Ideal-marginal score error allows unbounded finite-Euler Wasserstein error

For the separated-mixture question reproduced below, there are admissible fixed one-dimensional parameters with $L=2$, fixed positive score-error allowance ${\epsilon }_s$, and every fixed integer $J\ge 2$, for which the supremum of the displayed expected fidelity loss over the original finite-valued Borel score class is $+\mathrm{\infty }$, for every finite $M\ge 1$. Each individual field in the construction has a finite-first-moment Euler chain and obeys the required score bound at every time. Thus the original ideal-marginal $L^2$ hypothesis supplies no finite uniform bound for the requested loss, however large the fixed budget $JM$ is. The original question is retained verbatim under Original question.

Burns and Fridovich-Keil, full 2026 paper

Daras et al., full diffusion inverse-problems survey

Blechschmidt et al., full 2026 flow-matching paper

Prince, Chapter 18 full diffusion notes

Fix dimension $d\ge 1$, component count $L\ge 2$, noise scale $\sigma >0$, diffusion horizon $T>0$, and a continuous schedule $\beta :[0,T]\to (0,\mathrm{\infty })$. Assume the parameter choices below admit at least one component configuration and mixture weight vector. Consider a posterior density on ${\mathbb{R}}^d$ of the form $\pi (x)={\sum }_{\mathrm{\ell }=1}^L{w}_{\mathrm{\ell }}\mathcal{N}(x;{\mu }_{\mathrm{\ell }},{\mathrm{\Sigma }}_{\mathrm{\ell }}),{\sum }_{\mathrm{\ell }=1}^L{w}_{\mathrm{\ell }}=1,w_{\mathrm{\ell }}\ge w_{\min }>0,$ Here $\mathcal{N}(\cdot ;{\mu }_{\mathrm{\ell }},{\mathrm{\Sigma }}_{\mathrm{\ell }})$ denotes the normal density with mean ${\mu }_{\mathrm{\ell }}$ and covariance matrix ${\mathrm{\Sigma }}_{\mathrm{\ell }}$. The component means satisfy $\mathrm{\parallel }{\mu }_{\mathrm{\ell }}{\mathrm{\parallel }}_2\le M_0$ and $\mathrm{\parallel }{\mu }_{\mathrm{\ell }}-{\mu }_r{\mathrm{\parallel }}_2\ge \mathrm{\Delta }>0$ for $\mathrm{\ell }\mathrm{\ne }r$, and every eigenvalue of each positive-definite ${\mathrm{\Sigma }}_{\mathrm{\ell }}$ lies in $[v_{\min },v_{\max }]$ for fixed $0<v_{\min }\le v_{\max }<\mathrm{\infty }$. Assume $0<w_{\min }\le 1\mathrm{/}L$ and $\mathrm{\Delta }\ge 8\sqrt{d{v}_{\max }}$. For $t\in [0,T]$, let $p_t$ be the density of $X_t$ under $d{X}_t=-\frac{1}{2}\beta (t)X_{t}dt+\sqrt{\beta (t)}d{W}_t,X_0\sim \pi ,$ where $W_t$ is a standard $d$-dimensional Brownian motion, and let $s_t=\mathrm{\nabla }\log p_t$ be its exact score. Let ${\widehat{s}}_t$ be a supplied jointly Borel, finite-valued score field satisfying the integrated error bound ${\sup }_{0\le t\le T}{\int }_{{\mathbb{R}}^d}\mathrm{\parallel }{\widehat{s}}_t(x)-s_t(x){\mathrm{\parallel }}_2^2{p}_t(x)dx\le {\epsilon }_s^2\mathrm{.}$ Consider fields whose Euler chain has finite first moment. For $h=T\mathrm{/}J$, use the reverse-time Euler convention $Z_0\sim p_T$ and $Z_{k+1}=Z_k+h[\frac{1}{2}\beta (T-kh)Z_k+\beta (T-kh){\widehat{s}}_{T-kh}(Z_k)]+\sqrt{h\beta (T-kh)}{\zeta }_k,k=0,\dots ,J-1,$ with ${\zeta }_k\sim N(0,I_d)$ independent of $Z_0$ and of one another. Generate $M\ge 1$ independent terminal values $X^{(1)},\dots ,X^{(M)}$ and let ${\widehat{\pi }}_{J,M}=M^{-1}{\sum }_{i=1}^M{\delta }_{X^{(i)}}$; one score evaluation per reverse step gives online budget $B=JM$. Define disjoint mode regions $R_{\mathrm{\ell }}=\{x:\mathrm{\parallel }x-{\mu }_{\mathrm{\ell }}{\mathrm{\parallel }}_2<\mathrm{\Delta }\mathrm{/}4\}$, target masses $q_{\mathrm{\ell }}=\pi (R_{\mathrm{\ell }})$, and empirical masses ${\widehat{q}}_{\mathrm{\ell }}={\widehat{\pi }}_{J,M}(R_{\mathrm{\ell }})$. For probability measures on ${\mathbb{R}}^d$, let $W_1$ be the infimum of $E\mathrm{\parallel }X-Y{\mathrm{\parallel }}_2$ over all couplings with the specified marginals. Determine whether the integrated score-error assumption alone yields finite minimax-sharp nonasymptotic upper and lower bounds, uniformly over the stated score-field class, for $E{W}_1({\widehat{\pi }}_{J,M},\pi )+\mathrm{\Delta }{\sum }_{\mathrm{\ell }=1}^{L}E\mathrm{\mid }{\widehat{q}}_{\mathrm{\ell }}-q_{\mathrm{\ell }}\mathrm{\mid }+\mathrm{\Delta }P\{{\min }_{1\le \mathrm{\ell }\le L}{\widehat{q}}_{\mathrm{\ell }}=0\}$ as a function of $d,L,w_{\min },\mathrm{\Delta },v_{\min },v_{\max },M_0,T,\beta ,{\epsilon }_s,J,M$. In particular, characterize the compute regimes $B=JM$ that guarantee simultaneous control of distributional error and missed-mode probability uniformly over this family. This asks about score-based reverse diffusion with finite computation; it does not assume that the posterior is unimodal or that a good posterior mean certifies its mode weights.

The negative answer concerns every positive error allowance and every finite J>=2. It does not assert the same obstruction at J=1 or at exactly zero error. The mechanism is mismatch between ideal marginal mass and actual chain mass, not merely missing a mode by iid sampling. No numerical observation or source-specific failure experiment is used as the proof.