TL;DR  ACE diagnoses and corrects Marginal Path Collapse, a hidden failure mode in multi-constraint ratio-of-densities steering. Plug-and-play steering for Molecules and Images with no retraining required.

On the Collapse of Generative Paths:
A Criterion and Correction for Diffusion Steering

Ziseok Lee1*†, Minyeong Hwang2*, Wooyeol Lee1, Sanghyun Jo3,4, Jihyung Ko4, Young Bin Park5, Jae-Mun Choi5, Eunho Yang2,6†, Kyungsu Kim1,4,7†
1SNU Biomedical Sciences, 2KAIST AI, 3OGQ, 4SNU IPAI, 5Calici, 6AITRICS, 7SNU Transdisciplinary Innovations
*Equal Contribution, †Corresponding Author
ICML 2026
OpenReview Code arXiv

Abstract

Inference-time steering adapts pretrained diffusion and flow models to new tasks without retraining, often using ratio-of-densities constructions that reweight time-indexed marginals with fixed exponents. We identify Marginal Path Collapse, a failure mode in which the intermediate density defined by such compositions becomes non-normalizable despite valid endpoints. This collapse can arise when composing heterogeneous experts trained with mismatched noise schedules, negative exponents, or partial supports.

To address this, we provide a sharp sufficient Path Existence Criterion that certifies well-defined composed intermediate densities when \(C(t)>0\) and detects collapse when \(C(t)<0\), together with Adaptive Path Correction with Exponents (ACE). ACE generalizes Feynman–Kac steering to time-varying exponents and dynamically adjusts expert weights to keep the path valid. On flexible-pose scaffold decoration and compositional image generation, ACE prevents collapse and improves sample reliability over constant-exponent baselines.

What Problems Does ACE Solve?

ACE targets modular generation problems where a sample must satisfy several constraints at once. A simple template is a joint object \(X,Y\) with a local condition \(A\) on \(X\) and a joint condition \(B\) on \((X,Y)\). Under the conditional-independence structure used in the paper, Bayes' rule gives a ratio-of-densities target:

\[ p(X,Y\mid A,B) \propto p(X\mid A)\,\frac{p(X,Y\mid B)}{p(X)}. \]

This decomposition lets us combine pretrained experts instead of retraining a monolithic model. ACE solves the missing piece: making the entire intermediate generative path valid when these experts have different schedules, supports, or exponent signs.

Synthetic Checker

Two coordinates \(X,Y\) are generated from heterogeneous marginal and joint experts.

\(p(X,Y\mid A,B) \propto p(X\mid A)\,p(X,Y\mid B)/p(X)\)

This controlled setting exposes Marginal Path Collapse and tests the Path Existence Criterion.

Flexible-Pose Scaffold Decoration

Let \(X=M^{sc}\) be the scaffold and \(Y=M^R\) the R-groups. Topology acts locally, while protein-pocket binding acts globally.

\(p(M^{sc},M^R\mid A,B) \propto p(M^{sc}\mid A)\,p(M\mid B)/p(M^{sc})\)

ACE composes CONF, SBDD, and DN experts while repairing schedule-induced collapse.

Multi-Instance Image Generation

Region prompts \(c_i\) constrain local foregrounds \(F_i\), while a global prompt \(C\) constrains the full image.

\(p(X\mid c_{1:n},C) \propto p(X\mid C)^{\gamma_0(t)}\prod_i [p(F_i\mid c_i)/p(F_i)]^{\gamma_i(t)}\)

Here the path is homogeneous and valid, but ACE still improves concentration and prompt alignment.

Understanding ACE

ACE detects invalid composed paths with the Path Existence Criterion and corrects exponent schedules so that \(C_k(t)>0\) on the sampling timesteps.

Marginal Path Collapse & ACE Correction

Marginal Path Collapse and ACE Solution

Figure 1: Marginal Path Collapse breaks heterogeneous composition. In a 2D checkerboard built from two 1D priors and a 2D constraint, standard steering is biased and FKC fails when the path-existence criterion becomes negative. ACE dynamically adjusts exponents to ensure \(C(t)>0\), recovering the target path.

Understanding Path Collapse

Non-integrable Region in Gaussian Path

Figure 2: Non-integrable region in the ratio-of-Gaussians example. The path is well-defined at the endpoints, but the intermediate variance explodes, causing Marginal Path Collapse. The sampler may still run numerically, but it no longer follows the intended probability path.

Schedule Mismatch and Criterion Repair

Common noise schedules and the path existence criterion with and without ACE

Figure 3: Common noise schedules and Marginal Path Collapse. Many heterogeneous three-expert compositions enter a region where \(C(t)<0\), implying non-normalizable intermediate densities. The corrected exponents of ACE ensure \(C(t)>0\) for all sampled times.

Applications of ACE

One ratio-of-densities framework, multiple domains — from drug discovery to multi-instance image generation

Flexible-Pose Scaffold Decoration

Drug Discovery
Scaffold Decoration Results

Figure 6: ACE enables high-guidance steering for flexible-pose scaffold decoration. ACE generates highly dockable molecules by accurately modeling the ratio-of-density path, whereas FKC and NR produce suboptimal results due to ill-defined probability paths. The flexible-pose formulation relaxes the fixed-pose constraint of Delete and explores a larger search space.

Impact of Path Collapse on Molecules

Drug Discovery
Impact of Marginal Path Collapse

Figure E.9: Impact of Marginal Path Collapse on molecular generation. On scaffold decoration at \(\omega=1.3\), FKC enters a collapsed regime near the end of sampling where \(C(t)<0\), producing disjoint fragments. ACE applies the bump correction to keep \(C(t)>0\) throughout the trajectory and produces connected, chemically valid molecules.

Compositional Image Generation Vision

Figure E.18: Qualitative results of compositional image generation with ACE. Compared to the base Stable Diffusion 2.1 model (left), simulating the same model with ACE (right) yields better prompt alignment and layout guidance without additional training or external models. Table E.16 quantifies the improvement on COCO-MIG.

Key Contributions

Diagnosis: Path Existence Criterion

We derive an easy-to-compute criterion that certifies path existence when \(C_k(t)>0\) and detects collapse when \(C_k(t)<0\), using only schedules and exponents.

\( C_k(t) := \sum_{i: k \in I_i} \frac{\gamma_i(t)}{(\alpha_t^{(i)})^2} > 0 \)

Positive criterion certifies a valid path; negative criterion detects collapse.

Solution: ACE Framework

Building on this diagnosis, ACE extends Feynman–Kac steering to support time-varying exponents, dynamically adjusting expert weights throughout generation while preserving the final target distribution.

\( \tilde{\gamma}_i(t) = \gamma_i(t) + b_i(t)\)

Endpoint-preserving "bump" correction \(b_i(t)\) keeps the path-existence criterion positive (\(b_i(0)=b_i(t_{\mathrm{end}})=0\)). In the paper, we prove such corrections can always be constructed and empirically instantiate them with linear and quadratic bump components, selecting \(B_1,B_2\) to satisfy the criterion during sampling.

Key Insight: Marginal Path Collapse occurs when numerator and denominator experts shrink at incompatible rates, creating an intermediate imbalance where the combined density grows rather than decays at infinity. ACE repairs this by modifying the intermediate exponent path without changing the endpoint target.

Method Comparison

Method comparison

Table 1: Comparison of steering methodologies. Unlike heuristics or standard correctors, which assume homogeneity or do not verify path existence, ACE provides an explicit criterion and adaptive correction for path validity under heterogeneous conditions.

Quantitative Results

Comparison of NR, FKC, and ACE

Table 3: Performance comparison on CrossDocked, evaluating docking affinity (Vina Score) and drug-likeness (QED, SA, and Lipinski). Higher is better for OSR and drug-likeness; lower, more negative Vina scores are preferred. NR and FKC suffer from path-existence-criterion violations, while ACE remains valid across the tested configurations.

CrossDock Results Table

Table 4: Comparison against fixed-pose scaffold-decoration baselines on CrossDocked, evaluating docking affinity and drug-likeness. Higher is better for OSR and drug-likeness; lower, more negative Vina scores are preferred. Asterisks denote methods that require a reference ligand pose.

Computational cost of ACE, ACE-lite, and baselines

Table 5: Computational cost reported in terms of VRAM usage (GB) and runtime (min/molecule). ACE-lite significantly reduces the runtime overhead while maintaining performance, showing that the path-validity correction can be deployed efficiently alongside task-specific baselines.

COCO-MIG Benchmark Results for Compositional Image Generation

Table E.16: Instance attribute success ratio (%) on COCO-MIG, with the corresponding mIoU (%) shown in parentheses. L2–L6 denote tasks with 2 to 6 region-wise guidance conditions. CLIP is computed on the full image and prompt, while Local CLIP is computed on local crops and their corresponding local prompts.

BibTeX

@inproceedings{
    lee2026collapse,
    title={On the Collapse of Generative Paths: A Criterion and Correction for Diffusion Steering},
    author={Ziseok Lee and Minyeong Hwang and Wooyeol Lee and Sanghyun Jo and Jihyung Ko and Young Bin Park and Jae-Mun Choi and Eunho Yang and Kyungsu Kim},
    booktitle={Proceedings of the 43rd International Conference on Machine Learning},
    year={2026},
    url={https://openreview.net/forum?id=emv2qsi3TG}
}