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Ordered Diffusion Kernels

This paper introduces Ordered Diffusion Kernels (ODKs), a novel class of local kernels that approximate the infinitesimal generator of arbitrary Itô Stochastic Differential Equations by inferring data orderings rather than velocity fields, thereby enabling the accurate, uncoupled recovery of drift and diffusion coefficients in dynamical systems with limited prior information.

Original authors: Jack H. Soulsby, Andreas C. S. Jørgensen, Atiyo Ghosh, Vahid Shahrezaei

Published 2026-08-19

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CC BY 4.0

Original authors: Jack H. Soulsby, Andreas C. S. Jørgensen, Atiyo Ghosh, Vahid Shahrezaei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to understand the flow of a river by taking a single photograph of the water. You see the ripples, the foam, and the general direction, but you cannot see the current's speed, the hidden rocks beneath the surface, or the wind pushing the water from the side. In the world of data science, researchers often face this exact problem. They have vast collections of snapshots—measurements of cells, stars, or weather patterns—taken at random moments, but they lack the continuous video of how these systems actually move and change over time. For decades, scientists have tried to reconstruct these hidden movements using mathematical tools called kernels, which act like a lens to focus on the local relationships between data points. However, these traditional lenses have a blind spot: they struggle to separate the force that pushes a system in a specific direction from the random jiggling that happens along the way. Without knowing which is which, the reconstructed picture of the system's future remains blurry and often incorrect.

A team of researchers at Imperial College London and in France has developed a new way to look at these snapshots, one that successfully untangles the directed motion from the random noise. They call their method Ordered Diffusion Kernels. Instead of trying to guess the exact speed and direction of every single point in a dataset right from the start—a task that is often impossible without prior knowledge—they first ask a much simpler question: what is the order of things? They look at the data and determine which points are "earlier" and which are "later" in the system's journey, creating a simple map of progression. By focusing on this sequence first, they can build a mathematical model that accurately recovers both the steady drift of the system and the intensity of its random fluctuations, even when the data is sparse, noisy, or comes from a complex, high-dimensional world.

The core of this new approach lies in how it handles the concept of "ordering." In many natural systems, such as a cell growing and dividing, there is a clear path from a starting state to an ending state, but the exact timing or speed might be unknown. Traditional methods often try to fit a potential energy landscape to the data, assuming the system moves like a ball rolling down a hill. This works well for simple, steady systems but fails when the landscape is complex or when the data doesn't perfectly match the assumptions of a steady state. The new method relaxes this requirement. It does not demand a perfect map of the hill; it only needs a function that tells it which way is "up" the path. This ordering function acts as a guide, allowing the researchers to construct a kernel—a mathematical weighting tool—that respects the direction of the flow without being confused by the random noise.

Once this ordering is established, the researchers use it to build a model that approximates the infinitesimal generator of the system. In plain terms, this is the mathematical engine that describes how the system changes from one tiny moment to the next. The beauty of their construction is that it separates the two main components of this change: the drift, which is the predictable, directed movement, and the diffusion, which is the random, spreading motion. Previous methods often forced these two elements to be linked, meaning that if you changed the speed of the drift, you inadvertently changed the amount of random noise, or vice versa. This coupling made it difficult to study systems where the noise itself changes depending on where the system is. The new Ordered Diffusion Kernels break this link, allowing the researchers to estimate the drift and the diffusion independently. This is a significant advancement because it means they can now model systems where the randomness is not constant but varies across the landscape, a feature common in biological and physical processes.

To test their theory, the researchers applied their method to several synthetic datasets, creating artificial worlds with known rules to see if their tool could find them. In one experiment, they simulated data on a torus, a shape like a donut, which presents a unique challenge because the path loops back on itself. Traditional ordering methods struggle here because you cannot define a single "start" and "end" on a loop without creating a break in the logic. The researchers overcame this by using a local ordering function, which only compares nearby points rather than trying to order the entire loop at once. The results were striking: their model successfully reconstructed the exact mathematical rules governing the movement and the random spreading of the data, matching the known ground truth with high precision. They also demonstrated that their method could handle anisotropic diffusion, where the random spreading happens at different rates in different directions, a scenario that is notoriously difficult for standard tools.

The researchers also explored how to use this method when the data comes from a system where the timing of the snapshots is known, versus when it is not. In the case where the time of each measurement is unknown, they developed a way to infer the ordering and the diffusion strength by looking at the overall distribution of the data points. They found that even without knowing the exact time each point was captured, the geometry of the data contained enough information to recover the underlying dynamics. When the timing was known, they could use a more direct approach, fitting the model to the specific sequence of events. In both scenarios, the method proved robust, capable of recovering complex, non-linear patterns that other techniques missed.

One of the most compelling aspects of this work is its potential application to single-cell biology, a field where the destructive nature of measurement makes it impossible to track a single cell over time. Researchers can only take a snapshot of a cell's state and then destroy it, leaving them with a collection of independent moments rather than a continuous movie. The new method offers a way to reconstruct the "movie" from these static frames. By treating the progression of a cell's life as an ordering problem, the researchers showed that their tool could infer the gene regulatory networks that drive cell fate, separating the deterministic instructions of the cell from the random noise of biological variation. While the paper focused on synthetic data to prove the concept, the framework is explicitly designed to handle the messy, high-dimensional, and sparse nature of real-world biological data.

The researchers did not claim to have solved every problem in dynamical systems. They acknowledged that their method relies on the assumption that the data lies on a lower-dimensional surface within a high-dimensional space, a concept known as the manifold hypothesis. They also noted that near the boundaries of the data, where there are fewer neighbors to compare against, the accuracy of the reconstruction can degrade. However, they showed that even in these difficult regions, the method performs better than existing alternatives, particularly when the data contains some level of noise that blurs the boundary. They also discussed the challenge of identifying whether a system's behavior is driven by a deterministic force or by a gradient of random noise, a problem that remains difficult but is now more tractable with their new tool.

In the end, the work represents a shift in how we approach the reconstruction of hidden dynamics from static data. By prioritizing the simple, intuitive concept of order over the complex task of guessing the exact forces at play, the researchers have created a tool that is both flexible and powerful. It allows scientists to look at a scattered collection of points and see the river flowing beneath them, distinguishing the current from the eddies with a clarity that was previously out of reach. This is not just a mathematical refinement; it is a new way of seeing the world, one that turns a chaotic pile of snapshots into a coherent story of movement and change.

Technical Summary: Ordered Diffusion Kernels

1. Problem Statement

The paper addresses the challenge of inferring the infinitesimal generator of an Itô Stochastic Differential Equation (SDE) from data sampled from a dynamical system where little to no a priori dynamical information is available. While classical kernel methods (e.g., Diffusion Maps) can approximate the Laplace–Beltrami operator on a manifold, they often fail to capture arbitrary drift and diffusion terms without restrictive assumptions.

Existing approaches face specific limitations:

$\alpha$ -normalised Diffusion Maps: Introduce a drift term dependent on the sampling density, effectively coupling the drift to the stationary distribution ( $U(x) = -\log q(x)$ ). This restricts analysis to potential systems in steady state.
Target Measure Diffusion Maps (TMDmap): Can target arbitrary gradient flows but couple the drift to a unit diffusion, preventing the inference of state-dependent diffusion coefficients.
General Local Kernels (Berry & Sauer): Can approximate arbitrary second-order elliptic operators but require prior knowledge of the drift vector field $\mu(x)$ and diffusion tensor $D(x)$ , rendering the initial inference problem ill-posed for unknown systems.

The core difficulty lies in inferring a potential function $U(x)$ , which must simultaneously satisfy correct directionality and rate-of-change constraints, often requiring complex domain-specific inference machinery.

2. Methodology

2.1 The Ordering Function

To circumvent the difficulty of inferring a potential, the authors introduce the concept of an Ordering Function, $l(x)$ . Unlike a potential, an ordering function is only required to imply the correct directionality of the system (i.e., which neighbors are "further along" in the evolution), without needing to encode the specific rate of change or energy scale. This relaxes the inference problem significantly.

2.2 Ordered Diffusion Kernels (ODKs)

The authors define a novel class of local kernels, Ordered Diffusion Kernels (ODKs), designed to approximate the infinitesimal generator of a gradient-flow SDE with state-dependent isotropic diffusion. The kernel is defined as:

$K_\varepsilon(x, y) := h\left( \frac{\|y - x\|^2}{\varepsilon \rho(x)^2} + \frac{(l(y) - l(x) - \varepsilon \tau(x)\kappa(x))^2}{\varepsilon \rho(x)^2 \kappa(x)^2} - \frac{(l(y) - l(x))^2}{\varepsilon \rho(x)^2 \kappa(x)^2} \right)$

Where:

$h$ is a smooth, fast-decaying function (e.g., Gaussian).
$\rho(x)$ controls the diffusive strength.
$\tau(x)$ controls the speed of the process.
$\kappa(x)$ is a scale-regularisation term (often set to $\|\nabla l(x)\|$ ) to make the kernel agnostic to the scale of the ordering function.
The third term acts as a dimensionality correction to ensure the kernel behaves correctly on the underlying manifold.

Theoretical Convergence:
The paper proves that as $\varepsilon \to 0$ , the normalised Laplacian of the ODK converges to the infinitesimal generator $L$ of an Itô SDE:
$Lf(x) = \frac{\tau(x)}{\kappa(x)}\nabla l(x) \cdot \nabla f(x) + \frac{1}{2}\rho(x)^2 \Delta f(x)$
Crucially, this formulation decouples the drift (determined by the ordering) from the diffusion (determined by $\rho$ ), a limitation present in TMDmap.

2.3 Extensions

The framework is extended to handle more complex scenarios:

Arbitrary Drifts (lODK): By generalising the global ordering function $l(x)$ to a local ordering function $l(x, y)$ , the method can approximate arbitrary drift fields $\mu(x)$ , not just gradient flows. This handles non-linear orderings and complex topologies (e.g., cyclic orders on a torus).
Anisotropic Diffusions (anODK): The kernel is extended to approximate anisotropic diffusion tensors. However, this initially couples the drift and diffusion. The authors propose a Strang Splitting scheme to compose two kernels (one isotropic with drift, one symmetric anisotropic without drift) to remove this coupling and accurately reconstruct the generator.
Self-Tuning Bandwidths (stODK): The diffusion term $\rho(x)^2$ is replaced with $\rho_1(x)\rho_2(y)$ to handle non-uniformly sampled data. This introduces an effective drift term proportional to $\nabla \rho$ , which can be controlled via a $\beta$ -regularisation parameter.
Multi-Ordering (mODK): Supports multiple ordering functions $\{l_i\}$ to model systems with multiple potential wells or complex interactions.

2.4 Discrete to Continuous Convergence

The paper establishes that the discrete Laplacian constructed from $m$ independent samples converges in probability to the continuous operator as $m \to \infty$ , provided the bandwidth $\varepsilon$ scales appropriately with $m$ (specifically $\varepsilon \sim (\log m / m)^{2/(d+4)}$ ).

2.5 Inference via Optimisation

To recover unknown parameters from data, the authors propose two gradient-based optimisation objectives:

$L_q$ Objective: Used when sampling times are unknown. It minimises the KL-divergence between the empirical sampling distribution and the transient distribution predicted by the ODK model.
$L_p$ Objective: Used when sampling times are known. It maximises the likelihood of the joint distribution $p(x, t)$ , allowing for the simultaneous inference of state-dependent drift and diffusion.

3. Key Contributions

Novel Kernel Class: Introduction of Ordered Diffusion Kernels (ODKs) that approximate arbitrary gradient-flow SDE generators without coupling drift to diffusion.
Relaxed Inference: The shift from inferring a potential $U(x)$ to an ordering function $l(x)$ , significantly reducing the complexity of the initial inference problem.
Theoretical Guarantees: Proof of asymptotic convergence of ODK Laplacians to the infinitesimal generator of Itô SDEs, including extensions for local orderings, anisotropy, and self-tuning bandwidths.
Strang Splitting for Anisotropy: A method to decouple drift and diffusion in anisotropic settings using operator splitting, overcoming limitations of previous anisotropic kernel methods.
Non-Parametric Inference Framework: Development of $L_q$ and $L_p$ objectives to infer drift and diffusion parameters directly from static snapshots or time-stamped data without assuming a specific parametric form for the dynamics.

4. Numerical Results

The framework was validated on synthetic data:

Laplacian Convergence: Demonstrated accurate recovery of analytical Laplacians for both isotropic and anisotropic diffusions on a torus, with low $L_2$ error between numerical and analytical results.
ODE Dynamics: Applied to data from a spiral ODE, successfully extracting the extrinsic velocity field and the Mean Curvature Vector (MCV). The authors showed that a symmetric correction (subtracting a symmetric local kernel) effectively removes the MCV bias, recovering the true velocity.
SDE Parameter Inference:
- OU Process: Successfully inferred the ordering function (negative potential) and constant diffusion for a 2D Ornstein-Uhlenbeck process using the $L_q$ objective, achieving high correlation with ground truth.
- Non-Linear SDE: Applied the $L_p$ objective to a complex non-linear SDE with state-dependent diffusion. The method accurately reconstructed the ordering function. While the diffusion estimates were noisy, applying smoothness regularisation (gradient/Laplacian or Laplacian eigenbasis) allowed for the recovery of the general shape of the state-dependent noise, including subtle features like diffusion increases at specific radii.

5. Significance and Claims

The paper claims that ODKs provide a principled, non-parametric tool for recovering the dynamics of complex systems from static or sparsely sampled data where dynamical information is scarce.

Decoupling: The primary significance is the ability to approximate arbitrary gradient flows without the restrictive coupling between drift and sampling density (or diffusion) found in previous methods like TMDmap.
Scalability: By relaxing the inference of a potential to an ordering, the method makes the problem tractable for high-dimensional, noisy data (such as single-cell RNA-sequencing data, though the paper notes this specific application is not explored in the experiments).
Identifiability: The authors modestly note that while the $L_p$ objective allows for the inference of state-dependent noise, the problem is ill-conditioned. The signal for diffusion is weak and easily obscured by sampling density variations, requiring careful regularisation. They present empirical evidence of identifiability but acknowledge that proving it rigorously for state-dependent noise remains future work.

The paper positions ODKs as a bridge between local kernel methods and the rigorous theory of stochastic differential equations, offering a flexible framework for dynamical systems analysis where traditional parametric models are infeasible.