QUATERNION RECURRENT NEURAL NETWORKS
Recurrent neural networks (RNNs) are powerful architectures to model sequential data, due to their capability to learn short and long-term dependencies between the basic elements of a sequence. Nonetheless, popular tasks such as speech or images recognition, involve multi-dimensional input features that are characterized by strong internal dependencies between the dimensions of the input vector. We propose a novel quaternion recurrent neural network (QRNN), alongside with a quaternion long-short term memory neural network (QLSTM), that take into account both the external relations and these internal structural dependencies with the quaternion algebra. Similarly to capsules, quaternions allow the QRNN to code internal dependencies by composing and processing multidimensional features as single entities, while the recurrent operation reveals correlations between the elements composing the sequence. We show that both QRNN and QLSTM achieve better performances than RNN and LSTM in a realistic application of automatic speech recognition. Finally, we show that QRNN and QLSTM reduce by a maximum factor of 3.3x the number of free parameters needed, compared to real-valued RNNs and LSTMs to reach better results, leading to a more compact representation of the relevant information.
After the discussion with authors, I am happy to recommend acceptance.
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1.	In “Consequently, for each input vector of size N, output vector of size M, dimensions are split into four parts: the first one equals to r, the second is xi, the third one equals to yj, and the last one to zk to compose a quaternion Q = r1 + xi + yj + zk”, are you splitting dimension M or M\times N? And if you split M \times N (I believe that’s what you are doing), in which order you are splitting (row major right?) Please explain.
2.	I did not understand why authors didn’t go in the negative direction of the gradient in Eq. (10-11)?
3.	In section 3.4, authors mentioned “Moreover, an hyper-complex parameter cannot be simply initialized randomly and component-wise, due to the interactions between components.” which I strongly agree. But in Eq. (7) and (9) why the update rules and activation function are applied component wise?
4.	I really like the elegance in the parameter initialization. Couple of minor things here: (1) It’s better to mention in Eq. (16) why E(|W|) is 0 because of symmetry. (2) Reference should be 6.1 instead of 5.1.
5.	Another reasonable baseline will be using a complex network like (https://openreview.net/forum?id=H1T2hmZAb) and use the first two terms in Eq. (19) for representation. This will also possibly justify the usefulness of using higher order partials. 
6.	The authors mentioned multiple times about the achieved state-of-the-art results without giving any citation. As a reader not well versed in the acoustic domain, it will be nice to see some references to cross-validate the claim made.
General Comments:
1.	I understand the necessity of defining RNN/ LSTM model in the space of quaternions. But unit quaternions can be identified with other spaces where convolution is defined recently, e.g., with S^3 (https://arxiv.org/abs/1809.06211). I can see that this paper is contemporary, but at least can authors comment on the applicability of this general method in their case? Given that in NIPS’18 the following paper talked about RNN model on non-Euclidean spaces (https://arxiv.org/pdf/1805.11204.pdf), one can extend these ideas to develop an RNN model in the space of quaternions. Authors should look into it rigorously as future directions? But at least please comments on the applicability.
2.	The experimental results section is somewhat weak, the overall claim of using fewer parameters and achieving comparable results is only validated on TIMIT data. More experimentation is necessary. 
3.	In terms of technical novelty, though quaternion algebra is well-known, I like the parameter initialization algorithm. I can see the merit of this in ML/ vision community.  
Pros: 
1. Nice well grounded methodological development on well-known algebra. (simple but elegant, so that's good).
2. Nicely written and all the maths check out (that's good).
3. Experimental result on TIMIT dataset shows usefulness in terms of using fewer parameters (but still can achieve SOA results).
Cons:
1. See my comments above. I expect the authors to rebut/ address the aforementioned comments. Overall though simple but nice (and necessary) development of RNN/ LSTM framework in the space of quaternions. 
2. Lacks extensive experimental validation.
My reason for my rating is mainly because of (1) lack of experimental validation. (2) being aware of the recent development of general RNN model on non-Euclidean spaces, I want some comments in this direction (see detailed comment and reference above).