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Recurrent neural networks (RNNs) have been successfully used on a wide range of sequential data problems. A well known difficulty in using RNNs is the vanishing or exploding gradient problem. Recently, there have been several different RNN architectures that try to mitigate this issue by maintaining an orthogonal or unitary recurrent weight matrix. One such architecture is the scaled Cayley orthogonal recurrent neural network (scoRNN) which parameterizes the orthogonal recurrent weight matrix through a scaled Cayley transform. This parametrization contains a diagonal scaling matrix consisting of positive or negative one entries that can not be optimized by gradient descent. Thus the scaling matrix is fixed before training and a hyperparameter is introduced to tune the matrix for each particular task. In this paper, we develop a unitary RNN architecture based on a complex scaled Cayley transform. Unlike the real orthogonal case, the transformation uses a diagonal scaling matrix consisting of entries on the complex unit circle which can be optimized using gradient descent and no longer requires the tuning of a hyperparameter. We also provide an analysis of a potential issue of the modReLU activiation function which is used in our work and several other unitary RNNs. In the experiments conducted, the scaled Cayley unitary recurrent neural network (scuRNN) achieves comparable or better results than scoRNN and other unitary RNNs without fixing the scaling matrix.
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This content will become publicly available on November 19, 2025
Orthogonal Gated Recurrent Unit With Neumann-Cayley Transformation
In recent years, using orthogonal matrices has been shown to be a promising approach to improving recurrent neural networks (RNNs) with training, stability, and convergence, particularly to control gradients. While gated recurrent unit (GRU) and long short-term memory (LSTM) architectures address the vanishing gradient problem by using a variety of gates and memory cells, they are still prone to the exploding gradient problem. In this work, we analyze the gradients in GRU and propose the use of orthogonal matrices to prevent exploding gradient problems and enhance long-term memory. We study where to use orthogonal matrices and propose a Neumann series–based scaled Cayley transformation for training orthogonal matrices in GRU, which we call Neumann-Cayley orthogonal GRU (NC-GRU). We present detailed experiments of our model on several synthetic and real-world tasks, which show that NC-GRU significantly outperforms GRU and several other RNNs.
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- PAR ID:
- 10616223
- Publisher / Repository:
- Massachusetts Institute of Technology
- Date Published:
- Journal Name:
- Neural Computation
- Volume:
- 36
- Issue:
- 12
- ISSN:
- 0899-7667
- Page Range / eLocation ID:
- 2651 to 2676
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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