesis over the lifetime of an animal raised under different hypothetical rearing conditions. These growth profiles have several distinct features that form a theoretical prediction that could be tested experimentally. Finally, we show that place cells can emerge and refine in a realistic manner in our model as a direct result of the sparsification performed by the dentate gyrus layer.

Humans and animals are able to learn complex behaviors based on a massive stream of sensory information from different modalities. Early animal studies have identified learning mechanisms that are based on reward and punishment such that animals tend to avoid actions that lead to punishment whereas rewarded actions are reinforced. However, most algorithms for reward-based learning are only applicable if the dimensionality of the state-space is sufficiently small or its structure is sufficiently simple. Therefore, the question arises how the problem of learning on high-dimensional data is solved in the brain. In this article, we propose a biologically plausible generic two-stage learning system that can directly be applied to raw high-dimensional input streams. The system is composed of a hierarchical slow feature analysis (SFA) network for preprocessing and a simple neural network on top that is trained based on rewards. We demonstrate by computer simulations that this generic architecture is able to learn quite demanding reinforcement learning tasks on high-dimensional visual input streams in a time that is comparable to the time needed when an explicit highly informative low-dimensional state-space representation is given instead of the high-dimensional visual input. The learning speed of the proposed architecture in a task similar to the Morris water maze task is comparable to that found in experimental studies with rats. This study thus supports the hypothesis that slowness learning is one important unsupervised learning principle utilized in the brain to form efficient state representations for behavioral learning.

In the linear case, statistical independence is a sufficient criterion for performing blind source separation. In the nonlinear case, however, it leaves an ambiguity in the solutions that has to be resolved by additional criteria. Here we argue that temporal slowness complements statistical independence well and that a combination of the two leads to unique solutions of the nonlinear blind source separation problem. The algorithm we present is a combination of second-order independent component analysis and slow feature analysis and is referred to as independent slow feature analysis. Its performance is demonstrated on nonlinearly mixed music data. We conclude that slowness is indeed a useful complement to statistical independence but that time-delayed second-order moments are only a weak measure of statistical independence.

We present a model for the self-organized formation of place cells, head-direction cells, and spatial-view cells in the hippocampal formation based on unsupervised learning on quasi-natural visual stimuli. The model comprises a hierarchy of Slow Feature Analysis (SFA) nodes, which were recently shown to reproduce many properties of complex cells in the early visual system []. The system extracts a distributed grid-like representation of position and orientation, which is transcoded into a localized place-field, head-direction, or view representation, by sparse coding. The type of cells that develops depends solely on the relevant input statistics, i.e., the movement pattern of the simulated animal. The numerical simulations are complemented by a mathematical analysis that allows us to accurately predict the output of the top SFA layer.

Our nervous system can efficiently recognize objects in spite of changes in contextual variables such as perspective or lighting conditions. Several lines of research have proposed that this ability for invariant recognition is learned by exploiting the fact that object identities typically vary more slowly in time than contextual variables or noise. Here, we study the question of how this "temporal stability" or "slowness" approach can be implemented within the limits of biologically realistic spike-based learning rules. We first show that slow feature analysis, an algorithm that is based on slowness, can be implemented in linear continuous model neurons by means of a modified Hebbian learning rule. This approach provides a link to the trace rule, which is another implementation of slowness learning. Then, we show analytically that for linear Poisson neurons, slowness learning can be implemented by spike-timing-dependent plasticity (STDP) with a specific learning window. By studying the learning dynamics of STDP, we show that for functional interpretations of STDP, it is not the learning window alone that is relevant but rather the convolution of the learning window with the postsynaptic potential. We then derive STDP learning windows that implement slow feature analysis and the "trace rule." The resulting learning windows are compatible with physiological data both in shape and timescale. Moreover, our analysis shows that the learning window can be split into two functionally different components that are sensitive to reversible and irreversible aspects of the input statistics, respectively. The theory indicates that irreversible input statistics are not in favor of stable weight distributions but may generate oscillatory weight dynamics. Our analysis offers a novel interpretation for the functional role of STDP in physiological neurons.

In this letter, we introduce some mathematical and numerical tools to analyze and interpret inhomogeneous quadratic forms. The resulting characterization is in some aspects similar to that given by experimental studies of cortical cells, making it particularly suitable for application to second-order approximations and theoretical models of physiological receptive fields. We first discuss two ways of analyzing a quadratic form by visualizing the coefficients of its quadratic and linear term directly and by considering the eigenvectors of its quadratic term. We then present an algorithm to compute the optimal excitatory and inhibitory stimuli--those that maximize and minimize the considered quadratic form, respectively, given a fixed energy constraint. The analysis of the optimal stimuli is completed by considering their invariances, which are the transformations to which the quadratic form is most insensitive, and by introducing a test to determine which of these are statistically significant. Next we propose a way to measure the relative contribution of the quadratic and linear term to the total output of the quadratic form. Furthermore, we derive simpler versions of the above techniques in the special case of a quadratic form without linear term. In the final part of the letter, we show that for each quadratic form, it is possible to build an equivalent two-layer neural network, which is compatible with (but more general than) related networks used in some recent articles and with the energy model of complex cells. We show that the neural network is unique only up to an arbitrary orthogonal transformation of the excitatory and inhibitory subunits in the first layer.

We present an analytical comparison between linear slow feature analysis and second-order independent component analysis, and show that in the case of one time delay, the two approaches are equivalent. We also consider the case of several time delays and discuss two possible extensions of slow feature analysis.

In contrast to the equivalence of linear blind source separation and linear independent component analysis it is not possible to recover the origi- nal source signal from some unknown nonlinear transformations of the sources using only the independence assumption. Integrating the ob- jectives of statistical independence and temporal slowness removes this indeterminacy leading to a new method for nonlinear blind source sepa- ration. The principle of temporal slowness is adopted from slow feature analysis, an unsupervised method to extract slowly varying features from a given observed vectorial signal. The performance of the algorithm is demonstrated on nonlinearly mixed speech data.

Temporal slowness is a learning principle that allows learning of invariant representations by extracting slowly varying features from quickly varying input signals. Slow feature analysis (SFA) is an efficient algorithm based on this principle and has been applied to the learning of translation, scale, and other invariances in a simple model of the visual system. Here, a theoretical analysis of the optimization problem solved by SFA is presented, which provides a deeper understanding of the simulation results obtained in previous studies.

Invariant features of temporally varying signals are useful for analysis and classification. Slow feature analysis (SFA) is a new method for learning invariant or slowly varying features from a vectorial input signal. It is based on a nonlinear expansion of the input signal and application of principal component analysis to this expanded signal and its time derivative. It is guaranteed to find the optimal solution within a family of functions directly and can learn to extract a large number of decorrelated features, which are ordered by their degree of invariance. SFA can be applied hierarchically to process high-dimensional input signals and extract complex features. SFA is applied first to complex cell tuning properties based on simple cell output, including disparity and motion. Then more complicated input-output functions are learned by repeated application of SFA. Finally, a hierarchical network of SFA modules is presented as a simple model of the visual system. The same unstructured network can learn translation, size, rotation, contrast, or, to a lesser degree, illumination invariance for one-dimensional objects, depending on only the training stimulus. Surprisingly, only a few training objects suffice to achieve good generalization to new objects. The generated representation is suitable for object recognition. Performance degrades if the network is trained to learn multiple invariances simultaneously.

Computational models of neural map formation can be considered on at least three different levels of abstraction: detailed models including neural activity dynamics, weight dynamics that abstract from the neural activity dynamics by an adiabatic approximation, and constrained optimization from which equations governing weight dynamics can be derived. Constrained optimization uses an objective function, from which a weight growth rule can be derived as a gradient flow, and some constraints, from which normalization rules are derived. In this article, we present an example of how an optimization problem can be derived from detailed nonlinear neural dynamics. A systematic investigation reveals how different weight dynamics introduced previously can be derived from two types of objective function terms and two types of constraints. This includes dynamic link matching as a special case of neural map formation. We focus in particular on the role of coordinate transformations to derive different weight dynamics from the same optimization problem. Several examples illustrate how the constrained optimization framework can help in understanding, generating, and comparing different models of neural map formation. The techniques used in this analysis may also be useful in investigating other types of neural dynamics.

The system presented is a specialized version of a general object recognition system. Images of faces are represented as graphs, labeled with topographical information and local features. New graphs of faces are generated by an elastic graph matching procedure comparing the new face with a set of stored model graphs: the face bunch graph. The result of this matching process can be used to generate composite images of faces and to determine facial attributes represented in the face bunch graph, such as sex or the presence of glasses or a beard.

We present a system for recognizing human faces from single images out of a large database containing one image per person. Faces are represented by labeled graphs, based on a Gabor wavelet transform. Image graphs of new faces are extracted by an elastic graph matching process and can be compared by a simple similarity function. The system differs from Lades et al. (1993) in three respects. Phase information is used for accurate node positioning. Object-adapted graphs are used to handle large rotations in depth. Image graph extraction is based on a novel data structure, the bunch graph, which is constructed from a small set of sample image graphs.

We present a system for recognizing human faces from single images out of a large database containing one image per person. Faces are represented by labeled graphs, based on a Gabor wavelet transform. Image graphs of new faces are extracted by an elastic graph matching process and can be compared by a simple similarity function. The system differs from the preceding one (Lades et al., 1993) in three respects. Phase information is used for accurate node positioning. Object-adapted graphs are used to handle large rotations in depth. Image graph extraction is based on a novel data structure, the bunch graph, which is constructed from a small get of sample image graphs.

We present a system for the interpretation of camera images of scenes composed of several known objects with mutual occlusion. The scenes are analyzed by the recognition of the objects present and by the determination of their occlusion relations. Objects are internally represented by stored model graphs. These are formed in a semi-automatic way by showing objects against a varying background. Objects are recognized by dynamic link matching. Our experiments show that our system is very successful in analyzing cluttered scenes. The system architecture goes beyond classical neural networks by making extensive use of flexible links between units, as proposed in the dynamic link architecture. The present implementation is, however, rather algorithmic in style and is to be regarded as a pilot study that is preparing the way for a detailed implementation of the architecture.