Enhancing Foundation Models for Time Series Forecasting via Wavelet-based Tokenization

Thank you for your useful comments and suggestions that helped improve our paper. We have summarized our main revisions and answers in response to all reviewers in a separate comment above. Below we provide point-by-point answers to each section in your review.

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Regarding

In their qualitative analysis and their conclusion, the authors claim "excellent forecasting accuracy" as well as "superior generalization capabilities". However, the results in Figure 3 and 4 reveal that using wavelet-based features does not improve downstream performance with respect to weighted quantile loss (WQL) and mean absolute scaled error (MASE), in both in-distribution and zero-shot settings. In the former setting, wavelet-based tokenisation even hurts performance compaired to the adapted Chronos model. Only in the newly introduced visual relative squared error (VRSE) [7], the results indicate incremental improvements compared to the baselines.

The core motivation of our work was to develop a general purpose tokenizer that seamlessly captures global and local patterns in the time series. Our qualitative results (Figure 1 and Section 4.3) demonstrate the impressive performance of WaveToken on complex edge cases where existing foundation models fail, almost completely. These results highlight the remarkable ability of a wavelet-based tokenizer to capture nonstationary signals – which are pervasive in practical applications. Nevertheless, as outlined in the paper, WaveToken also improves quantitatively on Chronos 75% of the times and 83% of the times on the in-domain (15 datasets) and zero-shot (27 datasets) benchmarks, respectively, while employing a vocabulary one-quarter the size of Chronos. In addition, WaveToken achieves the best average rank across all metrics for both Benchmarks I and II (Figures 10 and 11).

We believe that in the natural progression of the field, different works make different types of contributions, and not every work brings (or needs to bring) a paradigm-shifting accuracy improvement. In the case of WaveToken, our findings position it as a promising avenue for developing general-purpose, information-efficient forecasting models, and it should therefore be evaluated independently in addition to its accuracy.

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Regarding

The authors do not report their results across multiple seeds to ensure robustness.

All results for WaveToken and Chronos are reported by averaging metrics across 3 different training runs (as mentioned in Appendix D), in order to draw reliable and reproducible conclusions. In hindsight, we realize we should have mentioned this in the main text as well. We have updated our manuscript accordingly.

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Regarding

The authors do not elaborate on the limitations of their work.

Thanks for pointing this out, we have added the paragraph below at the end of Section 5.

Specifically, WaveToken exploits an autoregressive model on wavelet coefficients. As such, it suffers from slower decoding and inference time relative to recently proposed alternatives, such as patch-based models. Applying WaveToken to patch-based architectures represents an interesting area for future research.

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Regarding

The authors neither provide configurations for hyperparamter tuning (including the final hyperparameter setting) nor share their code to support reproducibility

Section 4.1 provides all the details on our model, including parameter counts, architecture used, context and prediction length, training and evaluation strategies. In addition, Appendix F reports all the hyper-parameters used for each model, including all the baselines.

As for code, we plan to release a user-friendly research package complete with details on how to train and evaluate our tokenizer and models. Unfortunately, we could not share our code for review at this stage due to pending legal approvals.

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Regarding

How do the authors determine the "average rank" (l. 395) reported e.g. in Figure 10?

The average rank is computed by calculating the rank of each model on each dataset according to its forecasting performance relative to the other baselines, for a specific metric. The ranks are then averaged across all the datasets within a benchmark, i.e. separately for in-domain and zero-shot datasets.

Dear Reviewer qSp1,

Thank you again for the useful comments and questions that helped improve our paper. As the end of the discussion period is approaching, we would like to gently remind you to take into account our responses and the updated version of the manuscript into your evaluation.

We hope that we have satisfactorily addressed your questions and concerns. If so, we request you to consider raising your score to reflect that. If you have further questions, we will be happy to respond to them.

Thank you.

Best Regards, The Authors

Dear Reviewer qSp1,

Thank you again for your review. We kindly remind you to take into account our responses and the updated version of the manuscript into your evaluation.

We hope that we have satisfactorily addressed your questions and concerns. If so, we request you to consider raising your score to reflect that. If you have further questions, we will be happy to respond to them.

Thank you.

Best Regards,
The Authors

Dear Reviewer qSp1,

Thank you again for your helpful review. As you know, the discussion period will end in two days on Monday December 2nd. We would like to gently remind you to take into account our responses to your questions and concerns.

We hope our answers above have satisfactorily addressed your comments. If so, we request you to consider raising your score to reflect that. If you have further questions, we would be happy to respond to them.

Thank you.

Dear Reviewer qSp1,

The authors of submission 8429 have provided an extensive response to your review.

As the discussion period is almost over, please go over their response and determine which of your comments are addressed.

Please explain your decision to update (or not update) your score.

All the best,

The AC

Thank you very much for the response and the revised manuscript. I have carefully read both the rebuttal and the updated version of the study.

The authors have conducted an interesting investigation on whether wavelet-based features can enhance time series foundation models, such as Chronos [1].

While I find the contribution of this work to be limited, given that wavelet-based approaches have been extensively studied in the literature, including the field of time series analysis [2][3][4][5][6][7][8], this would not be a concern if the method demonstrated clear benefits for downstream performance. However, the experimental results indicate that the proposed approach does not offer clear advantages for downstream applications, especially when compared to the model it builds upon, i.e. Chonos [1].

Additionally, the rebuttal does not address my points related to other non-regressive forecasting models or general time series models, which could potentially benefit from wavelet-based features. The study also lacks an analysis of the learned vocabulary to explore whether patterns can be leveraged to improve the downstream performance.

Given these considerations, I cannot recommend accepting the study at this time, and thus leave my scores unchanged. However, I encourage the authors to refine and expand upon this work, as the conceptual idea of wavelet-based tokenisation is interesting.

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[1] Ansari et al. "Chronos: Learning the language of time series." arXiv (2024).

[2] Torrence et al. "A practical guide to wavelet analysis." Bulletin of the American Meteorological Society (1998).

[3] Percival, D. B. "Wavelet Methods for Time Series Analysis." Cambridge University Press (2000).

[4] Cazelles et al. "Wavelet analysis of ecological time series." Oecologia (2008).

[5] Pandey et al. "Intelligent hybrid wavelet models for short-term load forecasting." IEEE Transactions on Power Systems (2010).

[6] Wang et al. "Multilevel wavelet decomposition network for interpretable time series analysis." ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (2018).

[7] Sasal et al. "W-transformers: a wavelet-based transformer framework for univariate time series forecasting." IEEE international conference on machine learning and applications (2022).

[8] Posam et al. "DiffFind: Discovering Differential Equations from Time Series." Pacific-Asia Conference on Knowledge Discovery and Data Mining. Springer Nature Singapore (2024).

Thank you for your response and follow-up comments.

We agree with the reviewer that wavelet-based methods have been used previously in the context of time series forecasting. However, WaveToken uses it in the completely new context of tokenization for time series foundation models. These foundation models, such as Chronos, have received considerable attention from the time series community of late and our work addresses a critical component (tokenization) of these models.

We respectfully disagree with reviewer's assertion that "the proposed approach does not offer clear advantages for downstream applications". On the contrary, WaveToken improves quantitatively over Chronos 75% of the times and 83% of the times on the in-domain (15 datasets) and zero-shot (27 datasets) benchmarks, respectively. In the long-horizon experiments that we have added to the revision (Fig. 12) too, WaveToken shows its edge over Chronos. Furthermore, in the qualitative analysis (Figure 1 and Section 4.3), the difference between WaveToken and Chronos is stark, where Chronos completely fails on spiky and non-stationary data.

Additionally, the rebuttal does not address my points related to other non-regressive forecasting models or general time series models, which could potentially benefit from wavelet-based features.

Indeed, wavelet-based features could potentially help other time series models. However, our work is NOT about wavelet features in general, but about wavelet-based tokenization which is applicable to autoregressive methods that employ tokenization (such as Chronos) and not directly relevant to non-autoregressive models.

The study also lacks an analysis of the learned vocabulary to explore whether patterns can be leveraged to improve the downstream performance.

It is unclear what kind of analysis the reviewer is referring to here. In the context of WaveToken, the vocabulary are the bins of the wavelet coefficients obtained upon quantization. As such, the vocabulary is neither learned nor it is a vocabulary of patterns.

We hope that the reviewer reconsiders their score in the light of our response above.

Regarding

Learning autoregressive models on the wavelet coefficients, which do not preserve the temporal transitions, does not make sense to me. The author also noticed the counter-intuition. However, the insignificance of experimental results and lack of theoretical proof amplifies this irrationality. Can the author give more explanations and proofs in this respect? I am not denying the wavelet tokens, which have been extensively explored in previous works before. It might be more reasonable to learn the diffusion process on the decomposition-based coefficients. Whether the author has tried in this regard?

To clarify, temporal dependencies are preserved within each coefficient group. In other words, with a one-level wavelet decomposition, WaveToken is simply providing additional structure (namely, high-frequency versus low-frequency coefficients, divided in two concatenated groups) that the attention-based mechanism can leverage. The autoregressive model is still consistent within each coefficient group, and the attention mechanism allows it to determine when to switch from an autoregressive regime appropriate for low frequencies to an autoregressive regime appropriate for high frequencies.

This phenomenon is also highlighted in the attention maps presented in Figure 5: by decomposing the input sequence into approximation (low-frequency) and detail (high-frequency) coefficients, the model learns more easily to focus on the important portions to effectively forecast complex time series of practical relevance such as sparse spikes or non-stationary frequencies evolving over time, as can be seen in Figure 1.

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Regarding

About the experiments to support wavelet tokenization: As previously stated, the main evidence in the paper supporting wavelet tokenization comes from the performance of in-domain and zero-shot generalization. However, recent work has shown that there is a risk of overfitting these smaller time series datasets with the LLM [1]. It is recommended that the author provide more comparisons on diverse benchmarks.

WaveToken is trained on a large corpus of real and synthetic time series from diverse sources, NOT small individual time series datasets as studied in [1]. Our remarkable zero-shot results on 27 datasets showing that WaveToken performs better or on par with models trained on these datasets (e.g., DeepAR, TFT, PatchTST), despite never having seen these datasets, demonstrates that the risk of overfitting is non-existent. Furthermore, the models and evaluation settings considered in [1] have been superseded by the large-scale zero-shot evaluations in other recent works such as Chronos and TimesFM.

[1] Tan, M., Merrill, M. A., Gupta, V., Althoff, T., & Hartvigsen, T. (2024, June). Are language models actually useful for time series forecasting?. In The Thirty-eighth Annual Conference on Neural Information Processing Systems.

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Regarding

How does WaveToken perform in long-term forecasting? Recent work has shown that Chronos can be less effective at predicting long horizons than models that adopt patch tokens (such as TimesFM). It may be due to the multi-step error accumulation caused by its point-wise tokenization. I wonder if WaveToken can effectively solve this problem.

Thank you for suggesting this additional setting. We conducted new experiments (see the new Figure 12 and Appendix E in the paper) on a long-horizon benchmark constructed by increasing the forecast length $H$ of each dataset in Benchmark II (Zero-Shot) by a factor of 2 and 3 (see Table 9 for original forecast lengths). The datasets in Benchmark II without sufficient history in all time series to allow for longer horizons have been skipped. WaveToken outperforms other foundation models across all three metrics in the $H \times 2$ setting. When $H \times 3$, TimesFM performs better with respect to WQL only. All results for Chronos and WaveToken are averaged across three different seeds.

Thank you for your comments and questions that helped improve our paper. We have summarized our main revisions and answers in response to all reviewers in a separate comment above. Below we provide point-by-point answers to each section in your review.

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Regarding

There are some overclaims: WaveToken claimed to be a foundation model, however, being limited to a fixed context length is a pronounced defect.

WaveToken is only trained with a fixed context and prediction length, but during inference one is free to use arbitrary context and prediction lengths. This setup is not special to WaveToken and is also used by other foundation models for forecasting such as Chronos and TimesFM. We opted for training context and prediction lengths of 512 and 64, respectively, as they are sufficient for many practical use cases, as shown by our extensive evaluation on 42 datasets that cover different history lengths, prediction lengths, seasonalities, and domains.

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Regarding

Limited refinement over previous works: From the experimental results, the performance of the proposed model is not significant. The overall promotions (Table 3 and Table 4) are close to that of Chronos. Besides, the model seems to have a disadvantage in efficiency compared with the straightforward quantizing tokenization of Chronos.

Firstly, as noted in our response to a similar comment from reviewer zVkm, the core motivation of our work was to develop a general purpose tokenizer that seamlessly captures global and local patterns in the time series. Our qualitative results (Figure 1 and Section 4.3) demonstrate the impressive performance of WaveToken on complex edge cases where existing foundation models fail, almost completely. These results highlight the remarkable ability of a wavelet-based tokenizer to capture nonstationary signals – which are pervasive in practical applications. Nevertheless, as outlined in the paper, WaveToken also improves quantitatively on Chronos 75% of the times and 83% of the times on the in-domain (15 datasets) and zero-shot (27 datasets) benchmarks, respectively, while employing a vocabulary one-quarter the size of Chronos. In addition, WaveToken achieves the best average rank across all metrics for both Benchmarks I and II (Figures 10 and 11). Note that our comparison with Chronos, which uses a simple uniform quantization scheme directly on the time series values, readily serves as an ablation against a simpler tokenization scheme (the simplest possible scheme in this context).

We believe that in the natural progression of the field, different works make different types of contributions, and not every work brings (or needs to bring) a paradigm-shifting accuracy improvement. In the case of WaveToken, our findings position it as a promising avenue for developing general-purpose, information-efficient forecasting models, and it should therefore be evaluated independently in addition to its accuracy.

Secondly, WakeToken does not have a disadvantage in terms of efficiency as the wavelet tokenization is a convolution-and-downsampling operation which runs in linear time and the inference speed differences are not perceivable even when the wavelet transform is done in the CPU, as in our current implementation. Below we report running times (averaged over 10 repetitions) for both Chronos (Base) and WaveToken (Base) when forecasting a batch of 32 time series, with context length equal to 512 and prediction length equal to 64:

• Chronos: 6.56s +- 1.02ms
• WaveToken: 6.86s +- 1.14ms

The difference in inference times is negligible, and could be further reduced by applying the wavelet decomposition in parallel on the GPU.

Thank you for the detailed response and additional empirical experiments, which addressed my concerns regarding the performance brought by wavelet tokenization and the difference from previous LLM-based time series models. The results of long-term forecasting are inspiring. The response regarding the discussion of discrete/continuous tokenization is convincing to me, which highlights that discrete vocabulary can be more adaptive to the distributions of time series. Thus, the proposed wavelet-based approach may shield light on this community.

After reading all the comments and responses, I'd like to raise my score to the acceptance level.

Dear Reviewer KBzW,

Thank you again for the useful comments and questions that helped improve our paper. As the end of the discussion period is approaching, we would like to gently remind you to take into account our responses and the updated version of the manuscript into your evaluation.

We hope that we have satisfactorily addressed your questions and concerns. If so, we request you to consider raising your score to reflect that. If you have further questions, we will be happy to respond to them.

Thank you.

Best Regards, The Authors

Thank you for your useful comments and suggestions that helped improve our paper. We have summarized our main revisions and answers in response to all reviewers in a separate comment above. Below we provide point-by-point answers to each section in your review.

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Regarding

The practical impact of the thresholding decisions on model generalization is uncertain. For example, aggressive thresholding could obscure subtle patterns, which could limit the model's robustness across diverse datasets. So, how do you ensure such an effect would not happen?

We did not explicitly threshold the coefficients as thresholding did not significantly improve the forecasting performance (see Section 4.4). As noted in lines 509-513, this can be attributed to the quantization step of the tokenizer: all wavelet coefficients close to 0 are mapped to the bin centered at 0. This value is then used at inference time to forecast tokens mapped to this bin, thereby implicitly thresholding small coefficients. Nevertheless, we agree that aggressive thresholding, or quantization in our context, may obscure subtle patterns. However, when coupled with instance normalization (i.e., standardization), this is a limited cause of concern on most real-world datasets, as demonstrated by the excellent quantitative and qualitative performance of WaveToken on 42 benchmark datasets.

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Regarding

Using cross-entropy loss for next-token prediction in a continuous-data domain is unconventional, as it aligns more with the categorical outputs. So, how it guides learning for continuous variables is still ambiguous for me. Is it possible to compare with other loss functions, e.g. MSE?

While unconventional, using the cross-entropy loss for continuous data is not a new idea and is known in the literature as regression via classification [1, 2]. Previous works have shown the benefits of this approach across tasks, such as audio modeling (Wavenet [3]), Deep RL [4], and recently in the context of forecasting (Chronos [5]). Although the cross-entropy loss removes topological information, the model implicitly learns the topology from the data through large scale training. Using a categorical distribution allows the model to be flexible to accommodate multimodalities, strong asymmetries, sudden shifts, and more, as the distribution is not constrained to a specific shape.

Additionally, the use of cross-entropy loss allows to sidestep common issues with standard forecasting objectives (like MSE or quantile loss) such as their dependence on the scale of the inputs and on the presence of outliers, which can significantly degrade learning. Conversely, other approaches – e.g., DeepAR or more recent foundation models like Moirai – impose restrictive assumptions on the output forecasts via parametric distributions (usually Gaussian or Student’s-t), which are often not empirically valid. In our comparisons with these models, we already compare WaveToken against these alternative objective functions.

[1] Torgo, L., & Gama, J. (1997). Regression using classification algorithms. Intelligent Data Analysis, 1(4), 275-292

[2] Stewart, L., Bach, F., Berthet, Q., & Vert, J. P. (2023, April). Regression as classification: Influence of task formulation on neural network features. In International Conference on Artificial Intelligence and Statistics (pp. 11563-11582). PMLR

[3] Van den Oord A., Dieleman S., Zen H., Simonyan K., Vinyals O., Graves A., Kalchbrenner N., Senior A., Kavukcuoglu K. (2016), WaveNet: A Generative Model for Raw Audio, arXiv preprint arXiv:1609.03499

[4] Farebrother, J., Orbay, J., Vuong, Q., Taïga, A. A., Chebotar, Y., Xiao, T., ... & Agarwal, R. (2024). Stop regressing: Training value functions via classification for scalable deep RL. arXiv preprint arXiv:2403.03950

[5] Ansari, A. F., Stella, L., Turkmen, C., Zhang, X., Mercado, P., Shen, H., ... & Wang, Y. (2024). Chronos: Learning the language of time series. arXiv preprint arXiv:2403.07815

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Regarding

Will the code be released?

Yes, we plan to release a user-friendly research package complete with details on how to train and evaluate our tokenizer and models. Unfortunately, we could not share our code for review at this stage due to pending legal approvals.

Dear Reviewer uEP1,

Thank you again for the useful comments and questions that helped improve our paper. As the end of the discussion period is approaching, we would like to gently remind you to take into account our responses and the updated version of the manuscript into your evaluation.

We hope that we have satisfactorily addressed your questions and concerns. If you have further questions, we will be happy to respond to them.

Thank you.

Best Regards, The Authors

Thank you for your comments and questions that helped improve our paper. We have summarized our main revisions and answers in response to all reviewers in a separate comment above. Below we provide point-by-point answers to each section in your review.

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Regarding

Core idea of wavelet-based tokenization for large models has been explored in previous work, e.g., Wave-ViT (Zhu ETL. 2024), though in different domains. Insufficient discussion of how this approach differs fundamentally from or improves upon prior works in wavelet-based tokenizer

and

How does your wavelet-based tokenization fundamentally differ from prior approaches like Wave-ViT, particularly in handling temporal vs spatial data? Are concepts like "pixel-space token embedding" fundamentally different from the wavelet-based tokens in TS?

Thank you for bringing this very recent paper to our attention. We have added a citation to this work in the Related Work section. Nonetheless, this work has been developed with a completely different motivation than ours and addresses a different problem on another domain (as you have pointed out), i.e., image classification via Vision Transformers. As such, the conceptual similarity of WaveToken with the tokenizer introduced by Zhu. et al (2024) does not diminish the contribution or significance of our work. Furthermore, while conceptually similar, the image tokenizer in Zhu et al. (2024) is substantially different and cannot be directly compared to our work, which instead

1. proposes a tokenizer specifically tailored for time series forecasting with large foundation models based on the T5 architecture. The proposed design in Zhu et al. (2024) can’t be applied directly for T5 because 2D wavelet filters cannot be applied to time series;
2. leverages quantization of the wavelet coefficients in bins according to the coefficients’ distribution; and
3. preserves the length of the input and output sequences regardless of the decomposition level applied, thanks to the use of a maximally decimated wavelet transforms (see Appendix A.2).

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Regarding

The marginal improvement over Chronos raises questions about whether the gains justify the additional benefits of wavelet-based tokenization. An ablation study comparing against simpler tokenization approaches would help validate the necessity of wavelets.

and

Could you provide ablation studies comparing against simpler tokenization approaches to justify the added complexity of wavelets?

The core motivation of our work was to develop a general purpose tokenizer that seamlessly captures global and local patterns in the time series. Our qualitative results (Figure 1 and Section 4.3) demonstrate the impressive performance of WaveToken on complex edge cases where existing foundation models fail, almost completely. These results highlight the remarkable ability of a wavelet-based tokenizer to capture nonstationary signals – which are pervasive in practical applications. Nevertheless, as outlined in the paper, WaveToken also improves quantitatively on Chronos 75% of the times and 83% of the times on the in-domain (15 datasets) and zero-shot (27 datasets) benchmarks, respectively, while employing a vocabulary one-quarter the size of Chronos. In addition, WaveToken achieves the best average rank across all metrics for both Benchmarks I and II (Figures 10 and 11).

Note that our comparison with Chronos, which uses a simple uniform quantization scheme directly on the time series values, readily serves as an ablation against a simpler tokenization scheme (the simplest possible scheme in this context).

We believe that in the natural progression of the field, different works make different types of contributions, and not every work brings (or needs to bring) a paradigm-shifting accuracy improvement. In the case of WaveToken, our findings position it as a promising avenue for developing general-purpose, information-efficient forecasting models, and it should therefore be evaluated independently in addition to its accuracy.

Dear Reviewer zVkm,

Thank you again for the useful comments and questions that helped improve our paper. As the end of the discussion period is approaching, we would like to gently remind you to take into account our responses and the updated version of the manuscript into your evaluation.

We hope that we have satisfactorily addressed your questions and concerns. If so, we request you to consider raising your score to reflect that. If you have further questions, we will be happy to respond to them.

Thank you.

Best Regards, The Authors

Dear Reviewer zVkm,

Thank you again for your review. We kindly remind you to take into account our responses and the updated version of the manuscript into your evaluation.

We hope that we have satisfactorily addressed your questions and concerns. If so, we request you to consider raising your score to reflect that. If you have further questions, we will be happy to respond to them.

Thank you.

Best Regards,
The Authors

Dear Reviewer zVkm,

Thank you again for your time and engagement. As you know, the discussion period will end in two days on Monday December 2nd. We would like to gently remind you to take into account our responses to your additional concerns.

We hope our responses above have satisfactorily addressed your additional questions. If so, we request you to consider raising your score to reflect that. If you have further questions, we would be happy to respond to them.

Thank you.

Usage of cross-entropy loss when forecasting continuous data via next-token prediction

Some reviewers asked questions regarding the meaning, advantages and possible pitfalls of using the cross-entropy loss when forecasting continuous data.

While unconventional, using the cross-entropy loss for continuous data is not a new idea and is known in the literature as regression via classification [1, 2]. Previous works have shown the benefits of this approach across tasks, such as audio modeling (Wavenet [3]), Deep RL [4], and recently in the context of forecasting (Chronos [5]). Although the cross-entropy loss removes topological information, the model implicitly learns the topology from the data through large scale training. Using a categorical distribution allows the model to be flexible to accommodate multimodalities, strong asymmetries, sudden shifts, and more, as the distribution is not constrained to a specific shape.

Additionally, the use of cross-entropy loss allows to sidestep common issues with standard forecasting objectives (like MSE or quantile loss) such as their dependence on the scale of the inputs and on the presence of outliers, which can significantly degrade learning. Conversely, other approaches – e.g., DeepAR or more recent foundation models like Moirai – impose restrictive assumptions on the output forecasts via parametric distributions (usually Gaussian or Student’s-t), which are often not empirically valid. In our comparisons with these models, we already compare WaveToken against these alternative objective functions.

[1] Torgo, L., & Gama, J. (1997). Regression using classification algorithms. Intelligent Data Analysis, 1(4), 275-292

[2] Stewart, L., Bach, F., Berthet, Q., & Vert, J. P. (2023, April). Regression as classification: Influence of task formulation on neural network features. In International Conference on Artificial Intelligence and Statistics (pp. 11563-11582). PMLR

[3] Van den Oord A., Dieleman S., Zen H., Simonyan K., Vinyals O., Graves A., Kalchbrenner N., Senior A., Kavukcuoglu K. (2016), WaveNet: A Generative Model for Raw Audio, arXiv preprint arXiv:1609.03499

[4] Farebrother, J., Orbay, J., Vuong, Q., Taïga, A. A., Chebotar, Y., Xiao, T., ... & Agarwal, R. (2024). Stop regressing: Training value functions via classification for scalable deep rl. arXiv preprint arXiv:2403.03950

[5] Ansari, A. F., Stella, L., Turkmen, C., Zhang, X., Mercado, P., Shen, H., ... & Wang, Y. (2024). Chronos: Learning the language of time series. arXiv preprint arXiv:2403.07815

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Code Release

We plan to release a user-friendly research package complete with details on how to train and evaluate our tokenizer and models. Unfortunately, we could not share our code for review at this stage due to pending legal approvals.