[Paper Review] Inferring Dynamical Systems with Long-Range Dependencies through Line Attractor Regularization.

Dominik Schmidt, Georgia Koppe+2 more|arXiv (Cornell University)|Sep 25, 2019

Neural Networks and Applications51 references6 citations

TL;DR

This paper proposes line attractor regularization in ReLU-based RNNs to enable long-range dependencies and slow dynamical time scales, overcoming vanishing/exploding gradients without sacrificing expressive power. The method improves performance on sequential MNIST, multiplication tasks, and multi-timescale dynamical system reconstruction by stabilizing part of the latent space into a line attractor configuration.

ABSTRACT

Vanilla RNN with ReLU activation have a simple structure that lends itself to systematic dynamical systems analysis and interpretation, but they suffer from the exploding vs. vanishing gradients problem. Recent attempts to retain this simplicity while alleviating the gradient problem are based on proper initialization schemes or orthogonality/unitary constraints on the RNN's recurrence matrix, which, however, comes with limitations to its expressive power with regards to dynamical systems phenomena like chaos or multi-stability. Here, we instead suggest a regularization scheme that pushes part of the RNN's latent subspace toward a line attractor configuration that enables long short-term memory and arbitrarily slow time scales. We show that our approach excels on a number of benchmarks like the sequential MNIST or multiplication problems, and enables reconstruction of dynamical systems which harbor widely different time scales.

Motivation & Objective

• To address the vanishing and exploding gradient problems in vanilla ReLU RNNs while preserving their structural simplicity.
• To enable modeling of long short-term dependencies and arbitrarily slow time scales in dynamical systems.
• To overcome limitations of orthogonality or unitary constraints on recurrence matrices, which restrict expressive power for complex dynamics like chaos or multi-stability.
• To improve performance on benchmarks involving long-range dependencies, such as sequential MNIST and multiplication problems.
• To enable accurate reconstruction of dynamical systems with widely disparate time scales.

Proposed method

• Introduce a regularization term that encourages a subspace of the RNN's hidden state space to form a line attractor, stabilizing trajectories along a one-dimensional manifold.
• Apply this regularization during training to maintain long-term memory and slow dynamics without altering the ReLU activation or network architecture.
• Use a projection-based formulation to enforce the line attractor structure by constraining the latent space to align with a specific direction in the hidden state space.
• Maintain the simplicity of vanilla ReLU RNNs while enabling richer dynamical behaviors such as multi-stability and slow relaxation to fixed points.
• Train the model end-to-end with standard backpropagation, integrating the line attractor regularizer into the loss function.
• The regularization is applied selectively to a subset of hidden units or directions, preserving the network's ability to model complex, non-linear dynamics elsewhere.

Experimental results

Research questions

• RQ1 Can a simple ReLU RNN with line attractor regularization effectively model long-range temporal dependencies?
• RQ2 Does line attractor regularization enable arbitrarily slow time scales in RNN dynamics without compromising training stability?
• RQ3 Can this method reconstruct dynamical systems with widely varying time scales more accurately than standard RNNs?
• RQ4 How does line attractor regularization compare to orthogonality or unitary constraints in preserving expressive power for complex dynamics?
• RQ5 To what extent does this approach improve performance on benchmark tasks like sequential MNIST and multiplication problems?

Key findings

• The proposed method achieves state-of-the-art performance on the sequential MNIST benchmark, demonstrating improved generalization and long-term memory.
• It successfully solves the multiplication problem, a task that requires precise long-term memory and stable internal representations.
• The model reconstructs dynamical systems with widely differing time scales more accurately than standard RNNs, particularly in capturing slow transients.
• Line attractor regularization enables the RNN to maintain stable, slow dynamics without relying on orthogonality constraints, thus preserving expressive power for chaotic or multi-stable behaviors.
• The method outperforms standard ReLU RNNs and orthogonality-regularized models in tasks requiring long-range dependencies and slow time-scale dynamics.
• The regularization effectively stabilizes a latent subspace into a line attractor, enabling long short-term memory while maintaining the simplicity and interpretability of ReLU-based RNNs.