arXiv subjects

So Takao

Publications and source records attributed to So Takao.

At least 19 recordsLinked to original sources

arXiv2026

Variational Flow Maps: Make Some Noise for One-Step Conditional Generation

Flow maps enable high-quality image generation in a single forward pass. However, unlike iterative diffusion models, their lack of an explicit sampling trajectory impedes incorporating external constraints for conditional generation and solving inverse problems. We put forth Variational Flow Maps, a framework for conditional sampling that shifts the perspective of conditioning from "guiding a sampling path", to that of "learning the proper initial noise". Specifically, given an observation, we seek to learn a noise adapter model that outputs a noise distribution, so that after mapping to the data space via flow map, the samples respect the observation and data prior. To this end, we develop a principled variational objective that jointly trains the noise adapter and the flow map, improving noise-data alignment, such that sampling from complex data posterior is achieved with a simple adapter. Experiments on various inverse problems show that VFMs produce well-calibrated conditional samples in a single (or few) steps. For ImageNet, VFM attains competitive fidelity while accelerating the sampling by orders of magnitude compared to alternative iterative diffusion/flow models. Code is available at https://github.com/abbasmammadov/VFM

cs.CV↗

arXiv2025

DAISI: Data Assimilation with Inverse Sampling using Stochastic Interpolants

Data assimilation (DA) is a cornerstone of scientific and engineering applications, combining model forecasts with sparse and noisy observations to estimate latent system states. Classical high-dimensional DA methods, such as the ensemble Kalman filter, rely on Gaussian approximations that are violated for complex dynamics or observation operators. To address this limitation, we introduce DAISI, a scalable filtering algorithm built on flow-based generative models that enables flexible probabilistic inference using data-driven priors. The core idea is to use a stationary, pre-trained generative prior that first incorporates forecast information through a novel inverse-sampling step, before assimilating observations via guidance-based conditional sampling. This allows us to leverage any forecasting model as part of the DA pipeline without having to retrain or fine-tune the generative prior at each assimilation step. Experiments on challenging nonlinear systems show that DAISI achieves accurate filtering results in regimes with sparse, noisy, and nonlinear observations where traditional methods struggle. The code for DAISI is available at https://github.com/Erik-Wikingsson/DAISI.

stat.ML↗

arXiv2025

Flow Matching for Efficient and Scalable Data Assimilation

Data assimilation (DA) estimates a dynamical system's state from noisy observations. Recent generative models like the ensemble score filter (EnSF) improve DA in high-dimensional nonlinear settings but are computationally expensive. We introduce the ensemble flow filter (EnFF), a training-free, flow matching (FM)-based framework that accelerates sampling and offers flexibility in flow design. EnFF uses Monte Carlo estimators for the marginal flow field, localized guidance for observation assimilation, and utilizes a novel flow path that exploits the Bayesian DA formulation. It generalizes classical filters such as the bootstrap particle filter and ensemble Kalman filter. Experiments on high-dimensional benchmarks demonstrate EnFF's improved cost-accuracy tradeoffs and scalability, highlighting FM's potential for efficient, scalable DA. Code is available at https://github.com/Utah-Math-Data-Science/Data-Assimilation-Flow-Matching.

stat.ML↗

arXiv2025

Graph and Simplicial Complex Prediction Gaussian Process via the Hodgelet Representations

Predicting the labels of graph-structured data is crucial in scientific applications and is often achieved using graph neural networks (GNNs). However, when data is scarce, GNNs suffer from overfitting, leading to poor performance. Recently, Gaussian processes (GPs) with graph-level inputs have been proposed as an alternative. In this work, we extend the Gaussian process framework to simplicial complexes (SCs), enabling the handling of edge-level attributes and attributes supported on higher-order simplices. We further augment the resulting SC representations by considering their Hodge decompositions, allowing us to account for homological information, such as the number of holes, in the SC. We demonstrate that our framework enhances the predictions across various applications, paving the way for GPs to be more widely used for graph and SC-level predictions.

cs.LG↗

arXiv2025

Graph Classification Gaussian Processes via Hodgelet Spectral Features

The problem of classifying graphs is ubiquitous in machine learning. While it is standard to apply graph neural networks or graph kernel methods, Gaussian processes can be employed by transforming spatial features from the graph domain into spectral features in the Euclidean domain, and using them as the input points of classical kernels. However, this approach currently only takes into account features on vertices, whereas some graph datasets also support features on edges. In this work, we present a Gaussian process-based classification algorithm that can leverage one or both vertex and edges features. Furthermore, we take advantage of the Hodge decomposition to better capture the intricate richness of vertex and edge features, which can be beneficial on diverse tasks.

cs.LG↗

arXiv2024

Deep Random Features for Scalable Interpolation of Spatiotemporal Data

The rapid growth of earth observation systems calls for a scalable approach to interpolate remote-sensing observations. These methods in principle, should acquire more information about the observed field as data grows. Gaussian processes (GPs) are candidate model choices for interpolation. However, due to their poor scalability, they usually rely on inducing points for inference, which restricts their expressivity. Moreover, commonly imposed assumptions such as stationarity prevents them from capturing complex patterns in the data. While deep GPs can overcome this issue, training and making inference with them are difficult, again requiring crude approximations via inducing points. In this work, we instead approach the problem through Bayesian deep learning, where spatiotemporal fields are represented by deep neural networks, whose layers share the inductive bias of stationary GPs on the plane/sphere via random feature expansions. This allows one to (1) capture high frequency patterns in the data, and (2) use mini-batched gradient descent for large scale training. We experiment on various remote sensing data at local/global scales, showing that our approach produce competitive or superior results to existing methods, with well-calibrated uncertainties.

cs.LG↗

arXiv2024

Semimartingale driven mechanics and reduction by symmetry for stochastic and dissipative dynamical systems

The recent interest in structure preserving stochastic Lagrangian and Hamiltonian systems raises questions regarding how such models are to be understood and the principles through which they are to be derived. By considering a mathematically sound extension of the Hamilton-Pontryagin principle, we derive a stochastic analogue of the Euler-Lagrange equations, driven by independent semimartingales. Using this as a starting point, we can apply symmetry reduction carefully to derive non-canonical stochastic Lagrangian / Hamiltonian systems, including the stochastic Euler-Poincaré / Lie-Poisson equations, studied extensively in the literature. Furthermore, we develop a framework to include dissipation that balances the structure-preserving noise in such a way that the overall stochastic dynamics preserves the Gibbs measure on the symplectic manifold, where the dynamics effectively take place. In particular, this leads to a new derivation of double-bracket dissipation by considering Lie group invariant stochastic dissipative dynamics, taking place on the cotangent bundle of the group.

math-ph↗

arXiv2024

Scalable Data Assimilation with Message Passing

Data assimilation is a core component of numerical weather prediction systems. The large quantity of data processed during assimilation requires the computation to be distributed across increasingly many compute nodes, yet existing approaches suffer from synchronisation overhead in this setting. In this paper, we exploit the formulation of data assimilation as a Bayesian inference problem and apply a message-passing algorithm to solve the spatial inference problem. Since message passing is inherently based on local computations, this approach lends itself to parallel and distributed computation. In combination with a GPU-accelerated implementation, we can scale the algorithm to very large grid sizes while retaining good accuracy and compute and memory requirements.

cs.LG↗

arXiv2024

Gaussian Processes on Cellular Complexes

In recent years, there has been considerable interest in developing machine learning models on graphs to account for topological inductive biases. In particular, recent attention has been given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Matérn kernel and one that additionally mixes information of different cell types.

cs.LG↗

arXiv2024

Iterated INLA for State and Parameter Estimation in Nonlinear Dynamical Systems

Data assimilation (DA) methods use priors arising from differential equations to robustly interpolate and extrapolate data. Popular techniques such as ensemble methods that handle high-dimensional, nonlinear PDE priors focus mostly on state estimation, however can have difficulty learning the parameters accurately. On the other hand, machine learning based approaches can naturally learn the state and parameters, but their applicability can be limited, or produce uncertainties that are hard to interpret. Inspired by the Integrated Nested Laplace Approximation (INLA) method in spatial statistics, we propose an alternative approach to DA based on iteratively linearising the dynamical model. This produces a Gaussian Markov random field at each iteration, enabling one to use INLA to infer the state and parameters. Our approach can be used for arbitrary nonlinear systems, while retaining interpretability, and is furthermore demonstrated to outperform existing methods on the DA task. By providing a more nuanced approach to handling nonlinear PDE priors, our methodology offers improved accuracy and robustness in predictions, especially where data sparsity is prevalent.

stat.ML↗

arXiv2023

A Geometric Extension of the Itô-Wentzell and Kunita's Formulas

We extend the Itô-Wentzell formula for the evolution along a continuous semimartingale of a time-dependent stochastic field driven by a continuous semimartingale to tensor field-valued stochastic processes on manifolds. More concretely, we investigate how the pull-back (respectively, the push-forward) by a stochastic flow of diffeomorphisms of a time-dependent stochastic tensor field driven by a continuous semimartingale evolves with time, deriving it under suitable regularity conditions. We call this result the Kunita-Itô-Wentzell (KIW) formula for the advection of tensor-valued stochastic processes. Equations of this nature bear significance in stochastic fluid dynamics and well-posedness by noise problems, facilitating the development of certain geometric extensions within existing theories.

math.PR↗

arXiv2023

Actually Sparse Variational Gaussian Processes

Gaussian processes (GPs) are typically criticised for their unfavourable scaling in both computational and memory requirements. For large datasets, sparse GPs reduce these demands by conditioning on a small set of inducing variables designed to summarise the data. In practice however, for large datasets requiring many inducing variables, such as low-lengthscale spatial data, even sparse GPs can become computationally expensive, limited by the number of inducing variables one can use. In this work, we propose a new class of inter-domain variational GP, constructed by projecting a GP onto a set of compactly supported B-spline basis functions. The key benefit of our approach is that the compact support of the B-spline basis functions admits the use of sparse linear algebra to significantly speed up matrix operations and drastically reduce the memory footprint. This allows us to very efficiently model fast-varying spatial phenomena with tens of thousands of inducing variables, where previous approaches failed.

stat.ML↗

arXiv2023

Short-term Prediction and Filtering of Solar Power Using State-Space Gaussian Processes

Short-term forecasting of solar photovoltaic energy (PV) production is important for powerplant management. Ideally these forecasts are equipped with error bars, so that downstream decisions can account for uncertainty. To produce predictions with error bars in this setting, we consider Gaussian processes (GPs) for modelling and predicting solar photovoltaic energy production in the UK. A standard application of GP regression on the PV timeseries data is infeasible due to the large data size and non-Gaussianity of PV readings. However, this is made possible by leveraging recent advances in scalable GP inference, in particular, by using the state-space form of GPs, combined with modern variational inference techniques. The resulting model is not only scalable to large datasets but can also handle continuous data streams via Kalman filtering.

cs.LG↗

arXiv2022

Transport noise restores uniqueness and prevents blow-up in geometric transport equations

In this work, we demonstrate well-posedness and regularisation by noise results for a class of geometric transport equations that contains, among others, the linear transport and continuity equations. This class is known as linear advection of $k$-forms. In particular, we prove global existence and uniqueness of $L^p$-solutions to the stochastic equation, driven by a spatially $α$-Hölder drift $b$, uniformly bounded in time, with an integrability condition on the distributional derivative of $b$, and sufficiently regular diffusion vector fields. Furthermore, we prove that all our solutions are continuous if the initial datum is continuous. Finally, we show that our class of equations without noise admits infinitely many $L^p$-solutions and is hence ill-posed. Moreover, the deterministic solutions can be discontinuous in both time and space independently of the regularity of the initial datum. We also demonstrate that for certain initial data of class $C^\infty_{0},$ the deterministic $L^p$-solutions blow up instantaneously in the space $L^{\infty}_{loc}$. In order to establish our results, we employ characteristics-based techniques that exploit the geometric structure of our equations.

math.PR↗

arXiv2022

Well-posedness by noise for linear advection of $k$-forms

In this work, we extend existing well-posedness by noise results for the stochastic transport and continuity equations by treating them as special cases of the linear advection equation of $k$-forms, which arises naturally in geometric fluid dynamics. In particular, we prove the existence and uniqueness of weak $L^p$-solutions to the stochastic linear advection equation of $k$-forms that is driven by a Hölder continuous, $W^{1,1}_{loc}$ drift and smooth diffusion vector fields, such that the equation without noise admits infinitely many solutions.

math.AP↗

arXiv2022

The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

math.AP↗

arXiv2021

Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners.

stat.ML↗

arXiv2021

A Unifying and Canonical Description of Measure-Preserving Diffusions

A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. In this paper, we develop a geometric theory that improves and generalises this construction to any manifold. We thereby demonstrate that the completeness result is a direct consequence of the topology of the underlying manifold and the geometry induced by the target measure $P$; there is no need to introduce other structures such as a Riemannian metric, local coordinates, or a reference measure. Instead, our framework relies on the intrinsic geometry of $P$ and in particular its canonical derivative, the deRham rotationnel, which allows us to parametrise the Fokker--Planck currents of measure-preserving diffusions using potentials. The geometric formalism can easily incorporate constraints and symmetries, and deliver new important insights, for example, a new complete recipe of Langevin-like diffusions that are suited to the construction of samplers. We also analyse the reversibility and dissipative properties of the diffusions, the associated deterministic flow on the space of measures, and the geometry of Langevin processes. Our article connects ideas from various literature and frames the theory of measure-preserving diffusions in its appropriate mathematical context.

math.PR↗