Paper: Analyzing Deep Transformer Models for Time Series Forecasting via Manifold Learning (ICLR-2024)

Reviewer comment (Experiments):
**Lack of insights to timeseries.** The presented work, although bridges the gap of using manifold learning analysis to study TSF models, lacks insights to timeseries datasets. Specifically, timeseries datasets have their unique properties and challenges, e.g. sampling rate of the timeseries; tokenization layer (specifically, if the token is the aggregation of timeseries across different channels, as the channel-independence property studied in [1, 2, 3]; what is the proper length of the token as mentioned in [1]). The studied group of model architecture, which is transformer, is also not representative enough for timeseries applications, as many linear models [4] and previously, CNN-based models [5] have good performance on the studied task. Overall, this has limited the contribution of the proposed work.

Author response:
Our work focuses on studying the geometric properties of data manifolds arising in deep TSF models. As such, we were primarily interested in insights related to the architectures. This preference can be motivated by: i) the TwoNN and MAPC tools we employ reveal more on the architecture than on the particular data; and ii) we wanted to position our work in the global context of manifold learning approaches for deep neural networks. We would like to emphasize that our work does include a few insights related to the nature of datasets, as described by the end of page 6 where we compare between electricity and traffic vs. ETT and weather. Additionally, we also discuss the expressiveness of Autoformer and FEDformer with respect to the number of channels in time series data in page 7. However, in general, we opted for a more focused analysis and consistent observations on the geometric nature of data manifolds, and our exposition and study was designed accordingly. We believe our work is complementary to existing studies with insights on time series data [1, 2, 3], and we are happy to include these works in our discussion. In addition, we will add a discussion on the limitations of our work in providing insights on data in the final version.

Reviewer comment (Experiments):
**Lack of studied architectures.** The selected architectures Autoformer and Fedformer are out-of-dated and contain specific components. E.g. Fedformer uses frequency-analysis to decompose the frequency information (or seasonal trends) in timeseries. The Informer mentioned in appendix helps with this lack of architectures, yet I am not sure about the `Transformer’ mentioned in appendix. What are the specs of this transformer architecture? One important aspect of timeseries study is the spatiotemporal relationship in sequences (that are fed into the transformer); as well as the input dimension of the tokenization layer. How are those selected and how the analyzed properties differ across such choices?

Author response:
Please see our response above. In addition: There are several reasons for focusing on Autoformer and FEDformer architectures in our study. First, even though Autoformer and FEDformer were released in 2021 and 2022, respectively, they are among the best variants of the Transformer model for time series forecasting. Both models contain specific components and yet they share a similar structure which allows for a comparison.  Second, similar studies to ours on deep classification neural networks considered strong baselines such as VGG and ResNet. Following a similar reasoning, we opted for established TSF models that are still considered state-of-the-art and are commonly used in applications. Third, several derivative works are closely related to Autoformer and FEDformer due to their accurate TSF. Thus, while we can not guarantee that other transformer-based TSF models will admit similar observations as reported for Autoformer and FEDformer, it is reasonable to assume that derivative work will generalize. Finally, our results and insights extend beyond Autoformer and FEDformer and also range over Informer and Transformer, covering four architectures in total.

Reviewer comment (Novelty):
In the related work section, analysis of transformers: “While in general they question the effectivity of transformer for forecasting, new transformer-based approaches continue to appear (Zeng et al., 2023), consistently improving the state-of-the-art results on common forecasting benchmarks.” Zeng et al., 2023 claims that a linear-based modeling approach outperforms transformer-based modeling approaches, and does not present a transformer architecture by itself. It is unclear to me if the citation here is appropriate.

Author response:
We modified this sentence in the revised version as follows: "While in general Zeng et al. (2023) question the effectivity of transformer for forecasting, new transformer-based approaches continue to appear (Nie et al., 2023), consistently improving the state-of-the-art results on common forecasting benchmarks.”

Reviewer comment (Writing):
“Namely, we discard data from the red and blue trajectories” Please refer to Figure 1 in text for this sentence for clarity.

Author response:
Following the reviewer suggestion, we added a reference to Figure 1 in the revised text.

Reviewer comment (Experiments):
My first concern is why Autoformer and FEDformer were chosen as the models to be analyzed. The title of the paper is "Analyzing Deep Transformer Models for Time Series Forecasting via Manifold Learning", but both Autoformer and FEDformer go far beyond the standard transformer architecture. They are not even based on the standard auto-regressive decoder. Given the complexity of these models, it is hard to tell whether the observations are specific to these models due to their design choices or generalizable to other deep transformer models. I am not convinced that the observations are generalizable because of the high complexity of the chosen models.

Author response:
There are several reasons for focusing on Autoformer and FEDformer architectures in our study. First, while there are a few MLP models for TSF, the majority of successful TSF approaches are based on the transformer. Thus, investigating transformer architectures provides several models to study, many of which share important design choices. Second, similar studies to ours on deep classification neural networks considered strong baselines such as VGG and ResNet. Following a similar reasoning, we opted for established TSF models that are still considered state-of-the-art and are commonly used in applications. Third, while we generally agree that Autoformer and FEDformer are relatively complex models, we do not believe they are more complex than VGG and ResNet which were studied using a similar methodology. Finally, please see also the response below for an additional reason for mainly considering Autoformer and FEDformer.

Reviewer comment (Evaluation):
Given the high dimensionality of the latent representations, I am not sure whether the estimated intrinsic dimension and curvature information are reliable. The authors should provide some justification for the reliability of the estimates. For example, why is the estimated intrinsic dimension of the first layer close to 1 for Autoformer in Figure 3? This is surprising.

Author response:
Thank you for raising this point. We would like to note that both TwoNN and CAML were proven reliable for high dimensions when introduced in their respective original papers [1, 2], as well as in analysis works on NN [3, 4]. Specifically, TwoNN was shown to be reliable if the ID is smaller than 20 [1, 3], even for high-dimensional inputs. Similarly, CAML was shown to be robust for high-dimensional data on several manifolds with known curvature [2, 4]. In addition, we tested TwoNN and CAML for reliability on a synthetic dataset with known intrinsic dimension and curvature. We sampled $1000$ points from a sphere with radius $4$ and then embedded them into a $10000$ dimensional space. In this test, the extrinsic dimension of latent representations does not exceed $2048$. The estimated id was $1.96$, where the ground truth intrinsic dimension is $2$. Further, the curvature was $0.118$ with a ground truth Gaussian curvature of $0.111$. We added this example to the revised appendix.

    [1] "Estimating the intrinsic dimension of datasets by a minimal neighborhood information." by Elena Facco, et al. 

    [2] "Curvature-aware manifold learning." by Yangyang Li. 

    [3] "Intrinsic dimension of data representations in deep neural networks." by Alessio Ansuini, et al. 

    [4] "Data Representations' Study of Latent Image Manifolds." by Ilya Kaufman, and Omri Azencot.

Reviewer comment (Novelty):
The methodology part of this work is almost identical to Kaufman & Azencot, 2023. The content in Appendix D.2 & D.3 is almost a copy of Section 3 of Kaufman & Azencot, 2023. I am not sure whether this is OK.

Author response:
Thank you for pointing this out. We would like to note that our study employs TwoNN and MAPC, similarly to Ansuini et al. and Kaufman and Azencot. Thus, our methodology section in the appendix shares a lot in common with these works, and is provided mainly for completeness. We are happy to revise this section to highlight the important aspects of our work which are different from existing approaches in the final version.

Reviewer comment (Evaluation):
Given the claimed intrinsic dimension is so low, is it possible to visualize the latent representations in a 2D or 3D space? This may help us better understand the geometric properties of the latent representations.

Author response:
We will add a visualization of the latent representations in the next few days.

Reviewer comment (Novelty):
Theoretical contribution is limited in this paper. ID is computed by other paper (TwoNN method), and MAPC (we employ the curvature aware manifold learning (CAML) technique). Thus, both important tools of this paper is borrowed from other paper.

Author response:
Our work characterizes the geometric properties of latent data manifolds arising in time series forecasting transformer models. While our study is primarily empirical, our results and analysis have theoretical contributions. First, we suggest a qualitative characterization of latent manifolds in TSF where geometric profiles follow an encoding phase with decreasing or fixed ID and MAPC and a decoding phase with increasing geometric measures. Second, we position our analysis in the wider context of literature that aims to characterize the geometric features of deep neural networks. Third, we propose a new tool for comparing TSF models without accessing the test set. To the best of our knowledge, we are first to suggest such a tool for TSF tasks. Finally, we show that untrained TSF models exhibit a different behavior, and thus, the ID and MAPC profiles found after training are a unique characteristic of the learning process.

Reviewer comment (Theory):
What is the main theoretical contribution of this paper, if exists.

Author response:
Please see our response above.

Reviewer comment (Writing):
What is the main take-home message of the findings from this paper and why it is important.

Author response:
We show several important findings: deep transformer TSF models share a similar geometric behavior across layers, and that geometric features are correlated with model performance. Further, untrained models present different structures, which rapidly converge during training. Our geometric analysis may be used in designing new and improved deep forecasting neural nets. For instance, one may design models whose ID or MAPC follow a characteristic profile similar to what we found. There could be various ways to achieve this by e.g., parametrizing the learned representations to a certain distribution, or by e.g., penalizing unfavorable representation through the objective function. We leave further consideration and exploration of this direction to future work.