From: ~Xiao-Yang_Liu1
comment: The reviewer summarize our work quite accurate! Thanks!

1. The major novelty is using nonlinear transforms to capture the complex correlations in the tensor time series, which fits well for MDS.

3.From a computational aspect, our approach allows separability in the transform domain (thanks to the Convolution Theorem), leading to multiple independent LDS subproblems, and thus enables parallel implementation.

This transform approach does capture the sparsity of tensor time series and suits for spatiotemporal data. This may be the reason why those three approaches behave similarly.


The running time in Fig. 5 (the transformation running time is also counted in) may look a bit too optimistic at first. However, please notice the following two facts: (i) DFT and DWT are implemented really fast in MATLAB already; (ii) the transforms happen at the beginning and the inverse transforms at the end (in other words, they are one-time processes), while the major computation intensive task is parameter estimation. Therefore,  the much more accurate L-MLDS requires not too much computations over the much simpler linear model (LDS).


title: This transform approach does capture the sparsity of tensor time series and suits for spatiotemporal data.

From: ~Xiao-Yang_Liu1
comment: The reviewer summarize our work quite accurate! Thanks!

 The major novelty is using nonlinear transforms to capture the complex correlations in the tensor time series, which fits well for MDS. This transform approach does captures the sparsity of tensor time series and suits for spatiotemporal data.

We would like to express our gratitude to the reviewers for their valuable comments.

Comment (1) The accuracy results of L-MDS are impressive, compared with the original MDS and LDS. But it would be helpful to show the prediction visualizations and provide more insights.

Response: We agree that the visible prediction results and more insights will be helpful to understand and will add them in our revision.

Comment (2) The runtime complexity does not seem to take into account the transformation part. Operations like DFT and DWT usually are quite time-consuming, please comment on the applicability of the method to large-scale time series.

Response: Indeed we count the transformation running time in Fig. 5. It may look a bit too optimistic at first. However, please notice the following two facts: (i) DFT and DWT are implemented really fast in MATLAB already; (2) the transforms happen at the beginning and the inverse transforms at the end (in other words, they are one-time processes), while the major computation intensive task is parameter estimation.

Since our scheme requires less parameters (also much less running time) than the traditional MLDS scheme, and is comparable to the easiest LDS (essentially just multiple LDS subproblems), we believe our scheme is very scalable. Actually we did test relative large datasets (e.g., thousands of epochs)

Further, we provide supporting information by characterizing the computational complexity (by counting the multiplications). Assume the size of each epoch in a time series is $MN$. Then the complexity of the FFT for a epoch is $O(MNlog N)$ and that for DWT (one level of the transform) is $O(MN)$. To estimate the parameters of the L-MLDS, we divide the system into $N$ smaller LDSs. In each LDS, we have to do many $M\times M$ matrix inversions for the covariances in the training iterations and the complexity of each matrix inversion is $O(M^3)$. From this, we can see that the computational complexities of transformations are much lower than that of one iteration in training, and thus this part computational cost can be completely ignored.
title: The transform-based approach captures the essence.

From: NIPS.cc/2018/Workshop/Spatiotemporal/Paper5/AnonReviewer1
evaluation: The paper proposes an L-transformed multilinear dynamical system (MDS) for time series modeling. The L-transformation allows separability of the MDS into many linear dynamical systems, leading to more accurate and faster predictions. 

+ The L-transformation which transforms the input signals into the frequency domain is novel for MDS. 
+ The L-transformed MDS leverages the sparsity in multivariate time series and is suitable for spatiotemporal data.

- The accuracy results of L-MDS are impressive, compared with the original MDS and LDS. But it would be helpful to show the prediction visualizations and provide more insights.
- The runtime complexity does not seem to take into account the transformation part.  Operations like DFT and DWT usually are quite time-consuming, pls comment on the applicability of the method to large-scale time series.

From: NIPS.cc/2018/Workshop/Spatiotemporal/Paper5/AnonReviewer1
relevance: 5: very high
impact: 4: high
novelty: 4: high

From: NIPS.cc/2018/Workshop/Spatiotemporal/Paper5/AnonReviewer2
evaluation: The paper proposes a multilinear dynamical system (MLDS) that operates in the transform domain to predict tensor time series. The idea is to use nonlinear tranforms to capture complex correlations in the time series and decomposing the problem into several linear dynamical systems (LDS). This provides efficiencies in training (parallelizable, reduction in parameters). Whereas MDLS vectorizes tensors in a time series, the novelty of L-MLDS is in the use of nonlinear transforms to vectorize the tensors. These transforms capture some of the dependencies in the data rather than discarding them (as in LDS) or having to explicitly model the covariance tensors (as in MLDS).

Results are compelling, showing much lower error rates compared to LDS and traditional MLDS models. Interestingly, the model is robust to the choice of transform, with the three transforms considered performing about the same for all datasets. The running of the much more accurate L-MLDS is about the same as the much simpler linear model (LDS).

From: NIPS.cc/2018/Workshop/Spatiotemporal/Paper5/AnonReviewer2
relevance: 4: high
impact: 4: high
novelty: 4: high