Isometric Representations in Neural Networks Improve Robustness

Artificial and biological agents cannon learn given completely random and unstructured data. The structure of data is encoded in the metric relationships between data points. In the context of neural networks, neuronal activity within a layer forms a representation reflecting the transformation that the layer implements on its inputs. In order to utilize the structure in the data in a truthful manner, such representations should reflect the input distances and thus be continuous and isometric. Supporting this statement, recent findings in neuroscience propose that generalization and robustness are tied to neural representations being continuously differentiable. In machine learning, most algorithms lack robustness and are generally thought to rely on aspects of the data that differ from those that humans use, as is commonly seen in adversarial attacks. During cross-entropy classification, the metric and structural properties of network representations are usually broken both between and within classes. This side effect from training can lead to instabilities under perturbations near locations where such structure is not preserved. One of the standard solutions to obtain robustness is to add ad hoc regularization terms, but to our knowledge, forcing representations to preserve the metric structure of the input data as a stabilising mechanism has not yet been studied. In this work, we train neural networks to perform classification while simultaneously maintaining within-class metric structure, leading to isometric within-class representations. Such network representations turn out to be beneficial for accurate and robust inference. By stacking layers with this property we create a network architecture that facilitates hierarchical manipulation of internal neural representations. Finally, we verify that isometric regularization improves the robustness to adversarial attacks on MNIST.Comment: 14 pages, 4 figure

Population coding in sparsely connected networks of noisy neurons

This study examines the relationship between population coding and spatial connection statistics in networks of noisy neurons. Encoding of sensory information in the neocortex is thought to require coordinated neural populations, because individual cortical neurons respond to a wide range of stimuli, and exhibit highly variable spiking in response to repeated stimuli. Population coding is rooted in network structure, because cortical neurons receive information only from other neurons, and because the information they encode must be decoded by other neurons, if it is to affect behavior. However, population coding theory has often ignored network structure, or assumed discrete, fully connected populations (in contrast with the sparsely connected, continuous sheet of the cortex). In this study, we modeled a sheet of cortical neurons with sparse, primarily local connections, and found that a network with this structure could encode multiple internal state variables with high signal-to-noise ratio. However, we were unable to create high-fidelity networks by instantiating connections at random according to spatial connection probabilities. In our models, high-fidelity networks required additional structure, with higher cluster factors and correlations between the inputs to nearby neurons

The Non-Random Brain: Efficiency, Economy, and Complex Dynamics

Modern anatomical tracing and imaging techniques are beginning to reveal the structural anatomy of neural circuits at small and large scales in unprecedented detail. When examined with analytic tools from graph theory and network science, neural connectivity exhibits highly non-random features, including high clustering and short path length, as well as modules and highly central hub nodes. These characteristic topological features of neural connections shape non-random dynamic interactions that occur during spontaneous activity or in response to external stimulation. Disturbances of connectivity and thus of neural dynamics are thought to underlie a number of disease states of the brain, and some evidence suggests that degraded functional performance of brain networks may be the outcome of a process of randomization affecting their nodes and edges. This article provides a survey of the non-random structure of neural connectivity, primarily at the large scale of regions and pathways in the mammalian cerebral cortex. In addition, we will discuss how non-random connections can give rise to differentiated and complex patterns of dynamics and information flow. Finally, we will explore the idea that at least some disorders of the nervous system are associated with increased randomness of neural connections

RNNs Implicitly Implement Tensor Product Representations

Recurrent neural networks (RNNs) can learn continuous vector representations of symbolic structures such as sequences and sentences; these representations often exhibit linear regularities (analogies). Such regularities motivate our hypothesis that RNNs that show such regularities implicitly compile symbolic structures into tensor product representations (TPRs; Smolensky, 1990), which additively combine tensor products of vectors representing roles (e.g., sequence positions) and vectors representing fillers (e.g., particular words). To test this hypothesis, we introduce Tensor Product Decomposition Networks (TPDNs), which use TPRs to approximate existing vector representations. We demonstrate using synthetic data that TPDNs can successfully approximate linear and tree-based RNN autoencoder representations, suggesting that these representations exhibit interpretable compositional structure; we explore the settings that lead RNNs to induce such structure-sensitive representations. By contrast, further TPDN experiments show that the representations of four models trained to encode naturally-occurring sentences can be largely approximated with a bag of words, with only marginal improvements from more sophisticated structures. We conclude that TPDNs provide a powerful method for interpreting vector representations, and that standard RNNs can induce compositional sequence representations that are remarkably well approximated by TPRs; at the same time, existing training tasks for sentence representation learning may not be sufficient for inducing robust structural representations.Comment: Accepted to ICLR 201