Function-Space Diffusion for Motion Planning

• The paper introduces a diffusion-based motion planner that models trajectories as continuous functions, overcoming fixed-length waypoint limitations.
• It employs spectral decomposition with a Matérn-type covariance to generate smooth, resolution-invariant trajectories across varying discretizations.
• The framework enforces exact endpoint constraints via zero-end projection and DST-FNO, achieving zero-shot cross-resolution generalization in diverse environments.

Searching arXiv for the primary paper and closely related work to ground the article in current literature. Function-Space Diffusion for Motion Planning (FSD-MP) is a diffusion-based motion planner that models robot trajectories as continuous functions rather than fixed-length waypoint sequences, and performs diffusion directly in function space. In this formulation, a trajectory is treated as a map $x(\tau):[0,1]\to\mathbb{R}^d$, so the learned planner is not tied to a single discretization. The central objective is discretization-invariant trajectory generation, with the paper reporting zero-shot generalization across resolutions up to $16\times$ higher than the training resolution while preserving consistent planning behavior without retraining (Chang et al., 3 Jul 2026).

1. Definition, motivation, and planning formulation

FSD-MP is motivated by a limitation of existing diffusion-based motion planners that operate on fixed-length waypoint sequences. When a planner is trained on trajectories discretized at a particular resolution, such as $N=64$, the resulting model is resolution-dependent; the same model typically cannot be evaluated directly at much finer discretizations, such as $N=1024$, without retraining. FSD-MP addresses this by shifting the representation from sampled trajectory vectors to continuous trajectory functions (Chang et al., 3 Jul 2026).

The planning problem is formulated as the search for a smooth collision-free trajectory between start and goal configurations $q_s,q_g\in\mathbb{R}^d$:

$$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$

The paper also adopts a probabilistic view,

$$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$

where $p(x)$ is a prior over smooth feasible trajectories and $p(\mathcal O\mid x)$ is a likelihood encoding collision avoidance and other task constraints. In this sense, motion planning is cast as posterior sampling in trajectory function space.

The relevant function space is the Hilbert space

$\mathcal H=L^2([0,1],\mathbb R^d).$

This choice is not merely notational. It makes the diffusion process a process over trajectory functions themselves, rather than over a particular discretization of those functions. A common misconception is to view FSD-MP as standard waypoint diffusion followed by interpolation; the formulation instead places the stochastic process directly in $16\times$0, and the discretized trajectory is only an evaluation of the learned continuous object.

2. Function-space diffusion and spectral construction

The core generative mechanism is a linear stochastic differential equation in function space:

$16\times$1

where $16\times$2 is the noisy trajectory function at diffusion time $16\times$3, $16\times$4 is the noise schedule, $16\times$5 is the drift operator, $16\times$6 is the diffusion operator, and $16\times$7 is a cylindrical Wiener process. The covariance operator is $16\times$8 (Chang et al., 3 Jul 2026).

A central simplifying assumption is that $16\times$9 and $N=64$0 share an orthonormal eigenbasis $N=64$1, with

$N=64$2

Projecting onto this basis yields coefficients

$N=64$3

and the function-space SDE decouples into independent one-dimensional coefficient processes:

$N=64$4

This mode-wise decomposition is the mechanism by which diffusion becomes discretization-independent. Each spectral mode is diffused independently, and low-frequency and high-frequency content can be controlled separately through $N=64$5 and $N=64$6. The paper emphasizes that this is the key “function-space” design: the process is defined independently of discretization.

The coefficient-wise process has closed-form marginals,

$N=64$7

with

$N=64$8

and

$N=64$9

After reparameterization using $N=1024$0, the forward process takes the compact form

$N=1024$1

This mirrors standard diffusion, but with mode-dependent, function-space-compatible noise rather than finite-dimensional isotropic perturbations.

3. Matérn-type covariance and the validity of function-space noise

A major technical issue in infinite-dimensional diffusion is that white noise is not a valid Gaussian measure on $N=1024$2: the identity covariance is not trace-class, so $N=1024$3 is ill-defined in function space. FSD-MP addresses this by introducing a Matérn-type covariance operator,

$N=1024$4

where $N=1024$5 is the Laplacian under the chosen boundary condition, $N=1024$6 controls variance, $N=1024$7 controls length scale, and $N=1024$8 controls smoothness (Chang et al., 3 Jul 2026).

The paper states that this covariance is trace-class when $N=1024$9 for time-parameterized trajectories, ensuring a valid Gaussian measure in function space. It further notes that covariance eigenvalues decay like

$q_s,q_g\in\mathbb{R}^d$0

and that the trace is finite when $q_s,q_g\in\mathbb{R}^d$1; for trajectory functions, $q_s,q_g\in\mathbb{R}^d$2, giving the same threshold $q_s,q_g\in\mathbb{R}^d$3.

The use of Matérn-type noise has several consequences explicitly identified in the paper. It yields smooth, correlated perturbations rather than white noise; its spectrum decays with frequency, so high-frequency components are less noisy; and it is compatible with the spectral basis used for diffusion. The ablation replacing this trace-class covariance with identity covariance consistently underperforms FSD-MP, which the paper interprets as support for the claim that well-defined function-space perturbations matter.

This design choice is also conceptually important. In finite-dimensional diffusion, isotropic Gaussian noise is often taken for granted. In function-space diffusion, the noise model is part of the mathematical validity of the construction itself, not merely a hyperparameter choice.

4. Boundary compatibility, zero-end projection, and DST-FNO

A key challenge in continuous-trajectory generation is preserving the endpoint constraints

$q_s,q_g\in\mathbb{R}^d$4

across all resolutions. FSD-MP uses three coupled mechanisms to enforce this structure exactly (Chang et al., 3 Jul 2026).

First, constrained trajectories are converted into zero-boundary residuals by the zero-end projection

$q_s,q_g\in\mathbb{R}^d$5

which guarantees

$q_s,q_g\in\mathbb{R}^d$6

At inference, the original trajectory is reconstructed as

$q_s,q_g\in\mathbb{R}^d$7

The paper emphasizes that this gives exact endpoint satisfaction by construction, with no clamping or post-hoc correction.

Second, the model is conditioned on the start state, goal state, and diffusion time through

$q_s,q_g\in\mathbb{R}^d$8

Feature-wise linear modulation is then applied as

$q_s,q_g\in\mathbb{R}^d$9

with $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$0 generated by a layer-specific MLP from the conditioning vector.

Third, the reverse process is parameterized by a boundary-compatible Discrete Sine Transform-based Fourier Neural Operator (DST-FNO). Standard FNO uses FFT and assumes periodicity, which is not naturally aligned with zero-boundary residuals. FSD-MP instead uses the discrete sine transform with basis functions

$$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$1

which match the zero-boundary structure of the residualized trajectory.

The DST-FNO layer is defined as

$$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$2

where $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$3 and $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$4 are the DST and inverse DST, $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$5 is a learned spectral kernel, $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$6 is a pointwise linear map, and $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$7 is a nonlinearity.

The importance of boundary compatibility is reinforced by ablation results. Replacing DST-FNO with standard FFT-based FNO yields good performance near the training resolution but degrades as resolution increases; in Dense2D, the valid rate drops from $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$8 at $$x^\star=\arg\min_x J(x)\quad \text{s.t.}\quad x(0)=q_s,\ x(1)=q_g.$$9 to $$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$0 at $$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$1. Replacing DST-FNO with a UNet gives strong training-resolution performance but collapses under cross-resolution evaluation; in Dense2D, the valid rate goes from $$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$2 at $$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$3 to $$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$4 at $$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$5. These results support the claim that the reverse-process architecture must itself be resolution-agnostic, not merely the forward diffusion.

5. Training objective, reverse process, and collision guidance

Because the conditional distribution factorizes over spectral modes,

$$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$6

the score and the noise-prediction objective decompose mode-wise. The training loss is

$$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$7

By Parseval’s identity, this is equivalent to the function-space mean-squared error

$$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$8

The model therefore learns to predict function-space noise rather than waypoint-space noise (Chang et al., 3 Jul 2026).

The training procedure given in the paper consists of six steps: sampling an expert trajectory $$p(x\mid \mathcal O)\propto p(x)\,p(\mathcal O\mid x),$$9; extracting $p(x)$0; applying zero-end projection to obtain $p(x)$1; sampling diffusion time $p(x)$2 and Gaussian function-space noise $p(x)$3; diffusing in mode space via

$p(x)$4

and minimizing

$p(x)$5

At inference time, the method samples $p(x)$6 and runs reverse DDIM steps in spectral space. The paper describes a mode-wise DDIM-style update in which the current noisy coefficient is used to estimate a clean coefficient and then transported to a lower-noise step. The full sampler remains in spectral space until the final path is reconstructed through the affine start-goal restoration formula.

Obstacle avoidance is incorporated through classifier guidance. With collision cost $p(x)$7 and likelihood

$p(x)$8

the method approximates

$p(x)$9

because collision cost should be evaluated on the denoised trajectory. The guided update is

$p(\mathcal O\mid x)$0

where

$p(\mathcal O\mid x)$1

and $p(\mathcal O\mid x)$2 is a covariance preconditioner. This makes collision handling part of the reverse diffusion procedure rather than a purely post hoc filter.

6. Empirical results, zero-shot super-resolution, and relation to adjacent generative planners

FSD-MP is evaluated on four planning environments derived from the MPD setup: the 2D point-robot environments Simple2D and Dense2D, and the 7-DoF Franka manipulator environments Spheres3D and Warehouse. The paper also adds extra obstacles at inference time to test generalization to unseen obstacle layouts. Expert demonstrations are generated by RRTConnect followed by GPMP refinement. Baselines are MPD, MPD-Spline, GPMP, and MPD-Retrain. The reported metrics are success rate, defined as the percentage of start-goal pairs for which at least one generated trajectory is collision-free, and valid rate, defined as the percentage of generated trajectories that are collision-free. The evaluation protocol uses 120 random start-goal pairs per environment and 100 trajectory samples per pair (Chang et al., 3 Jul 2026).

At the training resolution $p(\mathcal O\mid x)$3, FSD-MP is reported as competitive with or better than baselines. In Simple2D, the valid rate is $p(\mathcal O\mid x)$4; in Dense2D, $p(\mathcal O\mid x)$5; in Spheres3D, success is $p(\mathcal O\mid x)$6 and valid rate is $p(\mathcal O\mid x)$7; and in Warehouse, success is $p(\mathcal O\mid x)$8 and valid rate is $p(\mathcal O\mid x)$9.

The paper’s headline empirical claim is zero-shot cross-resolution generalization. A model trained only at $\mathcal H=L^2([0,1],\mathbb R^d).$0 is evaluated without retraining at much higher resolutions, including $\mathcal H=L^2([0,1],\mathbb R^d).$1, and in some visualizations up to $\mathcal H=L^2([0,1],\mathbb R^d).$2, that is, $\mathcal H=L^2([0,1],\mathbb R^d).$3 higher resolution or beyond. Representative examples are summarized below.

Environment	Resolution	Result
Dense2D, extra obstacles	$\mathcal H=L^2([0,1],\mathbb R^d).$4	success $\mathcal H=L^2([0,1],\mathbb R^d).$5, valid $\mathcal H=L^2([0,1],\mathbb R^d).$6
Dense2D, extra obstacles	$\mathcal H=L^2([0,1],\mathbb R^d).$7	success $\mathcal H=L^2([0,1],\mathbb R^d).$8, valid $\mathcal H=L^2([0,1],\mathbb R^d).$9
Warehouse, extra obstacles	$16\times$00	success $16\times$01, valid $16\times$02
Warehouse, extra obstacles	$16\times$03	success $16\times$04, valid $16\times$05

These figures are presented as evidence that performance remains nearly constant across resolutions for FSD-MP, unlike the ablations. The method’s strengths are listed as resolution-agnostic generation, exact endpoint satisfaction via zero-end projection, boundary-aware spectral modeling with DST-FNO, smooth valid noise through Matérn-type covariance, and strong empirical generalization including super-resolution and unseen obstacles. The stated limitations are that the method still relies on a learned collision guidance mechanism and sampling-time optimization; it focuses on trajectory functions parameterized over time; very dense or highly constrained environments may require additional local refinement or adaptive resolution; and broader validation beyond the evaluated benchmark families would still be valuable.

Within the broader literature on generative motion planning, FSD-MP is closely related in spirit to flow-based planners that also model trajectory distributions rather than deterministic single paths. A particularly relevant comparison is Flow Motion Policy, an open-loop end-to-end neural planner for robotic manipulators that uses flow matching to model a conditional distribution over joint-configuration increments and supports inference-time best-of-$16\times$06 sampling (Soleymanzadeh et al., 8 Apr 2026). The two approaches differ in the space in which generation is defined: FSD-MP performs diffusion in function space and emphasizes discretization invariance, whereas Flow Motion Policy is framed as a flow-matching model over trajectory chunks and emphasizes stochastic multi-modality and post-generation best-of-$16\times$07 selection. This suggests a useful conceptual distinction within generative planning: one line of work focuses on continuous function-space representations and cross-resolution consistency, while another focuses on stochastic path diversity and efficient inference-time selection.