Marginal Path Collapse in Generative Modeling

• Marginal Path Collapse is a failure mode where intermediate probability measures become non-normalizable despite valid endpoints.
• It arises from combining expert models with mismatched noise schedules, leading to explosive density growth in specific scenarios.
• Adaptive Path Correction with Exponents (ACE) mitigates collapse by using time-varying exponents, ensuring robust and accurate trajectory sampling.

Marginal Path Collapse is a failure mode arising in generative modeling by path interpolation, where intermediate time-indexed probability measures along a constructed trajectory become non-normalizable, despite the endpoints remaining valid. This phenomenon systematically affects ratio-of-densities steering methods when combining multiple pretrained diffusion or flow models, particularly those with heterogeneous noise schedules or distinct dataset origins. Marginal Path Collapse has significant implications for inference-time controllable generation, notably manifesting in molecular design workflows requiring compositional generation from expert models trained under divergent regimes (Lee et al., 11 Dec 2025).

1. Formal Definition and Mathematical Characterization

For $n$ expert density paths $\{q^{(i)}_t\}$ on $\mathbb{R}^{d_i}$, mapped into a common state space $\mathbb{R}^d$ via projections $\pi_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$, a ratio-of-densities path with time-dependent exponents $\gamma_i(t)$ is defined as:

$$h_t(x) = \prod_{i=1}^n \left(q^{(i)}_t(\pi_i(x))\right)^{\gamma_i(t)}, \qquad x \in \mathbb{R}^d$$

The normalization constant is $Z_t = \int_{\mathbb{R}^d} h_t(x)\,dx$. A valid probability path requires $0 < Z_t < \infty$ for all $t \in [0,1]$. Marginal Path Collapse occurs when both endpoints are integrable ($0<Z_0<\infty, 0<Z_1<\infty$), but for some intermediate $t^* \in (0,1)$, $Z_{t^*} = \infty$, so the normalized density $p_{t^*}^*(x)$ is undefined (Lee et al., 11 Dec 2025).

2. Diagnostic Criterion for Collapse

The path existence criterion (Theorem 2.1, (Lee et al., 11 Dec 2025)) provides a necessary and sufficient test for collapse in typical stochastic interpolant constructions. Each expert path is generated by a linear or nonlinear time-dependent mixture of initial and terminal points, with noise schedule $\alpha^{(i)}_t$. For each coordinate $k$: $$C_k(t) := \sum_{i: k \in I_i} \frac{\gamma_i(t)}{(\alpha^{(i)}_t)^2}$$ Here, $I_i$ denotes the coordinates covered by the $i$th expert. The path $\{h_t\}$ is globally integrable for all $t$ if and only if $C_k(t) > 0$ for every $k$ and $t$. If any $C_k(t^*) < 0$, collapse is guaranteed at $t^*$. This reduces integrability to a check of the positive-definiteness of the aggregate quadratic form induced by the exponent-weighted volatility structure.

3. Archetypal Collapse Scenarios and Illustrative Examples

Marginal Path Collapse arises naturally when composing models with mismatched noise schedules or exponents, as commonly encountered in molecular generation applications. The Gaussian counterexample constructs a ratio-of-densities path with different variances:

$$h_t(x) = \frac{q^{(1)}_t(x) q^{(2)}_t(x)}{q^{(3)}_t(x) q^{(4)}_t(x)},\quad q^{(i)}_t = \mathcal{N}(0,\sigma_i^2(t)I)$$

Under certain exponents and noise schedules, $h_{t^*}(x)$ exhibits explosive growth, e.g., $h_{0.5}(x) \gtrsim \exp(+0.01\|x\|^2)$, rendering $Z_{0.5} = \infty$.

In compositional tasks with local and global constraints—such as combining local (e.g., $p(X|A)$) and global (e.g., $p(X,Y|B)$) expert distributions using inconsistent denoising schedules—collapse recurs. Empirically, with three heterogeneous schedules in synthetic 2D scenarios, collapse rates escalate with guidance scale ($w=1.0$: 41%; $w=2.0$: 66%; $w=15$: 80%) (Lee et al., 11 Dec 2025).

4. Resolution via Adaptive Path Correction with Exponents (ACE)

Adaptive path Correction with Exponents (ACE) is proposed to correct collapse and enable valid trajectory construction under the ratio-of-densities framework. ACE extends Feynman-Kac steering by incorporating time-varying exponents. For particles $(X_t, w_t)$, the joint SDE evolves as:

$$\begin{cases} dX_t = (v^*_t(X_t) + \tfrac{\sigma_t^2}{2} s^*_t(X_t))\,dt + \sigma_t\,dW_t\ d\log w_t = \nabla \cdot v^*_t(X_t)\,dt + \sum_{i=1}^n \dot{\gamma}_i(t) \log \tilde q_t^{(i)}(X_t)\,dt + \sum_{i=1}^n \gamma_i(t) D_t^{(i)}(X_t)\,dt \end{cases}$$

where $$s^*_t(x) = \sum_{i=1}^n \gamma_i(t)\, \tilde s^{(i)}_t(x)$$ are the aggregate score functions. Algorithmically, each timestep involves computing scores, propagating particles, updating log densities and weights, and performing resampling when effective sample size drops below a threshold. ACE is guaranteed (Thm 2.3) to yield the correct marginal law, provided the existence criterion $C_k(t)>0$ holds throughout (Lee et al., 11 Dec 2025).

5. Empirical Manifestations and Corrective Effectiveness

In synthetic benchmarks, conventional resampling and Feynman-Kac correction (FKC) methods show large distributional errors under collapse ($W_1 \approx 0.78$–2.13, MMD up to 1.43), while ACE yields $W_1 = 0.28\pm0.036$, MMD = 0.027$\pm0.0064$. For flexible-pose molecular scaffold decoration, ACE maintains $>96\%$ validity up to high guidance, compared to collapse-induced drops to 78.1% in FKC. Docking performance improves with ACE, achieving a mean score of $-5.72$ kcal/mol at $\omega = 1.3$ versus FKC's $-2.67$ kcal/mol. ACE matches or surpasses task-specific AutoFragDiff models ($>98\%$ validity; OSR 94.4%) (Lee et al., 11 Dec 2025).

Method	Collapse Frequency ($w=1.0$)	Validity ($\omega=1.4$)	Docking Score ($\omega=1.3$)
NR	41%	78.1%	N/A
FKC	41%	78.1%	$-2.67$ kcal/mol
ACE	0%	>96%	$-5.72$ kcal/mol

6. Broader Context and Practical Significance

Marginal Path Collapse is not intrinsic to score-based generative modeling, but arises from the interaction of heterogeneous expert compositions and guiding schedules. Its identification unifies and clarifies a range of empirically observed instabilities across conditional sampling, compositional generation, and molecular design. The existence criterion enables tractable diagnosis prior to execution, while ACE transforms ratio-of-densities steering from an unreliable heuristic to a provably correct, robust tool for inference-time control. Extension to arbitrary time-varying exponents further generalizes controllable guidance protocols (Lee et al., 11 Dec 2025). A plausible implication is that analogous collapse phenomena may emerge in other compositional or guided sampling settings where probability measures are manipulated via exponentiation of conditionally independent paths.