Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper introduces FiLM which stands for Frequency improved Legendre Memory Model for long-term time series forecasting. The authors leverage Legendre polynomials to obtain a fixed-sized representation of the cumulative history of the time series and combine it with Fourier analysis and low-rank approximation. The authors show that the method is competitive on long-term time series forecasting tasks and analyze the effect of several model choices and parameter sensitivities. 
strengths_and_weaknesses: The authors combine several known components (Legendre polynomials, Fourier analysis, low-rank approximation, mixture of experts) into a new time series forecasting method. While the individual components are not novel, the  non-trivial combination presented of the components presented in the paper in this paper is novel. Specifically, studying Legendre polynomials for time series forecasting could be an alternative to learning long-range dependencies in time series data. The authors provide proof for function approximation and error accumulation bounds. 

The proposed model is evaluated on a set of six real-world datasets in a long-term forecasting setting and the authors carefully analyze the impact of their components in ablation experiments. However, the plethora of results presented make it hard in some cases hard on what actually has been presented and why. More detailed explanation on what exactly is presented in each experiment would improve the paper. I will detail this in the Questions section. I would increase my score if these questions are addressed during the rebuttal. 
questions: 1. The results in Table 2, Table 3, and Table 4 (LPU layer effect, low-rank approximation and mode selection policy) and on different datasets. Why is this the case? How have the datasets been selected for each experiment? I suggest that the authors provide the full table (with all datasets) of each experiment in the Appendix and clarify why these specific datasets have been selected in each experiment. 

2. In Table 2 (LPU boosting results) the authors show the impact of LPUs (compared to linear layers). What does "comparable-sized linear layer" mean exactly here? Probably the biggest advantage of the LPU is to provide a meaningful representation of long time series history. However, using lagged values is a standard trick to capture long-range dependencies with standard NN architectures (like MLP layers or LSTM). I would suggest the authors include lags into this experiment to evaluate whether the LPU improves the performance over a standard architectural choice (for example, LSTM with lagged values) in forecasting. 

3. In Table 2, I'm unable to link the results in the "FilM" column in either Table 1 (multivariate results) or Table 10 (univariate results). Shouldn't be the results in the FiLM correspond to either of those (at least for electricity)? The error presented in this column is also much larger than the results in Table 1/10 and the error bars in Table 9. For Electricity these results are used also in Table 5 and 6 but the results in Table 1 are different. I would kindly ask the authors to clarify this. It seems to be as expected in Tables 3 and 4. 

4. Table 6: What is the relative improvement here? Relative over what? I again fail to link the results for FilM to either Table 1 or Table 10. I would kindly ask the authors to clarify the table. 
limitations: The limitations of the this model or the limitations of the evaluation are not discussed in the main text. I would kindly ask the authors to add a short limitations section.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper introduces two techniques to improve modeling of long time series - the Legendre Projection Unit (LPU), which compresses a time series with Legendre polynomials as basis; and the Frequency Enhanced Layer (FEL) which performs a low-rank approximation and selects a subset of Fourier transformation frequency modes. These layers are not domain or task specific and can be used in many time series modeling tasks. The authors provide both theoretical and empirical support for the effectiveness of the model. 
strengths_and_weaknesses: Strengths:
- Clear presentation. Figure 4 and 5 (and the code in appendix) are very informative in explaining how LPU and FEL are implemented. The extended list of ablation studies answered all of my questions about performance.  
- Strong performance compared to popular deep-learning methods for long-term time series forecasting.
- Theoretically sound methods for reducing noise and capturing structure in different timescales. 
- Merit in model design. One can change the basis function in LPU to other classes (Fourier / wavelets) according to data. Also love how they are modular - LPU and FEL are clear and easy to incorporate into many existing time series models to improve the problem of long horizon prediction performance deterioration. I can see wide use of methods introduced in this paper.  

Weakness: 
- This is not really a weakness. Both using orthogonal functions as basis to store features for time series (LPU) and using choosing Fourier frequencies / dimensional reduction to remove noise are well-explored idea in time series modeling. One may argue that the paper lacks novelty. Personally, I think it is nice to see them combined to achieve good performances and the ablation studies show the necessity of both to achieve the improvement. 


questions: Not being from the time series community, I was a little confused by the notations at first.  It might help clarity to, for example, introduce what subscripts and superscripts mean for $f(x)_{[t-\theta, t]}$,  $ g^{(t)}(x)$, and 

$ \mathbb{I}_{[t-\theta, t]}$ is before line 117.


limitations: N/A
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The paper proposes FiLM, a novel model based on Legendre projections, for long-horizon forecasting. The paper proposes two novel components: Legendre projection unit (LPU) and Fourier Enhanced Layer (FEL), which are combined in the FiLM architecture. Both LPU and FEL components can be used on multiple architectures. The authors tested the proposed approach on several benchmark datasets and compare it against recent Transformer-based models, and provide comprehensive ablation studies of the proposed components.
strengths_and_weaknesses: Strengths:
- The paper proposes two original components: LPU and FEL, which can be incorporated by many different architectures.
- FiLM achieves superior performance than baselines, including the recent FED-former model, and the proposed components improve the performance of other architectures.
- Authors provide some theoretical results which support the design of the components.
- FiLM is a simpler and faster model than Transformer based models.
- The paper is well written and clear.
- Long-horizon forecasting is a very relevant topic and an active area of research.

Weaknesses:
- Recent studies have shown that many transformer-based models do not achieve SoTA performance in this setting [1, 2]. In many cases, improvements over previous models are caused by flawed comparisons (such as not tunning baselines properly, I confirmed with the authors of the Autoformer that, for example, they did not tune baselines such as the N-BEATS) or by omitting stronger baselines. Authors should include non-Transformer methods as well, including simple models. For example, a simple seasonal naive outperforms the Autoformer and FED-Former in Exchange and ILI (based on own experiments).
- While the main experiments are performed on multivariate datasets, the paper does not discuss how (or if) FiLM models relations between the different time series. Based on the architecture description it seems forecasts are produced independently for each channel. Authors should make this clear.
- The results on training speed presented in appendix J are not convincing. First, the total training time is more relevant than per iteration time but is not included nor discussed in the paper. Second, authors should again include simpler (non-Transformer) baselines, to better assess the trade-off between computation complexity and performance.
- Authors do not discuss the complexity or training times against the number of time series (D). As seen in Figure 5, the components operate separately for each channel, which suggests poor scaling on D. Authors should provide comparisons in datasets with more time series, such as Traffic.
- Some missing references/baselines. The N-BEATS [3] is a popular model closely related to the proposed technique, and N-HiTS [1] is an extension of the N-BEATS tailored for long-horizon forecasting.

[1] N-HiTS: Neural Hierarchical Interpolation for Time Series Forecasting, https://arxiv.org/pdf/2201.12886.pdf
[2] FreDo: Frequency Domain-based Long-Term Time Series Forecasting, https://arxiv.org/pdf/2205.12301.pdf
[3] N-BEATS: Neural basis expansion analysis for interpretable time series forecasting, https://arxiv.org/pdf/1905.10437.pdf
questions: - Why is the performance of FiLM different in ablation tables (table 2, 5) than from the main result table?
- Can you provide an explanation on how the model learns relations between time series or is the model univariate?
- How did you choose the hyperparameters of the model for the main table, in particular, M and N? The sensitivity analysis in appendix F shows some approximate optimal rules for N. However, these results are performed on the test set. Ideally, hyperparameters should be chosen based on the validation set (for example on a small grid over different values of M, N, etc).

limitations: Limitations are discussed, and I do not identify potential negative social impacts.
review_title: null

Meta-review: metareview: Paper provides a time series modeling technique combining the use of Legendre polynomials for projections and Frequency based low rank approximation / selection.
The reviewers found the paper to be interesting, and the results convincing and possibly usable in other sequence modeling tasks.
Some questions were raised by nNQa  about the baselines / comparisons, that I felt were addressed appropriately by the authors. Other questions that were raised about the details of the experiments, including the datasets, the ablations performed, and comparisons to alternatives (such as lagged inputs in LSTMs, and comparisons to n-Hits) seem to have been well addressed by the authors. 

confidence: Certain
Decision: Accept