Unitary RNN: A Recurrent Neural Network Architecture with Simplified Parameters
Recurrent Neural Networks (RNNs) have been widely used in natural language processing, speech recognition, and image captioning due to their ability to capture sequential information. However, the vanishing and exploding gradient problems limit their performance in long sequences. Researchers have proposed several solutions to tackle these issues, including Long-Short Term Memory (LSTM) and Gated Recurrent Unit (GRU). Another alternative is the Unitary RNN, which uses a unitary hidden to hidden matrix to reduce the number of parameters.
The Dynamics of Unitary RNN
The dynamics of a Unitary RNN can be expressed as follows:
$$ h\_{t} = f\left(Wh\_{t−1} + Vx\_{t}\right) $$
The weight matrix $W$ is a unitary matrix, meaning that its conjugate transpose is equal to its inverse ($W^{†}W = I$). Unitary matrices preserve the length and angle of vectors, making them ideal for representing rotations and reflections in geometric transformations.
Unitary RNN parameterizes the weight matrix $W$ as a product of simpler unitary matrices:
$$ h\_{t} = f\left(D\_{3}R\_{2}F^{−1}D\_{2}PR\_{1}FD\_{1}h\_{t−1} + Vxt\right) $$
where:
- $D\_{1}$, $D\_{2}$, $D\_{3}$ are learned diagonal complex matrices that scale the magnitude of the complex values in the hidden state.
- $R\_{1}$, $R\_{2}$ are learned reflection matrices that can flip the sign of certain dimensions of the hidden state.
- $F$ and $F^{−1}$ are the discrete Fourier transformation and its inverse that rotate the complex values in the hidden state.
- $P$ is any constant random permutation that shuffles the hidden state dimensions.
- $f\left(h\right)$ applies a rectified linear unit with a learned bias to the modulus of each complex number in the hidden state.
Since only the diagonal and reflection matrices, $D$ and $R$, are learned, the Unitary RNN has fewer parameters than LSTMs with comparable numbers of hidden units.
Benefits of Unitary RNN
The Unitary RNN has several benefits over traditional RNNs:
- Simplified parameterization: By using a unitary hidden to hidden matrix, Unitary RNN can reduce the number of parameters and decrease the susceptibility to overfitting.
- Long-term memory: Since the unitary matrix preserves the angle and length of the input vectors, it can better capture the long-term dependencies of the sequential data.
- Efficient computation: The matrix multiplication operations needed in the Unitary RNN can be implemented efficiently using the Fast Fourier Transform (FFT) algorithm, making it faster to compute than other RNN architectures.
Applications of Unitary RNN
Researchers have found that Unitary RNN can improve the performance of various tasks, including:
- Language Modeling: Unitary RNNs achieved state-of-the-art perplexity scores on Penn Treebank and WikiText-2 datasets.
- Syntax Tree Prediction: Unitary RNNs outperformed LSTM and GRU on predicting the structure of a sentence based on the input words.
- Speech Recognition: Unitary RNNs achieved better recognition accuracy than LSTM and GRU on the TIMIT dataset.
- Image Captioning: Unitary RNNs improved the accuracy of image captioning compared to baselines in the MS COCO and Flickr30k datasets.
Limitations of Unitary RNN
Like any machine learning model, Unitary RNN has some limitations:
- Complexity: The complexity of the model depends on the number of hidden units and the length of the sequences. The diagonal and reflection matrices can also have a large number of parameters.
- Training: The training of Unitary RNN can be more challenging than other RNN architectures since it involves optimizing non-convex functions over complex numbers.
- Interpretability: The complex numbers used in the hidden state can be difficult to interpret, making it hard to understand the internal representation of the model.
Unitary RNN is a recurrent neural network architecture that uses a unitary hidden to hidden matrix to simplify the weight parameters. It can better capture the long-term dependencies in sequential data and has shown promising results in various tasks like language modeling, syntax tree prediction, speech recognition, and image captioning. However, the complexity of the model and the difficulty in training and interpreting it are some of its limitations.
Nevertheless, Unitary RNN is a promising approach to improve the performance of RNNs and advance the field of machine learning.
