Piotr Sokół

Il Memming Park, Ábel Ságodi, Piotr Aleksander Sokół

Neural dynamical systems with stable attractor structures, such as point attractors and continuous attractors, are hypothesized to underlie meaningful temporal behavior that requires working memory. However, working memory may not support useful learning signals necessary to adapt to changes in the temporal structure of the environment. We show that in addition to the continuous attractors that are widely implicated, periodic and quasi-periodic attractors can also support learning arbitrarily long temporal relationships. Unlike the continuous attractors that suffer from the fine-tuning problem, the less explored quasi-periodic attractors are uniquely qualified for learning to produce temporally structured behavior. Our theory has broad implications for the design of artificial learning systems and makes predictions about observable signatures of biological neural dynamics that can support temporal dependence learning and working memory. Based on our theory, we developed a new initialization scheme for artificial recurrent neural networks that outperforms standard methods for tasks that require learning temporal dynamics. Moreover, we propose a robust recurrent memory mechanism for integrating and maintaining head direction without a ring attractor.

Radosław Nowak, Adam Małkowski, Daniel Cieślak et al.

Graph embeddings have emerged as a powerful tool for representing complex network structures in a low-dimensional space, enabling the use of efficient methods that employ the metric structure in the embedding space as a proxy for the topological structure of the data. In this paper, we explore several aspects that affect the quality of a vertex embedding of graph-structured data. To this effect, we first present a family of flexible distance functions that faithfully capture the topological distance between different vertices. Secondly, we analyze vertex embeddings as resulting from a fitted transformation of the distance matrix rather than as a direct result of optimization. Finally, we evaluate the effectiveness of our proposed embedding constructions by performing community detection on a host of benchmark datasets. The reported results are competitive with classical algorithms that operate on the entire graph while benefitting from a substantially reduced computational complexity due to the reduced dimensionality of the representations.

Matthew Dowling, Piotr Sokół, Il Memming Park

We present the class of Hida-Matérn kernels, which is the canonical family of covariance functions over the entire space of stationary Gauss-Markov Processes. It extends upon Matérn kernels, by allowing for flexible construction of priors over processes with oscillatory components. Any stationary kernel, including the widely used squared-exponential and spectral mixture kernels, are either directly within this class or are appropriate asymptotic limits, demonstrating the generality of this class. Taking advantage of its Markovian nature we show how to represent such processes as state space models using only the kernel and its derivatives. In turn this allows us to perform Gaussian Process inference more efficiently and side step the usual computational burdens. We also show how exploiting special properties of the state space representation enables improved numerical stability in addition to further reductions of computational complexity.

Josue Nassar, Piotr Aleksander Sokol, SueYeon Chung et al.

Understanding the nature of representation in neural networks is a goal shared by neuroscience and machine learning. It is therefore exciting that both fields converge not only on shared questions but also on similar approaches. A pressing question in these areas is understanding how the structure of the representation used by neural networks affects both their generalization, and robustness to perturbations. In this work, we investigate the latter by juxtaposing experimental results regarding the covariance spectrum of neural representations in the mouse V1 (Stringer et al) with artificial neural networks. We use adversarial robustness to probe Stringer et al's theory regarding the causal role of a 1/n covariance spectrum. We empirically investigate the benefits such a neural code confers in neural networks, and illuminate its role in multi-layer architectures. Our results show that imposing the experimentally observed structure on artificial neural networks makes them more robust to adversarial attacks. Moreover, our findings complement the existing theory relating wide neural networks to kernel methods, by showing the role of intermediate representations.

Piotr A. Sokol, Il Memming Park

Recently mean field theory has been successfully used to analyze properties of wide, random neural networks. It gave rise to a prescriptive theory for initializing feed-forward neural networks with orthogonal weights, which ensures that both the forward propagated activations and the backpropagated gradients are near $\ell_2$ isometries and as a consequence training is orders of magnitude faster. Despite strong empirical performance, the mechanisms by which critical initializations confer an advantage in the optimization of deep neural networks are poorly understood. Here we show a novel connection between the maximum curvature of the optimization landscape (gradient smoothness) as measured by the Fisher information matrix (FIM) and the spectral radius of the input-output Jacobian, which partially explains why more isometric networks can train much faster. Furthermore, given that orthogonal weights are necessary to ensure that gradient norms are approximately preserved at initialization, we experimentally investigate the benefits of maintaining orthogonality throughout training, from which we conclude that manifold optimization of weights performs well regardless of the smoothness of the gradients. Moreover, motivated by experimental results we show that a low condition number of the FIM is not predictive of faster learning.