Does a diffusion model remember where it started?

Based on Per-Starting-Point Plausibility and Diversity Bounds for Score-Based Diffusion Models (NeurIPS 2026, with Hady W. Lauw)

Scope: the formal bounds concern the unguided reverse SDE under the paper's assumptions, with constant-noise OU and VP forward processes. They do not directly cover guided inverse-problem samplers or probability-flow ODEs.

The setup everyone knows

A diffusion model generates data by reversing a noising process. Training teaches a network the score, the direction that makes a noisy sample look more like real data. To generate, you draw pure Gaussian noise and follow the score back, step by step, until an image appears.

The theory behind this is mature. Existing convergence results say: if you start from random noise, the distribution of what comes out is close to the data distribution. But notice what that statement averages over: all possible starting noises at once.

Mixture-level bounds average over all starting noises. Per-start pipelines (SDEdit, posterior sampling, OOD detection) need a statement about one start.

The gap

Many real pipelines don't start from fresh random noise. They fix a starting point and run the reverse process from there:

SDEdit noises an input image partway, then denoises it.
Posterior sampling for inverse problems starts from something tied to a measurement.
OOD detection can look at what the reverse process does from one specific input.

For these, "the mixture over all starts is fine" does not by itself bound the error from a particular start. I need to know what happens from this start. I call this setting diffusion from deterministic starting points (DDS), and as far as I know it had no non-asymptotic analysis. Two questions come up:

Plausibility. From one fixed start, are the outputs still close to the data distribution?
Diversity. From two different starts, are the output distributions still different, or did the noise wash the start away?

They pull against each other. If the start is completely forgotten, outputs look like generic data but carry no information about where you began. If it's remembered too strongly, the outputs may not look like data at all.

Part 1: plausibility, split into three costs

The first theorem bounds the Wasserstein-1 distance between the data distribution q and the output law H from start x. The trick is to walk from q to H through two intermediate distributions, so the triangle inequality splits the error into pieces you can name:

The triangle inequality along this chain splits the distance between q and H into three costs, each with its own cause.

W_init, the conditioning cost. Fixing the endpoint of the forward noising process tilts the data distribution a bit (a Bayesian posterior). This tilt shrinks as the forward process runs longer, because the noise attenuates the signal exponentially.
W_stop, the early-stopping cost. The score blows up near zero noise, so every sampler stops slightly early. For a fixed start this costs roughly the square root of the stopping time.
W_gen, the generation error. This is the familiar part: learned score vs. true score, plus discretization. I run the true reverse process and the learned one on the same Brownian noise, so the noise cancels in their difference and what remains is a clean inequality. Contraction in the true score damps the error over time.

The first two terms don't exist in mixture-level analyses, which is what makes the per-start case new. Also, I only require the score network to be accurate in a ball around the start, not everywhere, so the bound stays meaningful for starts that are far from typical training data.

Part 2: diversity from the same coupling trick

Run two copies of the sampler from different starts x(i) and x(j), driven by the same noise. Because the noise is shared, it cancels in the difference of the two trajectories. What is left is a deterministic recursion for the gap between them, so I can track how far apart their means stay.

Project both outputs onto the direction in which their means differ most. That gives two 1D distributions with a mean gap δ and standard deviations σ_V, σ_U. A Cantelli-style argument then lower-bounds the total variation distance between the two output laws. The bound is positive exactly when the mean gap beats the spread, with a threshold at δ² = 4σ_Vσ_U.

This is a certificate: when the condition holds, the two starts are provably distinguishable at that sampling horizon. It is not a claim that the start is always remembered.

What it looks like

Schematic of the 4-mode Gaussian-mixture experiment (not the real plot). Both properties hold at once: outputs look like the data overall, yet each start leaves an imprint.

On a 4-mode Gaussian mixture with the exact score, outputs from every start roughly recover the full data distribution (plausibility). But the densest half of each output's mass sits on the mode closest to where it started (diversity). Both are true at once. I see the same qualitative picture on 2D toy sets, MNIST and CIFAR-10.

One more finding: the deterministic probability-flow ODE sends each start to a single point (output std below 1e-10 on MNIST), while the stochastic SDE produces many different digits from the same start. The analysis is about the SDE.

Limits

Only the unguided reverse SDE. Guidance and ODE-style samplers are out of scope.
Formal bounds cover OU and VP with constant noise scale. Other forward processes are illustrative.
Real inverse problems add a measurement operator, which this analysis doesn't handle yet. That is the main open direction.