Humans and animals can recognize latent structures in their environment and apply this information to efficiently navigate the world. Several recent works argue that the brain supports these abilities by forming neural representations that encode such latent structures in flexible, generalizable ways. However, it remains unclear what aspects of neural population activity are contributing to these computational capabilities. Here, we develop an analytical theory linking the mesoscopic statistics of a neural population's activity to generalization performance on a multi-task learning problem. To do this, we rely on a generative model in which different tasks depend on a common, unobserved latent structure and predictions are formed from a linear readout of a neural population's activity. We show that three geometric measures of the population activity determine generalization performance in these settings. Using this theory, we find that experimentally observed factorized (or disentangled) representations naturally emerge as an optimal solution to the multi-task learning problem. We go on to show that when data is scarce, optimal codes compress less informative latent variables, and when data is abundant, optimal codes expand this information in the state space. We validate predictions from our theory using biological and artificial neural network data. Our results therefore tie neural population geometry to the multi-task learning problem and make normative predictions of the structure of population activity in these settings.
Recently, growth in our understanding of the computations performed in both biological and artificial neural networks has largely been driven by either low-level mechanistic studies or global normative approaches. However, concrete methodologies for bridging the gap between these levels of abstraction remain elusive. In this work, we investigate the internal mechanisms of neural networks through the lens of neural population geometry, aiming to provide understanding at an intermediate level of abstraction, as a way to bridge that gap. Utilizing manifold capacity theory (MCT) from statistical physics and manifold alignment analysis (MAA) from high-dimensional statistics, we probe the underlying organization of task-dependent manifolds in deep neural networks and macaque neural recordings. Specifically, we quantitatively characterize how different learning objectives lead to differences in the organizational strategies of these models and demonstrate how these geometric analyses are connected to the decodability of task-relevant information. These analyses present a strong direction for bridging mechanistic and normative theories in neural networks through neural population geometry, potentially opening up many future research avenues in both machine learning and neuroscience.
The representations of neural networks are often compared to those of biological systems by performing regression between the neural network responses and those measured from biological systems. Many different state-of-the-art deep neural networks yield similar neural predictions, but it remains unclear how to differentiate among models that perform equally well at predicting neural responses. To gain insight into this, we use a recent theoretical framework that relates the generalization error from regression to the spectral bias of the model activations and the alignment of the neural responses onto the learnable subspace of the model. We extend this theory to the case of regression between model activations and neural responses, and define geometrical properties describing the error embedding geometry. We test a large number of deep neural networks that predict visual cortical activity and show that there are multiple types of geometries that result in low neural prediction error as measured via regression. The work demonstrates that carefully decomposing representational metrics can provide interpretability of how models are capturing neural activity and points the way towards improved models of neural activity.
Self-supervised Learning (SSL) provides a strategy for constructing useful representations of images without relying on hand-assigned labels. Many such methods aim to map distinct views of the same scene or object to nearby points in the representation space, while employing some constraint to prevent representational collapse. Here we recast the problem in terms of efficient coding by adopting manifold capacity, a measure that quantifies the quality of a representation based on the number of linearly separable object manifolds it can support, as the efficiency metric to optimize. Specifically, we adapt the manifold capacity for use as an objective function in a contrastive learning framework, yielding a Maximum Manifold Capacity Representation (MMCR). We apply this method to unlabeled images, each augmented by a set of basic transformations, and find that it learns meaningful features using the standard linear evaluation protocol. Specifically, we find that MMCRs support performance on object recognition comparable to or surpassing that of recently developed SSL frameworks, while providing more robustness to adversarial attacks. Empirical analyses reveal differences between MMCRs and representations learned by other SSL frameworks, and suggest a mechanism by which manifold compression gives rise to class separability.
Understanding how the statistical and geometric properties of neural activations relate to network performance is a key problem in theoretical neuroscience and deep learning. In this letter, we calculate how correlations between object representations affect the capacity, a measure of linear separability. We show that for spherical object manifolds, introducing correlations between centroids effectively pushes the spheres closer together, while introducing correlations between the spheres' axes effectively shrinks their radii, revealing a duality between neural correlations and geometry. We then show that our results can be used to accurately estimate the capacity with real neural data.
Understanding the asymptotic behavior of gradient-descent training of deep neural networks is essential for revealing inductive biases and improving network performance. We derive the infinite-time training limit of a mathematically tractable class of deep nonlinear neural networks, gated linear networks (GLNs), and generalize these results to gated networks described by general homogeneous polynomials. We study the implications of our results, focusing first on two-layer GLNs. We then apply our theoretical predictions to GLNs trained on MNIST and show how architectural constraints and the implicit bias of gradient descent affect performance. Finally, we show that our theory captures a substantial portion of the inductive bias of ReLU networks. By making the inductive bias explicit, our framework is poised to inform the development of more efficient, biologically plausible, and robust learning algorithms.
Adversarial examples are often cited by neuroscientists and machine learning researchers as an example of how computational models diverge from biological sensory systems. Recent work has proposed adding biologically-inspired components to visual neural networks as a way to improve their adversarial robustness. One surprisingly effective component for reducing adversarial vulnerability is response stochasticity, like that exhibited by biological neurons. Here, using recently developed geometrical techniques from computational neuroscience, we investigate how adversarial perturbations influence the internal representations of standard, adversarially trained, and biologically-inspired stochastic networks. We find distinct geometric signatures for each type of network, revealing different mechanisms for achieving robust representations. Next, we generalize these results to the auditory domain, showing that neural stochasticity also makes auditory models more robust to adversarial perturbations. Geometric analysis of the stochastic networks reveals overlap between representations of clean and adversarially perturbed stimuli, and quantitatively demonstrates that competing geometric effects of stochasticity mediate a tradeoff between adversarial and clean performance. Our results shed light on the strategies of robust perception utilized by adversarially trained and stochastic networks, and help explain how stochasticity may be beneficial to machine and biological computation.
Adversarial defenses train deep neural networks to be invariant to the input perturbations from adversarial attacks. Almost all defense strategies achieve this invariance through adversarial training i.e. training on inputs with adversarial perturbations. Although adversarial training is successful at mitigating adversarial attacks, the behavioral differences between adversarially-trained (AT) models and standard models are still poorly understood. Motivated by a recent study on learning robustness without input perturbations by distilling an AT model, we explore what is learned during adversarial training by analyzing the distribution of logits in AT models. We identify three logit characteristics essential to learning adversarial robustness. First, we provide a theoretical justification for the finding that adversarial training shrinks two important characteristics of the logit distribution: the max logit values and the "logit gaps" (difference between the logit max and next largest values) are on average lower for AT models. Second, we show that AT and standard models differ significantly on which samples are high or low confidence, then illustrate clear qualitative differences by visualizing samples with the largest confidence difference. Finally, we find learning information about incorrect classes to be essential to learning robustness by manipulating the non-max logit information during distillation and measuring the impact on the student's robustness. Our results indicate that learning some adversarial robustness without input perturbations requires a model to learn specific sample-wise confidences and incorrect class orderings that follow complex distributions.
Backpropagation (BP) uses detailed, unit-specific feedback to train deep neural networks (DNNs) with remarkable success. That biological neural circuits appear to perform credit assignment, but cannot implement BP, implies the existence of other powerful learning algorithms. Here, we explore the extent to which a globally broadcast learning signal, coupled with local weight updates, enables training of DNNs. We present both a learning rule, called global error-vector broadcasting (GEVB), and a class of DNNs, called vectorized nonnegative networks (VNNs), in which this learning rule operates. VNNs have vector-valued units and nonnegative weights past the first layer. The GEVB learning rule generalizes three-factor Hebbian learning, updating each weight by an amount proportional to the inner product of the presynaptic activation and a globally broadcast error vector when the postsynaptic unit is active. We prove that these weight updates are matched in sign to the gradient, enabling accurate credit assignment. Moreover, at initialization, these updates are exactly proportional to the gradient in the limit of infinite network width. GEVB matches the performance of BP in VNNs, and in some cases outperforms direct feedback alignment (DFA) applied in conventional networks. Unlike DFA, GEVB successfully trains convolutional layers. Altogether, our theoretical and empirical results point to a surprisingly powerful role for a global learning signal in training DNNs.
Invariant object recognition is one of the most fundamental cognitive tasks performed by the brain. In the neural state space, different objects with stimulus variabilities are represented as different manifolds. In this geometrical perspective, object recognition becomes the problem of linearly separating different object manifolds. In feedforward visual hierarchy, it has been suggested that the object manifold representations are reformatted across the layers, to become more linearly separable. Thus, a complete theory of perception requires characterizing the ability of linear readout networks to classify object manifolds from variable neural responses. A theory of the perceptron of isolated points was pioneered by E. Gardner who formulated it as a statistical mechanics problem and analyzed it using replica theory. In this thesis, we generalize Gardner's analysis and establish a theory of linear classification of manifolds synthesizing statistical and geometric properties of high dimensional signals. [..] Next, we generalize our theory further to linear classification of general perceptual manifolds, such as point clouds. We identify that the capacity of a manifold is determined that effective radius, R_M, and effective dimension, D_M. Finally, we show extensions relevant for applications to real data, incorporating correlated manifolds, heterogenous manifold geometries, sparse labels and nonlinear classifications. Then, we demonstrate how object-based manifolds transform in standard deep networks. This thesis lays the groundwork for a computational theory of neuronal processing of objects, providing quantitative measures for linear separability of object manifolds. We hope this theory will provide new insights into the computational principles underlying processing of sensory representations in biological and artificial neural networks.