# MercerFlow: Flow Matching in a Kernel-Induced Latent Space for Probabilistic Forecasting

arXiv: https://arxiv.org/abs/2610.06039 · HTML: https://mlsanity.com/paper/2610.06039/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 16 of its 34 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 16 of 16)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2410.03024/) (2024)
- [Time Series Diffusion in the Frequency Domain](https://mlsanity.com/paper/2402.05933/) (2024)
- [DCTdiff: Intriguing Properties of Image Generative Modeling in the DCT Space](https://mlsanity.com/paper/2412.15032/) (2024)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)
- [Adam Exploits ℓ_∞-geometry of Loss Landscape via Coordinate-wise Adaptivity](https://mlsanity.com/paper/2410.08198/) (2024)
- [DMamba: Decomposition-enhanced Mamba for Time Series Forecasting](https://mlsanity.com/paper/2602.09081/) (2026)

## Abstract (arXiv)

Recent work has shown that probabilistic flow matching for time series forecasting benefits from a data-matched prior. The resulting prior introduces local correlations, which a sequential architecture usually absorbs: a recurrent neural network (RNN), a structured state-space model (S4), or a Transformer. However, such a backbone costs GPU memory and time per epoch. A cheaper alternative is MLP-based latent-space flow matching: embed the time series via an invertible map to a single latent vector and learn the flow there, so a tabular MLP can treat the series as a set of features. The relationship between the prior and the choice of linear latent map is understudied in conditional flow matching (CFM) forecasting, yet we found it strongly affects performance. Fixed transforms such as Fourier or discrete cosine (DCT) are only well-conditioned for Ornstein--Uhlenbeck priors, while a principal-component (PCA) map fit to the data is a strong but training-set-dependent reference sensitive to train--test shift. Instead, we propose to use the Mercer eigenbasis of the prior kernel: it diagonalises the centred covariance exactly, decouples from training data, and adapts to non-stationary and periodic priors. On five GluonTS benchmarks (ETTh1, ETTh2, Weather, Electricity, Traffic) under a shared protocol with TSFlow, the resulting MLP matches or beats it on CRPS at about 4.7× less training memory and 3.5×--4.4× less time per epoch.