This letter deals with neural networks as dynamical systems governed by finite difference equations. It shows that the introduction of k -many skip connections into network architectures, such as residual networks and additive dense networks, defines k th order dynamical equations on the layer-wise transformations. Closed-form solutions for the state-space representations of general k th order additive dense networks, where the concatenation operation is replaced by addition, as well as k th order smooth networks, are found. The developed provision endows deep neural networks with an algebraic structure. Furthermore, it is shown that imposing k th order smoothness on network architectures with d -many nodes per layer increases the state-space dimension by a multiple of k , and so the effective embedding dimension of the data manifold by the neural network is k · d -many dimensions. It follows that network architectures of these types reduce the number of parameters needed to maintain the same embedding dimension by a factor of k 2 when compared to an equivalent first-order, residual network. Numerical simulations and experiments on CIFAR10, SVHN, and MNIST have been conducted to help understand the developed theory and efficacy of the proposed concepts.

Estimation of a generating partition is critical for symbolization of measurements from discrete-time dynamical systems, where a sequence of symbols from a (finite-cardinality) alphabet may uniquely specify the underlying time series. Such symbolization is useful for computing measures (e.g., Kolmogorov-Sinai entropy) to identify or characterize the (possibly unknown) dynamical system. It is also useful for time series classification and anomaly detection. The seminal work of Hirata, Judd, and Kilminster ( 2004 ) derives a novel objective function, akin to a clustering objective, that measures the discrepancy between a set of reconstruction values and the points from the time series. They cast estimation of a generating partition via the minimization of their objective function. Unfortunately, their proposed algorithm is nonconvergent, with no guarantee of finding even locally optimal solutions with respect to their objective. The difficulty is a heuristic nearest neighbor symbol assignment step. Alternatively, we develop a novel, locally optimal algorithm for their objective. We apply iterative nearest-neighbor symbol assignments with guaranteed discrepancy descent, by which joint, locally optimal symbolization of the entire time series is achieved. While most previous approaches frame generating partition estimation as a state-space partitioning problem, we recognize that minimizing the Hirata et al. ( 2004 ) objective function does not induce an explicit partitioning of the state space, but rather the space consisting of the entire time series (effectively, clustering in a (countably) infinite-dimensional space). Our approach also amounts to a novel type of sliding block lossy source coding. Improvement, with respect to several measures, is demonstrated over popular methods for symbolizing chaotic maps. We also apply our approach to time-series anomaly detection, considering both chaotic maps and failure application in a polycrystalline alloy material.