Analyzing Deep Transformer Models for Time Series Forecasting via Manifold Learning
Authors: Ilya Kaufman
Omri Azencot
Venue: Accepted by TMLR ()

Abstract: Transformer models have consistently achieved remarkable results in various domains such as natural language processing and computer vision. However, despite ongoing research efforts to better understand these models, they still lack a comprehensive understanding. This is particularly true for deep time series forecasting methods, where analysis and understanding work is relatively limited. Time series data, unlike image and text information, can be more challenging to interpret and analyze. To address this, we approach the problem from a \emph{manifold learning} perspective, assuming that the latent representations of time series forecasting models lie next to a low-dimensional manifold. In our study, we focus on analyzing the geometric features of these latent data manifolds, including intrinsic dimension and principal curvatures. Our findings reveal that deep transformer models exhibit similar geometric behavior across layers, and these geometric features are correlated with model performance. Additionally, we observe that untrained models initially have different structures, but they rapidly converge during training.
By leveraging our geometric analysis and differentiable tools, we can potentially design new and improved deep forecasting neural networks. This approach complements existing analysis studies and contributes to a better understanding of transformer models in the context of time series forecasting.

Published in Transactions on Machine Learning Research (10/2024)
Analyzing Deep Transformer Models for Time Series Fore- casting via Manifold Learning
Iya Kaufman
Syakke@@post.guu.cm
Ben- Con-Gurion University of the Negev
Omri Azencot
azencol@cs..o www.:eeesbbbgg.. agw.ac.i..................
Ben-Gurion University of the Negev
Reviewed on OpenReview: Attps: //
// operr/veewwwww..m. net/ pormmr id- zRZe930Zho
Abstract
Transformer models have consistently achieved remarkable results in various domains such as natural language processing and computer vision. However, despite ongoing research efforts to better understand these models, the field still lacks a comprehensive understand- ing. This is particularly true for deep time series forecasting methods, where analysis and understanding work is relatively limited. Time series data, unlike image and text informa- tion, can be more challenging to interpret and analyze. To address this, we approach the of
problem from manifold learning perspective, assuming that the latent represent time series forecasting models lie next low-dimensional manifold. In our study, we focus analyzing the goometric features of these latent data manifolds, including intrinsic di- mension and principal curvatures. Our findings reveal that deep transformer models exhibit similar geometric behavior across layers, and these geometric features are correlated with model performance. Additionally, we observe that untrained models initially have different structures, but they rapidly converge during training. By leveraging our geometric analy- sis and differentiable tools, we can potentially design new and improved deep forecasting neural networks. This approach complements existing analysis studies and contributes better understanding of transformer models in the context of time series forecasting. Code is released at https: www.tthub.cooeeeeeeeeemmeeeeeeee http://///gctubbbooooo/////teee/ htt/://zecccttooooee gacencootggrouu/GATTTY
1 Introduction
Over the past decade, modern deep learning has shown remarkable results on multiple challenging tasks in computer vision (Krizhevsky et al. 2012), natural language procossing (NLP) (Pennington et al. 2014). and speech recognition (Graves al. 2013 among other domains (Berman et al. 2023 Naiman et al. 2024 Goodfellow et al. 2016). Recently, the transformer (Vaswani et al. 2017) has revolutionized NT.P by allowing neural networks to capture long-range dependencies and contextual information effectively. In addition, transformer-based architectures have been extended to non-NLP fields, and they are among the state-of-the art (SOTA) models for vision (Dosovitskiy et al. 2020) as well as time series forecasting (TSF) (Wu et al. 2021 Zhou et al. 2022). Unfortunatcly, while previous works, c.g., Zeiler & Fergus (2014); Karpathy et al. (2015 ;
Tsai al. (2019) among others, altempted explain the underlying mechanisms of neural networks, deep transformer models are still considered not well understood.
The majority of approaches analyzing the inner workings of vision and NLP transformer models investigate their attention modules (Bahdanau el al. 2015) and salient inputs (Wallace el al. 2019). Unfortunately, time series forecasting methods have received significantly less attention. This may be in part due to their relatively recent appearance as strong contenders on TSF comparison to non-dcep and hybrid techniques (Oreshkin et al. 2020). Further, while vision and NLP modalities may be "natural" interpret and analyze (Naiman & Azencol 2023 time series requires analysis tools which may be challenging to develop for deep models.

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For instance, N-BEATS Oreshkin et al. 2020) designed a method that promotes the learning of trend and seasonality parts, however, their model often recovers latent variables whose relation to trend and seasonality is unclear (Challu et al. 2022). Moreover, there is already a significant body of work of SOTA TSF that warrants analysis and understanding. Toward bridging this gap, we investigate in this work the geometric properties of latent representations of transformer-bassed TSF techniques via manifold learning tools.
Manifold learning is the study of complex data encodings under the manifold hypothests where high- dimensional data is assumed to lie close to low-dimensional manifold (Coifman & Lafon 2006). This assumption is underlying the development of numerous machine learning techniques, akin to considering independent and identically distributed (i.i.d.) samples (Goodfellow et al. 2016). Recent examples include works on vision (Nguyen et al. 2019), NLP (Hashimoto et al. al.] 2016)), and time series forecasting [Papaioannou le al. ). However, to the best of our knowledge, there is no systematic work that analyzes transformer- based TSF deep neural networks from a manifold learning perspective. In what follows, we advocate the study of goometric features of Riemannian manifolds (Lee) 2006) including their intrinsic dimension (ID) and mean absolule principal curvalure (MAPC). The ID) is the minimal degrees of freedom needed for lossless encoding of the data, and MAPC measures the deviation of manifold from being flat.
Previously, geometric features of data manifolds were considered in the context of analyzing deep convo- lutional neural worksrks (CNN) (Ansuini et al.) 2019 Kaufman & Azencot 2023). Motivated by these recent works, we extend their analysis on image classification to the time series forecasting setting, focusing on SOTA TSF models (Wu et al.) 2021 Zhou et al., 2022) evaluated on several multivariate time series datasets. We aim at characterizing the dimension and curvature profiles of latent representations along layers of deep transformer models. Our study addresses the following questions: (i) how do dimensionality and curvature change across layers? are the resulting profiles similar for different architectures and datasets? (ii) is there a correlation between geometric features of the data manifold to the performance of the model? (iii) how do untrained manifolds differ from trained ones? how do manifolds evolve during training?
Our results show that transformer forecasting manifolds undergo two phases: during encoding, curvature and dimensionality either drop or stay fixed, and then, during the decoding part, both dimensionality and curvature increase significantly. Further, this behavior is shared across several architectures, datasets and forecast horizons. In addition, we find that the MAPC estimate is correlated with the test mcan squared error, allowing one to compare models without access to the test set. Moreover, this correlation is unlike the one found in deep neural networks for classification. Finally, untrained models show somewhat random dimension and curvature patterns, and moreover, geometric manifolds converge rapidly (within a few epochs) to their final geometric profiles. This finding may be related to studies on the neural tangent kernel (Li & Liang 2018: Jacot et al., 2018)) and linear models for forecasting ŞZeng et al., 2023).. We believe that our geometric insights, results and tools may be used to design new and improved deep forecasting tools.
2 Related Work
Our research lies the intersection of understanding deep transformer- based models, and manifold learning for analysis and time series. We focus our discussion on these topics.
Analysis of transformers. Large transformer models have impacted the field of NLP and have led to works such as Vig (2019) that analyze the multi-head attention patterns and found that specific attention heads can be associated with various grammatical functions, such as co-reference and noun modifiers. Several works (Clark et al., 2019 Tenney et al., 2019 [oggess et al., 2021) study the BERT model (Devlin et al., 2019) and show that lower layers handle lexical and syntactic information such as part of speech, while the upper layers handle increasingly complex information such semantic roles and co-reference. In Dosovitsskiy ct al. 2020), the authors inspect patch-based vision transformers (ViT) and find that the models globally atlend to image regions that are semantically relevant for classification. Caron et al. (2021) show that a self-super trained VIT produces explicit representations of the semantic location of objects within natural images. [Chffer et al. (2021) compute relevancy score for self-attention layers that is propagatod throughout the network, yielding visualization that highlights class-specific salient image regions. Nie al. 2023) recently studied the effectiveness of transformer in TSF in terms of their ability to extract temporal
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relations, role of self-attention, temporal order preservation, embedding strategies, and their dependency train set size. 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.
Manifold learning analysis. Motivated by the ubiquitous manifold hypothesis, several existing ap- proaches investigate geometric features of data representations across different layers. In Hauser & Ray (2017), the authors formalize a Riemannian geometry theory of deep neural networks (DNN) and show that residual neural networks are finite difference approximations of dynamical systems. Yu et al. (2018) compare Iwo neural networks by inspecting the Riemann curvature of the learned representations ions in fully connected layers. Cohen et al.) (2020) examine the dimension, radius and capacity throughout the training process, and they suggest that manifolds become lincarly separable towards the end of the layer's hicrarchy. Doimo et al. (2020) analyzed DNNs trained on ImageNet and found that the probability density of neural representations across different layers exhibits hierarchical clustering pattern that aligns with the semantic hierarchy of concepts. Stephenson et al. (2021} conclude that data memorization primarily occurs in deeper layers, due to decreasing object manifolds' radius and dimension, and that generalization can be restored by reverting the weights of the final layers to an earlier epoch. Perhaps closest to our approach are the works by Ansuiii et al.] (2019) and Kaufman & Azencot (2023), where the authors estimate the intrinsic dimension and Riemannian curvature, respectively, of popular deep convolutional neural networks. Both works showed characteristic profiles and strong correlation between the estimated geometric measure and the general- ization error. Recently, Valeriani et al. (2023) investigated the intrinsic dimension and probability density of large transformer models in the context of classification tasks on protein and genetic sequence datasets. Complementary to previous works, our study focuses on the setting of regression time series forecasting problems using multivariate real-world time series datasets.
Manifold learning in time series forecasting. Unfortunately, latent representations of deep TSF re- ceived less attention in the literature, and thus we discuss works that generally investigate TSF from a manifold learning perspective. Paapaioannou et al. (2022) embed high-dimensional time series into a lower- dimensional space using nonlinear manifold learning techniques to improve forecasting. Similarly, Han et al. (2018) proposed a novel framework, which performs nonuniform embedding, dynamical system revealing, and lime-series prediction. In [i et al. (2021). the authors exploit. manifold learning to extract the low- dimensional intrinsic patterns of electricity loads, to be used as input to recurrent modules for predicting low-dimensional manifolds. Lin et al. (2006) employ a dynamic Bayesian network to learn the underlying nonlinear manifold of time series data, whereas Shnitzer et al. (217)) harness diffusion maps to recover the states of dynamical systems. Kaufman & AAzencot (2024) approximate the data manifold to sample synthetic examples for regression problems. Finally, manifold-based distance functions for time series were proposed in Roddigues et al. (2018): : O'Reilly et al. (2017).
3 Background and Method
Time series forecasting. Given a dataset of multivariate time series sequences D := {of {=f: where F1:T-h = .I,...,,TTH c RP, the goal in time series forecasting (TSF) is to accurately forecast the series TT+1 based on the sequence Fi.T, where we omit j for brevity. The values 7 and h are typically referred to as lookback and horizon, respectively. The forecast accuracy can be mcasured in several ways of which the mean squared error (MSE) is the most common. We denote by 5T+1TT+ =[((r)) the outpui of certain - -.112 is the forecast error. In our study,
forecast model, e.g., a neural network, then EMSE := 1. TT ||,,- we consider T 96, and h = 96, 192, 336 and 720, and standard benchmark datasets including Electricity, Traffic, FTTml, ETTm2, ETTh1, ETTh2, and weather (Wu al. 2021p. Tn App. A we provide detailed description of the datasets and their properties.
Transformer-based TSF deep neural networks Statc-of-tho-art (SOTA) deep time series forccasting models appeared only recently (Orehhkin et al. 2020). enjoying rapid development of transformer baaso based architectures, e.g., Zhou et al. (2021): Wu et al. 2022 Liu et al. 22021 Zhou et al. (2022) 0g Nie et al. (2023)
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Encoder Input
Attention
Decompastion
Feed Forward
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Sessonal
Decompositon
Cross Alleniin
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Formard
Decomposiion
Output
Trend init
Decoder
We study Transformer-based architectures (Wu et al. 2021 Zhou et al. 2022 that include two encoders and one decoder, and an output linear layer. We sample geometric features the output of sequence decomposition layers, depicted as solid blue blocks.
Figure 1:
among many others. In what follows, will focus on Autoformer (Wu et al. 2021 and FEDformer (Zouu et al. 2022 as they are established architectures that are still considered SOTA (Nochumsohn & Azencot 2024]. In App. C we also mention additional TSF models and their analysis. Please see Fig. for a schematic illustration of the architecture we investigate. The network composed two encoder blocks and a single decoder block, where the encoder and decoder blocks include two and three sequence dccomposition layens, respectively. Both Autoformer and FEDformer utilize these decomposition layers extract trend and seasonality information. The network includes two inputs, the input signal X which is fed to the encoder and the scasonality information X, of X that serves as the input to the decoder. Note that the cross-attention modulc in the decoder receives two independent data streams, one is the output of the encoder while the other is a function of X,. Our goal is to analyze the geometrical features of the data propagation along the layers of the network in a serial manner (with respect to depth), therefore, we focus on the path from the encoder through the dccoder to the output (shown in red Fig. D Namely, we follow data along the rod trajectory, while ignoring the black trajectory. In particular, our analysis based sampling geometric properties of the data manifold after every decomposition module and after the final linear layer of the network. We chose the output of the decomposition blocks rather than the attention blocks since the Fourier Cross-correlation layer of the FEDformer model outputs almost identical values for all samples in the series, yielding zero curvature estimates.
Scries decomposition. While modern data analysis considers disentangled representations Naiman et al. 2023 Berman et al.] 2024), it is common in time series analysis to decompose signals to trend and seasonal parts [Anderson Kendall 1976 Cleveland et al. 1990). This decomposition facilitates learning of complex temporal patterns, allowing to better capture global properties. In the time scries forccasting domain, prior Autoformer, time series decomposition was mainly used pre-processing tool, applied on the input data, Both Autoformer and FFDformmr utilize decomposition as core component, enabling to progressively decompose the hidden time series throughout the entire forecasting process. This includes both historical (input) data and the predicted (output) intermediate results. The series decomposition block (noted as Decomposition in Fig. D receives a time series signal as an input and outputs the seasonality (right arrow) and the trend (bottom arrow) such that their summation results in the original signal.
Encoder and decoder. The encoder utilized for modeling the seasonal parts as it discards the trend output from the decomposition block. The encoder's output, which includes past seasonal information, will serve cross-information to assist the decoder in refining the prodiction results. The decoder serves two purposes, it accumulates the trend components and propagates the scasonal component. The final prediction is the of the seasonal and trend components.
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Figure 2: Intrinsic dimension and mean absolute principal curvature along the layers of Aut- oformer and FEDformer on traffic dataset for multiple forecasting horizons. Тop) intrinsic dimension. Bottom) mean absolute principal curvature. For each model, both ID and MAPC share a similar profile across different forecasting horizons.
Geometric properties of data manifolds. The fundamental assumption in our work is that data repre-
sentations computed across layers of transformer-based models lie on Riemannian manifolds (Lee) 2006). We
are interested in computing the intrinsic dimension (ID) and the mean absolute principal curvature (MAPC)
of the manifold, following recent work on deep CNNs (Ansuini et al., 2019 Kaufman & Azencol 2023
We compute the ID using the TwoNN method (Facco et al. 2017) that utilizes the Pareto distribution of the ratio between the distances to the two closest neighbors estimate the dimension. For the MAPC,
employ the curvature aware manifold learning (CAML) technique LLi 2018) that parametrizes the manifold via ils second-order Taylor expansion, allowing to estimate curvalures via the eigenvalues of local Hessian matrices, More details related to ID and MAPC are provided in App. D
Data collection. In this study, every archilecture is trained on all datasets and horizons, using 10 different seed numbers. For every combination model, dataset, horizon and seed, we extract the latent data representations across layers, and we compute the ID and MAPC. The intrinsic dimension estimated on 500k point samples from D, resulting in single scalar value d. The estimated ID used as input to the CAMT, algorithm that uses 100k samples, and it returns d(D) d) principal curvatures per point, where D is the extrinsic dimension. The MAPC of manifold is calculated by computing the mean absolute value for each point and taking the mean over all points.
4 Results
4.1 Data manifolds share similar geometric profiles
In our first empirical result, compute the intrinsic dimension (TD) and mean absolute principal curvature (MAPC) across the layers of Autoformer and FEDformer models on the traffic dataset. In Fig. 2 we plot the ID (top row) and MAPC (bottom row) for Autoformer (left column) and FEDformer (right column) on multiple forecast. horizons 96, 192, 336, 720. The r-labels refer to the layers we sample, where labels to refer two sequence dccomposition layers per encoder block (and thus four in total), labels to 6 denote the decoder decomposition layers, and label is the linear output layer, see Fig. [ for the network scheme. Our results indicate that during the encoding phase, the ID and the MAPC are relatively fixed for Autoformer and decrease for FEDformer, and during the decoder module, these values generally increase with depth. Specifically, the ID values change from min(ID) = 1.2 to max(ID) = 8.1, showing a relatively small variation across layers. comparison, the mean absolute principal curvature values present. a larger deviation as they range from min MMAMM) 0.2 to max(MAPC) = 19.2.
Remarkably, it can be observed from Fig. 2 that both Autoformer and FEDformer feature similar ID and MAPC profiles in terms of values. Further, strong similarity in trend be viewed across different
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Figure 3: ID profiles across layers of Autoformer and FEDformer electricity, traffic, weather and ETTml datasets for multiple forecasting horizons. Each panel includes ID profiles per datasel, for several horizons (left to right) and architectures (top to bottom).
forecast horizons per method. Moreover, Autoformer and FEDformer differ during the encoding phase (layers 1 to 4), but match quite well during the decoding and output phases (layers 5 to 7). Our intrinsic dimension estimations stand in contrast to existing results on classification tasks with CNN and transformer architectures that observed a "hunchback" ID profile (Ansuini et al. 2019 Valeriani et al. 2023). That is, prior work found the intrinsic dimension to incrcase significantly at the first few layers, and then, presented sharp decrease with depth. This qualitative difference in ID can be attributed to the differences ween classification models as in Ansuini et al. (019 in comparison to regression TSF networks we consider here, In particular, deep classification neural networks essentially recover the related low-dimensional manifold, to facilitate linear separation of classes (Cohen et al., 2020), and thus one may expect low ID toward the final layers of the network. On the other hand, forecast regression models aim to encode the statistical distribution imput data, to facilitate forecast of the horizon windows and it is typically of a higher dimension due spurious variations. In particular, regression TSF models are expected dimensional and simple representation while encoding. Then, more complex manifold that better reflects the properties of the data. is learned during decoding. Importantly, while intrinsic dimension profiles do not exhibit the "hunchback" shape identified in Ansuini et al. (2019), our estimated ID d is significantly smaller than the extrinsic dimension D = 512, in correspondence with existing work. Finally, our MAPC profiles attain a "step-like" appearance, similar to the results in Kaufman & Azencot (2023), where they identify a sharp increase in curvature in the final layer, and we observe such a jump in the decoder.
learn an underlying low-
To extend our analysis, we present in Fig. 3 and 4 the TD and MAPC profiles, respectively, for Autoformer (top) and FEDffrmer (bottom) for several horizons using multiple different datasets. For all Autoformer configurations, the IDs in Fig. generally increase with depth, and the IDs of FEDformer present "v"-shape for electricity and traffic and "step"-like behavior for weather and ETTml. Interestingly, ETTml (and other ETT* datasets, please see Fig. g) shows a hunchback trend, however, the drop of ID) in the final layer is due to ETT datasets consisting of a total of seven features, and thus we do not consider this behavior to be characteristic to the network. As in Fig. 2 and existing work (Ansuini et al. 2019 the intrinsic dimension d is much lower than its extrinsic counterpart
Our MAPC results Fig. indicate a shared step-like behavior in general for all models, horizons, and datasets, where the main difference is where the curvature increase occurs. For electricity and traffic, observe sharp increase at the beginning of the decoder block, whereas for weather and ETTml, the increase often appears at the final layer. Additionally, the maximal curvature values for weather and ETTml tend to be higher than those of electricity and traffic. Overall, our results suggest that weather and ETTml are associaled with manifolds whose geometric features match. This observation can be justified by the known correlation between electricity transformer temperature (ETT) and climate change (Hashmi et al.) 2013 Gao al. 2018). Similarly, electricity consumption (clectricity) and road occupancy (traffic) attain shared behavior that may be explained due the strong seasonality component. in these datasets Zeng et al. 2023

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Figure MAPC profiles across layers of Autoformer and FEDformer on electricity, traffic, weather and ETTml for multiple horizons. Each panel includes MAPC profiles per dataset, for several horizons (left right) and architectures (top bottom).
4.2 Final MAPC correlated with performance
We further investigate whether geometric properties of the learned manifold are related to inherent features of the model. For instance, previous works find a strong correlation between the ID (Ansuini et al., 2019) and MAPC (Kaufman Azencot 2023) with model performance and generalization. Specifically, the intrinsic dimension in the last hidden layer correlated with the top-5 score on image classification, i.c., lower ID is associated with lower error. Similarly, large normalized MAPC gap between the penultimate and final layers of CNNs is related to high classification accuracy. These corrclations are important they allow developers and practitioners to evaluate and compare deep neural networks based on statistics obtained directly from the train set. This is crucial in scenarios where, e.g., the test set is unavailable during model design.
We show in Fig. 5 plots the test mean squared error (eMSE) vs. the MAPC in the final layer of Autoformer and FEDformer models Irained on ETTml, ETTm2, ETTh1, ETTh2, weather, and electricity. For each dataset, we plot four colored circles corresponding to the four different horizons, where each circle scaled proportionally the horizon length. The dashed graphs are generated by plotting the eMSE with respect to the MAPC, representing the best linear fit for each dataset. Due to different scales, an inlay of the electricity dataset is added to the bottom right corner of each architecture. We find positive slope in all Autoformer and FEDformer models with an average correlation coefficient of 0.76 and 0.7, respectively (see full results in Tab. D In all cases, we observe a correlation between the test MSE and final MAPC, namely, the model performs better as curvature decreases, as shown Fig.
Autoformer
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weather electricity
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Figure 5: MAPC is correlated with model performance. Each color represents different dataset while the size of the dot is determined by the forccast horizon (longer horizon results in a larger dot). The test mean squared error proportional to the MAPC on multiple datasets.

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Table 1: Correlation coefficients of test MSE vs. final MAPC for several datasets and architectures.
ETTml
ETTm2
ETTh1
ETTh2
Weather
Electricity
Autoformer
0.46
0.81
0.97
0.67
0.93
0.71
FEDformer
0.74
0.86
0.63
0.50
0.58
0.70
As in Sec. 4.1 we identify different characteristics for TSF models with respect to classification neural networks. While popular CNNs show better performance when the MAPC gap high (Kaufman Azencot 2023), we report an opposite trend, namely, better models are associated with lower MAPC. We believe this behavior may be altributed to elassification networks requiring final high curvatures to properly classify the statistical distribution of the input information and its large variance. In addition, note the relatively flat slope profiles presented across all datasets and architectures. Essentially, these results indicate that while the manifolds become more complex in terms of curvature, transformers yield a relatively fixed MSE, regardless of the underlying MAPC. Thus, our results may hint that Autoformer and FEDformer are not expressive enough. Indeed, the datasets we consider consist of many features (on the range of hundreds for electricity and traffic). Therefore, while TSF approaches may need highly expressive networks model these datasets due their complex statistics, it might be that current approaches local minimum. We hypothesize that significantly deeper and more expressive TSF models as in Nie et al. (2023) and is common in classification (He el. al. 2016) may yield double descent forecasting architectures [Belkin et al. 2019).
not achieve better representations
in these cases, and they tend to get "stuck"
4.3 Manifold dynamics during training
Our analysis above focuses fully trained deep neural networks and the geometric properties of the learned manifolds. In addition, we also investigate below the evolution of manifolds during training and their struc- ture at initialization. In prior works, Anssini et al. (019 observed that randomly initialized architectures exhibit a constant ID profile, and further, there is an opposite trend in ID in intermediate layers vs. final layers convolutional neural networks during training. Kaufman & Azencot (2023) find that untrained models have different MAPC profiles than trained networks, and they observe that the normalized MAPC gap consistently increases with model performance during training. Moreover, ID and MAPC profiles con- verge consistently to their final configuration training proceeds. Motivated by their analysis, study the general convergence and trend of ID and MAPC of TSF models during training evolution.
We show in Fig. f the ID and MAPC profiles for Autoformer and FEDformer during training, where each plot is colored by its sampling epoch using the hot colormap. First, the untrained ID and MAPC profiles (dashed black) are somewhat random comparison to the other geometric profiles. Second, the overall convergence to the final behavior extremely fast, requiring approximately five epochs converge in all
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Figure 6: Training dynamics of the ID and MAPC on traffic dataset. The plot shows how the ID and MAPC change during training, colored by the training epoch.

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the configurations which is consistent with the results of Bonheme & Grzes (2022 where they show that the ID does not change much after the first epoch. Moreover, the encoder in the Autoformer converges within two epochs, whereas the FEDfformer model needs more epochs for the encoder to converge. Third, the decoder shows a slower convergence for both methods, suggesting that "most" learning takes place in the decoder component of transformer-bused forecasting models. The previous observation aligns with the works of Bonheme & Grzes ( Raghu et al. (2017), showing that representations of layers closer the input tend to stabilize quicker. More specifically, Bonheme & Grzes (2023) show that encoders' representations are generic while decoders' are specific, resulting in slight change of the encoders' representations during training. Finally, except for the untrained profiles, the TD and MAPC curves during training are generally similar across different epochs. The latter observation may mean that Autoformer and FEDformer mainly perform fine-tuning training as their underlying manifolds do not change much during training.
4.4 Distribution of principal curvatures
Li 2018 we employ for estimating the principal curvatures produces d(D d) values per point, yielding a massive amount of curvature information for analysis. Following Kaufman & Azencot (2023); we compute and plot in Fig. the distribution of principal curvatures for every layer, shown as smooth histogram for Autoformer and FEDformer models electricity and traffic datasets with horizons 192, 336. The histogram plots are colored by the network depth using the hot colormap. These distributions strengthen our analysis in Sec. 4.1 where we observe a "step"-like pattern in MAPC, where the sharp jump in curvature occurs at the beginning of the decoder. Indeed, the curves in Fig. 7 related lo layers to 4 span a smaller range in comparison to the curves layers to 7. Further, the histograms show that the distribution of curvature is relatively fixed across the encoder blocks, and similarly, different but rather fixed profile appears in the decoder.
The CAML algorithm
5 Discussion
Deep neural networks are composed of several computation layers. Each layer receives inputs from a preceding layer, applies (nonlinear) transformation, and feeds the outputs subsequent layer. The overarching theme in this work is the investigation of data representations arising during the computation of deep models. To study these latent representations, we adopt a ubiquitous ansatz, commonly known as the manifold hypothesis (Coifman & Lafon 2006): We assumne that while such data may be given in a high-dimensional and complex format, it lies on or next to a low-dimensional manifold. The implications of this inductive bias
are paramount; manifolds are rich mathematical objects that are actively studied theory Lee 2006 and
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Figure 7: Principal curvature distribution of Autoformer and FEDformer. Each plot shows the histogram profiles of principal curvatures per layer, colored by their relative depth.

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practice (Chaudhry et al. 2018), allowing one to harness the abundant classical tools and recent developments to study deep representations.
Our study aligns with the line of works that aims at better understanding the inner mechanisms of deep neural networks. Indeed, while modern machine learning has been dominating many scientific and engineering disciplines since the appearance of AlexNet (Krizhevsky et al., 2012), neural net architectures are still considered not well understood by many. In this context, the manifold ansatz is instrumental existing analysis works investigate geometric features of latent manifolds and their relation to the underlying task and model performance. For instance, [Assuini et al. (2019) compute the intrinsic dimension of popular convolutional neural networks. Following their work, Kaufman & Azencot (2023) estimate the mcan absolute principal curvatures. However, while CNNs are relatively studied from the manifold viewpoint, impactful sequential transformer models (Vaswani et al. 2017), received less attention (Valeriani et al. 2023 The lack of analysis is even more noticeable for the recent state-of-the-art transformer-basee time series forecasting works, c.g., Wu al. (202I). The main objective of our work is to help bridge this gap and study doep forecasting models trained on common challenging datasets from a manifold learning viewpoint.
We compute the WD and MAPC of data representations from several different deep architectures, forecasting tasks and datasets. To this end, we employ differentiable tools (Facco et al. 2017 Li 2018) that produce a single scalar ID) and many principle curvat ures combined to a single scalar MAPC, per manifold. Our results raise several intriguing observations, many of them are in correspondence with existing work. First, the ID is much smaller than the extrinsic dimension, reflecting that learned manifolds are indeed low-dimensional. Second, the ID and MAPC profiles across layers are similar for many different architectures, tasks, and datasets. In particular, fixed, and in the decoder, both geometric features increase with depth. Third, the MAPC in the final layer is strongly correlated with model performance, presenting a correlation, i... error is lower when MAPC is lower. Fourth, we observe that related but different datasets attain similar manifolds, whereas unrelated datasots are associated to manifolds with different characteristics. Finally, untrained models present random ID and MAPC profiles that converge lo their final configuration within a few epochs.
identify two phases, where in the encoder, ID and MAPC are decreasing stay
Our analysis and observations lie at the heart of the differences between classification and regression tasks; rescarch avenue that only recently had started to be addressed more frequently (Muthukumar et al. 2021 Yao el al. 2022). Our results indicate a fundamental difference between image classilication and time series forecasting regression models: while the former networks shrink the ID significantly to extract a meaningful representation that is amenable for linear separation, TSF models behave differently. Indeed, the ID generally increases with depth, perhaps to properly capture the large variance of the input domain where regression networks predict. Moreover, while a high MAPC gap seems to be important for classification, we find an opposite trend in TSF regression problems. In conclusion, we believe that our work sets the stage for a deeper investigation of classification believe that fundamental advancements this front will lead to powerful machine learning models, better suited for solving the task at hand.
regression from manifold learning and other perspectives. We
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Acknowledgements
This rescarch partially supported by the Lynn and William Frankel Center of the Computer Science Department, Ben-Gurion University of the Negev, an ISF grant 668/21, ISF equipment grant, and by the Israeli Council for Higher Education (CHE) via the Data Science Research Center, Ben-Gurion University of the Negev, Isracl.
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A TSF datasets
Blow we provide a detailed description of the datascts used in the paper. A summery of the datasets can be found in Tab. 2
Electricity Transformer Temperature (ETT) Zhou et al. (2021): The ETT contains electricity power load (six features) and oil temperature collected over a period of two years from two countries China. The dataset is versatile and exhibits short-term periodical patterns, long-term periodical patterns, long-term trends, and irregular patterns. The dataset is further divided two granularity levels: ETTh1, ETTh2 for one hour level and ETTm1, ETTm2 for 15 minutes level.
Weather
D The dataset contains 21 meteorological sensors for range of year in Germany.
Electricity Consuming Load (ECL) It contains the hourly electricity consumption (Kwh) of 321 clients.
Traffic The dataset consists of hourly data spanning 24 months (2016-2018) obtained from the California Department of Transportation. This data provides information on road occupancy rates, measured by 862 sensors on freeways in the San Francisco Bay area, ranging between 0 and
Table 2: We detail several statistics regarding the datasets considered in this work.
Dataset
Number of features
Number of train samples
Granularity
ETTml, ETTm2
7
34369
15 minutes
ETTh1, ETTh2
7
34369
1 hour
Weather
21
36696
hour
ECL
321
18221
1 hour
Traffic
862
12089
hour
B Additional results
B.1 ETT* datasets analysis
To complement the results in the main article we add a comparison of the ID and MАPĆ of three different ETT datasets: ETTm1, ETTh1 and ETTh2. We notice that ETTm1, ETTh1 and ETTh2 show hunchback trend for Autoformer while for FEDformer the hunchback trend appears for larger forecast horizons as shown in Fig. g
Our MAPC results in Fig. | show that MAPC are relatively fixed and start to rise at the decoder.
C
TSF models
Here we will provide a supplementary analysis of additional TSF models: vanilla Transformer /Vaswani
et al. 2017). Informer [Zhou et al. 2021 and PatchTST (Nic al. 2023
Transformer and Infromer: These models have similar architecture to the Autoformer and FEDformer as shown in Fig.[] The models are composed of two encoder layers and one decoder layer, however, in contrast to Autoformer and FEDformer, Transformer and Infromer do not contain scries decomposition layors. The analysis of Transformer and Infromer inspects the output of the encoder layers, decoder layer and the last linear layer. Tn Fig. 10 we observe trends similar to ones shown for the Autoformer and FEDformer. When comparing different datasets, we see similar trends, a monotonic increase in ID for the Transformer and saw-like behaviour for Informer (see Fig 11 The MAPC across datasets for the Transformer exhibits monotonic increase trend while the Informer has a "v"-shape (see Fig 12
https://ww..gc-jjee http://jjjsaannp.dedeeettette de/wetter/
https://archive.tt
bttps://pans.det http://dshcttttcccccc.ceeeseeedd
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96
192
336
720
ETTm1 ETTh1
ETTh2
Autaformer
2-
FEDformer
layer depth
Figure 8: ID profiles across layers of Autoformer and FEDformer ETT datasets for multiple forecasting horizons. Each panel includes separate ID profiles per dataset, for several horizons (left to right) and architectures (top to bottom).
96
192
336
720
107
FTTm1 FFThi
ETTN2
FEDromee Autoformer
MAPC
201
20a
MAPC
102
104
207
layer depth
Figure 9: MAPC profiles across layers of Autoformer and FEDformer on ETT datasets for multiple forecasting horizons. Each panel includes separate MAPC profiles per dataset, for several horizons (left to right)) and architectures (top to bottom).
PatchTST: PatchTST is a transformer- based model composed of vanilla Transformer encoders and a linear layer as a decoder. PatchTST differs from all other architecture mentioned the paper by dividing a
Transformer
Informer
336
8
192
720
6
4-
10a
10
MAPC
10a
10a
layer depth
Figure 10: Intrinsic dimension and mean absolute principal curvature along the layers of Trans- former and Informer on traffic dataset for multiple forecasting horizons. Top) intrinsic dimension. Bottom) mean absolute principal curvature. For each model, both ID and MAPC share a similar profile across different forecasting horizons.
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192
336
720
electricity traffic
weather ErTm1
Informer Transformer
10
layer depth
Figure 11: ID profiles across layers of Transformer and Informer electricity, traffic, weather and ETTml datasets for multiple forecasting horizons. Each panel includes separate MAPC profiles per dataset, for several horizons (left to right) and architectures (top to bottom).
sequence into patches and using them inputs to the network. Each patch is then embedded, and from that point on, all sequential information becomes inaccessible, i.e., we cannot manually extract temporal information for each time stamp in the patch. Our analysis focuses on time series manifolds, where each element represents a single point in time, while the latent representation product manifolds where each element is a sequence points time. Similar to the analysis performed on the Transformer and Informer, we inspect the output of the encoder layers and the decoder layer which the last linear layer. We notice that the ID remains relatively stable during the encoding phase and increases in the last layer, behavior consistent across datasets and horizons. A similar trend is observed for the MAPC during the encoding phase where on some datasets the MAPC increases on the last layer and on other it decreases. See Figs. and 14
the PatchTST model lie on
Intrinsic dimension and mean absolute principal curvature
D.1 Intrinsic dimension
To estimate the ID of data representations in TSF neural networks, we the TwoNN (accoo al. 2017) global id estimator. The IDDestimator utilizes the distances only to the first two nearest neighbors of each point. This minimal selection helps reduce the impact of inconsistencies in the dataset during the estimation process.
192
336
electricity
weather
Transfomer
MAPC
2.0-
Informer
MAPC
20a
2
layer depth
Figure 12: MAPC profiles across layers of Transformer and Informer on electricity, traffic, weather and ETTml datasets for multiple forecasting horizons. Each panel includes separate MAPC profiles per dataset, for several horizons (left to right) and architectures (top bottom).
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3 10
10 96
336
192
720
MAPC
x 10
8
x 10
layer depth
2
3
Figure 13: ID and MAPC along the layers of PatchTST on weather dataset for multiple fore- casting horizons. Left) intrinsic dimension. Right) mean absolute principal curvature. ID and MAPC share similar profile across different forecasting horizons.
... ,Nn a points uniformly sampled on a manifold with intrinsic dimension Method Let X (x #2,
d For each point 24, we find the two shortest distances r1, from clements in X \ {2;) and compute the ratio 1r 22 It can be shown that μ4, S i N follow Pareto dist.ibuttin with parameter d on 11
[1, oc), that is (Pa d) = du, (d+11) While d can be estimated by maximizing the likelihood:
N
P2 AN d) = d
14
(d+1)
(1)
we follow the method proposed by Facco et al. 2017 based the cumulative distribution F(u) - The idea is to estimate by linear regression on the empirical estimate of F(u). This is done by sorting the values of μ in ascending order and defining pemp (14) n straight. line is then fitted on the points of the plane {(log ps. log (1 F(mP)))] The slope of the line the estimated ID.
96
192
336
720
ETTmZ
Electricity weather
ETTh1
10a
MAPC
10-2
layer depth
Figure 14: ID and MAPC profiles across layers of PatchTST on ETT* electricity and weather datasets for multiple forecasting horizons. Each panel includes separate ID (top) and MAPC (bottom) profiles per datasot, for several horizons (left right).
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Autoformer
FEDformer
123
twomn
electricity traffic
10.0
TLE
5IN22 ML.
5.-
0
8-
10.0
5.-
25
Figure 15: Comparison of several ID cstimation tools. Most estimators agree with the trend observed by the TwoNN ID estimator.
D.2 Comparison of intrinsic dimension estimators
ID estimators operate basod on a variety of principles. They are developed using specific features such as the number of data points within a fixed-radius sphere, linear separability, the expected normalized distance to the nearest neighbor. Consequently, different ITD estimation methods yield varying ID values. In Fig. 15 proivde comparison of several ID estimators from Scikit-dimension (Bac et al.] 2021). The results indicate that the ID profile obtained by the TwoNN estimator used in this work is simillar to other ID estimation tools. We observe different trend from the LPCA estimator which we attribute to the linear nature of the algorithm. LPCA is grounded in the observation that for data situated within a lincar subspace, the dimensionality corresponds to the count of nou-zero eigenvalues of the covariance matrix. We note that the inconsistency appears in the layers where the curvature increases and thus the manifold deviates from a flat (lincar) surface.
D.3 Data density
In comparison to the intrinsic dimension, which is characteristic of the entire manifold, curvature in- formation is local. Morcover, curvatures caleulated using second-order derivatives of the manifold. Consequently, our study assumes that the data is dense enough to compute curvatures. However, the la- tent representations of data, are both high-dimensional and sparse, which presents significant difficulties in calculating local differentiable values on such as curvature.
The typical characteristics of data used in machine learning require a large number nearby points to create a stable neighborhood. One commonly used tool for this k-Nearest-Neighbourr (KNN). However, KNN can sometimes generate non-local and sparse neighborhoods, where the "neighbors' are effectively far apart in Euclidean sense. Another approach is to use domain-specific augmentations, such window cropping, window warping or slicing. However, this approach only explores a specific aspect of the data manifold and may overlook other important parts. A more effective approach, regardless of the domain, is to compute the Singular Value Decomposition (SVD) for each time series. This generates close neighborhood by filtering out small amounts of noise in the data. This approach well-motivated from a differential geometry standpoint, as it approximates the manifold at point and samples the neighborhood.
Neighborhood generation. To improve the local density of time sereis samples, we use proce- dure similar to Yu et al. (2018) to generate artificial new samples by reducing the "noise" levels of the original data. Specifically, given a d dimensional time series Fır E R let ø1. = UEVT be its SVD, where U E T V E Rdxd and Σ E Rxxd a rectangular diagonal matrix with singular values (21,02,.... ,aa) (a)) on the diagonal descending order such that r is the rank ZI.T. Let m the smallest index such that the explained varince a of the m'th mode less than or equal to 1e We define Σ,07
Σ'=(01,02 "2. Mm m- ud-m+10d such that u, L.d. W(O 1) and construct क1. UVVT, this process is repeated 64 times for each time series, generating 64 new time series.
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D.4 Curvature estimation
There are several methods available for estimating curvature quantities of data representations, discussed in papers such as Brahma et al. (2015); (Shao et al. (2018). For our purposes, we have chosen to use the algorithm described in Li] (2018), which is called Curvature Aware Manifold Learning (CAML). We opted for this algorithm because it is supported by theoretical foundations and is relatively efficient. In order to use CAML, we need to provide the neighborhood information of a sample and an estimate of the unknown ID. The ID is estimated using the TwoNN algorithm, as described in D.3] similarly to Ansuini et al. (2019): Kaufman & Azencol (2023).
In order to estimate the curvature of data Y {yi, 92. yNN C RD, we make the assumption that the data lies on a d-dimensional manifold M embedded in RD, where d is much smaller than D. Consoquently, M can be considered as a sub-manifold of RDD The main concept behind CAML is to compute a local approximation of the embedding map using second-order information.
T:R" + .RD - - f(x)) tG, - N
(2)
where X = [I1 ;NN C Rd are low-dimensional representations of Y, and fei, C2, eN are the noises. In the context of this paper, the embedding map f is the transformation that maps the low- dimensional dynamics to the sampled features for cach time stamp t that might hold redundant information.
In order to estimate curvature information at a point yi E Y, we follow the procedure described above to define its neighborhood. This results in a set of nearby points (y....... Fi where K represents the number of neighbors. Using this set along with the point yt, we utilize SVD to construct a local natural orthonormal coordinate frame J.T.... Ha Ty ~ This coordinate frame consists of basis for the tangent space (first d elements) and basis for the normal space. To be precise, we denote the projection of and Vi, for j = ,,...,K K onto the tangent space spanned by 0/Om",..., 0/0xn as Ij and Ui respectively. It is important to note that the neighborhood of y, must have a rank of r > d. If the rank is less than d, then SVD cannot accurately encode the normal component at Fy, leading to poor approximations of f at xz. Therefore, we verify that Vig has a rank of d + 1 or higher.
9 a
The map f can be expressed in the alternative coordinate frame as f(x' ..... ) =x=" ..... .... f'....... ..... ff 4. The second-order Taylor expansion of fo at Wi, with respect to I4, with an error of Oluu, B). is represented by
( R (1:) + 2 + AA
where a = 1,..,, D d, A.; = (ui n.)) and Bi an elemnt in the neighborhood of xy. The gradient of 8² is its Hessian. We have a neighborhood (y...... Fix of yo for is denoted by Vf", and H = O2
and their corresponding tangent representations }. J. Using equation we can form a system of linear equations, as explained in D.5] The principal curvatures are the eigenvalues of 11", so estimating curvature information involves solving a linear regression problem followed by an eigendecomposition. Each Hessian has d eigenvalues, so each sample will have (D d) x d principal curvatures. Additionally, one can compute the Riemannian curvature tensor using the principal curvatures, but this requires high computational resources duc to its large number of clements. Morcover, as the Riemannian curvature tensor is fully determined by the principal curvatures, we focus our analysis on the eigenvalues of the Hessian. To evaluate the curvature of manifolds, we estimate the mean absolute principal curvature (MAPC) by taking the mean of the absolute values of the eigenvalues of the estimated Hessian matrices.
D.5 Estimating the Hessian Matrix
In order to estimate the Hessian of the embedding mapping f where a = 1,..., D - d, we build a set of linear equations that solves Eq. g We approximate fa by solving the system f = VXY,, where X, holds the unknown elements of the gradient V and the hessian Ha We define fa = f (u,)).. ... fo (uxx) ) where ui are points in the neighborhood of z,, in the local natural orthogonal coordinates. The local natural hogonll coordinates are set of coordinates that are defined at specific point p of the manifold. They
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Published in Transactions on Machine Learning Research (10/2024)
are constructed by finding basis for the tangent space and normal space at point p by applying Principal Component Analysis, such that the first coordinates (associated with the most significant modes, i.e., largest singular values) represent the tangent space, and the rest represent the normal space. We define V - Wixl. where Vi, is given via
1/ [1] U/, n (ut)
(u (u), n².))
(ut xxxxx)
The set of linear equations fa VXX, is solved by using the least square estimation resulting in X; vija, where X; = V Vfrd Ho Hodd Hal.2 ..... Hod-l.d) In practice, we estimate only the upper triangular part 4 since is a symmetric matrix. The gradient values V are ignored since they are not required for the CAML algorithm. We refer the reader for more comprehensive and detailed analysis in [Li 2018)
22


Reviews:
[{"review_id": "fWCCCgiQ6K", "reviewer": "Reviewer_7DNQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper is positioned very clearly, building on prior work while still presenting interesting new results.\n- Lots of experimental results, which is important for an analysis based paper.\n- Very clear overview of experimental setup.\n- In general, good commentary/analysis/discussion of observed results. Especially comparing differences to geometric properties of image classification features. Interesting observations regarding training dynamics -> representations from encoder settle before decoder.\n\nWeaknesses:\nThe discussion in 4.2 seems contradictory:\n- The second paragraph analyzes the results stating that lower final MAPC is correlated with lower error (ex. “the model performs better as curvature decreases”)\n- Then the third paragraph states that “we note the relatively flat slope profiles presented across all datasets and architectures” and that “these results indicate that while the manifolds become more complex in terms of curvature, transformers yield a relatively fixed MSE, regardless of the underlying MAPC. Thus, our results may hint that Autoformer and FEDformer are not expressive enough.”\nParagraph two states that the observed correlation (low MAPC -> lower error) is meaningful, but then paragraph three states that the error doesn’t change much with curvature, and that more expressive models are needed to benefit from increased curvature. Either I am misunderstanding this section (possible!), or these statements are contradictory (ie. relationship is meaningful or not).\nGiven the first interpretation (ie. correlation is meaningful), I also think some of the claims would benefit from being toned down ex. “is strongly correlated with model performance” in the discussion.\n\nAdding more background on transformers for TSF would be helpful ex. What is a sequence decomposition layer? What information is present in each of the three data flow paths in Figure 1? All of this can be found in the associated papers, but since understanding where you measure the geometric properties is central to the paper, I think it makes sense to include it.\n- Ex. “we discard data from the red and blue trajectories” -> what are the red and blue trajectories? What does it mean to discard this data?\n\nFigure 1 could be made more clear:\n- What do the data paths represent? \n- What are the inputs and outputs?\n- Why is only one layer type shown? Can you adopt a more standard transformer diagram?\n- Show that you also sample after the final linear layer.\n\nIn general, the figure captions could be more descriptive:\n- Figure 5: what do the dots and lines represent?\n- Figure 6: what is the dashed line?\n\nSmall nits:\n- Figure ylabels: “Id -> ID”\n- In appendix D.3: “as described inD.2” -> “as described in D.2”\n- Section 4.3: “previous observation alignes” -> “previous observation aligns”", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper conducts an analysis of the geometric properties (intrinsic dimension and mean absolute principal curvature) of activations in transformers trained for time series forecasting tasks. Although there have been similar investigations into the geometric properties for other tasks (ex. classification), this is the first paper (that I am aware of) that tackles the same problem for TSF. The authors investigate 1) How these properties vary by layer in the models, 2) How these properties evolve over training, 3) The correlation between geometric properties and overall performance. Importantly, the authors highlight the differences in these observations between classification and TSF models, and provide intuitions for the reasons behind these differences.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall there are no major changes or experiments that I think need to be run. I think it is important that the discussion in Section 4.2 is clarified (see earlier comment). I also believe that the other exposition/figure changes would strengthen the paper.", "broader_impact_concerns": "I do not have any concerns about ethical implications of the work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "qIcRzq57d2", "reviewer": "Reviewer_BtEZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n1. As far as I'm aware, the results presented in this paper analyzing the geometric properties of transformers trained in the context of time series forecasting are new, and these experiments have not shown up before.\n\n## Weaknesses\n\n1. The overall novelty of the paper is quite limited. This kind of manifold analysis has been done for deep networks before and the authors explicitly admit that they reuse the approaches from Ansuini et al. (2019) and Kaufman & Azencot (2023). The paper reuses these methods to analyze transformers trained in the context of time series forecasting and produces an analogous analysis as what was given in Kaufman & Azencot (2023), closely following the original set-up. There is some analysis given that compares and contrasts the results in the paper to previous literature, but the novelty is still rather limited and the motivation for this specific application over any other application is not entirely clear.\n\n## Verdict\n\nThis paper presents a geometric analysis of transformers trained in the context of time series forecasting. It reuses manifold analysis approaches, borrowing heavily from Kaufman & Azencot (2023), who perform this analysis for CNNs as opposed to transformers. As such, the methodological novelty of the paper is quite limited. It essentially performs a geometric analysis of transformers trained in a particular context using existing tools. The contribution is of a highly empirical nature and the central results are more so a geometric analysis of the transformers in this context and a post-hoc analysis of how they compare to the geometric analysis available in prior work. I believe the claims made in the paper are reasonably well-supported by the empirical results. I also believe this paper is of interest to at least some portion of TMLR's audience, specifically those who have performed/perform this analysis on related models in related contexts. As a consequence, I recommend an accept rating for this paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper performs a geometric analysis of transformers trained for time series prediction via latent manifolds. Previously, geometric features of data manifolds were analyzed in the context of deep convolutional neural networks (CNNs) by Ansuni et al. (2019) and Kaufman & Azencot (2023). Motivated by these works, the authors of this paper extend the analysis to state-of-the-art transformers trained for time series prediction. They investigate both the intrinsic dimension (ID) and the mean absolute principal curvature (MAPC) of several latent manifolds. They find that manifolds of transformer forecasting models undergo two phases: during encoding, curvature and dimensionality either drop or stay fixed, and then, during decoding, both dimensionality and curvature increase significantly. Further, this behavior is shared across several architectures, datasets, and forecast horizons. In addition, they find that MAPC is correlated with the test mean squared error, which, at least on an empirical level, allows one to compare models without access to the test set. Finally, they find that untrained models show somewhat random dimension and curvature patterns, and moreover, the latent manifolds converge rapidly to their final geometric curvature profiles.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The writing in this paper can use some work. The phraseology was poor or awkward in some places, although the paper was overall understandable. Additionally, I would like the figures in the paper to be much closer to where they are referenced in text. For example, Figure 2 is a figure from the experiments section and appears at the top of page 3, yet the first reference to this figure is at the bottom of page 5 in the experiments section. In general, the figures from the experimental section are scattered throughout the paper, yet they should be condensed in some way to be closer to the relevant section.\n\nAdditionally, I would like to give a non-exhaustive list of minor corrections below:\n\n- Abstract: \"they still lack a comprehensive understanding\" -> \"we still lack a comprehensive understanding\" or \"the field still lacks a comprehensive understanding\"\n\n- Section 1, first paragraph: Please use \\citet in the final sentence for in-text citations. This mistake reoccurs several times throughout the paper, although I will only point it out the first time. Please be careful when distinguishing in-text vs. parenthetical citations.\n\n- Section 2, second paragraph: Please use \"effectiveness\" instead of \"effectivity\"; this is far more standard terminology.\n\n- Section 4.3, second paragraph: \"alignes\" -> \"aligns\"", "broader_impact_concerns": "The work deals with analyzing the geometry of transformers trained in the context of time series forecasting. I believe that this analysis does not require a broader impact statement and also does not require a statement addressing potential ethical implications of the presented research.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "2xYJ59XRz0", "reviewer": "Reviewer_7Bjy", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n\nThe authors tackle an interesting problem and use proper methodology for analyzing the latent manifolds of the networks. \n\n# Weaknesses\n\nThe paper repeatedly tries to claim that the patterns observed are consistent across both Autoformer and FEDformer architectures, and as a result, the analysis can be carried out on other transformer models. However, I do not share this observation. I believe that basing any analysis on these hypotheses is not well-grounded. For instance, in Figure 2, which focuses solely on the traffic dataset, the ID estimate trends vary significantly across the two architectures. Furthermore, discrepancies are evident in other instances, such as the MAPC trends for ETTM1, which do not align across AutoFormer and FEDformer in Figure 4 (with a 336 horizon), as well as in most of the ID trends in Figure 3.\n\nI have significant concerns about the statistical association between MSE (performance) and MAPC in Section 4.2. While the authors demonstrate some degree of correlation between the two, correlation alone does not guarantee that having MAPC estimates beforehand helps us deploy more performant models. I recommend adopting a causal approach to substantiate this hypothesis rather than relying solely on correlation. For instance, the authors could implement a regularization technique to deliberately lower MAPC and then observe if this leads to better generalization performance. Alternatively, they could train multiple networks, discard those with unusually high MAPC estimates, and demonstrate that the remaining models perform better. This would provide practical value to the observation. Additionally, I find that Table 1 does not show a consistent pattern and is therefore unconvincing. I believe further evidence is necessary to support the claims in this section.\n\nThe authors have chosen a single method of ID estimation throughout their paper and there is a high possibility that the ID analysis overfits to the TwoNN method that has been adopted.\n\nFinally, I had a high-level comment. While I acknowledge that transformers are a good method for modelling time-series data, I would not argue that they are prominent (at least yet) or they will ever be! In vision and NLP, ViTs and transformer architectures are indeed the go-to methods, but in time-series forecasting, transformers are not yet definitively the preferred architecture. For instance, ForecastNet [A] and neural operator networks, which extend beyond transformers, demonstrate very strong performance. Therefore, although I appreciate the efforts of this paper in elucidating the inner workings of transformers for time-series forecasting, there is still no concrete evidence that they are the best method for time-series analysis, and as a result, their geometric analysis might not be as well-motivated. On the other hand, conducting geometric analysis on a broader range of time-series models such as the ones argued, and observing similar trends, could be a much more compelling argument for the goal of this research.\n\n## References\n[A] Pathak, Jaideep, et al. \"Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators.\" arXiv preprint arXiv:2202.11214 (2022).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper seeks to elucidate the inner workings of time-series transformer models by analyzing the manifolds derived from their latent embeddings. Among the various methods to analyze latent manifolds, the authors focus on estimating intrinsic dimension (ID) and curvature which provide a high-level summary of the manifold characteristics that can be studied. Their key findings include:\n\n1. The ID and curvature estimates exhibit a V-like pattern across the transformer's layers, decreasing (or staying constant) during the encoding phase and increasing during the decoding phase.\n2. Higher curvature correlates with poorer performance in terms of MSE on the test set.\n3. The evolution of ID and curvature estimates is tracked throughout the training process.\n4. These geometric observations are extensively discussed and compared with analogous models in vision and NLP.\n5. The ID and curvature trend sometimes show similar behaviour across different transformer models, showing that some of the observations here can easily be carried out in other models.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The ID analysis might be overfitting to the chosen TwoNN method. Could you consider incorporating ID estimates using other methods available in scikit-dimension [A]? For instance, ESS and LPCA are well-regarded for ID estimation. Additionally, model-based approaches such as LIDL [B] and FLIPD [C], which leverage density estimators, can provide valuable insights into the data manifold. I would suggest that for this study, it is crucial to include **at least one other estimator**. For example, the authors can use ESS ID estimates to ensure that the observed patterns are consistent across different estimation methods. I would argue the same for MAPC estimates, but I am not sure if there are many methods of estimating curvature in the literature.\n\nIn section 4.1, the authors have mentioned *\"deep classification neural networks essentially recover the related low-dimensional manifold, to facilitate a linear separation of classes, and thus one may expect a low ID towards the final layers of the network\"*. I do not follow the logic and I think it should be written more elaborately or rigorously. How does linear separability relate to low ID? Data can be ambient dimensional, but still be linearly separable.\n\nIn section 4.2, the authors have mentioned *\"While popular CNNs show better performance when the MAPC gap is high, we report an opposite trend, namely, better models are associated with a lower MAPC. We believe this behaviour may be attributed to classification networks requiring final high curvatures to properly classify the statistical distribution of the input information and its large variance\"*. I do not follow the reasoning! Can the authors elaborate on how high curvature corresponds to better classification?\n\n## References\n\n[A] Bac, Jonathan, et al. \"Scikit-dimension: a python package for intrinsic dimension estimation.\" Entropy 23.10 (2021): 1368.\n\n[B] Tempczyk, Piotr, et al. \"Lidl: Local intrinsic dimension estimation using approximate likelihood.\" International Conference on Machine Learning. PMLR, 2022.\n\n[C] Kamkari, Hamidreza, et al. \"A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models.\" arXiv preprint arXiv:2406.03537 (2024).", "broader_impact_concerns": "No broader impact concerns come to mind for this paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper studies geometric properties of intermediate activations of transformer models for time series forecasting. In particular, the authors estimate the intrinsic dimension and mean principal curvature of the manifolds of activations, and study how these quantities vary by layer and through training, and how they correlate to model performance. The resulting insights are well supported by evidence, and will be of interest to some members of TMLR's audience. I thus recommend acceptance.
Decision: Accept (poster)