Xaq Pitkow

Nikos Karantzas, Emma Besier, Josue Ortega Caro et al.

Despite the enormous success of artificial neural networks (ANNs) in many disciplines, the characterization of their computations and the origin of key properties such as generalization and robustness remain open questions. Recent literature suggests that robust networks with good generalization properties tend to be biased towards processing low frequencies in images. To explore the frequency bias hypothesis further, we develop an algorithm that allows us to learn modulatory masks highlighting the essential input frequencies needed for preserving a trained network's performance. We achieve this by imposing invariance in the loss with respect to such modulations in the input frequencies. We first use our method to test the low-frequency preference hypothesis of adversarially trained or data-augmented networks. Our results suggest that adversarially robust networks indeed exhibit a low-frequency bias but we find this bias is also dependent on directions in frequency space. However, this is not necessarily true for other types of data augmentation. Our results also indicate that the essential frequencies in question are effectively the ones used to achieve generalization in the first place. Surprisingly, images seen through these modulatory masks are not recognizable and resemble texture-like patterns.

Ben Baker, Richard D. Lange, Andrew Richmond et al.

Representations play a central role in the study of both biological and artificial intelligence, as well as philosophy of mind. Across neuroscience, computer science, and philosophy, a recurring theme is that representations not only carry information but should be ``useful'' for or ``usable'' by an agent in some sense. Here, we review how the ``usefulness'' of representations has been conceptualized and how it figures into different conceptions of representation. We identify and explore four aspects of use and usability: representations generally carry \textit{information}; that information may or may not be \textit{useful} and it may or may not be encoded in a usable \textit{format}; and the representations may or may not be \textit{used downstream}. Building on these four aspects of information and use, we then organize existing perspectives on neural representations into three levels: Representations as Information (Level 1); Representations as Usable (Level 2); and Representations as Used (Level 3). Our account is meant to give readers an appreciation for the diversity of notions of ``neural representation,'' help them navigate the vast and multi-disciplinary literature on the topic, and help them clarify the appropriate notion of representation for their own investigations.

Ralf M. Haefner, Jeff Beck, Cristina Savin et al.

This perspective piece is the result of a Generative Adversarial Collaboration (GAC) tackling the question `How does neural activity represent probability distributions?'. We have addressed three major obstacles to progress on answering this question: first, we provide a unified language for defining competing hypotheses. Second, we explain the fundamentals of three prominent proposals for probabilistic computations -- Probabilistic Population Codes (PPCs), Distributed Distributional Codes (DDCs), and Neural Sampling Codes (NSCs) -- and describe similarities and differences in that common language. Third, we review key empirical data previously taken as evidence for at least one of these proposal, and describe how it may or may not be explainable by alternative proposals. Finally, we describe some key challenges in resolving the debate, and propose potential directions to address them through a combination of theory and experiments.

Rajkumar Vasudeva Raju, Zhe Li, Scott Linderman et al.

Patterns of microcircuitry suggest that the brain has an array of repeated canonical computational units. Yet neural representations are distributed, so the relevant computations may only be related indirectly to single-neuron transformations. It thus remains an open challenge how to define canonical distributed computations. We integrate normative and algorithmic theories of neural computation into a mathematical framework for inferring canonical distributed computations from large-scale neural activity patterns. At the normative level, we hypothesize that the brain creates a structured internal model of its environment, positing latent causes that explain its sensory inputs, and uses those sensory inputs to infer the latent causes. At the algorithmic level, we propose that this inference process is a nonlinear message-passing algorithm on a graph-structured model of the world. Given a time series of neural activity during a perceptual inference task, our framework finds (i) the neural representation of relevant latent variables, (ii) interactions between these variables that define the brain's internal model of the world, and (iii) message-functions specifying the inference algorithm. These targeted computational properties are then statistically distinguishable due to the symmetries inherent in any canonical computation, up to a global transformation. As a demonstration, we simulate recordings for a model brain that implicitly implements an approximate inference algorithm on a probabilistic graphical model. Given its external inputs and noisy neural activity, we recover the latent variables, their neural representation and dynamics, and canonical message-functions. We highlight features of experimental design needed to successfully extract canonical computations from neural data. Overall, this framework provides a new tool for discovering interpretable structure in neural recordings.

Ryan Liu, Dilip Arumugam, Cedegao E. Zhang et al.

While contemporary large language models (LLMs) are increasingly capable in isolation, there are still many difficult problems that lie beyond the abilities of a single LLM. For such tasks, there is still uncertainty about how best to take many LLMs as parts and combine them into a greater whole. This position paper argues that potential blueprints for designing such modular language agents can be found in the existing literature on cognitive models and artificial intelligence (AI) algorithms. To make this point clear, we formalize the idea of an agent template that specifies roles for individual LLMs and how their functionalities should be composed. We then survey a variety of existing language agents in the literature and highlight their underlying templates derived directly from cognitive models or AI algorithms. By highlighting these designs, we aim to call attention to agent templates inspired by cognitive science and AI as a powerful tool for developing effective, interpretable language agents.

Renjie Liao, Yuwen Xiong, Ethan Fetaya et al.

In this paper, we revisit the recurrent back-propagation (RBP) algorithm, discuss the conditions under which it applies as well as how to satisfy them in deep neural networks. We show that RBP can be unstable and propose two variants based on conjugate gradient on the normal equations (CG-RBP) and Neumann series (Neumann-RBP). We further investigate the relationship between Neumann-RBP and back propagation through time (BPTT) and its truncated version (TBPTT). Our Neumann-RBP has the same time complexity as TBPTT but only requires constant memory, whereas TBPTT's memory cost scales linearly with the number of truncation steps. We examine all RBP variants along with BPTT and TBPTT in three different application domains: associative memory with continuous Hopfield networks, document classification in citation networks using graph neural networks and hyperparameter optimization for fully connected networks. All experiments demonstrate that RBPs, especially the Neumann-RBP variant, are efficient and effective for optimizing convergent recurrent neural networks. Code is released at: \url{https://github.com/lrjconan/RBP}.

Lane Lewis, Zhixin Wang, David Schwab et al.

If a person can solve a task, can measuring their brain make it easier to train a model to solve that task too? Recent NeuroAI work suggests that supplementing task training with neural recordings can modestly improve model performance and robustness. However, it is unclear when there should be a benefit from using neural data and how much benefit to expect. We formulate this question mathematically, and begin to address it theoretically using a simple, analytically tractable linear gaussian model of task targets and neural recordings. For a multimodal estimator trained on both brain data and task labels, we derive scaling laws for how performance scales with the numbers of brain and task samples. From these laws we derive relative value and exchange rates between brain samples and task samples, quantifying how much extra task samples neural data is worth as a function of task-brain alignment, neural and task noise, latent dimension, and brain data sample size. We also analyze test distribution shift, to identify conditions where brain-regularized learning can produce substantial robustness gains through learned invariances. Finally, under a fixed collection budget, we characterize the regimes in which brain data is worth collecting. Our results provide a foundation for understanding how valuable brain data could be for improving machine learning.

Lokesh Boominathan, Yizhou Chen, Matthew McGinley et al.

Being attentive to task-relevant features can improve task performance, but paying attention comes with its own metabolic cost. Therefore, strategic allocation of attention is crucial in performing the task efficiently. This work aims to understand this strategy. Recently, de Gee et al. conducted experiments involving mice performing an auditory sustained attention-value task. This task required the mice to exert attention to identify whether a high-order acoustic feature was present amid the noise. By varying the trial duration and reward magnitude, the task allows us to investigate how an agent should strategically deploy their attention to maximize their benefits and minimize their costs. In our work, we develop a reinforcement learning-based normative model of the mice to understand how it balances attention cost against its benefits. The model is such that at each moment the mice can choose between two levels of attention and decide when to take costly actions that could obtain rewards. Our model suggests that efficient use of attentional resources involves alternating blocks of high attention with blocks of low attention. In the extreme case where the agent disregards sensory input during low attention states, we see that high attention is used rhythmically. Our model provides evidence about how one should deploy attention as a function of task utility, signal statistics, and how attention affects sensory evidence.

Reece Keller, Alyn Kirsch, Felix Pei et al.

Autonomy is a hallmark of animal intelligence, enabling adaptive and intelligent behavior in complex environments without relying on external reward or task structure. Existing reinforcement learning approaches to exploration in reward-free environments, including a class of methods known as model-based intrinsic motivation, exhibit inconsistent exploration patterns and do not converge to an exploratory policy, thus failing to capture robust autonomous behaviors observed in animals. Moreover, systems neuroscience has largely overlooked the neural basis of autonomy, focusing instead on experimental paradigms where animals are motivated by external reward rather than engaging in ethological, naturalistic and task-independent behavior. To bridge these gaps, we introduce a novel model-based intrinsic drive explicitly designed after the principles of autonomous exploration in animals. Our method (3M-Progress) achieves animal-like exploration by tracking divergence between an online world model and a fixed prior learned from an ecological niche. To the best of our knowledge, we introduce the first autonomous embodied agent that predicts brain data entirely from self-supervised optimization of an intrinsic goal -- without any behavioral or neural training data -- demonstrating that 3M-Progress agents capture the explainable variance in behavioral patterns and whole-brain neural-glial dynamics recorded from autonomously behaving larval zebrafish, thereby providing the first goal-driven, population-level model of neural-glial computation. Our findings establish a computational framework connecting model-based intrinsic motivation to naturalistic behavior, providing a foundation for building artificial agents with animal-like autonomy.

Itzel Olivos-Castillo, Paul Schrater, Xaq Pitkow

A key challenge in advancing artificial intelligence is achieving the right balance between utility maximization and resource use by both external movement and internal computation. While this trade-off has been studied in fully observable settings, our understanding of resource efficiency in partially observable environments remains limited. Motivated by this challenge, we develop a version of the POMDP framework where the information gained through inference is treated as a resource that must be optimized alongside task performance and motion effort. By solving this problem in environments described by linear-Gaussian dynamics, we uncover fundamental principles of resource efficiency. Our study reveals a phase transition in the inference, switching from a Bayes-optimal approach to one that strategically leaves some uncertainty unresolved. This frugal behavior gives rise to a structured family of equally effective strategies, facilitating adaptation to later objectives and constraints overlooked during the original optimization. We illustrate the applicability of our framework and the generality of the principles we derived using two nonlinear tasks. Overall, this work provides a foundation for a new type of rational computation that both brains and machines could use for effective but resource-efficient control under uncertainty.

Kandan Ramakrishnan, R. James Cotton, Xaq Pitkow et al.

Recent work on object-centric world models aim to factorize representations in terms of objects in a completely unsupervised or self-supervised manner. Such world models are hypothesized to be a key component to address the generalization problem. While self-supervision has shown improved performance however, OOD generalization has not been systematically and explicitly tested. In this paper, we conduct an extensive study on the generalization properties of contrastive world model. We systematically test the model under a number of different OOD generalization scenarios such as extrapolation to new object attributes, introducing new conjunctions or new attributes. Our experiments show that the contrastive world model fails to generalize under the different OOD tests and the drop in performance depends on the extent to which the samples are OOD. When visualizing the transition updates and convolutional feature maps, we observe that any changes in object attributes (such as previously unseen colors, shapes, or conjunctions of color and shape) breaks down the factorization of object representations. Overall, our work highlights the importance of object-centric representations for generalization and current models are limited in their capacity to learn such representations required for human-level generalization.

Zhe Li, Andreas S. Tolias, Xaq Pitkow

Many complex systems are composed of interacting parts, and the underlying laws are usually simple and universal. While graph neural networks provide a useful relational inductive bias for modeling such systems, generalization to new system instances of the same type is less studied. In this work we trained graph neural networks to fit time series from an example nonlinear dynamical system, the belief propagation algorithm. We found simple interpretations of the learned representation and model components, and they are consistent with core properties of the probabilistic inference algorithm. We successfully identified a 'graph translator' between the statistical interactions in belief propagation and parameters of the corresponding trained network, and showed that it enables two types of novel generalization: to recover the underlying structure of a new system instance based solely on time series observations, or to construct a new network from this structure directly. Our results demonstrated a path towards understanding both dynamics and structure of a complex system and how such understanding can be used for generalization.

Richard D. Lange, Ari Benjamin, Ralf M. Haefner et al.

Sampling and Variational Inference (VI) are two large families of methods for approximate inference that have complementary strengths. Sampling methods excel at approximating arbitrary probability distributions, but can be inefficient. VI methods are efficient, but may misrepresent the true distribution. Here, we develop a general framework where approximations are stochastic mixtures of simple component distributions. Both sampling and VI can be seen as special cases: in sampling, each mixture component is a delta-function and is chosen stochastically, while in standard VI a single component is chosen to minimize divergence. We derive a practical method that interpolates between sampling and VI by solving an optimization problem over a mixing distribution. Intermediate inference methods then arise by varying a single parameter. Our method provably improves on sampling (reducing variance) and on VI (reducing bias+variance despite increasing variance). We demonstrate our method's bias/variance trade-off in practice on reference problems, and we compare outcomes to commonly used sampling and VI methods. This work takes a step towards a highly flexible yet simple family of inference methods that combines the complementary strengths of sampling and VI.

Kijung Yoon, Emin Orhan, Juhyun Kim et al.

Neurons in the brain are complex machines with distinct functional compartments that interact nonlinearly. In contrast, neurons in artificial neural networks abstract away this complexity, typically down to a scalar activation function of a weighted sum of inputs. Here we emulate more biologically realistic neurons by learning canonical activation functions with two input arguments, analogous to basal and apical dendrites. We use a network-in-network architecture where each neuron is modeled as a multilayer perceptron with two inputs and a single output. This inner perceptron is shared by all units in the outer network. Remarkably, the resultant nonlinearities often produce soft XOR functions, consistent with recent experimental observations about interactions between inputs in human cortical neurons. When hyperparameters are optimized, networks with these nonlinearities learn faster and perform better than conventional ReLU nonlinearities with matched parameter counts, and they are more robust to natural and adversarial perturbations.

Yicheng Fei, Xaq Pitkow

Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.

Shell Xu Hu, Md Rifat Arefin, Viet-Nhat Nguyen et al.

To extract the voice of a target speaker when mixed with a variety of other sounds, such as white and ambient noises or the voices of interfering speakers, we extend the Transformer network to attend the most relevant information with respect to the target speaker given the characteristics of his or her voices as a form of contextual information. The idea has a natural interpretation in terms of the selective attention theory. Specifically, we propose two models to incorporate the voice characteristics in Transformer based on different insights of where the feature selection should take place. Both models yield excellent performance, on par or better than published state-of-the-art models on the speaker extraction task, including separating speech of novel speakers not seen during training.

Mengye Ren, Eleni Triantafillou, Kuan-Chieh Wang et al.

Despite impressive progress in deep learning, generalizing far beyond the training distribution is an important open challenge. In this work, we consider few-shot classification, and aim to shed light on what makes some novel classes easier to learn than others, and what types of learned representations generalize better. To this end, we define a new paradigm in terms of attributes -- simple building blocks of which concepts are formed -- as a means of quantifying the degree of relatedness of different concepts. Our empirical analysis reveals that supervised learning generalizes poorly to new attributes, but a combination of self-supervised pretraining with supervised finetuning leads to stronger generalization. The benefit of self-supervised pretraining and supervised finetuning is further investigated through controlled experiments using random splits of the attribute space, and we find that predictability of test attributes provides an informative estimate of a model's generalization ability.

Minhae Kwon, Saurabh Daptardar, Paul Schrater et al.

A fundamental question in neuroscience is how the brain creates an internal model of the world to guide actions using sequences of ambiguous sensory information. This is naturally formulated as a reinforcement learning problem under partial observations, where an agent must estimate relevant latent variables in the world from its evidence, anticipate possible future states, and choose actions that optimize total expected reward. This problem can be solved by control theory, which allows us to find the optimal actions for a given system dynamics and objective function. However, animals often appear to behave suboptimally. Why? We hypothesize that animals have their own flawed internal model of the world, and choose actions with the highest expected subjective reward according to that flawed model. We describe this behavior as rational but not optimal. The problem of Inverse Rational Control (IRC) aims to identify which internal model would best explain an agent's actions. Our contribution here generalizes past work on Inverse Rational Control which solved this problem for discrete control in partially observable Markov decision processes. Here we accommodate continuous nonlinear dynamics and continuous actions, and impute sensory observations corrupted by unknown noise that is private to the animal. We first build an optimal Bayesian agent that learns an optimal policy generalized over the entire model space of dynamics and subjective rewards using deep reinforcement learning. Crucially, this allows us to compute a likelihood over models for experimentally observable action trajectories acquired from a suboptimal agent. We then find the model parameters that maximize the likelihood using gradient ascent.

Zhe Li, Wieland Brendel, Edgar Y. Walker et al.

Despite impressive performance on numerous visual tasks, Convolutional Neural Networks (CNNs) --- unlike brains --- are often highly sensitive to small perturbations of their input, e.g. adversarial noise leading to erroneous decisions. We propose to regularize CNNs using large-scale neuroscience data to learn more robust neural features in terms of representational similarity. We presented natural images to mice and measured the responses of thousands of neurons from cortical visual areas. Next, we denoised the notoriously variable neural activity using strong predictive models trained on this large corpus of responses from the mouse visual system, and calculated the representational similarity for millions of pairs of images from the model's predictions. We then used the neural representation similarity to regularize CNNs trained on image classification by penalizing intermediate representations that deviated from neural ones. This preserved performance of baseline models when classifying images under standard benchmarks, while maintaining substantially higher performance compared to baseline or control models when classifying noisy images. Moreover, the models regularized with cortical representations also improved model robustness in terms of adversarial attacks. This demonstrates that regularizing with neural data can be an effective tool to create an inductive bias towards more robust inference.

Saurabh Daptardar, Paul Schrater, Xaq Pitkow

Continuous control and planning remains a major challenge in robotics and machine learning. Neuroscience offers the possibility of learning from animal brains that implement highly successful controllers, but it is unclear how to relate an animal's behavior to control principles. Animals may not always act optimally from the perspective of an external observer, but may still act rationally: we hypothesize that animals choose actions with highest expected future subjective value according to their own internal model of the world. Their actions thus result from solving a different optimal control problem from those on which they are evaluated in neuroscience experiments. With this assumption, we propose a novel framework of model-based inverse rational control that learns the agent's internal model that best explains their actions in a task described as a partially observable Markov decision process (POMDP). In this approach we first learn optimal policies generalized over the entire model space of dynamics and subjective rewards, using an extended Kalman filter to represent the belief space, a neural network in the actor-critic framework to optimize the policy, and a simplified basis for the parameter space. We then compute the model that maximizes the likelihood of the experimentally observable data comprising the agent's sensory observations and chosen actions. Our proposed method is able to recover the true model of simulated agents within theoretical error bounds given by limited data. We illustrate this method by applying it to a complex naturalistic task currently used in neuroscience experiments. This approach provides a foundation for interpreting the behavioral and neural dynamics of highly adapted controllers in animal brains.

A. Emin Orhan, Xaq Pitkow

Training recurrent neural networks (RNNs) is a hard problem due to degeneracies in the optimization landscape, a problem also known as vanishing/exploding gradients. Short of designing new RNN architectures, previous methods for dealing with this problem usually boil down to orthogonalization of the recurrent dynamics, either at initialization or during the entire training period. The basic motivation behind these methods is that orthogonal transformations are isometries of the Euclidean space, hence they preserve (Euclidean) norms and effectively deal with vanishing/exploding gradients. However, this ignores the crucial effects of non-linearity and noise. In the presence of a non-linearity, orthogonal transformations no longer preserve norms, suggesting that alternative transformations might be better suited to non-linear networks. Moreover, in the presence of noise, norm preservation itself ceases to be the ideal objective. A more sensible objective is maximizing the signal-to-noise ratio (SNR) of the propagated signal instead. Previous work has shown that in the linear case, recurrent networks that maximize the SNR display strongly non-normal, sequential dynamics and orthogonal networks are highly suboptimal by this measure. Motivated by this finding, here we investigate the potential of non-normal RNNs, i.e. RNNs with a non-normal recurrent connectivity matrix, in sequential processing tasks. Our experimental results show that non-normal RNNs outperform their orthogonal counterparts in a diverse range of benchmarks. We also find evidence for increased non-normality and hidden chain-like feedforward motifs in trained RNNs initialized with orthogonal recurrent connectivity matrices.

Arun Kumar, Zhengwei Wu, Xaq Pitkow et al.

Animal behavior is not driven simply by its current observations, but is strongly influenced by internal states. Estimating the structure of these internal states is crucial for understanding the neural basis of behavior. In principle, internal states can be estimated by inverting behavior models, as in inverse model-based Reinforcement Learning. However, this requires careful parameterization and risks model-mismatch to the animal. Here we take a data-driven approach to infer latent states directly from observations of behavior, using a partially observable switching semi-Markov process. This process has two elements critical for capturing animal behavior: it captures non-exponential distribution of times between observations, and transitions between latent states depend on the animal's actions, features that require more complex non-markovian models to represent. To demonstrate the utility of our approach, we apply it to the observations of a simulated optimal agent performing a foraging task, and find that latent dynamics extracted by the model has correspondences with the belief dynamics of the agent. Finally, we apply our model to identify latent states in the behaviors of monkey performing a foraging task, and find clusters of latent states that identify periods of time consistent with expectant waiting. This data-driven behavioral model will be valuable for inferring latent cognitive states, and thereby for measuring neural representations of those states.

Zhengwei Wu, Paul Schrater, Xaq Pitkow

Complex behaviors are often driven by an internal model, which integrates sensory information over time and facilitates long-term planning. Inferring an agent's internal model is a crucial ingredient in social interactions (theory of mind), for imitation learning, and for interpreting neural activities of behaving agents. Here we describe a generic method to model an agent's behavior under an environment with uncertainty, and infer the agent's internal model, reward function, and dynamic beliefs. We apply our method to a simulated agent performing a naturalistic foraging task. We assume the agent behaves rationally --- that is, they take actions that optimize their subjective utility according to their understanding of the task and its relevant causal variables. We model this rational solution as a Partially Observable Markov Decision Process (POMDP) where the agent may make wrong assumptions about the task parameters. Given the agent's sensory observations and actions, we learn its internal model and reward function by maximum likelihood estimation over a set of task-relevant parameters. The Markov property of the POMDP enables us to characterize the transition probabilities between internal belief states and iteratively estimate the agent's policy using a constrained Expectation-Maximization (EM) algorithm. We validate our method on simulated agents performing suboptimally on a foraging task currently used in many neuroscience experiments, and successfully recover their internal model and reward function. Our work lays a critical foundation to discover how the brain represents and computes with dynamic beliefs.

KiJung Yoon, Renjie Liao, Yuwen Xiong et al.

A fundamental computation for statistical inference and accurate decision-making is to compute the marginal probabilities or most probable states of task-relevant variables. Probabilistic graphical models can efficiently represent the structure of such complex data, but performing these inferences is generally difficult. Message-passing algorithms, such as belief propagation, are a natural way to disseminate evidence amongst correlated variables while exploiting the graph structure, but these algorithms can struggle when the conditional dependency graphs contain loops. Here we use Graph Neural Networks (GNNs) to learn a message-passing algorithm that solves these inference tasks. We first show that the architecture of GNNs is well-matched to inference tasks. We then demonstrate the efficacy of this inference approach by training GNNs on a collection of graphical models and showing that they substantially outperform belief propagation on loopy graphs. Our message-passing algorithms generalize out of the training set to larger graphs and graphs with different structure.

A. Emin Orhan, Xaq Pitkow

Skip connections made the training of very deep networks possible and have become an indispensable component in a variety of neural architectures. A completely satisfactory explanation for their success remains elusive. Here, we present a novel explanation for the benefits of skip connections in training very deep networks. The difficulty of training deep networks is partly due to the singularities caused by the non-identifiability of the model. Several such singularities have been identified in previous works: (i) overlap singularities caused by the permutation symmetry of nodes in a given layer, (ii) elimination singularities corresponding to the elimination, i.e. consistent deactivation, of nodes, (iii) singularities generated by the linear dependence of the nodes. These singularities cause degenerate manifolds in the loss landscape that slow down learning. We argue that skip connections eliminate these singularities by breaking the permutation symmetry of nodes, by reducing the possibility of node elimination and by making the nodes less linearly dependent. Moreover, for typical initializations, skip connections move the network away from the "ghosts" of these singularities and sculpt the landscape around them to alleviate the learning slow-down. These hypotheses are supported by evidence from simplified models, as well as from experiments with deep networks trained on real-world datasets.