Denoising Landing Flow Optimization

• DLF is a continuous-time, inference algorithm that recovers implicit manifold geometry from Gaussian-smoothed data distributions.
• It links score estimation with projected descent by incorporating a landing term that pulls iterates toward the manifold.
• The method demonstrates empirical success in optimizing over the orthogonal group and in control tasks, approximating classical Riemannian projections.

Searching arXiv for the named method and closely related work. Denoising Landing Flow (DLF) is a continuous-time, inference-time algorithm for Riemannian optimization on manifolds that are not given analytically but only implicitly through a data distribution concentrated near a low-dimensional manifold $\mathcal M \subset \mathbb R^d$. In the formulation introduced in "Landing with the Score: Riemannian Optimization through Denoising" (Kharitenko et al., 27 Sep 2025), DLF recovers approximate manifold geometry from the Gaussian-smoothed data distribution learned by diffusion-model score estimation, and then combines projected descent with a landing term that pulls iterates toward the manifold. The method is designed for problems of the form $\min_{x\in \mathcal M} f(x)$ when classical manifold operations such as tangent projection, closest-point projection, and retractions are unavailable.

1. Optimization on an implicit data manifold

The problem setting is Riemannian optimization over a manifold that is known only through samples from a data distribution $\mu_{\mathrm{data}}$, rather than through an explicit constraint representation or a closed-form atlas. The objective is

$\min_{x \in \mathcal M} f(x),$

with $f:\mathbb R^d \to \mathbb R$ smooth. In classical Riemannian optimization, one typically assumes access to operations such as the orthogonal projection $P_{T_x\mathcal M}$ onto the tangent space, the closest-point projection $\pi(x)$ onto $\mathcal M$, and a retraction or exponential map. The central difficulty addressed by DLF is that these operations are unavailable for implicitly defined data manifolds (Kharitenko et al., 27 Sep 2025).

The framework replaces explicit geometry by the Gaussian-smoothed data distribution

$$p_\sigma = \mathcal N(0,\sigma^2 I) * \mu_{\mathrm{data}}.$$

This substitution is consequential because $p_\sigma$ is precisely the type of object approximated by score-based diffusion models. A plausible implication is that DLF turns pretrained or newly trained score estimators into geometric surrogates for manifold optimization, rather than using them only for generation or denoising.

2. Link function and recovery of manifold geometry

The core construction is the link function

$\min_{x\in \mathcal M} f(x)$0

This function connects data density to geometric operations. The paper derives the identities

$\min_{x\in \mathcal M} f(x)$1

and

$\min_{x\in \mathcal M} f(x)$2

where $\min_{x\in \mathcal M} f(x)$3 denotes the posterior distribution over clean data points $\min_{x\in \mathcal M} f(x)$4 given a noisy observation $\min_{x\in \mathcal M} f(x)$5. The paper abbreviates these quantities as

$\min_{x\in \mathcal M} f(x)$6

Their geometric meaning emerges in the small-$\min_{x\in \mathcal M} f(x)$7 regime. Under manifold regularity assumptions, $\min_{x\in \mathcal M} f(x)$8 approximates the closest-point projection $\min_{x\in \mathcal M} f(x)$9, while $\mu_{\mathrm{data}}$0 approximates the tangent-space projector $\mu_{\mathrm{data}}$1. The paper gives the uniform approximation bounds

$\mu_{\mathrm{data}}$2

and

$\mu_{\mathrm{data}}$3

for $\mu_{\mathrm{data}}$4 in a tubular neighborhood of $\mu_{\mathrm{data}}$5. On the manifold, this yields

$\mu_{\mathrm{data}}$6

In effect, the link function provides approximations to the two operations that classical Riemannian methods require most: projection back to the manifold and projection of ambient gradients onto the tangent space (Kharitenko et al., 27 Sep 2025).

3. Score-based formulation and DLF dynamics

The framework is explicitly tied to the diffusion-model score function $\mu_{\mathrm{data}}$7. Using the Gaussian smoothing relation $\mu_{\mathrm{data}}$8, the paper derives the score-posterior identities

$\mu_{\mathrm{data}}$9

or equivalently

$\min_{x \in \mathcal M} f(x),$0

The score network is trained through conditional score matching, with loss

$\min_{x \in \mathcal M} f(x),$1

whose unique minimizer is $\min_{x \in \mathcal M} f(x),$2.

DLF itself is defined from a learned map $\min_{x \in \mathcal M} f(x),$3 satisfying $\min_{x \in \mathcal M} f(x),$4 and $\min_{x \in \mathcal M} f(x),$5. Its dynamics are

$\min_{x \in \mathcal M} f(x),$6

with landing gain $\min_{x \in \mathcal M} f(x),$7. The first term,

$\min_{x \in \mathcal M} f(x),$8

acts as projected descent, approximating tangent-space descent on the manifold. The second term,

$\min_{x \in \mathcal M} f(x),$9

is the landing term, which pulls the state toward the manifold surrogate.

If $f:\mathbb R^d \to \mathbb R$0 exactly, then DLF is the gradient flow of

$f:\mathbb R^d \to \mathbb R$1

since

$f:\mathbb R^d \to \mathbb R$2

In the limit $f:\mathbb R^d \to \mathbb R$3, with $f:\mathbb R^d \to \mathbb R$4, this reduces to ordinary Riemannian gradient flow on $f:\mathbb R^d \to \mathbb R$5. A practical Euler discretization uses

$f:\mathbb R^d \to \mathbb R$6

with $f:\mathbb R^d \to \mathbb R$7, $f:\mathbb R^d \to \mathbb R$8, and $f:\mathbb R^d \to \mathbb R$9. The Jacobian-vector product is computed by one forward pass and one backward pass through the score network (Kharitenko et al., 27 Sep 2025).

4. Feasibility, stationarity, and landing behavior

The analysis assumes uniform approximation error bounds

$P_{T_x\mathcal M}$0

where $P_{T_x\mathcal M}$1 is a tubular neighborhood of $P_{T_x\mathcal M}$2. Under smoothness conditions, if $P_{T_x\mathcal M}$3 is sufficiently small and

$P_{T_x\mathcal M}$4

then the DLF solution exists globally and remains in the tube,

$P_{T_x\mathcal M}$5

The asymptotic conclusions are twofold. First, every accumulation point $P_{T_x\mathcal M}$6 satisfies the distance bound

$P_{T_x\mathcal M}$7

Second, the projected point $P_{T_x\mathcal M}$8 has small Riemannian gradient norm: $P_{T_x\mathcal M}$9 is bounded by a constant multiple of $\pi(x)$0, with the exact constant structure given in the paper. These results are the basis for the paper’s claims of approximate feasibility and optimality.

The exact $\pi(x)$1 case clarifies the term “landing.” The paper shows that

$\pi(x)$2

Hence the distance-like quantity $\pi(x)$3 is non-increasing, the trajectory stays in the tubular neighborhood for all time, and if $\pi(x)$4, it lands on the manifold asymptotically. Accumulation points are then critical points on $\pi(x)$5 (Kharitenko et al., 27 Sep 2025).

5. Empirical demonstrations and application domains

The paper validates the framework on two classes of problems. The first is optimization over the orthogonal group $\pi(x)$6, using Brockett’s cost

$\pi(x)$7

The setup uses $\pi(x)$8, uniform data sampled from $\pi(x)$9, a score network trained with denoising score matching, Euler discretization of DLF, landing gain $\mathcal M$0, and step size $\mathcal M$1. The reported findings are that the optimized objective drops below the best value present in the training set, performance improves as $\mathcal M$2, and DLF approximates the exact landing flow well (Kharitenko et al., 27 Sep 2025).

The second domain is finite-horizon data-driven control on implicit behavior manifolds. The systems are a double pendulum and a unicycle car. The feasible set is the input-output trajectory manifold

$\mathcal M$3

and the objective is

$\mathcal M$4

for a reference trajectory $\mathcal M$5. The paper reports that a diffusion model is trained on measured trajectories and then DRGD and denoising optimization are run. The optimized trajectories track the reference significantly better than the best training example, the trajectory obtained by simulating the actual dynamics remains close to the optimized trajectory, and the method generalizes beyond the training set. The paper also notes that DRGD, rather than DLF, is the main method used in the control experiments. This suggests that DLF primarily serves as the continuous-time landing mechanism within a broader score-based Riemannian optimization framework.

6. Relation to DRGD and distinction from similarly named methods

DLF and Denoising Riemannian Gradient Descent (DRGD) are companion algorithms in the same framework, but they occupy different algorithmic roles. DLF is the continuous-time landing flow

$\mathcal M$6

where feasibility is controlled dynamically by the landing term. DRGD is the discrete update

$\mathcal M$7

which is closer to classical Riemannian gradient descent: a tangent-like step followed by an approximate retraction. The paper characterizes DLF as a robust landing mechanism and DRGD as a more direct discrete optimizer (Kharitenko et al., 27 Sep 2025).

Several distinct uses of denoising-flow terminology can be confused with DLF. In robot manipulation, DVAC, introduced in "Denoising Tells When to Replan: Denoising-Variance Adaptive Chunking for Flow-Based Robot Policies" (Feng et al., 2 Jun 2026), uses denoising variance in flow-based robot policies to adapt execution horizon at test time; it is a training-free adaptive chunking rule for action chunks, not a manifold-optimization algorithm. In image processing, the damped flow model of "Damped second order flow applied to image denoising" (Baravdish et al., 2018) is a second-order nonlinear PDE for variational denoising, not a learned or score-based landing flow. In generative modeling, "The Generation Phases of Flow Matching: a Denoising Perspective" (Gagneux et al., 28 Oct 2025) develops a denoising interpretation of flow matching and studies temporal phases of generation, but it does not define or benchmark a method explicitly called DLF.

A common misconception is therefore to treat any denoising-related flow, or any “landing” behavior in a denoising trajectory, as DLF. In the specific technical sense established in (Kharitenko et al., 27 Sep 2025), DLF denotes a score-based continuous-time optimization flow on implicit data manifolds, built from the link between Gaussian-smoothed data density and Riemannian geometry.