A Differential Geometry Perspective on Orthogonal Recurrent Models
This work addresses the exploding and vanishing gradients problem in RNNs for machine learning applications, but it is incremental as it builds on existing orthogonal RNNs.
The paper tackled the problem of learning long-term dependencies in recurrent neural networks by proposing a new recurrent model based on differential geometry, achieving comparable or better results than state-of-the-art orthogonal RNNs on benchmark tasks.
Recently, orthogonal recurrent neural networks (RNNs) have emerged as state-of-the-art models for learning long-term dependencies. This class of models mitigates the exploding and vanishing gradients problem by design. In this work, we employ tools and insights from differential geometry to offer a novel perspective on orthogonal RNNs. We show that orthogonal RNNs may be viewed as optimizing in the space of divergence-free vector fields. Specifically, based on a well-known result in differential geometry that relates vector fields and linear operators, we prove that every divergence-free vector field is related to a skew-symmetric matrix. Motivated by this observation, we study a new recurrent model, which spans the entire space of vector fields. Our method parameterizes vector fields via the directional derivatives of scalar functions. This requires the construction of latent inner product, gradient, and divergence operators. In comparison to state-of-the-art orthogonal RNNs, our approach achieves comparable or better results on a variety of benchmark tasks.