AdaPTS: Adapting Univariate Foundation Models to Probabilistic Multivariate Time Series Forecasting

Abdelhakim Benechehab; Albert Thomas; Bal\'azs K\'egl; Giuseppe Paolo; Maurizio Filippone; Vasilii Feofanov

REVIEW 4 major objections 8 minor 51 references

AdaPTS: Adapting Univariate Foundation Models to Probabilistic Multivariate Time Series Forecasting

T0 review · 4 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that frozen univariate time-series foundation models can be extended to probabilistic multivariate forecasting by wrapping them in trainable feature-space adapters, and shows the recipe works on the Moment model…

desk verdict Useful adapter framework for multivariate forecasting with a frozen univariate FM, but the headline 5/8 result rests on best-of-many selection without a fixed rule, and the theory doesn't cover the actual nonlinear FM. read the letter →

arxiv 2502.10235 v1 pith:AKVZYJEB submitted 2025-02-14 stat.ML cs.LG

Abdelhakim Benechehab , Vasilii Feofanov , Giuseppe Paolo , Albert Thomas , Maurizio Filippone , Balázs Kégl This is my paper · ORCID

classification stat.MLcs.LG MSC 62M1068T0762F15

keywords timeseriesforecastingfoundationmodelsadaptersmultivariateprobabilisticuncertaintyquantificationvariationalinferenceMonteCarlodropout

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Pre-trained time-series foundation models are usually univariate: they ingest one channel at a time, so applying them to multi-channel data means either processing channels independently, which ignores cross-feature structure, or fine-tuning the whole model. AdaPTS proposes to insert a trainable feature-space transformation, called an adapter, before a frozen univariate foundation model and invert it after forecasting, so the multivariate problem is solved in a learned latent space with no change to the backbone. The paper reports that this procedure improves the forecasting accuracy of Moment on 5 of the 8 tasks tested, matches it on 2, and degrades on 1, while stochastic versions of the adapters add probabilistic predictions and uncertainty estimates. The contribution is a modular recipe: keep the foundation model frozen, learn only the adapter, and thereby reuse univariate foundation models for multivariate probabilistic forecasting.

What carries the argument

The carrying object is the adapter, formally defined as an invertible feature-space transformation $\varphi: \mathbb{R}^D \to \mathbb{R}^{D'}$ applied before the foundation model and inverted after it, so that predictions are obtained as $\mathrm{dec}(f_{FM}(\mathrm{enc}(X)))$. The probabilistic extension makes the encoder stochastic: a VAE adapter maximizes the ELBO $\mathbb{E}_{q_\varphi(Z|X)}[\log p_\theta(Y|X, f_{FM}(Z))] - \mathrm{KL}(q_\varphi(Z|X) \| p(Z))$, and a dropout linear autoencoder treats dropout as approximate variational inference over adapter weights. The foundation model's parameters are frozen throughout; only the encoder and decoder are trained.

What would settle it

Run the synthetic linear experiment of Section 3.3 with a known linear foundation model and channel-dependent biases so that $B$ has full rank, compute $W^*_\varphi$ from Eq. (4), and verify it attains the global minimum of the loss and beats the identity adapter; if the pseudo-inverse stabilization changes the optimum, the claimed global optimality is not established. For the nonlinear setting, train AdaPTS with the encoder removed and only a linear decoder; if accuracy gains persist, the paper's feature-transformation explanation is not the operative mechanism.

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Extended reading notes

Core claim

Under linearity assumptions—linear adapter and linear foundation model—the paper derives the closed-form optimal adapter $W^*_\varphi = (B^\top A)^+ B^\top B$, where $A = Y - W_{FM}^\top X$ and $B = b_{FM} 1^\top$, and proves that it beats the identity adapter on the forecasting objective. For the actual nonlinear case, AdaPTS instantiates adapters as linear and deep autoencoders and as probabilistic variants (VAE and dropout linear autoencoder) trained with an ELBO-style objective, and the experiments show MSE improvements on ETTh1 with $H=96$, Illness with $H=24$ and $H=60$, and Weather with $H=96$, matched performance on ETTh1 with $H=192$ and ExchangeRate with $H=96$, and degradation on ExchangeRate with $H=192$. The paper also claims that the learned latent representations overlap much better between training and test windows than the raw features, which it links to robustness against distribution shift, and that the probabilistic adapters are reasonably calibrated at short horizons.

Load-bearing premise

The load-bearing premise is that a linear model of the foundation model faithfully represents the setting where adapters help, but the actual foundation model in every experiment, Moment, is a deep nonlinear transformer, so the closed-form optimality proof does not directly cover the regime it motivates.

Editorial extensions

If this is right

Any frozen univariate time-series foundation model can be lifted to multivariate forecasting by learning an adapter at its input and output, without retraining the backbone.
Reducing the latent dimension below the number of observed channels can preserve or improve accuracy while lowering inference cost; on Illness the VAE adapter reaches its best MSE with only two latent components.
Deterministic foundation models can return predictive distributions by using a stochastic adapter, with calibration that the paper shows is reasonable at short horizons.
The decoder is the main source of accuracy gain in the deterministic linear adapter, so learning to invert the forecast back into the original feature space matters more than the encoder's compression.
The benefit is not universal: on ExchangeRate with $H=192$ every adapter variant degrades relative to the no-adapter baseline, so gains depend on dataset and horizon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's theory covers linear foundation models only; the proof of adapter optimality does not transfer unchanged to nonlinear Moment, so the explanation for the empirical gains in the real experiments remains an open question.
If the decoder is the main source of gain, then adapters might be replaceable by a lightweight learned mapping on the FM's output alone, which would be cheaper still; the paper does not test this.
Because calibration degrades as the horizon grows, a deployment of AdaPTS would likely need horizon-dependent likelihoods or post-hoc recalibration to avoid overconfident long-range forecasts.
The latent-space overlap result suggests a testable extension: adapters should help most on multivariate tasks with strong train/test distribution shift, and could be evaluated on cross-domain forecasting benchmarks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The thing to know about AdaPTS is that it packages a sensible idea — put a trainable invertible encoder–decoder around a frozen univariate foundation model and treat the latent space as stochastic to get probabilistic forecasts — and shows it can beat the plain per-channel Moment baseline on several tasks. That is genuinely useful for practitioners who want uncertainty estimates without fine-tuning a large model. The paper is also honest in places: it reports standard errors, includes ablations on latent dimension and β/σ, and admits the calibration is overconfident at longer horizons, which qualifies the abstract's uncertainty claim.

The new ingredient is the adapter family itself (linear AE, dropout LAE, linear VAE, VAE) applied to multivariate forecasting; the prior PCA adapters from the same group were for classification and RL. That framing is clear, and the appendix at least names the hyperparameters.

Now the soft spots, in decreasing importance.

First, the central empirical claim — "improves on Moment in 5 of 8 tasks" — is an oracle count. Each task is scored by the best of several adapter variants, each tuned with its own hyperparameters. There is no pre-specified rule for choosing the adapter family on validation data. With five families and HPO, some variant beating the baseline on a task is weak evidence that the framework as a decision procedure helps. The code is only released after acceptance, so the selection process cannot be audited. This is the main load-bearing weakness.

Second, the theory. Proposition 3.4 gives a closed-form optimal adapter under a linear-FM assumption, and the proof's convexity claim is shaky (the loss is convex in W_phi^{-1}, not W_phi; the gradient argument treats the inverse as the variable). The paper does not claim the result transfers to Moment, only labels it "for ease of derivation," so it is a motivating example rather than a false claim. Fine, but then it shouldn't be presented as "theoretical foundations of adapters." A short paragraph saying "the linear case motivates the architecture; the nonlinear case is empirical" would be more accurate.

Third, the baselines. Only Moment with and without adapters, plus PCA. No comparison to other multivariate forecasting methods (iTransformer, PatchTST, etc.) or to established probabilistic baselines (DeepAR, etc.), so the practical gain is not benchmarked against the field.

Minor: the "8% improvement" for ETTh1 H=96 is not what Table 1 shows (0.411 to 0.395 is ~4%). And in the MAE table, LinearAE on Illness H=24 lists 2.624±0.035, the same number as the MSE table, likely a copy-paste error.

Who is this for? Researchers and engineers who want a cheap way to reuse a frozen univariate FM for multivariate probabilistic forecasting. The framework is simple and modular; if the code and a fixed selection rule appear in revision, the empirical claims could be verified.

My recommendation: send to peer review. The idea is useful, the authors are clear-eyed about limitations, and the flaws are fixable. A referee should push for a pre-specified adapter-selection rule, a release of code, a comparison against at least one strong multivariate baseline, and toning down the theory claims.

Referee Report

4 major / 8 minor

Summary. The paper introduces AdaPTS, a framework of feature-space adapters that map multivariate time series into a latent space, apply a frozen univariate foundation model (Moment) per channel, and map predictions back to the original space. Several adapter families are proposed (linear autoencoder, dropout linear autoencoder, linear VAE, VAE, plus PCA as a baseline), together with a theoretical analysis for the linear-adapter/linear-FM case and experiments on four multivariate benchmarks (ETTh1, Illness, Weather, ExchangeRate). The main empirical claim is that AdaPTS improves Moment's MSE in 5 of 8 tasks, matches it in 2, and degrades in 1, while providing probabilistic outputs and reduced-dimensional representations.

Significance. If the claims are correct, AdaPTS would be a simple, modular way to extend univariate time-series foundation models to multivariate probabilistic forecasting without fine-tuning the FM, with potential computational savings from dimensionality reduction. The paper is clearly written, the problem is well motivated, and the idea of using invertible or encoder-decoder feature transformations is interesting. The strengths are the breadth of adapter variants and the effort to provide both theory and experiments. However, the theoretical result in Proposition 3.4 contains a serious mathematical flaw (the claimed closed-form solution is singular), and the empirical '5/8' headline rests on an oracle-selection protocol rather than a single reproducible decision procedure. As presented, the evidence does not yet support the central claims; the concept may be salvageable with substantive corrections.

major comments (4)

[§3.2 / Proposition 3.4 and Appendix A.1] The closed-form solution W* = (B^T A)^+ B^T B is singular for D > 1 whenever B = b_FM 1^T, because B^T B has rank one and therefore rank(W*) <= 1. This violates Assumption 3.2, which requires W_phi to be in GL(D), and contradicts the invertibility requirement in Definition 3.1. The proposition therefore does not solve the constrained optimization problem over invertible adapters. The lambda-regularized form in Remark 3.5 solves a different objective, so it does not repair the theoretical claim.
[Appendix A.1] The proof's assertion that L(W_phi) is convex in W_phi is incorrect: the map W -> W^{-1} is not affine, and the squared Frobenius norm of A - B W^{-1} is not convex in W over GL(D). For example, in the scalar case f(w) = (a - b/w)^2 the second derivative is (6b^2 - 4abw)/w^4, which can be negative. Consequently, the stationary-point condition derived in the proof does not establish global optimality, and the 'global solution' wording of Proposition 3.4 is unsupported.
[§5.1 / Table 1] The summary 'AdaPTS improves the forecasting accuracy of Moment in 5 out of the 8 considered tasks' is an oracle-selection count: for each task the best-performing adapter among PCA, LinearAE, dropoutLinearAE, LinearVAE, and VAE is chosen post hoc, with no pre-specified rule for selecting the adapter family from the validation set alone. Because each family has its own tuned hyperparameters, the probability that at least one variant beats the baseline on a task by chance is high, so the 5/8 count does not describe the behavior of a single reproducible AdaPTS procedure. Please report the test performance of a validation-selected adapter (with the selection rule stated), or fix the adapter family across all tasks, and also correct the '8% improvement' claim for ETTh1 H=96, since Table 1 shows a change from 0.411 to 0.395, about 4%.
[Assumption 3.3] The theory assumes a linear foundation model f_FM(X) = W_FM^T X + b_FM 1^T, but all experiments use Moment, a deep nonlinear transformer. The paper labels the assumption 'for ease of derivation' but does not provide a nonlinear analogue or an argument that the linear result transfers to the actual experimental setting. Appendix D.1 replaces the closed form with gradient-based optimization for Moment, but this is an empirical observation, not a theoretical guarantee. If the theory is meant only as intuition, that should be stated clearly; if it is claimed as a foundation, a nonlinear analysis or an explicit scope limitation is needed.

minor comments (8)

[§3.1 / Eq. (1)] The notation f_FM(X) is used both for the channel-wise application of a univariate model to a multivariate input and for the matrix form in Assumption 3.3; the dimensions of X, Y, and the adapter outputs should be stated more carefully.
[§5.1] The '8% improvement' for ETTh1 H=96 is inconsistent with Table 1: the MSE changes from 0.411 to 0.395, which is approximately a 4% improvement. Please correct this number.
[Fig. 5] The reliability diagram's caption does not explain the color or shading used for different prediction horizons, and the figure reports only the first feature of one dataset, which is a limited basis for the general calibration claim.
[Reproducibility statement] The abstract states 'We release the code', while the reproducibility statement says the code 'will be released' after acceptance; these statements should be made consistent.
[Appendix C.2] There is a typo: 'mean squred error' should be 'mean squared error'.
[Appendix A.1] The chain rule expression for the gradient with respect to W_phi has a sign error: the correct expression is +2 W^{-T}(B^T A - B^T B W^{-1}) W^{-T}, not the negative. The stationary condition is unaffected, but the error should be fixed.
[Table 3 / Fig. 6] The main experiments use beta=0.5 and sigma=1.0 according to Table 3, while the ablation in Fig. 6 explores beta up to 4.0 and log sigma^2 up to 3.0; it would be helpful to state explicitly which hyperparameter values were used for each row of Table 1.
[§3.3] The synthetic experiment appears to compute the loss on the same data used to derive W*; please specify whether the reported MSE is in-sample or on a held-out test set.

Circularity Check

2 steps flagged · score 4.0 of 10

Supporting 'optimal adapter' demonstrations fit the reported MSE, so their improvement over identity is by construction; the central held-out benchmark claim is not circular, though the 5/8 summary is a best-per-task selection.

fitted input called prediction [Section 3.3 (Working example, Fig. 2); Eqs. (3)-(4)]
"We then compute the closed-form solution W∗φ (Eq. (4)) on raw data X, and compare the resulting loss value with the baseline (using the identity matrix I as adapter) and the PCA-only adapter."

W∗φ is defined by Eq. (4) as the minimizer of L(Wφ)=||Y−(W⊤FMXWφ+bFM1⊤)W−1φ||2F in Eq. (3), i.e. the very MSE in which Y is the target. The identity adapter I is feasible in GLD(R), so L(W∗)≤L(I) is guaranteed by the optimization. The paper uses the same (X,Y) for both computing W∗ and reporting the MSE, with no held-out split described, so Fig. 2's 'optimality' is a restatement of the minimization rather than a predictive validation of adapters.
fitted input called prediction [Appendix D.1 (Moment applied to synthetic data, Eqs. (14)-(15), Fig. 8)]
"To approximately solve this optimization problem, we instantiate Wφ as a single-linear-layer encoder denoted encθ, and respectively the inverse transformation W−1φ as a single-linear-layer decoder denoted decθ. We then use gradient-based optimization of the parameters θ using the Adam optimizer, aiming at solving the following optimization problem: θ∗ = arg minθ ||Y − decθ(fMoment(encθ(X)))||2F (15). Fig. 8 shows the performance gain obtained by optimizing a linear adapter on Moment-small foundation model."

Here θ∗ minimizes exactly the MSE ||Y−decθ(fMoment(encθ(X)))||2F that is then plotted in Fig. 8, and the identity encoder/decoder is a feasible solution, so the optimized adapter is guaranteed to match or beat the baseline on the same synthetic data. The appendix presents this as validating the 'adapter optimality condition' for a nonlinear FM, but no held-out split is described; the reported performance gain is a training-fit artifact, not a predictive result.

full rationale

The main benchmark comparison in Table 1 is not circular: adapters are trained on training splits, hyperparameters are selected on validation data, and test MSE is reported on held-out test sets. The 5/8 summary, however, is an oracle-style best-over-variants count, since different adapter families win on different tasks and no fixed selection rule is specified; this is a selection/statistical concern rather than an input-output tautology. The self-citations to Feofanov et al. (2024) and Benechehab et al. (2025) are background context and not load-bearing, and the linear-FM assumption (3.3) limits the theory's applicability to the nonlinear Moment model without being circular. The actual circularity is confined to the supporting 'optimal adapter' demonstrations in Sections 3.3 and D.1, where the adapter parameters are fit by minimizing the same MSE that is later reported, with identity in the feasible class, so the claimed improvement is forced by construction. Section 5.1's '8% improvement' for ETTh1 H=96 also does not match Table 1 (0.411 to 0.395 is about 4%), but that is a numerical-consistency issue rather than circularity. Because the central empirical claim has independent held-out content, the overall circularity score is moderate rather than high.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The 'adapter' is a trainable parameterized function, not a postulated entity, and the stochastic latent space is a standard modeling device. Free parameters are the VAE and dropout hyperparameters; the linear closed form adds an unspecified regularization lambda. The key axioms are the linear-FM assumption, which does not hold for the evaluated Moment model, the full-rank invertibility condition, the cited BNN density estimation theorem, and standard VAE assumptions.

free parameters (5)

beta (beta-VAE KL weight) = 0.5
Chosen by hand and tuned on validation; Fig. 6 shows it controls MSE and ECE.
sigma (likelihood noise) = 1.0 (fixed)
Noise variance in the Gaussian likelihood; fixed during training, with ablations over log sigma^2 from 0.5 to 3.0.
dropout probability = 0.1
Dropout rate for the dropoutLinearAE adapter; listed as a default in Table 3.
latent dimensionality D' = varies (2 to 14)
Number of latent components; selected by validation in Fig. 3, where 2 components suffice for the VAE on Illness.
lambda (regularization in Eq. 4) = not specified
Positive constant added to the diagonal in Remark 3.5 to stabilize inversion; no numeric value is given.

assumptions (4)

ad hoc to paper The foundation model is linear: f_FM(X) = W_FM^T X + b_FM 1^T (Assumption 3.3).
Used to derive Proposition 3.4. Moment is not linear, so the closed-form adapter solution does not apply to the experiments.
domain assumption W_phi has full rank (Assumption 3.2).
Needed for invertibility of the adapter; learned adapters are not guaranteed full rank and the implementation uses pseudo-inverses.
domain assumption Partially stochastic BNNs give universal conditional density estimation (Sharma et al., 2023) if stochasticity is early enough and stochastic units match or exceed output dimension.
Cited to justify that a stochastic encoder plus frozen FM can represent any conditional density; not proved or tested in this paper.
standard math The VAE posterior is approximated by a Gaussian variational family q_phi(Z|X) and prior p(Z)=N(0,I).
Standard VAE assumptions used in the ELBO derivation of Proposition 4.1.

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Cite this review

Pith. "Pith review of AdaPTS: Adapting Univariate Foundation Models to Probabilistic Multivariate Time Series Forecasting." pith.science (2026). https://pith.science/paper/AKVZYJEB

@misc{pith2026250210235,
  author       = {Pith},
  title        = {Pith review of: AdaPTS: Adapting Univariate Foundation Models to Probabilistic Multivariate Time Series Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKVZYJEB}},
  note         = {Machine review of arXiv:2502.10235}
}

read the original abstract

Pre-trained foundation models (FMs) have shown exceptional performance in univariate time series forecasting tasks. However, several practical challenges persist, including managing intricate dependencies among features and quantifying uncertainty in predictions. This study aims to tackle these critical limitations by introducing adapters; feature-space transformations that facilitate the effective use of pre-trained univariate time series FMs for multivariate tasks. Adapters operate by projecting multivariate inputs into a suitable latent space and applying the FM independently to each dimension. Inspired by the literature on representation learning and partially stochastic Bayesian neural networks, we present a range of adapters and optimization/inference strategies. Experiments conducted on both synthetic and real-world datasets confirm the efficacy of adapters, demonstrating substantial enhancements in forecasting accuracy and uncertainty quantification compared to baseline methods. Our framework, AdaPTS, positions adapters as a modular, scalable, and effective solution for leveraging time series FMs in multivariate contexts, thereby promoting their wider adoption in real-world applications. We release the code at https://github.com/abenechehab/AdaPTS.

Figures

Figures reproduced from arXiv: 2502.10235 by the authors.

**Figure 1.** (a) Augmenting Moment time series foundation model with the AdaPTS framework provides probabilistic and more accurate predictions. (b) The AdaPTS framework: The input time series is transformed through a feature space transformation φ that maps into a stochastic latent space. The prediction is then conducted using a pre-trained FM before transforming back the predicted, now distribution, to the original feature sp… view at source ↗

**Figure 2.** Optimality of W∗ φ. Comparing the MSE obtained with W∗ φ against the baseline, for 1000 randomly generated linear FM [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Impact of the number of components on model per [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Visualization of the latent representation obtained by different adapters (with number of components equal to 2) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Reliability diagram for the first feature of the ETTh1 (H = 96) dataset using LinearVAE. To evaluate the calibration of our adapter-based probabilistic forecasters, we use quantile calibration as depicted in the reliability diagram in [PITH_FULL_IMAGE:figures/full_f…

**Figure 7.** Figure 7: LinearAE components ablation. Nevertheless, as shown in our previous experiments, particularly [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 6.** Figure 6: 0.5 1.0 1.5 2.0 3.0 auto log 2 0.5 1.0 2.0 4.0 2.46 2.43 2.43 2.45 2.43 2.58 2.45 2.42 2.44 2.41 2.40 2.59 2.42 2.44 2.41 2.40 2.39 2.56 2.44 2.39 2.42 2.43 2.41 2.68 MSE 0.5 1.0 1.5 2.0 3.0 auto log 2 0.28 0.28 0.28 0.28 0.27 0.28 0.28 0.28 0.28 0.27 0.27 0.28 0.28 0.…

**Figure 8.** Figure 8: Moment on simulated independent data. Training parameters. After the pre-processing phase, we proceed to split the data into a train-validation-test sets, where the validation set serves as a tool to select the best hyperparameters for the adapter. The resulting adapte…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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