Laurenz Wiskott

Eddie Seabrook, Laurenz Wiskott

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 utilize 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.

Jan Rathjens, Shirin Reyhanian, David Kappel et al.

Understanding the mechanisms underlying deep neural networks remains a fundamental challenge in machine learning and computer vision. One promising, yet only preliminarily explored approach, is feature inversion, which attempts to reconstruct images from intermediate representations using trained inverse neural networks. In this study, we revisit feature inversion, introducing a novel, modular variation that enables significantly more efficient application of the technique. We demonstrate how our method can be systematically applied to the large-scale transformer-based vision models, Detection Transformer and Vision Transformer, and how reconstructed images can be qualitatively interpreted in a meaningful way. We further quantitatively evaluate our method, thereby uncovering underlying mechanisms of representing image features that emerge in the two transformer architectures. Our analysis reveals key insights into how these models encode contextual shape and image details, how their layers correlate, and their robustness against color perturbations. These findings contribute to a deeper understanding of transformer-based vision models and their internal representations. The code for reproducing our experiments is available at github.com/wiskott-lab/inverse-tvm.

Moritz Lange, Noah Krystiniak, Raphael C. Engelhardt et al.

Real-world reinforcement learning (RL) environments, whether in robotics or industrial settings, often involve non-visual observations and require not only efficient but also reliable and thus interpretable and flexible RL approaches. To improve efficiency, agents that perform state representation learning with auxiliary tasks have been widely studied in visual observation contexts. However, for real-world problems, dedicated representation learning modules that are decoupled from RL agents are more suited to meet requirements. This study compares common auxiliary tasks based on, to the best of our knowledge, the only decoupled representation learning method for low-dimensional non-visual observations. We evaluate potential improvements in sample efficiency and returns for environments ranging from a simple pendulum to a complex simulated robotics task. Our findings show that representation learning with auxiliary tasks only provides performance gains in sufficiently complex environments and that learning environment dynamics is preferable to predicting rewards. These insights can inform future development of interpretable representation learning approaches for non-visual observations and advance the use of RL solutions in real-world scenarios.

Frederik Baucks, Robin Schmucker, Laurenz Wiskott

Curriculum analytics (CA) studies curriculum structure and student data to ensure the quality of educational programs. An essential aspect is studying course properties, which involves assigning each course a representative difficulty value. This is critical for several aspects of CA, such as quality control (e.g., monitoring variations over time), course comparisons (e.g., articulation), and course recommendation (e.g., advising). Measuring course difficulty requires careful consideration of multiple factors: First, when difficulty measures are sensitive to the performance level of enrolled students, it can bias interpretations by overlooking student diversity. By assessing difficulty independently of enrolled students' performances, we can reduce the risk of bias and enable fair, representative assessments of difficulty. Second, from a measurement theoretic perspective, the measurement must be reliable and valid to provide a robust basis for subsequent analyses. Third, difficulty measures should account for covariates, such as the characteristics of individual students within a diverse populations (e.g., transfer status). In recent years, various notions of difficulty have been proposed. This paper provides the first comprehensive review and comparison of existing approaches for assessing course difficulty based on grade point averages and latent trait modeling. It further offers a hands-on tutorial on model selection, assumption checking, and practical CA applications. These applications include monitoring course difficulty over time and detecting courses with disparate outcomes between distinct groups of students (e.g., dropouts vs. graduates), ultimately aiming to promote high-quality, fair, and equitable learning experiences. To support further research and application, we provide an open-source software package and artificial datasets, facilitating reproducibility and adoption.

Shirin Reyhanian, Laurenz Wiskott

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.

Eddie Seabrook, Laurenz Wiskott

Slow feature analysis (SFA) is an unsupervised method for extracting representations from time series data. The successor representation (SR) is a method for representing states in a Markov decision process (MDP) based on transition statistics. While SFA and SR stem from distinct areas of machine learning, they share important properties, both in terms of their mathematics and the types of information they are sensitive to. This work studies their connection along these two axes. In particular, both SFA and SR are explored analytically, and in the setting of a one-hot encoded MDP, a formal equivalence is demonstrated in terms of the grid-like representations that occur as solutions/eigenvectors. Moreover, it is shown that the columns of the matrices involved in SFA contain place-like representations, which are formally distinct from place-cell models that have already been defined using SFA.

Jan Rathjens, Laurenz Wiskott

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 take a critical look at how classifying and reconstructing interact in deep learning architectures. Our approach utilizes a purposefully designed family of model architectures reminiscent of autoencoders, each equipped with an encoder, a decoder, and a classification head featuring varying modules and complexities. We meticulously analyze the extent to which classification- and reconstruction-driven information can seamlessly coexist within the shared latent layer of the model architectures. Our findings underscore a significant challenge: Classification-driven information diminishes reconstruction-driven information in intermediate layers' shared representations and vice versa. While expanding the shared representation's dimensions or increasing the network's complexity can alleviate this trade-off effect, our results challenge prevailing assumptions in predictive coding and offer guidance for future iterations of predictive coding concepts in deep networks.

Frederik Baucks, Robin Schmucker, Conrad Borchers et al.

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.

Pavlos Rath-Manakidis, Frederik Strothmann, Tobias Glasmachers et al.

Interpretation and visualization of the behavior of detection transformers tends to highlight the locations in the image that the model attends to, but it provides limited insight into the \emph{semantics} that the model is focusing on. This paper introduces an extension to detection transformers that constructs prototypical local features and uses them in object detection. These custom features, which we call prototypical parts, are designed to be mutually exclusive and align with the classifications of the model. The proposed extension consists of a bottleneck module, the prototype neck, that computes a discretized representation of prototype activations and a new loss term that matches prototypes to object classes. This setup leads to interpretable representations in the prototype neck, allowing visual inspection of the image content perceived by the model and a better understanding of the model's reliability. We show experimentally that our method incurs only a limited performance penalty, and we provide examples that demonstrate the quality of the explanations provided by our method, which we argue outweighs the performance penalty.

Jan Rathjens, Robin Schiewer, Laurenz Wiskott et al.

Mamba has attracted widespread interest as a general-purpose sequence model due to its low computational complexity and competitive performance relative to transformers. However, its performance can degrade when inference sequence lengths exceed those seen during training. We study this phenomenon using a controlled vision task in which Mamba reconstructs images from sequences of image patches. By analyzing reconstructions at different stages of sequence processing, we reveal that Mamba qualitatively adapts its behavior to the distribution of sequence lengths encountered during training, resulting in strategies that fail to generalize beyond this range. To support our analysis, we introduce a length-adaptive variant of Mamba that improves performance across training sequence lengths. Our results provide an intuitive perspective on length generalization in Mamba and suggest directions for improving the architecture.

Tim Schwabe, Moritz Lange, Laurenz Wiskott et al.

Gradient-based causal discovery shows great potential for deducing causal structure from data in an efficient and scalable way. Those approaches however can be susceptible to distributional biases in the data they are trained on. We identify two such biases: Marginal Distribution Asymmetry, where differences in entropy skew causal learning toward certain factorizations, and Marginal Distribution Shift Asymmetry, where repeated interventions cause faster shifts in some variables than in others. For the bivariate categorical setup with Dirichlet priors, we illustrate how these biases can occur even in controlled synthetic data. To examine their impact on gradient-based methods, we employ two simple models that derive causal factorizations by learning marginal or conditional data distributions - a common strategy in gradient-based causal discovery. We demonstrate how these models can be susceptible to both biases. We additionally show how the biases can be controlled. An empirical evaluation of two related, existing approaches indicates that eliminating competition between possible causal factorizations can make models robust to the presented biases.

Moritz Lange, Raphael C. Engelhardt, Wolfgang Konen et al.

Reasoning about the trajectories of multiple, interacting objects is integral to physical reasoning tasks in machine learning. This involves conditions imposed on the objects at different time steps, for instance initial states or desired goal states. Existing approaches in physical reasoning generally rely on autoregressive modeling, which can only be conditioned on initial states, but not on later states. In fields such as planning for reinforcement learning, similar challenges are being addressed with denoising diffusion models. In this work, we propose an object-centric denoising diffusion model architecture for physical reasoning that is translation equivariant over time, permutation equivariant over objects, and can be conditioned on arbitrary time steps for arbitrary objects. We demonstrate how this model can solve tasks with multiple conditions and examine its performance when changing object numbers and trajectory lengths during inference.

Merlin Schüler, Eddie Seabrook, Laurenz Wiskott

Slow Feature Analysis is a unsupervised representation learning method that extracts slowly varying features from temporal data and can be used as a basis for subsequent reinforcement learning. Often, the behavior that generates the data on which the representation is learned is assumed to be a uniform random walk. Less research has focused on using samples generated by goal-directed behavior, as commonly the case in a reinforcement learning setting, to learn a representation. In a spatial setting, goal-directed behavior typically leads to significant differences in state occupancy between states that are close to a reward location and far from a reward location. Through the perspective of optimal slow features on ergodic Markov chains, this work investigates the effects of these differences on value-function approximation in an idealized setting. Furthermore, three correction routes, which can potentially alleviate detrimental scaling effects, are evaluated and discussed. In addition, the special case of goal-averse behavior is considered.

Merlin Schüler, Laurenz Wiskott

This work presents a novel probabilistic interpretation of Slow Feature Analysis (SFA) through the lens of variational inference. Unlike prior formulations that recover linear SFA from Gaussian state-space models with linear emissions, this approach relaxes the key constraint of linearity. While it does not lead to full equivalence to non-linear SFA, it recasts the classical slowness objective in a variational framework. Specifically, it allows the slowness objective to be interpreted as a regularizer to a reconstruction loss. Furthermore, we provide arguments, why -- from the perspective of slowness optimization -- the reconstruction loss takes on the role of the constraints that ensure informativeness in SFA. We conclude with a discussion of potential new research directions.

Raphael C. Engelhardt, Marcel J. Meinen, Moritz Lange et al.

In previous research, we developed methods to train decision trees (DT) as agents for reinforcement learning tasks, based on deep reinforcement learning (DRL) networks. The samples from which the DTs are built, use the environment's state as features and the corresponding action as label. To solve the nontrivial task of selecting samples, which on one hand reflect the DRL agent's capabilities of choosing the right action but on the other hand also cover enough state space to generalize well, we developed an algorithm to iteratively train DTs. In this short paper, we apply this algorithm to a real-world implementation of a robotic task for the first time. Real-world tasks pose additional challenges compared to simulations, such as noise and delays. The task consists of a physical pendulum attached to a cart, which moves on a linear track. By movements to the left and to the right, the pendulum is to be swung in the upright position and balanced in the unstable equilibrium. Our results demonstrate the applicability of the algorithm to real-world tasks by generating a DT whose performance matches the performance of the DRL agent, while consisting of fewer parameters. This research could be a starting point for distilling DTs from DRL agents to obtain transparent, lightweight models for real-world reinforcement learning tasks.

Robin Schiewer, Anand Subramoney, Laurenz Wiskott

Hierarchical model-based reinforcement learning (HMBRL) aims to combine the benefits of better sample efficiency of model based reinforcement learning (MBRL) with the abstraction capability of hierarchical reinforcement learning (HRL) to solve complex tasks efficiently. While HMBRL has great potential, it still lacks wide adoption. In this work we describe a novel HMBRL framework and evaluate it thoroughly. To complement the multi-layered decision making idiom characteristic for HRL, we construct hierarchical world models that simulate environment dynamics at various levels of temporal abstraction. These models are used to train a stack of agents that communicate in a top-down manner by proposing goals to their subordinate agents. A significant focus of this study is the exploration of a static and environment agnostic temporal abstraction, which allows concurrent training of models and agents throughout the hierarchy. Unlike most goal-conditioned H(MB)RL approaches, it also leads to comparatively low dimensional abstract actions. Although our HMBRL approach did not outperform traditional methods in terms of final episode returns, it successfully facilitated decision making across two levels of abstraction using compact, low dimensional abstract actions. A central challenge in enhancing our method's performance, as uncovered through comprehensive experimentation, is model exploitation on the abstract level of our world model stack. We provide an in depth examination of this issue, discussing its implications for the field and suggesting directions for future research to overcome this challenge. By sharing these findings, we aim to contribute to the broader discourse on refining HMBRL methodologies and to assist in the development of more effective autonomous learning systems for complex decision-making environments.

Moritz Lange, Raphael C. Engelhardt, Wolfgang Konen et al.

Visual navigation requires a whole range of capabilities. A crucial one of these is the ability of an agent to determine its own location and heading in an environment. Prior works commonly assume this information as given, or use methods which lack a suitable inductive bias and accumulate error over time. In this work, we show how the method of slow feature analysis (SFA), inspired by neuroscience research, overcomes both limitations by generating interpretable representations of visual data that encode location and heading of an agent. We employ SFA in a modern reinforcement learning context, analyse and compare representations and illustrate where hierarchical SFA can outperform other feature extractors on navigation tasks.

Zahra Fayyaz, Aya Altamimi, Sen Cheng et al.

Many different studies have suggested that episodic memory is a generative process, but most computational models adopt a storage view. In this work, we propose a computational model for generative 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 information based on general semantic information in a process we call semantic completion. As episodes we use images of digits (MNIST) augmented by different backgrounds representing context. Our model is based on a VQ-VAE which generates a compressed latent representation in form of an index matrix, which still has some spatial resolution. We assume that attention selects some part of the index matrix while others are discarded, this then represents the gist of the episode and is stored as a memory trace. At recall the missing parts are filled in by a PixelCNN, modeling semantic completion, and the completed index matrix is then decoded into a full image by the VQ-VAE. 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. Due to the combinatorics in the index matrix, the model 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.

Robin Schiewer, Laurenz Wiskott

In a multi-task reinforcement learning setting, the learner commonly benefits from training on multiple related tasks by exploiting similarities among them. At the same time, the trained agent is able to solve a wider range of different problems. While this effect is well documented for model-free multi-task methods, we demonstrate a detrimental effect when using a single learned dynamics model for multiple tasks. Thus, we address the fundamental question of whether model-based multi-task reinforcement learning benefits from shared dynamics models in a similar way model-free methods do from shared policy networks. Using a single dynamics model, we see clear evidence of task confusion and reduced performance. As a remedy, enforcing an internal structure for the learned dynamics model by training isolated sub-networks for each task notably improves performance while using the same amount of parameters. We illustrate our findings by comparing both methods on a simple gridworld and a more complex vizdoom multi-task experiment.

Hlynur Davíð Hlynsson, Laurenz Wiskott

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.

Stefan Richthofer, Laurenz Wiskott

A Sturm-Liouville problem ($λwy=(ry')'+qy$) is singular if its domain is unbounded or if $r$ or $w$ vanish at the boundary. Then it is difficult to tell whether profound results from regular Sturm-Liouville theory apply. Existing criteria are often difficult to apply, e.g. because they are formulated in terms of the solution function. We study the special case that the potential $q$ is zero under Neumann boundary conditions and give simple and explicit criteria, solely in terms of the coefficient functions, to assess whether various properties of the regular case apply. Specifically, these properties are discreteness of the spectrum (BD), self-adjointness, oscillation ($i$th solution has $i$ zeros) and that the $i$th eigenvalue equals the SFA delta value (the total energy) of the $i$th solution. We further prove that stationary points of each solution strictly interlace with its zeros (in singular or regular case, regardless of the boundary condition, for zero potential or if $q < λw$ everywhere). If $\frac{r}{w}$ is bounded and of bounded variation, the criterion simplifies to requiring $\frac{|w'|}{w} \to \infty$ at singular boundary points. This research is motivated by Slow Feature Analysis (SFA), a data processing algorithm that extracts the slowest uncorrelated signals from a high-dimensional input signal and has notable success in computer vision, computational neuroscience and blind source separation. From [Sprekeler et al., 2014] it is known that for an important class of scenarios (statistically independent input), an analytic formulation of SFA reduces to a Sturm-Liouville problem with zero potential and Neumann boundary conditions. So far, the mathematical SFA theory has only considered the regular case, except for a special case that is solved by Hermite Polynomials. This work generalizes SFA theory to the singular case, i.e. open-space scenarios.

Hlynur Davíð Hlynsson, Merlin Schüler, Robin Schiewer et al.

Deeply-learned planning methods are often based on learning representations that are optimized for unrelated tasks. For example, they might be trained on reconstructing the environment. These representations are then combined with predictor functions for simulating rollouts to navigate the environment. We find this principle of learning representations unsatisfying and propose to learn them such that they are directly optimized for the task at hand: to be maximally predictable for the predictor function. This results in representations that are by design optimal for the downstream task of planning, where the learned predictor function is used as a forward model. To this end, we propose a new way of jointly 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 search on a graph in a viewpoint-matching task and the representation learned to maximize predictability is found to outperform a pre-trained representation. Our approach is shown to be more sample-efficient than standard reinforcement learning methods and our learned representation transfers successfully to dissimilar objects.

Laurenz Wiskott, Fabian Schönfeld

Many problems in machine learning can be expressed by means of a graph with nodes representing training samples and edges representing the relationship between samples in terms of similarity, temporal proximity, or label information. Graphs can in turn be represented by matrices. A special example is the Laplacian matrix, which allows us to assign each node a value that varies only little between strongly connected nodes and more between distant nodes. Such an assignment can be used to extract a useful feature representation, find a good embedding of data in a low dimensional space, or perform clustering on the original samples. In these lecture notes we first introduce the Laplacian matrix and then present a small number of algorithms designed around it.

Hlynur Davíð Hlynsson, Alberto N. Escalante-B., Laurenz Wiskott

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.

Jan Melchior, Mehdi Bayati, Amir Azizi et al.

We present a computational model based on the CRISP theory (Content Representation, Intrinsic Sequences, and Pattern completion) of the hippocampus that allows to continuously store pattern sequences online in a one-shot fashion. Rather than storing a sequence in CA3, CA3 provides a pre-trained sequence that is hetero-associated with the input sequence, which allows the system to perform one-shot learning. Plasticity on a short time scale therefore only happens in the incoming and outgoing connections of CA3. Stored sequences can later be recalled from a single cue pattern. We identify the pattern separation performed by subregion DG to be necessary for storing sequences that contain correlated patterns. A design principle of the model is that we use a single learning rule named Hebbiand-escent to train all parts of the system. Hebbian-descent has an inherent forgetting mechanism that allows the system to continuously memorize new patterns while forgetting early stored ones. The model shows a plausible behavior when noisy and new patterns are presented and has a rather high capacity of about 40% in terms of the number of neurons in CA3. One notable property of our model is that it is capable of `boot-strapping' (improving) itself without external input in a process we refer to as `dreaming'. Besides artificially generated input sequences we also show that the model works with sequences of encoded handwritten digits or natural images. To our knowledge this is the first model of the hippocampus that allows to store correlated pattern sequences online in a one-shot fashion without a consolidation process, which can instantaneously be recalled later.

Jan Melchior, Laurenz Wiskott

In this work we propose Hebbian-descent as a biologically plausible learning rule for hetero-associative as well as auto-associative learning in single layer artificial neural networks. It can be used as a replacement for gradient descent as well as Hebbian learning, in particular in online learning, as it inherits their advantages while not suffering from their disadvantages. We discuss the drawbacks of Hebbian learning as having problems with correlated input data and not profiting from seeing training patterns several times. For gradient descent we identify the derivative of the activation function as problematic especially in online learning. Hebbian-descent addresses these problems by getting rid of the activation function's derivative and by centering, i.e. keeping the neural activities mean free, leading to a biologically plausible update rule that is provably convergent, does not suffer from the vanishing error term problem, can deal with correlated data, profits from seeing patterns several times, and enables successful online learning when centering is used. We discuss its relationship to Hebbian learning, contrastive learning, and gradient decent and show that in case of a strictly positive derivative of the activation function Hebbian-descent leads to the same update rule as gradient descent but for a different loss function. In this case Hebbian-descent inherits the convergence properties of gradient descent, but we also show empirically that it converges when the derivative of the activation function is only non-negative, such as for the step function for example. Furthermore, in case of the mean squared error loss Hebbian-descent can be understood as the difference between two Hebb-learning steps, which in case of an invertible and integrable activation function actually optimizes a generalized linear model. ...

Hlynur Davíð Hlynsson, Laurenz Wiskott

Several methods of estimating the mutual information of random variables have been developed in recent years. They can prove valuable for novel approaches to learning statistically independent features. In this paper, we use one of these methods, a mutual information neural estimation (MINE) network, to present a proof-of-concept of how a neural network can perform linear ICA. We minimize the mutual information, as estimated by a MINE network, between the output units of a differentiable encoder network. This is done by simple alternate optimization of the two networks. The method is shown to get a qualitatively equal solution to FastICA on blind-source-separation of noisy sources.

Merlin Schüler, Hlynur Davíð Hlynsson, Laurenz Wiskott

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.

Stefan Richthofer, Laurenz Wiskott

Extended Predictable Feature Analysis (PFAx) [Richthofer and Wiskott, 2017] is an extension of PFA [Richthofer and Wiskott, 2015] that allows generating a goal-directed control signal of an agent whose dynamics has previously been learned during a training phase in an unsupervised manner. PFAx hardly requires assumptions or prior knowledge of the agent's sensor or control mechanics, or of the environment. It selects features from a high-dimensional input by intrinsic predictability and organizes them into a reasonably low-dimensional model. While PFA obtains a well predictable model, PFAx yields a model ideally suited for manipulations with predictable outcome. This allows for goal-directed manipulation of an agent and thus for local navigation, i.e. for reaching states where intermediate actions can be chosen by a permanent descent of distance to the goal. The approach is limited when it comes to global navigation, e.g. involving obstacles or multiple rooms. In this article, we extend theoretical results from [Sprekeler and Wiskott, 2008], enabling PFAx to perform stable global navigation. So far, the most widely exploited characteristic of Slow Feature Analysis (SFA) was that slowness yields invariances. We focus on another fundamental characteristics of slow signals: They tend to yield monotonicity and one significant property of monotonicity is that local optimization is sufficient to find a global optimum. We present an SFA-based algorithm that structures an environment such that navigation tasks hierarchically decompose into subgoals. Each of these can be efficiently achieved by PFAx, yielding an overall global solution of the task. The algorithm needs to explore and process an environment only once and can then perform all sorts of navigation tasks efficiently. We support this algorithm by mathematical theory and apply it to different problems.

Stefan Richthofer, Laurenz Wiskott

Predictable Feature Analysis (PFA) (Richthofer, Wiskott, ICMLA 2015) is an algorithm that performs dimensionality reduction on high dimensional input signal. It extracts those subsignals that are most predictable according to a certain prediction model. We refer to these extracted signals as predictable features. In this work we extend the notion of PFA to take supplementary information into account for improving its predictions. Such information can be a multidimensional signal like the main input to PFA, but is regarded external. That means it won't participate in the feature extraction - no features get extracted or composed of it. Features will be exclusively extracted from the main input such that they are most predictable based on themselves and the supplementary information. We refer to this enhanced PFA as PFAx (PFA extended). Even more important than improving prediction quality is to observe the effect of supplementary information on feature selection. PFAx transparently provides insight how the supplementary information adds to prediction quality and whether it is valuable at all. Finally we show how to invert that relation and can generate the supplementary information such that it would yield a certain desired outcome of the main signal. We apply this to a setting inspired by reinforcement learning and let the algorithm learn how to control an agent in an environment. With this method it is feasible to locally optimize the agent's state, i.e. reach a certain goal that is near enough. We are preparing a follow-up paper that extends this method such that also global optimization is feasible.

Varun Raj Kompella, Laurenz Wiskott

A compact information-rich representation of the environment, also called a feature abstraction, can simplify a robot's task of mapping its raw sensory inputs to useful action sequences. However, in environments that are non-stationary and only partially observable, a single abstraction is probably not sufficient to encode most variations. Therefore, learning multiple sets of spatially or temporally local, modular abstractions of the inputs would be beneficial. How can a robot learn these local abstractions without a teacher? More specifically, how can it decide from where and when to start learning a new abstraction? A recently proposed algorithm called Curious Dr. MISFA addresses this problem. The algorithm is based on two underlying learning principles called artificial curiosity and slowness. The former is used to make the robot self-motivated to explore by rewarding itself whenever it makes progress learning an abstraction; the later is used to update the abstraction by extracting slowly varying components from raw sensory inputs. Curious Dr. MISFA's application is, however, limited to discrete domains constrained by a pre-defined state space and has design limitations that make it unstable in certain situations. This paper presents a significant improvement that is applicable to continuous environments, is computationally less expensive, simpler to use with fewer hyper parameters, and stable in certain non-stationary environments. We demonstrate the efficacy and stability of our method in a vision-based robot simulator.

Björn Weghenkel, Asja Fischer, Laurenz Wiskott

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.

Alberto N. Escalante-B., Laurenz Wiskott

Slow feature analysis (SFA) is an unsupervised-learning algorithm that extracts slowly varying features from a multi-dimensional time series. A supervised extension to SFA for classification and regression is graph-based SFA (GSFA). GSFA is based on the preservation of similarities, which are specified by a graph structure derived from the labels. It has been shown that hierarchical GSFA (HGSFA) allows learning from images and other high-dimensional data. The feature space spanned by HGSFA is complex due to the composition of the nonlinearities of the nodes in the network. However, we show that the network discards useful information prematurely before it reaches higher nodes, resulting in suboptimal global slowness and an under-exploited feature space. To counteract these problems, we propose an extension called hierarchical information-preserving GSFA (HiGSFA), where information preservation complements the slowness-maximization goal. We build a 10-layer HiGSFA network to estimate human age from facial photographs of the MORPH-II database, achieving a mean absolute error of 3.50 years, improving the state-of-the-art performance. HiGSFA and HGSFA support multiple-labels and offer a rich feature space, feed-forward training, and linear complexity in the number of samples and dimensions. Furthermore, HiGSFA outperforms HGSFA in terms of feature slowness, estimation accuracy and input reconstruction, giving rise to a promising hierarchical supervised-learning approach.

Alberto N. Escalante-B., Laurenz Wiskott

Slow feature analysis (SFA) is an unsupervised learning algorithm that extracts slowly varying features from a time series. Graph-based SFA (GSFA) is a supervised extension that can solve regression problems if followed by a post-processing regression algorithm. A training graph specifies arbitrary connections between the training samples. The connections in current graphs, however, only depend on the rank of the involved labels. Exploiting the exact label values makes further improvements in estimation accuracy possible. In this article, we propose the exact label learning (ELL) method to create a graph that codes the desired label explicitly, so that GSFA is able to extract a normalized version of it directly. 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, a higher classification rate is obtained compared to a graph equivalent to nonlinear Fisher discriminant analysis. The method is versatile, directly supports multiple labels, and provides higher accuracy compared to current graphs for the problems considered.

Nan Wang, Jan Melchior, Laurenz Wiskott

We present a theoretical analysis of Gaussian-binary restricted Boltzmann machines (GRBMs) from the perspective of density models. The key aspect of this analysis is to show that GRBMs can be formulated as a constrained mixture of Gaussians, which gives a much better insight into the model's capabilities and limitations. We show that GRBMs are capable of learning meaningful features both in a two-dimensional blind source separation task and in modeling natural images. Further, we show that reported difficulties in training GRBMs are due to the failure of the training algorithm rather than the model itself. Based on our analysis we are able to propose several training recipes, which allowed successful and fast training in our experiments. Finally, we discuss the relationship of GRBMs to several modifications that have been proposed to improve the model.

Nan Wang, Dirk Jancke, Laurenz Wiskott

Spontaneous cortical activity -- the ongoing cortical activities in absence of intentional sensory input -- is considered to play a vital role in many aspects of both normal brain functions and mental dysfunctions. We present a centered Gaussian-binary Deep Boltzmann Machine (GDBM) for modeling the activity in early cortical visual areas and relate the random sampling in GDBMs to the spontaneous cortical activity. After training the proposed model on natural image patches, we show that the samples collected from the model's probability distribution encompass similar activity patterns as found in the spontaneous activity. Specifically, filters having the same orientation preference tend to be active together during random sampling. Our work demonstrates the centered GDBM is a meaningful model approach for basic receptive field properties and the emergence of spontaneous activity patterns in early cortical visual areas. Besides, we show empirically that centered GDBMs do not suffer from the difficulties during training as GDBMs do and can be properly trained without the layer-wise pretraining.

Stefan Richthofer, Laurenz Wiskott

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 sub-signals 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.

Jan Melchior, Asja Fischer, Laurenz Wiskott

This work analyzes centered binary Restricted Boltzmann Machines (RBMs) and binary Deep Boltzmann Machines (DBMs), where centering is done by subtracting offset values from visible and hidden variables. We show analytically that (i) centering results in a different but equivalent parameterization for artificial neural networks in general, (ii) the expected performance of centered binary 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 binary RBMs/DBMs, and (iv) using the enhanced gradient is equivalent to setting the offset values to the average over model and data mean. Furthermore, numerical simulations suggest that (i) optimal generative performance is achieved by subtracting mean values from visible as well as hidden variables, (ii) centered 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 diverging, (v) centered RBMs/DBMs reach higher LL values than normal RBMs/DBMs while having a smaller norm of the weight matrix, (vi) centering leads to an update direction that is closer to the natural gradient and that the natural gradient is extremly efficient for training RBMs, (vii) centering dispense the need for greedy layer-wise pre-training of DBMs, (viii) furthermore we show that pre-training often even worsen the results independently whether centering is used or not, and (ix) centering is also beneficial for auto encoders.