$K^2$VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting

Bin Yang; Chenjuan Guo; Hongfan Gao; Jilin Hu; Xiangfei Qiu; Xingjian Wu

REVIEW 3 major objections 4 minor 2 cited by

$K^2$VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A variational autoencoder that linearizes its latent dynamics with a Koopman operator and filters them with a Kalman recursion claims state-of-the-art probabilistic forecasting on both short and long horizons at lower cost.

desk verdict Solid engineering paper with a novel Koopman-Kalman VAE for probabilistic forecasting; the empirical claim is overstated and the convergence theorem is circular, but the architecture and experimental breadth deserve proper review. read the letter →

arxiv 2505.23017 v3 pith:UXGCIJNM submitted 2025-05-29 cs.LG cs.AI

Xingjian Wu , Xiangfei Qiu , Hongfan Gao , Jilin Hu , Bin Yang , Chenjuan Guo This is my paper

classification cs.LGcs.AI

keywords probabilistictimeseriesforecastinglong-termvariationalautoencoderKoopmanoperatorKalmanfiltererroraccumulationuncertaintyquantificationgenerativemodels

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

The paper claims that long-term probabilistic time series forecasting fails because the nonlinear, non-stationary dynamics of real series make generative models both inaccurate and expensive as the prediction horizon grows. To fix this, K²VAE maps patched series tokens through a learned measurement function into a space where their evolution is approximately a linear dynamical system governed by a Koopman operator, then runs a learned Kalman filter over that system so that each forecast step refines the state and carries a covariance forward as the uncertainty of the prediction. The design gives the variational autoencoder's latent space a concrete meaning: the posterior mean is the filtered trajectory and the posterior covariance is the process uncertainty of the linear system. The authors report that this one-step generative model beats eleven baselines on most of the tested short- and long-term datasets, with the largest gains on long horizons, while using far less inference time and GPU memory than diffusion- or flow-based models. If the claim holds, probabilistic forecasting can stay accurate and cheap out to horizons of hundreds of steps instead of collapsing as the horizon grows.

What carries the argument

The load-bearing object is the pairing of a Koopman operator with a Kalman recursion inside a variational autoencoder, so that the latent space becomes the state space of a filtered linear system. The KoopmanNet provides the linearization: a learnable measurement function $\psi$ maps patch-embedded tokens into measurement space, and the transition operator $\mathbf{K} = K_{\mathrm{loc}} + K_{\mathrm{glo}}$ is assembled from the one-step eDMD formula $K_{\mathrm{loc}} = X_{\mathrm{fore}}(X_{\mathrm{back}})^\dagger$ plus a learnable global correction, so iterating $\mathbf{K}$ on the first token produces the whole context reconstruction and horizon forecast in one pass. The KalmanNet performs the refinement: its process model predicts $\hat{z}_k = A z_{k-1} + B u_k$ and $\hat{P}_k = A P_{k-1} A^\top + Q$, its observation model produces a Kalman gain $K_k = \hat{P}_k H^\top (H \hat{P}_k H^\top + R)^{-1}$, and the update $z_k = \hat{z}_k + K_k(\hat{x}^H_k - H\hat{z}_k)$ blends the linear forecast with the integrated nonlinear residual; the covariance update is rewritten in a numerically stable dual form to preserve positive definiteness. The refined states and covariances directly define the variational posterior, giving the VAE latent semantics, and a proven convergence statement says the KalmanNet's transition reduces to the Koopman operator as the residual control input vanishes.

What would settle it

Retrain K²VAE on a long-horizon benchmark with the update step $z_k = \hat{z}_k + K_k(\hat{x}^H_k - H\hat{z}_k)$ replaced by $z_k = \hat{z}_k$ (or with the previous refined state as the observation) while keeping the control input, the KoopmanNet, and all training settings identical; if long-horizon CRPS is essentially unchanged, the Kalman observation mechanism is not what produces the reported gains. A second check is to measure the empirical coverage of the 95% prediction intervals on held-out long-horizon test sets, since the paper visualizes intervals but never scores their calibration.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that probabilistic forecasting can be recast as Bayesian filtering over a linearized measurement space, and that this removes both the accuracy loss and the cost blow-up that appear when the forecast horizon extends. The KoopmanNet learns a measurement function $\psi$ that projects patch-embedded tokens into a space where consecutive tokens advance under a Koopman operator $\mathbf{K} = K_{\mathrm{loc}} + K_{\mathrm{glo}}$, with $K_{\mathrm{loc}}$ fit in closed form by one-step extended dynamic mode decomposition and $K_{\mathrm{glo}}$ learned; iterating $\mathbf{K}$ from the first token generates the context reconstruction and the horizon forecast. Because this learned linearization is imperfect, the authors call the system 'biased' and pass it through the KalmanNet, which integrates the nonlinear residual as a control input and treats the Koopman forecast as the observation. Iterated predict-and-update steps then yield refined states $z_k$ and covariance matrices $P_k$, which define the variational posterior $Q(Z|X) = \mathcal{N}(Z', P)$; the decoder acts as the inverse measurement function and outputs a Gaussian target distribution. The authors report that the resulting model surpasses all compared baselines in most settings, including a 7.3% CRPS reduction over the best short-term baseline and a 20.9% reduction over the best long-term baseline, and their ablations show that both the KoopmanNet and the KalmanNet are needed for the gains.

Load-bearing premise

The load-bearing premise is that feeding the KoopmanNet's own forecast back into the Kalman update as though it were a measurement of the true future actually corrects the state and curbs error accumulation; in a standard Kalman filter such correction is justified only when the observation comes from the real system, so the claimed mechanism treats one model output as evidence about the world.

Editorial extensions

If this is right

One-step generation replaces iterative sampling: because the whole horizon is drawn from a single Gaussian posterior, inference time and memory scale with the model size rather than with the number of denoising or flow steps, which is what the efficiency experiments measure.
Long horizons become tractable for probabilistic use: the reported advantage over point-forecasting models equipped with distributional heads grows as the horizon extends to 96–720 steps, the regimes used in energy, traffic, and public-health planning.
The latent space gains semantics: with the posterior mean as the filtered state trajectory and the posterior covariance as process uncertainty, the model outputs per-token prediction intervals as a by-product of the filtering recursion.
The ablations indicate that linearization and filtering are complements: removing either the KoopmanNet or the KalmanNet degrades scores substantially, and the mixed operator $K_{\mathrm{loc}} + K_{\mathrm{glo}}$ is what keeps long-horizon training numerically stable.

Reading between the lines

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

An editorial extension: the same predict–update recursion could run online after deployment, assimilating newly arriving observations into the state without retraining, a streaming-forecasting use the paper does not discuss.
The paper displays 95% confidence intervals but does not score their calibration; an editorially suggested check is measuring empirical coverage on the long-horizon test sets, which would separate genuine uncertainty quantification from a well-shaped predictive distribution.
The filtered-state representation is portable: because every token carries a mean and a covariance, the same encoder could be reused for imputation, anomaly detection, or scenario generation, not only for forecasting.
A cheap diagnostic suggested by the architecture is freezing the Kalman gain at zero while keeping the residual control input, to test how much of the long-horizon gain is filtering versus residual integration.

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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

Patrick—

This is a serious engineering contribution. The architecture—Koopman linearization via eDMD plus a learnable global correction, a learnable Kalman filter inside a VAE, all as a one-step generative forecaster—is new and well-suited for long-horizon probabilistic forecasting. The experimental effort is substantial: five seeds per run, ablations for every module, efficiency measurements, and released code and data. On the most demanding long-horizon setting (L=720), K2VAE beats all baselines on most datasets, and its memory/compute footprint is genuinely attractive (0.094GB vs 1.411GB for CSDI on Electricity-96).

That said, the paper overclaims. The abstract says K2VAE “outperforms state-of-the-art methods in both short- and long-term PTSF,” but the full per-horizon tables tell a more nuanced story. On Weather-L, CSDI has lower CRPS at L=96 (0.068 vs 0.080) and L=192 (0.068 vs 0.079); on Exchange-L at L=96, GRU-MAF, Trans-MAF, and CSDI all beat K2VAE. The main text only highlights L=720, where K2VAE wins most often, and no average ranks or significance tests are reported. So the blanket claim isn’t falsifiable as stated.

The theoretical analysis is the weakest part. Theorem 3.2 essentially assumes what it needs to prove: it sets U→0 and then cherry-picks H→I, A→0, R→0 so the Kalman update reduces to the Koopman prediction. That shows consistency, not convergence. Theorem 3.1 is a standard Joseph-stabilized update and is fine. The Kalman mechanism itself is also conceptually off: Eq. 20 treats the model’s own prediction as the observation, so the filter is aligning one model output with another rather than correcting with real data. It may still work as a learned refinement, but the “reducing error accumulation” story is not the classical Kalman one. The paper also misses the KalmanNet literature, which would sharpen the novelty claim about the Kalman component.

Overall, this is a paper worth engaging with. The architecture is novel, the empirical study is broad, and the reproducibility is good. The flaws are fixable: tone down the headline claim, add per-horizon significance or rank-based comparisons, rewrite Theorem 3.2 honestly, and clarify what the KalmanNet is actually doing. That’s exactly the kind of revision a serious referee should request, not a reason to desk-reject. I’d send it to review.

Referee Report

3 major / 4 minor

Summary. K^2VAE is a VAE-based probabilistic forecaster that patches multivariate series, projects tokens into a learned Koopman measurement space, fits a linear Koopman operator (local eDMD plus a global learnable part), and then runs a learnable Kalman filter/update to produce refined states and covariance matrices that serve as a variational posterior. A decoder maps samples to an isotropic Gaussian predictive distribution. The paper claims state-of-the-art performance in both short- and long-term PTSF and reports efficiency gains over diffusion/flow baselines, with experiments on ProbTS benchmarks, 5-seed runs, ablations, and qualitative interval plots.

Significance. The empirical contribution is potentially useful: a one-step generative forecaster that is lightweight and shows strong results on several datasets is a practically meaningful addition to LPTSF. Strengths include the external ProbTS benchmark, 5-seed error bars, full ablation tables, efficiency measurements, and a public code/data repository. However, the two load-bearing supports for the claims—the convergence theorem and the blanket 'outperforms SOTA' statement—are not established by the presented evidence. The theoretical result relies on an unproved assumption and a special-case construction, and the full per-horizon tables contain clear counterexamples to the unqualified empirical claim. These issues are correctable in revision, but they are substantive enough to require reworking the paper's claims and proof.

major comments (3)

[Abstract, §4.2, Table 9/10] The statement that K^2VAE 'outperforms state-of-the-art methods in both short- and long-term PTSF' is not supported by the complete per-horizon results. In Table 9, Weather-L at L=96 gives K^2VAE CRPS 0.080±0.007 against CSDI 0.068±0.008, and at L=192 0.079±0.009 against 0.068±0.006. On Exchange-L at L=96, K^2VAE (0.031±0.002) is worse than GRU MAF (0.026±0.010), Trans MAF (0.028±0.002), and CSDI (0.028±0.003). Even in the short-term Table 2, CSDI has lower CRPS on Electricity-S (0.051±0.001 vs 0.053±0.002). No average rank, aggregated CRPS, or significance test is reported, so the main-table emphasis on L=720 does not establish the blanket claim. Please qualify the claim or add an appropriate aggregate comparison with uncertainty.
[Appendix A.2, Theorem 3.2] Theorem 3.2 (and the corresponding Theorem A.2 in the appendix) is not proven. The proof begins by assuming U→0 without any argument that the training objective drives the Integrator residual to zero; Section 3.1.2 asserts this, and the assertion is not analyzed. More importantly, the 'general case' merely states that A can be treated as a fine-tuned Koopman operator, and convergence is only demonstrated for the special parameter selection H→I, A→0, R→0, under which Eq. (20) becomes z_k = x_hat^H_k by construction. This does not establish that the KalmanNet state transition converges to the Koopman operator for the learned matrices used in practice. Please either provide a genuine proof under stated assumptions or explicitly relabel this as a limiting-case remark rather than a theorem.
[§3.1.2, Eqs. (16)–(20)] The observation model in Eq. (16) defines o_k = H z_k + v_k, but the update in Eq. (20) uses x_hat^H_k, the KoopmanNet's own forecast, as the observation. This is not an observation of the true future state, so the standard Kalman-filter justification for state correction does not apply; the update is a learned alignment between two model outputs. Since the paper attributes the reduction of error accumulation to this Kalman mechanism, the mechanism needs a formal justification or a controlled experiment (e.g., replacing the update with a generic learned correction, or ablating the Kalman structure while keeping parameters) to show what the Kalman form actually contributes.

minor comments (4)

[Throughout] There are many typographical errors ('excell', 'consitute', 'roubustness', 'weightsumed', 'foreacasting', 'sytem', 'distengle', 'pionner') that should be corrected during copyediting.
[Appendix C.3] The metric is inconsistently written as 'CPRS' instead of 'CRPS'.
[Table 8] The code repository for TimeGrad is listed as the PatchTST repository, which appears to be an incorrect URL; please verify all baseline links.
[Table 9] Several long-term baseline cells are marked '-' (e.g., CSDI on Electricity-L and Traffic-L in Table 9); please state explicitly why these runs are missing and whether the omitted baselines change the comparisons that support the conclusion.

Circularity Check

1 steps flagged · score 5.0 of 10

Empirical results are externally benchmarked, but the key convergence theorem reduces by construction to the Koopman operator it claims to recover.

self definitional [Section 3.2.2 / Appendix A.2, Theorem 3.2 (A.2), Eqs. (11), (35)-(36)]
"We then consider the special case that KalmanNet fully relys on the observation ˆxH k from the linear system constructed by Koopman Operator K, thus H → I, A→ 0, R→ 0, the Predict and Update Steps are converted to: Predict: ˆzk = 0 (35) Update: zk = ˆxH k (36) The system constructed by KalmanNet can be treated as zt = Kzt−1 equivalent to the original Koopman Operator."

The 'observation' fed into the update is not an external measurement: Eq. (11) defines x_H_i = (K)^(i+n-1) x_P*_1, so it is generated by iterating the Koopman operator K itself. Choosing H -> I, A -> 0, R -> 0 in Eq. (34) forces z_k = x_H_k, and the sequence x_H_k satisfies z_k = K z_{k-1} by the definition of powers of K. The claimed convergence is therefore imposed by the degenerate parameter choice, not derived from the Kalman recursions. The premise U -> 0 is also asserted from LRec optimization in Sec. 3.1.2 rather than established, making the theorem's conclusion identical to its construction.

full rationale

The paper's central empirical claim is grounded in external benchmarks (ProbTS datasets and public baselines), so it is not circular; any overstatement of the per-horizon results (e.g., Table 9 shows CSDI beating K2VAE on Weather-L at L=96/192 and on Exchange-L at L=96) is a correctness and reporting concern, not a circularity one. There is no load-bearing self-citation chain: the Koopman and Kalman citations are to classic and independent prior work, and no uniqueness theorem is imported. The main circular step is the formal convergence result: Theorem 3.2's proof reaches its conclusion by selecting filter parameters (H->I, A->0, R->0) that make the Kalman update output exactly the KoopmanNet's own forecast, which is already defined as a power of K. This does not invalidate the empirical comparisons, but it means the theoretical justification for the KalmanNet 'refining' the Koopman system is partially self-referential. A moderate score of 5 reflects one construction-level circular step in an otherwise externally tested architecture.

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

The central contribution is an end-to-end learned forecasting architecture. The free parameters are the learned linear-dynamics and Kalman components; the axioms are the modeling assumptions needed for the Koopman and Kalman narratives, including two ad hoc assumptions that are not independently verified.

free parameters (5)

K_glo (global Koopman matrix) = learned (no closed form)
Added to eDMD-derived K_loc to make the linear transition stable and global; fitted by backpropagation (Section 3.1.2).
Kalman matrices A, B, H = learned (no closed form)
State transition, control, and observation matrices of KalmanNet are learnable; they are not derived from a physical model (Eqs. 14-16).
Noise covariance factors L_Q, L_R = learned, initialized as identity
Process and observation noise are parameterized via lower-triangular Cholesky factors to keep positivity (Section 3.1.2).
Prior P(Z|X) = N(0, I)
A standard normal prior is chosen for the latent measurement space; it is not derived from Koopman theory.
Weighting of L_ELBO and L_Rec = not specified
The objective is called a weighted sum, but the weighting coefficients are not given (Section 3.1.4).

assumptions (5)

domain assumption A finite-dimensional MLP psi with fixed dimension d accurately approximates a Koopman measurement function that linearizes the dynamics.
Koopman theory gives an infinite-dimensional linear space; the paper assumes a learned finite projection is sufficient (Section 2, Section 3.1.2).
domain assumption The patch-level dynamics in measurement space are approximately linear between consecutive patches.
eDMD computes K_loc as a linear map between successive patch embeddings, which requires local linearity (Eqs. 6-8).
ad hoc to paper The Integrator residual U vanishes during training.
Used in Theorem 3.2 to claim convergence to the Koopman operator; no empirical evidence is provided (Section 3.2.2).
ad hoc to paper The Koopman prediction x_hat_H_k can serve as the Kalman observation o_k.
Standard Kalman filtering requires real sensor observations; here the model's own forecast is used as the measurement (Eqs. 16, 20).
domain assumption The target distribution P(Y|Z) is isotropic Gaussian.
The decoder outputs a Gaussian likelihood; real multivariate series may have skewness or fat tails (Section 3.1.3).

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Cite this review

Pith. "Pith review of $K^2$VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting." pith.science (2026). https://pith.science/paper/UXGCIJNM

@misc{pith2026250523017,
  author       = {Pith},
  title        = {Pith review of: $K^2$VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXGCIJNM}},
  note         = {Machine review of arXiv:2505.23017}
}

abstract

Probabilistic Time Series Forecasting (PTSF) plays a crucial role in decision-making across various fields, including economics, energy, and transportation. Most existing methods excell at short-term forecasting, while overlooking the hurdles of Long-term Probabilistic Time Series Forecasting (LPTSF). As the forecast horizon extends, the inherent nonlinear dynamics have a significant adverse effect on prediction accuracy, and make generative models inefficient by increasing the cost of each iteration. To overcome these limitations, we introduce $K^2$VAE, an efficient VAE-based generative model that leverages a KoopmanNet to transform nonlinear time series into a linear dynamical system, and devises a KalmanNet to refine predictions and model uncertainty in such linear system, which reduces error accumulation in long-term forecasting. Extensive experiments demonstrate that $K^2$VAE outperforms state-of-the-art methods in both short- and long-term PTSF, providing a more efficient and accurate solution.

Figures

Figures reproduced from arXiv: 2505.23017 by the authors.

**Figure 1.** We compare three native probabilistic forecasting models including GRU MAF, TimeGrad, and CSDI with three point forecasting models equipped with distributional heads including FITS, PatchTST, and iTransformer on ETTh1. Longer forecasting horizons lead to rapid collapse of the CRPS metric (lower is better) on probabilistic forecasting models, even worse than point forecasting models. provides significant support fo… view at source ↗

**Figure 2.** The data flow of K2VAE. It models time series through the KoopmanNet, which constructs a biased linear system. Then the linear system is refined through the KalmanNet while the uncertainty is modeled. Finally, the target distributions over the horzion are predicted through the Decoder. from multiple sensors to enhance estimation accuracy while modeling the uncertainty of the system. VAE for Probabilistic Time Series… view at source ↗

**Figure 3.** The architecture of K2VAE. Input Token Embedding Module patchifys the time series into tokens and applies Embedding. Encoder Module consists of KoopmanNet and KalmanNet, transforming the tokens into linear system in the space of measurement function, refining it and modeling the process uncertainty as the variational distribution. After resampling from the variational distribution, Decoder module constructs the like… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Model efficiency comparison. All the statistical data is obtained on the Electricity-L (T = L = 96). Sample-wise inference time and max gpu memory is obtained with batch size equals 1. Lower values of CRPS indicate better performance. is that K2VAE utilizes the VAE arc…

**Figure 5.** Figure 5: Visualization of input-24-predict-24 results on the Solar-S dataset. 0 25 50 75 100 125 150 175 200 0.5 0.0 0.5 1.0 groundtruth prediction 95% CI 0 25 50 75 100 125 150 175 200 0.5 0.0 0.5 1.0 1.5 2.0 groundtruth prediction 95% CI 0 25 50 75 100 125 150 175 200 0.5 0.0…

**Figure 6.** Figure 6: Visualization of input-96-predict-96 results on the ETTm1-L dataset. 0 25 50 75 100 125 150 175 200 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 groundtruth prediction 95% CI 0 25 50 75 100 125 150 175 200 2 1 0 1 groundtruth prediction 95% CI 0 25 50 75 100 125 150 175 200 1 0 1 2…

**Figure 7.** Figure 7: Visualization of input-96-predict-96 results on the Electricity-L dataset. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

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2023 arXiv
[2023]

Triformer: Triangular, variable-specific attentions for long sequence multivariate time series forecasting

Cirstea, R., Guo, C., Yang, B., Kieu, T., Dong, X., and Pan, S. Triformer: Triangular, variable-specific attentions for long sequence multivariate time series forecasting. In IJCAI, pp. 1994–2001, 2022a. Cirstea, R.-G., Yang, B., Guo, C., Kieu, T., and Pan, S. Towards spatio-t...

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[2024]

Timevae: A variational auto-encoder for multivariate time series generation

Desai, A., Freeman, C., Wang, Z., and Beaver, I. Timevae: A variational auto-encoder for multivariate time series generation. arXiv preprint arXiv:2111.08095,

arXiv
[2025]

I., Hyndman, R

Godahewa, R., Bergmeir, C., Webb, G. I., Hyndman, R. J., and Montero-Manso, P. Monash time series forecasting archive. arXiv preprint arXiv:2105.06643,

arXiv

Pith tools

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

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A com- prehensive survey of deep learning for multivariate time series forecasting: A channel strategy perspective

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Entire space counterfactual learning for reliable content recommendations

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Score: Story coherence and re- trieval enhancement for ai narratives

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Ginar: An end-to-end multivariate time se- ries forecasting model suitable for variable missing

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ProbTS: Benchmarking point and distributional fore- casting across diverse prediction horizons

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IMDPrompter: Adapting SAM to Image Manipulation Detection by Cross-View Automated Prompt Learning

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fine-tuned

14 A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting A. Theoretical Analyses A.1. The Stability of KalmanNet Since the proposed KalmanNet works in a data-driven manner, the floating-point operation error may cause the covariance matrix P losing positive definiteness, which often occurs in the step (21). To mitigat...

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However, these methods still require manual feature engineering and model design

gain popularity for handling nonlinear patterns. However, these methods still require manual feature engineering and model design. Recently, deep learning has made impressive progress in natural language processing (Chen et al., 2024; Zhang & Qi, 2024; Wang et al., 2024e; Wu et al., 2024a; 2025a), computer vision (Zhang et al., 2025b; 2024b; Wu et al., 20...

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