Learning long-range spatial dependencies with horizontal gated-recurrent units (1805.08315v4)

Abstract: Progress in deep learning has spawned great successes in many engineering applications. As a prime example, convolutional neural networks, a type of feedforward neural networks, are now approaching -- and sometimes even surpassing -- human accuracy on a variety of visual recognition tasks. Here, however, we show that these neural networks and their recent extensions struggle in recognition tasks where co-dependent visual features must be detected over long spatial ranges. We introduce the horizontal gated-recurrent unit (hGRU) to learn intrinsic horizontal connections -- both within and across feature columns. We demonstrate that a single hGRU layer matches or outperforms all tested feedforward hierarchical baselines including state-of-the-art architectures which have orders of magnitude more free parameters. We further discuss the biological plausibility of the hGRU in comparison to anatomical data from the visual cortex as well as human behavioral data on a classic contour detection task.

• The paper demonstrates that Euler’s method effectively discretizes Mély's continuous system through optimal parameter selection.
• It reformulates complex differential equations into a discrete model resembling convolutional RNNs with ReLU nonlinearity.
• This approach enhances computational efficiency and scalability, paving the way for advanced neural network applications.

Discretization of Dynamical Systems via Euler's Method: An Application to Mély's Model

This paper presents a detailed examination of the discretization process of continuous dynamical systems using Euler's method, focusing specifically on its application to a variant of Mély's model. In the context of numerical methods, Euler's method is a fundamental technique used for obtaining approximate solutions to ordinary differential equations (ODEs). The paper leverages Euler's method to derive a discrete approximation of Mély's continuous dynamical system, which is known for its application within the domain of recurrent neural networks (RNNs).

Euler's Method and Its Application

The manuscript begins by succinctly summarizing Euler's method. Given an ODE of the form $\dot{x} = f(x, t)$, Euler's method approximates the solution by iteratively updating $x(t)$ using the equation $x(t+h) \approx x(t) + h f(x(t), t)$. The paper anchors this classical technique in the task at hand, transforming a continuous-time model into a discrete framework.

Reformulation of Mély's System

Mély's dynamical system is initially presented in its continuous form, expressed through differential equations incorporating parameters $\eta$, $\epsilon$, $\xi$, $\alpha$, $\mu$, $\sigma$, and $\tau$. The paper strategically selects parameter equivalences, $\eta = \tau$ and $\sigma = \epsilon$, to simplify the model for computational efficiency, aligning it more closely with the architecture of hierarchical Gated Recurrent Units (hGRUs).

Through a systematic application of Euler's method, the continuum model is discretized with a focus on optimizing parameter selection, such as choosing $h = \frac{\eta}{\epsilon^2}$. This choice ensures the cancellation of specific terms in the discretized equations, highlighting the precision with which the method can approximate the dynamics of the original model.

Implications and Future Developments

The discrete-time representation derived from Mély's system bears notable structural resemblances to a convolutional RNN equipped with a ReLU nonlinearity. Such a connection implies potential applications in areas where discrete convolutions and non-linear transformations are prevalent, like computer vision and signal processing. Moreover, the paper suggests computational advantages in terms of simplicity and efficiency, afforded by the transition from a continuous to a discrete framework.

Theoretically, this work provides insights into the interplay between continuous dynamical systems and discrete-time neural architectures. The precise discretization not only preserves the intrinsic properties of the original system but also offers a scalable model adaptable to various computational tasks.

In terms of future exploration, the paper prompts considerations on optimizing other dynamical systems using similar parameter simplifications and discretization techniques. As the field of AI and neural computation progresses, such methodologies might continually enhance the computational tractability of complex models while maintaining their theoretical robustness.

In conclusion, this research underscores the utility of Euler's method in discreetly approximating Mély's dynamical system, demonstrating its potential broader applicability within computational neuroscience and AI-focused domains.