Denoising Diffusions with Optimal Transport: Localization, Curvature, and Multi-Scale Complexity
Authors: Tengyuan Liang
Kulunu Dharmakeerthi
Takuya Koriyama
Venue: Accepted by TMLR ()

Abstract: Adding noise is easy; what about denoising? Diffusion is easy; what about reverting a diffusion? Diffusion-based generative models aim to denoise a Langevin diffusion chain, moving from a log-concave equilibrium measure $\nu$, say an isotropic Gaussian, back to a complex, possibly non-log-concave initial measure $\mu$. The score function performs denoising, moving backward in time, and predicting the conditional mean of the past location given the current one. We show that score denoising is the optimal backward map in transportation cost. What is its localization uncertainty? We show that the curvature function determines this localization uncertainty, measured as the conditional variance of the past location given the current. We study in this paper the effectiveness of the diffuse-then-denoise process: the contraction of the forward diffusion chain, offset by the possible expansion of the backward denoising chain, governs the denoising difficulty. For any initial measure $\mu$, we prove that this offset net contraction at time $t$ is characterized by the curvature complexity of a smoothed $\mu$ at a specific signal-to-noise ratio (SNR) scale $r(t)$. We discover that the multi-scale curvature complexity collectively determines the difficulty of the denoising chain. Our multi-scale complexity quantifies a fine-grained notion of average-case curvature instead of the worst-case. Curiously, it depends on an integrated tail function, measuring the relative mass of locations with positive curvature versus those with negative curvature; denoising at a specific SNR scale is easy if such an integrated tail is light. We conclude with several non-log-concave examples to demonstrate how the multi-scale complexity probes the bottleneck SNR for the diffuse-then-denoise process.

Published in Transactions on Machine Learning Research (01/2026)
Denoising Diffusions with Optimal Transport: Localization, Curvature, and Multi-Scale Complexity
Tengyuan Liang
edu
University of Chicago, Booth School of Business
Kulunu Dharmakcerthi
kukkanad(huchiccago.cc
University of Chicago
Takuya Koriyama
thortyam@ehicaagoooolo
University of Chicago, Booth School of Business
Reviewed on OpenReview: https: // opw.reviee.r.......ommoo net/ formm? id= sd=siiuU6gB
Abstract
Adding noise easy; what about denoising? Dilfusion easy; what about reverting a diffusion? Diffusion-basee based generative models aim to denoise a Langevin diffusion chain, moving from a log-concave equilibrium measure v, say an isotropic Gaussian, back to complex, possibly non-log-concave initial measure μ.. The score function performs denoising, moving backward in time, and predicting the conditional mean of the past location given the current one. We show that score denoising is the optimal backward map in transportation cost. What is its localization uncertainty? We show that the curvature function determines this localization uncertainty, measured as the conditional variance of the past location given the current. We study this paper the effectiveness of the -effese-then-eenoiiee process: the contraction of the forward diffusion chain, offset by the possible expansion of the backward denoising chain, governs the denoising difficulty. For any initial measure μ4, prove that this offset net contraction at time & is characterized by the curvature complexity of smoothed at a specific signal-to-noise ratio (SNR) scale r(f). We discover that the multi-scale curvature complexity collectively determines the difficulty of the denoising chain. Our multi-scale complexity quantifies a fine-grained notion of everage-case curvature instead the worst- case. Curiously, depends on an integrated tail function, mcasuring the relative mass of locations with positive curvature versus those with negative curvature; denoising at specific SNR scale is easy if such integrated tail is light. We conclude with several non- log-concave examples to demonstrate how the multi-scale complexity probes the bottleneck SNR for the diffuse-then-denois process.
1 Introduction
Empiricallyy diffusion models exhibit compelling performance as probabilistic generative models for complex, multi-dimensional probability measures (Ho al.) 2020 Sohl-Dicksttiin et al. 2015 Song Ermon, 2019 Song et al.] 2021: Karras al. 2022). They are often employed when traditional sampling methods suffer, such when the probability measure is multi-modal and supported on unknown manifold that is hard to mathematize, such as the probability distribution of (pixels of) images. Despite their impressive performance in practice, several fundamental theoretical questions regarding denoising quality remain unanswered (Bockk
et al. 2020 Chen et al. 2022 Lee et. al. 2022).
In a nutshell, diffusion models aim to revert a Langevin diffusion chain, moving from log-concave equilibrium measure v, say an isotropic Gaussian, back toward complex, possibly non-log-concave initial measure μ. A Langevin diffusion is a forward chain in the space of probability measures, implemented iteratively with

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Published in Transactions on Machine Learning Research (01/2026)
stepsize as in k.
where is the equilibrium measure and fl denotes the forward transition mar at step
4 Forward Diffusion μ= puu Mu
S pkn
K=000
MKn
Time Reversal == 140 f. My by
Man f
f MKn
(2)
by
bj
This forward chain is Markovian and thus time-reversible as in 2 where byg denotes the backward transition mar a for the y-chain at step k. For sampling, diffusion models propose starting the time reversal process at K + 2o, namely the equilibrium measure v, and aim to reverse back to the initial mcasure μ.. However, this is problematic both theoretically (in terms of probability) and conceptually (in terms of optimization). Theoretically, the time reversal of Markov Chain starting from equilibrium (invariant measure) will get stuck (Norris 1998). The backward chain v bb will stay as the invariant measure and never reach PKy. Conceptually, hoping to trace back the initial condition μ starting from is infeasible. If two forward chains with initials * nz both end up al v, the time-reversal starting from v (and solely using the future information) cannot recover the past and distinguish these two chains.
Therefore, framing the diffusion model as a time-reversal Markov chain-one that recovers the past from the future- not fully resolve the underlying conceptual issues. In contrast, we propose studying diffusion models form of sensitivity analysis. It is clear that starting from MKy and traversing back following transitions b% ... bj" ... by will identify μ.. But what will happen if we start from perturbed version V =: =N # PKn and follow the same traversing path b'K ... by ... by as illustrated in !
Time Reversal J =: pi 4- My f-
Man
MKn bxx
(3)
by
by
Backward Denoising i 2 vo t- Un
t-
Mkn
VKn :=
(4)
by
by
b%
The traversing path carries the past information, namely the signature of the initial measure μ,, but starts with an easy-to-sample v as surrogate, replacing MKy. This mismatch renders the backward denoising process 4 non-Morkovian, thereby making it plausible to recover the past from the future.
Sensitivity Analysis: Score, Curvature, and Localization. How can the traversing operator by" be estimated? We show in Proposition that the optimal backward operator bf-in terms of transportation cost depends on the score function logPPs, where Pru, is the probability density function of Pиu.: A key observation in diffusion models is that score estimation can be cast as supervised prediction problem (Saremi & Hyvärinen 2019h using the forward diffusion chain, where the goal is to predict the previous location given the current one, in terms of conditional expectation, seen in Proposition Z
The fundamental question is whether the perturbation ūKn zo MKy will get amplified along the backward denoising chain. Will Vky as in 33-14 stay close to Pkn, for k = K.... ,0 This paper provides precise study of diffusion models from the viewpoint of denoising quality. We unveil in Proposition g that the curvature function log controls the key aspects of this sensitivity analysis. In other words, the curvature function governs the score function's denoising capability, which we refer to as localization. Localization quantifics the uncertainty of the previous location given the current, in terms of conditional covariance.
Most of the current theoretical literature (Lee et al. 2023 Chen et al. 2022 2023 treats this curvature as muisance, for example, by focusing on "non-expansion" metrics d, and leveraging a form of data- lata-processing inequality,
(1))=1 Fx) < 1, for d E (drv,dki.)
Consequently, d(p, Bp) S v), and d(us Po) can be determined by the contraction of the forward diffusion chain alone. However, even for simple Gaussians, the backward denoising could either be (i) an
The rigorous definition will follow Section
2 Agaai,, the rigorous definition will follow in Section
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expansion (ii) contraction much faster than the forward diffusion, under the Wasserstein-2 metric, W. Consider the forward diffusion (5) with initialization j ~ (N, and temperature 3 = Recall, the forward diffusion contracts in W at a rate 1- - 77. Our Corollary implies that the one-step backward denoising at effective time = kn with μi ~ NNee" hon eeeee 2km + 2kr) satisfics the equality
(1) 1
if s² > 1, if s² < 2 and
W( 1k-1) W(i,, vk)
(ii) < 1 - 1
log (2)1-
In either case, treating the backward denoising simply non-expansive under certain metrics is losing significant information about the teaiss-then-denooee process.
In contrast, we study the process under the Wasscrstein-2 metric and provide a fino-grained analysis of how a complexity measure based on the curvature, V Pog.r.. completely governs the expansion or contraction of the backward denoising step.
Diffuse- Seeeettthe-DDoiiis Process: Offset Contraction/Exxaansii lent to studying the effectiveness of a liffuse-hhen-denole process. To simplify the exposition, we consider a one-step version here Given any two measures, μ0 # vo, run one step of forward diffusson as in [ with = 1, with Z isotropic Gaussian and 3 E R4, the temperature,
The sensitivity analysis is equiva-
X,, (1 Xo + 228-1n Xo ~ po, and Y, =
+ 23 Yo~mo 10
Denote the measure associated with Y, ~ By, then run one step of backward denoising with the optimal b" (as in 4 with K 1) to obtain V// tu Vy. A natural question is whether the cumulative effect of this diffuse-then-denoise process is contraction. Namely, is W(Ho 7/) smaller than W u))?
At first sight, it may seem unnatural that the diffluse then-deoosse process will yield an improvement. How can adding noise and then denoising be better? Consider setting g 0, and let no = 5, and 1/ 6y be two Dirac measures supported at # y. One can verify that
Forward W(H: vn) = || - yll, Backward W(m, B,y) = (1 7))l - y Forward-Lhen-Backwar W (Ho, zp) = (1 --)) W Byy = 1|x- yl = WWW up)
Here, the backward expansion offsets the forward contraction, resulting no net effect for the forward-then- backward process.
Curiously, we shall show in Theorems I and roughly speaking, when 8-1 * 0 and V log has certain curvature complexity quantified by R, there is a net effect of the diffuse-then-denoise process
W (Mm: Bn)
Forward Diffusion:
$ -
W (μo, vo)
W (no, Po)
Backward Denoising:
< 11 -
WHu Dy))
W W(Ho, Po)
Diffuse-then-Denoil
< 1 -
W W(Hn, vn)
This contrasts sharply with the 3-1 = 0 case: curiously, adding non-trivial noise then denoising could be beneficial and result in an effective net contraction B-'c, provided the curvature Ç 0. We emphasize that these inequalities become equalities for simple log-concave po' 's thus establishing the sharpness of our characterization, see Corollary ! depends on the curvature loa.turrr//lcalzzationooo function defined in Proposition 3
In summary, the success or failure of the diffuse then-denoise process solely
Chaining the argument, we show in Corollary 2 that, rather than log-concavity, it is a multi-scale complexity along all time scales that controls the net effect of the diffuse-then-denoise process (4) with K steps
W Po) W(Ho un) < exp
( -3-1
Σ
K
Skn 0(1²)) 0
k=1
3

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where Sky is some curvature/locaaliation complexity of Mky at time t t= kn. For general initial mcasure μ. the diffused version jt could be non-log-concave, depending on the time scale t. In Section 4 we introduce a notion of multi-scale complexity that describes the localization difficulty for generic μ and is new to the literature. The multi-scale corresponds to the effective signal-to-noise ratio for the denoising step at cach time scale I.
Multi-Scale Complexity: Beyond Log-Concavity, With any initial measure H, the Ornstein- Uhlenbeck diffusion process at different time scales t is closely tied to multi-scale smoothing, at different signal-to-noise ratio r(t) (defined in 12)).
Y, =x + (X ~ (O,1).
We introduce in Section 4 the following multi-scale complexity, defined based on the localization function (v(y)= = CoufrX = y voloop
/ P(L,(Y,) u)du 8 (0, 1-5
h (5(
Here PY,, > u) is the tail probability of random variable || Cou |Ylllop (Y,]rl its integrated tail. We show in Theorem 4 that the growth of this integrated tail function 8 + (6, r) governs the effectiveness of the diffuse-then-denoise process at any time i, with the corresponding SNR scale r(t). Our characterization holds for any generic μ that extends far beyond log-concavity.
and therefore hy ().)) controls
This multi-scale complexity at every SNR scale 2 O, conceptually, is quantifying the curvature V log PY, (y, in certain average sense weighted by (v). This notion, rather than the worst-case curvature, governs the localization accuracy of the backward denoising chain at every scale. Our analysis is fine-grained in two senses. First, for any X ~ μt, across different scales of the non-log-concavity of yy, changes in a complex, multi-resolution way. Non-log-concavity across all scales of r collectively determines the effect of the backward denoising chain. Second, unlike in the worst-case analysis, where the worst point with the largest positive curvature V yologyy, (y, (4) dictales the analysis, we leverage the following observation. If point y with a large positive curvature V2 log (y) is unlikely to oceur with (y, small, the overall non-log-concavity is still benign. This is true many examples; see Section
This multi-scale complexity may appear mysterious; we present a concrete, non-log-concave example to clarify intuitions. Consider the simplest one-dimensional non-log-concave measure 1 10+1-
Localization and Curvature: L,(v) S 1 - 8 if and only if log 3 - Ia. that is, (y,
strongly log-concave at y. These y's are good locations with strong curvature: accurate denoising and localization from y is easy. These are locations where diffuse-then--ennisse beneficial. See locations outside the shaded areas in Figure
Survival Function: provided we have samples Y,, the survival function (1 ((L()) > > 1-6) tells the probability of bad locations with possible non-log- non-log-concavits concavity where the backward denoising
is hard. It quantifies the mass that may induce large expansion in the diffuse-then-dennise process. See shaded arcas in Figure 1
Integrated Tail: slow growth in the integrated tail function - h., r) implies that one can take an effectively large 8 such that the bad locations with positive curvature have negligible expansion effect, and good locations with strong negative curvature induce a contraction ellect, offsetting the expansion. This complexity quantifies an overall notion of curvature.
Figure 1 shows: (a) low 0.71, h2 (0, r) (0.5,, O, no growth, (b) mid = 1.50, (0,
0.13, (0.5, r) = 0.24, rapid growth of integrated tail as increases from 0 to 0.5, and (c) high r= 3.00, (0, r) = 0.02, (0.5, r) = 0.03, a very slow growth. Curiously, the complexity is non- monotonic SNR the mid presents the hardest non-log-concavitty for localization.
How is this multi-scale complexity useful? Here we plot the survival function (u),, indexed by the different SNR Curiously, the growth rate of the integrated tail complexity h, r) is non-monotonic in SNR r:
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Adl Co
(a) r= 0.71
(b) r= 1.50
(c) 3.00
Figure We plot the density py, py,(), (-), for three SNR r's. Red shaded area corresponds to non-log-concave region with V logpy..
8 with 8 and Orange shaded area corresponds 6 0.5. For each ö, we report the integrated tail (,, and survival function s, - d) for ó E (0,0.5). (a) low 0.71, sr(1) (0.5) 0; (b) mid 1.50, (1) 0.18, (0.5) 0.27, non-trivial mass of bad locations; (c) high 3.00, s1( = 0.01, (0.5) = 0.01, though bad locations do exist, samples Y, rarely end up there.
both low SNR S and high 2 2 have extremely slow growth in the integrated tail; the hardest non-log- concavity happens when E (1,2), Conceptually, for any given initial measure the multi-scale complexity tells us precisely at what time scale r(t) the backward denoising chain suffers the most. See Section 5 for more comprehensive examples.
1.0
1.0
survival Survival function ((()
0.010
survival function s ((u)
1.010
0.230
1.341
0.6
0.560
0.450
Survival-uuntionn
0.6
1.672
0.670
2.003
0.4
0.780
0.4
2.169
0.890
2.334
1.000
0.0
0.0
1.0
2.5
1.0
2.5
(a)
(b)
Figure 2: (a) se(u) Low SNR. (b) s.(u) High SNR.
1.1 Preliminaries and Notations
Notation Throughout this paper, we consider E P2(X), the space of probability measures with a bounded second moment with X Rd. PX(X) c P2(X) denotes probabilily measures absolutely continuous the Lebesgue measure. For a measure E PX))), denote == p. _d the Radon-Nikodym derivative w.r.t Lebesgue measure, where Pa (X) the density function.. We reserve F, for functionals P2(X) R, and h. 9. for real-valued functions X R. Let E (x;; denote smooth vector fields; i denotes the identity map, t, b denotes the optimal transport maps, in Definition 2 We use X, Y 2, B to denote random vectors. For a matrix M, ||M denotes the operator norm, tr(M) denotes the trace of the matrix. For a vector v, ||o||| denotes the Euclidean norm.
The forward Langevin diffusion process stochastic differential equation the form,
dx, -Vff + 2 V E (X,X)
(5)

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Here 3-1 serves the temperature parameter, which controls the magnitude of the injected noise dBe. The probability measure of Xx, denoted as Me with density Pr == Pac evolves according to the Fokker-Planck partial differential equation
O.p: - V
(6)
Forward diffusson can also be viewed as a gradient flow in the Wasserstein space. For two measures Jt, V E (X)), define the Wasserstein metric as
(fllc- vll uIl 1/2
V)( == min =ell(p,))
Jordan el al. 1998) showed that the forward Fokker-Planck PDE, M4 - Meton can be viewed as the steepest descent with respect to the Wasserstein metric in the infinitesimal limit, as +
1 Pttn == arg min - W2 W2 + G(v), PX() R.. (X) 2n)
Here the functional =(v)=((v) = functional &
+ 3- E(v) consists of two parts, a potential functional F and an entropy
F(v) = / fdv= / (())
E(V): = / log dv
du = / Pog (x) dz
dx
1.2 Related Work
Probabilistic generative models, including generative adversarial networks, flow-based generative models, and
catusioo-bassed generative models, have recently received broad research interest, both empirically (Good-
fellow al.] 2020
Song & Ermon 2019
2020
Song al.] 2021 Karras al.] 2022
Nichol & Dhariwal
2021 Kingma et al. 2021
Huang et al. 2023 and theoretically
Papamakarios et al. 2021 Liang 2021
Hur al. 2024 Oko et 2023 Lübeck et al. 2022 Chen et al. 2024: Block et al.) 2020
DeBortoli ct al.
2021 Lee et al.
2022 Montanari 2023). Generative models with diffusion-based sampling can be traced
back to three influential formulations: Denoising Diffusion Probabilistic Models (DDPMs) (Ho et al. 2020:
Sohl-Dickstein et al. 2015), Score-based based Generative Models (SGMs) (Song & Ermon 2019 and Scoor-based
Stochastic Differential Equations (Song et al., 2021: Karras et al. 2022). The latter universalizes the frame- works by discretizing a particular Stochastic Differential Equation specific signal-to-noise ratios (SNR).
Sampling can be viewed as Langevin dynamics for the time-reversed SDE (Anderson 1982 Ho et al. 2020:
Dockhorn et al. 2021). or deterministic transports via probability flow ODE (Song et al. 2020 Maoutsa
et al. 2020 Guo et al. 2022). The deterministic denoising outlined in [Song et al (2020) is optimal in terms
of transportation of measure in the Wasserstein metric. This was previously noted by Chen et al. (0024)
Jordan et al. (998) in the infinitesimal limit + O. We show in Proposition [ that for a fixed stepsize the score denoising map is the optimal transport map in the backward chain.
Rigorous justification for diffuse-then-dennisee can be found in the seminal work of [Saremi & Hyvärinen (2019) who unified two distinct schemes: (1) smoothing via Gaussian convolution Y= X [N Ia), and (2) denoising via the score function XX y] y + VoggP)))), to design a single machine for sampling X. Their approach is motivated by fundamental result in the concentration of measure literature /Vershynin, 2018); that d-dimensional Gaussian vectors concentrate on a uniform sphere of radius vd in higl-dimenssoos. And so, while X may be non-log-concave or confined to a low-dimensional manifold, Y may not be. Thus, diffuse (X + Y) then denoise (Y + X) presents a way to sample when the distribution of X is misbchavod. Exploiting the machinery of measure transportation, we show that Gaussian convolution up to an appropriate signal-to-nois ratio promotes log-concavity, yielding desirable contractive properties
for denoising, Y X.

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2020 Song al. 2021 Karras et al. 2022 demonstrates dramatic improvements in sample quality and log- lhool metries for diffusions by employing rich scheme of noise scheduling, namely the effective SNR + r(t), a map between time and the corresponding SNR. Several authors consider jointly learning the schedule alongside diffusion network parameters NNichol & Dhariwal 2021: Kingma el al. (02TT. While state-of-the-art application ions suggest that convolving at multiple noise scales is advantagoous (Dhariwal & Nichol] 2021: Austin et al., 2021), , its theoretical benefit remains to be understood. Restricting to the Ornstein- Cheebinn-hhlneckk process, we discover a multi-scale complexity measure that controls the effective contraction of diffuse-then-deoise at each SNR Particularly for non- log-concave distributions, a wide regime of SNR schedule r(t) may be necessary as diffusing at specific SNR scale r pushes the problem into an effective near-log-concaye setting with strong curvature. Our multi-scale complexity determines the effective curvalure, thereby providing a vehicle for probing the noise-scheduling question.
Recent empirical work (Song & Ermon 2019
Theorctical investigation of diffusion- sampling mostly focuses on the non-expansive properties of the denoising process under f-divergence, say total variation drv or Kullback-Leibler dkL,; where data processing inequities hold. The current theory falls short in addressing the behavior of the backward denoising process under the Wasserstein metric, where the backward denoising chain does incur expansion even for simple Gaussian distributions (Chen et al.) 2022). Typically, the problem is coupled with the additional burden of having access to only an estimated score function (Block et al. 2020 DeBortoli et al. 2021 Lee et al. 2022). This presents a significant challenge; however, it is not the focus of the current paper. We isolate treatment of the backward denoising question: initializing at pr, and the equilibrium measure, 2, suppose we observe {14 b² ... bb pk), (%o b, .... bb pK= == v}. How does d(uk, ...-duu (0) evolve? the backward denoising chain expansive or contractive at each time scale 1 = kny What geometric complexities govern the quality of the denoising at each time scale? We view our focus as complementary to the current literature, which conducts careful sensitivity analyses of score estimation and time discretization.
(x- +
Provided the score functions are accurate in an L2 sense, Block et al. (2000); Lee et al. 2022 established
results in the Wasserstein metric for unimodal distributions, for example, those satisfying strong dissipativity or a log-Sobolev inequality. These results are extended in Lee et al. (2023) : Chen et al. 2022 2023 to allow substantial non-log-concavity for reverse SDE sampling, and in Chen et al. (2024) for the probability flow ODE. However, these analyses are conducted in drv or dKL and avoid the detailed curvature information in the backward map. For drv dRL, and finite K, one can always appeal to data processing inequality to bound d(Ho, Po) I d(ux, 1),, solely determined by the forward diffusion chain. This is not fine-grainet enough to understand denoising. It overlooks the curvature information and the amount of non-log-concavin at any specific SNR r. As in the introduction, the backward denoising step could expand significantly or contract much faster than the forward diffusion under the Wasserstein metric.
More recently, Bruno et al. (0233 ; Gao et al. (2025) establish results in the Wasserstein metric under log- concavity assumptions. The concurrent work /GentiloniSSilveri & Ocello, 2025) tackles the problem assuming weak log-concavity of the target density, property with origins in dissipativity that the target density is essentially log-concave outside region of compact support. Our work makes such assumptions. Similar to Gentiloni-Silveri & Ocello (2025), we note that the denoising process can exhibit non-contractive behavior. In our analysis, the fine-grained behavior of the denoising process is governed by a notion of average curvature, defined as an integrated tail function measuring the relative mass of locations with positive curvature versus those with negative curvature. Notions of average curvature have been used derive improved convergence bounds for diffusion models with respect to dkL (Chen et al. 2023 Benton et al., 2023). Our paper shows that different notion of average curvature also plays role in the sensitivity analysis when the metric is the Wasserstein distance. Our perspective helps bridge the parallel theoretical analyses of diffusion models under f-divergence and Wasserstein distance.
More importantly, previous approaches disconnect denoising from diffusion when evaluating the success of score-based sampling. We show that the net benefit of the tiff-ee-then--eeoisee process matters: the contraction of the forward diffusion, offset by the possible backward expansion, results in net benefit as long as there is enough negative curvature on average. This paradigm shift is enabled by new notion of
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multi-scale complexity governing the effective curvature and "localization" effect of the diffuse-then-denene process. This allows an analysis that extends beyond log-concavity assumptions.
2 Denoising and Localization: Score and Curvature
2.1 Score Function: Denoising and Optimal Transport
The score function arises naturally the Fokker-Planck PDE G For valid probability density Ps the vector field V po + gd defines the score function.
Definition 1 (Score Function). For measure v E (X), denote its densily with respect to the Lebesgue measure as Pv where
L The score function is + V log (x))
It turns out, even for non-vanishing 7; the score function Vlog Pt-r Pe-n induces an optimal plan to localize and denoise from Petp defined in 77, back to Mt, in the sense of optimal transport (OT). In the 7j 0 case, the induced by the OT map studied in Chen al. (2024): Song et al. (2020) Jordan et al. 1998 We first introduce the OT map.
Definition 2 (OT Map, (Brenerr 1987 For n, E Pg(X), there exists a unique optimal transport map
ty X + X that solves the Monge problem
1/2
(flv-t(w)) ))()))))))))))))
t = arg min tt
and attains the minimum of the Kantorovich problem
1/2
W2(n, v) - (f
Proposition (Score Function and Backward OT Map). Consider the Wasserstein gradient descent. as with lower-semicontinuus = F Assume that there erists Tn, > 0, such that for all n (0, m)), Me4- in (7) admils well-defonee minimiizer. Then for any 1) (0, the oplimal transport map tii in Definition 2 takes the form,
(t - -))()) vf(x)))((()) Vf(x) 8 log uetn (I), for a..r--.ee Rdd 1
Recall that discretized Langevin diffusion reads X(47 - X, -V=-NVVXX+++++>>>))
22-
Another interpretation
of the score function is that it induces backward denoising step for the diffusion, namely, quantifying the barycenter E fXX,) Xt+n y]. In other words, score estimation can be cast as prediction problem based on the diffusion process, where one aims to predict X, - (XV((())) based on Xeto Saremi & Hyvärinen 2019
Proposition 2 (Score and Backward Denoising). Consider Y X + oZ, where X ~~ p and Z N'O0, Ia) and X, Z are independent. Let py denole the densily function associaled with the random variables Y. Then
V Vlogy((U) = - o -
The score function is the optimal transport map to denoise the diffusion process, but how accurate is the denoising stop? Concoptually, the denoising quality depends on the localization - X v]- As we shall see next, the localization quality of the score function as backward denoising step depends on the curvature function, I + (x))
2.2 Curvature Function: Backward Localization
The curvature function, namely, the derivative the score function, governs whether the backward denoising step is localized. The following proposition describes the variability of the backward denoising, Cool X fX,,) Xx+, = 2)- Intuitively, a large positive curvature of the log density function results in a large conditional covariance and in turn, makes the denoising process hard.
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Proposition 3 (Curvature and Localization). Consider Y = = oZ, where X ~ P and Z ~ (( Ia) and X, Z are independent. Let py denote the densily function associaled with the random variables Y. Then
1 (Ta -o-Cool C = vl} Voogpy(y) g²
The validity of the diftusion-then-denoise process depends on the spectrum of the conditional covariance function. Motivated by this, we shall define multi-scale complexity measure in Section Before concluding this section, we show that the average-case curvature is always negative, although the worst-case curvature is positive for non-log-concave measures. In other words, average curvature tr| 2 Po)) weighed by (v). is strictly negative for any u. This will be useful later. Proposition 4 (Curvature and Score).
f tr| logn -
1 log po
VV
An immediate implication that the average localization radius is bounded.
E Cool X
Yllo S E fr [Cool Y] S d.
As shall see in Definition in Section the diffuse-then-denoise process.
4.2 the tail behavior of Cool Ylllop governs the complexity of
We conclude this section by noting that the score function Vlog the curvature function V p(V). and higher-order generalizations play a central role in characterizing the quality of distributional denoising, In particular, Liang (2025a) analyzes the universality of distributional denoisers constructed from score and curvature functions, while Liang (2025b) proposes a hierarchy of denoisers based on higher-order scores that, in the limit, converges the optimal map transporting Py to Px.
3 Forward Contraction and Backward Expansion
In this section, we study how the curvature governs the effectiveness of the diffuse-then-denoise process in diffusion models.
3.1 Contraction of Forward Chain
Consider two forward diffusion chains Pr, Vi at a given time t, with an infinitesimal time increment Recall the Fokker-Planck PDE in (G); the measures are implemented by
Pe4n = fl Hr, where fP." = i (V + 8 v P..)
(8)
# fu vs, where f = (V 8 1 log Pv.)
Pt4 =
Again, these are the Fuler discretization of the Wasserstein gradient flow as in F(v) + -8-1E(p).
7 with functional G(v)
Theorem 1 (Forward Contraction). Assume for some À E R. v²f(2) m X las Vo E X. For any
# E PX(X) and any B E Rt, the one-step forward process
g
satisfies
1 WW M(+,) W (Me, (Me, Mc)
lim sup 70
-2
"
W2 vt)
Remark 1. The above Theorem shows that as 17 + 0, for any Med 1t
Waluerm Ut+g) < nni
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With the choice V., the invariant measure, have Vetn Me: and thus
W2(peen v+) V2(
S nn.
Conceptually, the one-step contraction rate for the forward diffusion process governed by This contrac-
(ion rate is sharp and holds equality for some μe, v's, for example, Lake μa, v Dirac measures.
3.2 Expansion of Backward Chain
Recall the backward OT map as in Proposition
b = it VV V log
(9)
This backward OT map encapsulates the plan to revert the chain b Pt - Mt-m which we will formally prove in Theorem in next section. In this section, we conduct sensitivity analysis on the backward map: consider a measure y that is a small perturbation to Pe, will by v stay close to b K. p4? This question concerns the expansion rate the backward chain.
We first derive warm-up result in the simple case, employing a notion of worst-case curvature. Later, we will generalize the result. in Section 4 elucidating how average-case curvature gives rise to multi-resolution complexity measure that governs the effectiveness of the diffuso-then-denne
process.
Theorem 2 (Backward Expansion). Consider fx)()), V (olog Puu Puc( (z) with bounded eigenvalues over E X. Assume for some K E Rt, f(x 3 K. Ia. Vx E X. Assume V Vlog Pp: E (x, and for some Ç E RR
lo Pro (x) -- Ia, Vr E X.
Then for any Rt, the one-step backward map
satisfies
( R-n b ") - W (Mt.
limsup sup 7 (X) 17
< 2
W2
Remark 2. We emphasize that this theorem operates even when Ç 0, namely when μ4 non-log-concans The above theorem shows that, for any E (X), n +
W2(b Pe, bl."v) < 11+
W2(pe,
Conceptually, the one-step expansion rate for the backward OT map is governed by k - 3-1C. We call this an expansion because it positive when 3 is large enough, regardless of the sign of Ç.
This backward expansion inevitable, even for certain log-concave μr. As a simple corollary of Theorem we show the expansion upper bound is tight.
Corollary 1 (Lower Bound). Assume for some Ks Rt,
f=(()= Ia. V logPu, - Ia.
Then for any 3 > S/R, the one-step backward map 9 satisfies for all
4. 1e, WW
by." v) #
W3(m, 2(k- > 0.
lim
(pr,
3.3 Diffuse-then
One-Step Improvement and Chaining
Consider the special case of an Ornstein-Uhlenbeck process where f(x) = ||a||///. Take any two chains and 24, as + O, then Theorem cllimms the forward diffusion satisfies contraction
W2(P (P0.)) Pr, P2)
Waluer Metg) - - ~1-n+0 0(7))
v2(H,,
We(Hu,
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The backward denoising in Theorem 2 satisfies expansion at most
(2 H. ( pu Me, by " Wabb"""utn Mutm vttn)
S1+ (11+ 0
W2 fu. Pe, v1)
W2 V(4n)
We shall show next in Theorem 3 that
W2 " Me, = 0(1)²)
Then compared Vi, the diffuse- Suffuse-then-cee then-denoise version bu.npee-' # #
will get closer Pe, in the following sense
3. vs) W2(n,, "
$1- $1-18-'5001
W2(u,, 1/:)
The one-step diffuse-then-denoise process b4. f has a net contraction 8 B when Ç > 0, namely, the forward contraction, offset by the possible backward expansion, presents a net gain in localization.
Theorem 3. Define the one-step diffuse-Ueen-denoise
fu. := i meiinn(// "VV 8 v logPa); My My "
1 = : i- 1) (V 5 log Pog
Assume V,, V Pog V llogpn, E C (X, then
W2 # " 0(1²)
A direct corollary concerning the consider finite K-step liffse-theen-denise chains:
chain now follows from Theorem 1
2 and 3
We
Forward diffuse: for the a chain, set po = n, and define - :
'/V/ 3-1 log Pp(-))))))) and
Pkg = (Ch"") #((0-))) recursively for k = ..., K. Same for the v chain.
Backward denoise: for the v chain, recursively apply the backward OT map bb = i (VS+ V Ppg.)..) to MKns for k = K,K- -,...,,,,
Corollary 2
Denoise: Chaining). Let f(x) ||x||//2 /2. Define the diffuse- thene---seen--eeiseee map
f(k] pV.1) f- f/
bix] := blth by b
Assume there crists sequence of Skn IR such that
7 log Ppn, (x) I - Ckyy Id, Wr E X, Vk = 1,2,..,,
Then the diffuse-thers denooiiiss process on v with fixee K steps, denoted as (bfk) fl(a) u, satisfies
W2(M, (bfkk o (fkk) ") S exp 2m Sky + 067))
K
(10)
(2(,vv)
The above Corollary states the denoising quality of the difluss-Chen-deenosse chain is collectively determined by the curvature at each step Sky- For can-log-concave μ4, along the chain, some the Skn will be positive, and some will be negative. Therefore, we need fine-grained understanding of the curvature each time scale t = loy understand the whole chain behavior. This calls for fine-resolution analysis of the curvature of ,t, going beyond the worst-case curvature, the focus of the next section.
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4 Beyond Log- Concavioc:: A Multi-Scale Complexity
In this section, we restrict to the Trnstcin-Uhlenbee process and discover multi-scale complexity measure that controls the effective contraction contraation/expansiiiii eexanssoon of the diftuse-then-denoise process. The key is that for the Chlenbee process, jr is a smoothed version of at a particular signal-to- to-noise ratio scale = r(t).
4.1 OU Process
Definition 3 (Ornstein-Uhlenber Process). Define the Drastein-UUllenbeck Process with inilialization Xo H, and potential f(x) = ||x11//2
dx, = dx,t + /2
Then the distribution of Xs, Vt E R+ admits the representation
X, e Xo (Le Z 1a)
(11)
For any measure M P2(X), the above representation motivates us to consider sequence of problems indexed by a multi-scale signal-to-noise ratio. Recall Proposition 3
(1) et
e s(1) = VB- e-2))
(12)
V Pog (x) = st) {Ia - Coulrrtt = x|}
The curvature at each scale quantifies and, collectively, determines the measure's complexity.
If is log-concave measure, then by Prékopa-Leindler Inequality, pa's 's are convolutions of a log-concays measure with a Gaussian measure, and therefore, log-concave. Hence, for any f, we know that V log (r) = 0. As a direct consequence of Equation 110, we know that the diffuse-then-denoisee chain is effective.
However, for non-log- conccve concave measure μ4, the V log (z) may have positive eigenvalues. Given Equation [2), studying the positive eigenvalues of V log Pr. is equivalent understanding the tail behavior of the localization quantity Con(r(t)X,)] X.], which motivates the following section.
4.2 Multi-Scale Complexity: Non-Log- Non-Log-Con and Survival Function
We define sequence of functions at various signal-to-noise ratio (SNR) scales. In nutshell, the different scales of SNR probe the tail behavior of the localization quantity, /o, In turn, the localization quantity determines the landscape of the curvature of the mcasure H4, smoothed version of the original measure at a given SNR scale r(t), as shown in Equations 11 12 This multi-scale corruption also occurred in stochastic localization; see [ontanari (2023) for the connection between stochastic localization and diffusions. We give precise complexity measures of the denoising problem, cast in quantities determined by the survival function of the random variable Coc[r(t)X,]X.]] As result, these complexity measures illustrate the effective contraction expansion rate for the chel-se-teen-denoiie process, covering the case of non-log-coneave pts.
Two direct consequences of the newly proposed complexity measures follow: (1) It gives rise to fine-grained analysis for the
process at any SNR scale, for any non-log-concave measures, to be shown in Section 4.3 (2) motivates a simulation-based numerical tool to visualize the bottlencck SNR scale for
the denoising problem, as we shall demonstrate in Section Several curious phenomena are unveiled and rationalized by the new multi-scale complexity.
Definition 4 (SNR, Localization and Survival Function). Given an instial mensure μ E P2(RR)) we define, for any SNR E Rz0
: XZ, (X,Z) ((0,
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where X ~ μ and Z ~ Ia) are independent.
Define the localization function and the associated random variable
(v) Coo|rX| ylllop and L, || |
and denote the survival function of L, as s,(-) : -R+ + [0, sy(u) = F(L, > u)
A lew remarks follow from this definition: (i) When the measure μ = (() is log-concave, then by Prékopa- Leindler Inequality, for all r, the measure L(Y,) is log-concave as well. Proposition 3 implies that the random variable L, S Thercfore, the survival function s, 0. (ii) When the mcasure μ ((X) is non-log-concave, then the measure L(Y,) will be non-log-concave for some For these r's, know that Al, (,) 1 and in turn implies s,( = P(L, >)>0. 0.
This property is crucial and distinguishes the difficult settings for the backward transport. The validity and effectiveness of reverting the diffusion model depends on the integrated tail of the survival function sr(u), u E [0, 1), which we define now.
Definition 5 (Multi-Scale Complexity). For 8 E (0, 1], define
(ō,r) = 1 1-8
s.(uddu (, == (y(5,r)
5
Define
EE [0, 1] e arg mig mu(0.))
(+) max
6E[0.1]
and the minimal value
m²(r) == min mp(6,r) 6E(0.1]
A few observations follow for the mp(st) 8 + r6) function, which ensures (+) and m²(r) are well-dofined.
Proposition 5. The following properties for my( hold:
1. Log-concave cose: If s+(1) 0, then mp((st) is non-decreasing function in 8 E [0, 1];
Non-log-cone con-cvvee case: If (1) O, then mp(:rr is either (i) non-increasing in E [0. 1), or (ii
U-shaped in 8 [0, 1), namely first non-increasing then non-decreasing.
At high level, if for some 8 E [0, 1] the integrated tail f sy (u)du is small, then the effective contraction of the difluse-then-denoiee process will be governed by this 8. By Proposition 5 m (r) 2 0 is well-defined and,
as we shall see in the next section, this will quantify the type of perturbation that the diffuse- thifuse-theendeeoisee step can tolerate and still effectively contract.
4.3 Backward Expansion: Refined Analysis
In this section, we give a proof of why the multi-scale survival function at each SNR schedule r(1) = c-
(ターi(1--ーr) for t [0, oo] controls the backward expansion of the diffusion-then-denoii model. The result generalizes Theorem to the case of arbitrary non-log-concawe measures.
Definition For measure μ E (X),,, define class of measures in reference to p.
sUpzeDom(H) (1t:) 12 $ M)
M(P, (v PX(X)
J 1(t%% =)
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If M 1, the set M(J, M) consists of simple measures that are location shift of μ. If M oc, the set M(U M) = (X) all Wasserstein space. Here M controls the richness of perturbations ions around
Theorem 4 (Backward Expansion: Beyond Log-Concavityy Recall the hy hy(6,1 r) function in Definition
Assume V log Pus (x) has bounded eigenvalues over E X. Consider the backward OT map for the OU process as in Definition S then for any å E [0,
W Pe, b4 - (2(,))
sup 9-0
lim sup
S - I-e- 2 M (6, r(t))]
(w.
MEMin, p
Here the SNR schedule r(t) is defined in
12
To build intuition for this result, we will isolate the following term,
Cās (1)
sup 6E(0.1]
-- 18 M (hh r())).
We shall extensively explore how the multi-scale complexity affects this effective curvature complexity SM (t) with several non-log-concave examples in the next section. The curious reader may wonder whether this multi-scale complexity recovers the result obtained in the previous section for the log-concave case. The answer is yes.
In the log-concave case, namely, there exists 2 O, 2 log (x) v - Ia- For č(t) = (1 - -e-22 and r= r(), we have V log (1) 3 - la. this case, (z) S 51-5 and 8, - 2 = Then, (,, r) = for all ò [o, []. and
(r) =0, ('(r)=
(t) lim 1
22 [5*(r) - M h,
e e-2
In the con-log-concanys case, for any M < m it is possible to obtain She (t) > 0, and thus, net contraction for the diffuse-then--eenisse process. We will explore this extensively in the next section and delineate the behavior of Shr (t) for host of fundamental non-log-concav distributions.
5 Examples
Theorem 4 depends on an integrated survival function (6, r) whose shape is difficult to guess. Luckily, its complete behavior can be captured by understanding the following quantities.
(u) PL, u), m²(r) min (5. r), 8*(r) max { [0. 1] : arg min m.. sE(0,1]
sE(o.1]
We will methodically calculate and plot these objects for range of fundamental distributions. Given target distribution, empirical or with explicit form, we can use Monte Carlo simulation method to visualize the functions above.
For any empirical measure μo: 15 defines an expression for the empirical version of Ly)...
One simulate Y, by sampling rX +,,,),), for X no-
Then one can obtain the empirical survival function, s()). m² (r), (*()) and (t)(t) follow.
Capturing this behavior precise time scales will allow us to ascortain (1) when contraction easy, and if not, (2) what SNR, r, the process transitions a non-log-concass setting. In the following, we will restrict ourselves to the OU process, with 1, and in one dimension.
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5.1 Warm-UP Log- Cog-Concave
Suppose the larget measure, Mo is log-concave. Via Prékopa-Leindler Inequality, the full chain of measures generated by the OU process admits log-concavity. That is, there exists a sequence of non-negative constants ((r) > 0 such that
V Io) -((), Vr 0.
We know from the previous section that this implies,
(r(r) =0 (*(r) ((r), (2)(1) 8)(r(() - c-
We visualize and validate this result for three log- log-concave distributions. Example 1 (Point Mass). Consider the simplest case where the target measure μ4 is a Dirac measure, ōg- We assess the localization of the backward transport by investigating the random covariance, L,.
V2 logpy. (v) L, L, 0.
In this simple case; L, is a point mass at 0. The backward transport localizes completely. Tt follows that sr( 0. Therefore,
m²(r)= = 0, 5*(r) 1, (t) 1 26'
1.5
0.3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0
0.2
0.4
0.5
0.8
1.0
1.2
(a)
(b)
Figure (a) (t) (b) (t)..
Example 2 (Normal). Consider that No is normal distribution N N(m, o²). Again, we calculate the
localization quantity,
d2p2 L
2 log (v)
222
r2r2 +1
L, is a point mass with location depending the SNR, r. In turn, the survival function is not 0, but step function with an explicit threshold,
su)) = { 1, Vu E [0, 222
g
0, Yu E oc)
We plot this in Figure (a). The backward transport localizes completely r 0 (or equivalently t 3 oo). For any r, can find 8 e [0, such that (5, r) 0. It follows,
m² (r) = 0, 8*(r)= +1' (1) 1 e 2 (0
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We confirm variety of o² choices in Figure 4 (b) and (c).
1.0
111200
1-2.000
3=0000
1.50
r+4.500
1.25
6=0.00
1.00
m- 70800
0.4
r=
0.2
0.25
0.00
0.0
0.6
1.0
(a)
(b)
2.00
0
1.75
0.331
1.50
0
1.25
0-9.44
1.00
0:1.37
56
0=79
0.000
0.50
0.25
0.00
1.4
(c)
Figure 4: (a) su()). (b) m (r)()) (r). (c) (t).
Example 3 (Uniform). Consider the case when the target distribution is uniform Au = Unif(-1, As before, we gauge the localization via the random covariance the backward transport, L,-
2 logPy.()) = [ (u + r)ply + (v- - -r_ Φly + oly -
}
+1y + D -
4(Y)) = 1+ 2 log
Based on this characterization we simulate st(,,, m (r), 8* (r),,, (t))). As expected we see,
s((1) = 0, m m²(r) 0, Co (1) > 0.
We recover the same qualitative behavior as in the preceding examples (point mass, Gaussian). However, the fine-grained behavior is remarkably different; see Figure (c). Note that though (%) > 0, we have the non-monotonic behavior the effective contract at different time scales. The slowest effective contraction is happening in some small time scales.
The uniting technicality that allows this is:
L 41,, 1, 1 (1) 0, Vr 2 0.
By Proposition 3 this is exactly saying that the curvature, 2 22 (v) 5.
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1.0
r-1.200
2.5
0.8
T-2.300
T-4.00
2.0
0.6
506
6.000
1.5
7-000
0.4
10 000
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
14
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(a)
(b)
220
2.0
1.5
0.0
0.4
1.0 1.2
1.4
(c)
Figure 5: (a) sy(u). (b) m (r(r)) (c) (*)
5.2 Beyond Log-Come cogcaviyyy
With general non-log-concave as initialization, the Ornstein-Uhlenbedd process at lime may not have log.concavity. Indeed, V (v) is not uniformly upper bounded by 0, or equivalently, the random covariance L, cannot be uniformly bounded from above by 1. We scparate from the rogime of the previous three examples, of which exploited this property to show contraction at all scales
It important understand (i) when the backward chain enters this non-log-concawy concave regime, and (ii) the expansion behavior in this regime. To that end, we make use of the following notation:
maxfr sr(1) = 0J.
Example 4 (Two Point Mass). We consider Xo _6 + 6r for some > 0. We assess the localization via the random variance, L,
δ(y ru) ply + ) ). sup L,()) L,(0) (rp)².
L+(1) = (rw)²(1
δ(y ru) + Φly
We see immediately, r" r== 11 We can explicitly calculate (w).
(u) = 1 + V(r))² + +rp + +
1
log rp + / (rp)2 "
2(rm) ru / (ru)2 - su) = 0, ((ru)2,00)
log
2(rw) rp / (rp)) ~
r4 -1
(13)
(14)
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(1) (r ()= In this setting, expect log- ogg-concave behavior. By
14
(r(r= = 0. ("(r)= = (r4), co(0) - (r(t)))2 21
5 1.
(ii) (r-> oo): For extremely high SNR, we can see s,(u) 0. This phenomenon is qualitatively the same log-concavity and can be rationalized essentially recovering the single-point mass behavior in the limit. In particular,
m (r) - 0, 8"(r) - 1, (t) +
(iii Contraction should be hardest at mid-range SNR, where we longer have log-concavity. The behavior is simulated and displayed in Figure (c). In this setting, we can tolerate complexity M S m < oo
in order for non-expansion (3r(1) 0. This is visualized in Figure i (d). Note the non-monotonic behavior of the curvature complexity S3 (t): there seems to be mid-range of time at which the chain could expand, when the type of perturbation is complex with large M.
The conclusion is not pessimistic. For most of the process, regardless of M, (t) is positive and contraction occurs. We remind the reader that M is a proposed upper bound Definition 6 In our numerical examples, this is typically small. Refer to Figure to see the succoss of diffuso-thord tuffus--then-denosie in this case.
1.0
rw0.100
1.0
r=1.00
r-0.200
r-1.111
r=0.400
m=173
T-0.500
0.6
0.6
1-2.5
0=0,00
T=23667
0.4
r=0,900
t.7.89
1=10000
3-30000
0.2
0.2
0.0
0.0
1.2
(a)
(b)
2.00
175
5-
150
4-
4=
3
Git
2-
0.75
1.
0.50
0-
025
-1.
2-
3.0
0.2
0.4
0.6
0.8
1.0
1.2
(c)
(d)
Figure 6: (a) (u) Low SNR. (b) su(u) High SNR. (c) m² (r), 8+). (d) (t).
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25-
00
25-
(a) Forward Diffusion via OU
(b) Backward Transport via OT
Figure 7: (a) Forward diffusion initialized at μo 0.56 - 0.50 Po 60. T 100, 1 0.01. (b) Backward
transport initialized at pr, Vr, and applying the backward OT map bd..,
re0.10
r=1.100
rw0,200
0.8
0.8
r=0.400
r=1.733
0.6
0-0.000
0.6
r-2.156
0=0,700 700
r=2.667
mm0 9.0
2.778
50.000
0.2
0.2
0.0
0.0
0.6
1.2
(a)
(b)
2.00
3.0
M-
175
2.5
3
2,0
=
1.5
0.75
0.5
050
0.0
025
-1.0
0.0
0.2
0.4
0.6
0.8
1.2
3.0
0.2
0.0
0.8
0.5
1.0
1.2
(c)
(d)
Figure 8: (a) su( Low SNR. (b) s.(u) High SNR. (c) m² (r), (r). (d) (3,
Example 5 (Mixture of Point Mass and Gaussian). We now consider srffuse-then-dennee for the target distribution; 3.0 + 1). Let v I. We first calculate the covariance,
(y2/02 - 1) (w/v) + (y² 1)o()) u (y/v) + p(y)
v " (y/ uyb())
wQ(v))
(
up (1/o)+0(y))
We simulate the survival function s(u) (L, u) via Monte Carlo, Figures (a) and (b). lack of log-concavity is clear for high SNR.
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①1) (rr) We expect mr(r) O. While don't have explicit form for r", simulations in Figure validate this. See also Monte Carlo-based simulations for 8", Soe-
(ii) (r > r): We move away from the log-concave regime for large SNR (small t). We conjecture that as - 00, (r(r) + 2- To see this, note that st() approaches a step function form with a step at 0.5. In this setting, the behavior of SM is captured Figure g (d). We note that regardless of M, the range of time when the process effectively expands minimal compared to the full process.
Refer to Figure 9 to see the success of diffuse-then-denoie in this case.
29.
25+
00
22-
26.
35-
ab
0.45
0368
0d97
0668
15
0.80
0d87
8590
ala
601
(a) Forward Diffusion via OU
(b) Backward Transport via OT
Figure (a) Forward diffusion initialized Mo 0.58, + 0.5A(O, 1), v/ 8g. T 100, n 0.01. (b) Backward transport initialized al Jr,, MT, and applying the backward OT map bu...
Example 6 (Mixture of Gaussian, Heterogencous Varianco). Consider the general Gaussian Mixture given by;
Xo Pi N(p of)
In Figure 10
we consider a specific formulation: 0-1W(-4,1) + + 0.44(2, 0.3N 1).
Defining 1/- / Po I, 4 === -I( r44), for i [m], derive the conditional covariance:
L,(3) = 1+ E. peu -11-))0(4) E.pro:((t.)
---
(-En ((b()) Exprrio(t)
(15)
The survival function su) P(L, u) is simulated via Monte Carlo and is displayed in Figures 10 (a) and (b). As soon as the SNR is not small, we are no longer in the log-concave setting.
(i) (r S r*): We expect m (r) = 0. Simulations in Figure 10 show a flat m for sufficiently large r. Even for mid-range SNR, we diverge from this log-concave setting.
(ii) (r>1 r"): The shape of Ç3 is captured in Figure 10 (d). contrast to the previous two cases, this region is not so narrow. For a chosen noise schedule, liffuse-then-denoie still seems successful for sampling in this setting, Figure 11
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1.0
1.0
r-0.200
m-13111
T-0,500
1-1.444
0.6
0-0.000 800
0.6
506
r=0.00
500
-22367
000000
t-2.578
0.4
0.4
110000
53000
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(a)
(b)
175
0.6
1.0
0.4
125
0.2
0.0
-0.2
==1
-0.4
3
0.2
0.00
0.25
0.50
0.75
1.00
1.75
2.00
1.75
2.00
(c)
(d)
Figure 10: (a) sr(u) Low SNR. (b) su( High SNR. (c) m² ( (r). (d) (t).
65n
1:90
1
sis
(a) Forward Diffusion via OU
(b) Backward Transport via OT
Figure 11: (a) Forward diffusion initialized at μ40 = 0.1V(-4,1) + (.2N((2,,08) + 0.4N 0.55) + 043 1), dg. T T= = 100, 0.01. (b) Backward transport initialized at pe,, and applying the backward OT map bu..
Acknowledgements
TL acknowledges the support from the NSF Carcer Award (DMS-2042473) and the Wallman Society of Fellows at the University of Chicago.
21

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Published in Transactions on Machine Learning Research (01/2026)
References
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008
Brian DO Anderson. Reverso-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313 326, 1982.
Jacob Austin, Daniel D Johnson, Jonathan Ho, Daniel Tarlow, and Rianne Van Den Berg. Structured denoising diffusion models in discrete state-spaces. Advances in Newral Information Processing Systems, 34:17981-17993, 2021.
Joe Benton, Valentin De Bortoli, Arnand Doucet, and George Deligiannidis. Nearly d-linear convergence bounds for diffusson models via stochastic localization. arXiv preprint arXiv:2308.0300 2023.
Adam Block, Youssef Mroueh, and Alexander Rakhlin. Generative modeling with denoising auto-encoders and langevin sampling. arXiv preprint wXXw:2002.00107 2020.
Yann Brenier. Décomposition polaire et réarrangement monotone des champs de vecteurs. CR Acad. Sci. Paris Sér. Malh., 305:805 808, 1987.
Stefano Bruno, Ying Zhang, Dong- Young Lim, Ömer Deniz Akyildiz, and Sotirios Sabanis. On diffusion- based generative models and their error bounds: The log-concave case with full convergence estimates. arXiv preprint Xr::22211 19584, 2023.
Hongrui Chen, Holden Lee, and Jianfeng Lu. Improved analysis of score-based generative modeling: user- friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learn- ing, pp. 4735-4763. PMLR, 2023.
Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru Zhang. Sampling is as easy learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint 2rXi::2209. 1 11215, 2022.
Sitan Chen, Sinho Chewi, Holden Lee, Yuanzhi Li, Jianfeng Lu, and Adil Salim. The probability flow ode is provably fast. Advances in Neural Information Processing Systems, 36, 2024.
Valentin DeBortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34: 17695-17709, 2021.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-877 8794, 2021.
Tim Dockhorn, Arash Vahdat, and Karsten Kreis. based generative modeling with critically-damped langevin diffusion. arXiv preprint crXiv:2112.00066 07.666 2021.
Xuefeng Gao, Hoang M Nguyen, and Lingjiong Zhu. Wasserstein convergence guarantees for a general class of score-based gencrative models. Journal of machine learning research, 26(43):1-54, 2025.
Marta Gentiloni-Silveri and Antonio Ocello. Beyond log-concavity and score regularity: Improved conver- gence bounds for score-based generative models in w2-distance. arXiv e-prints, pp. arXiv-2501, 2025.
Ian Goodfellow, Jean Pouget-Abadic, Mchdi Mirza, Bing Xu, David Warde-Farloy, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. Communications of the ACM, 63(11): 139-144, 2020.
Wenxuan Guo, YoonHaeng Hur, Tengyuan Liang, and Chris Ryan. Online learning to transport via the minimal selection principle. In Po-Ling Loh and Maxim Raginsky (ods.), Proceedings of Thirty Fifth Conference on Learning Theory, volume 178 of Proceedings of Machine Learning Research, pp. 4085-4109. PMLR, 02-05 05 Jul 2022.
22

---

Published in Transactions on Machine Learning Research (01/2026)
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in Neural Information. Processing Systems, 33:6840- 6851, 2020.
Rongjie Huang, Jiawei Huang, Dongchao Yang, Yi Ren, Luping Liu, Mingze Li, Zhenhui Ye, Jinglin Liu, Xiang Yin, and Zhou Zhao. Make-an-audio: Text-to-audin generation with prompt-enhanced diffusion models. In International Conference on Machine Learning, pp. 13916 13932. PMLR, 2023.
YoonHaeng Hur, Wenxuan Guo, and Tengyuan Liang. Reversible gromov monge sampler for simulation- based inference. SIAM Journal on Mathematics of Data Science, 6(2):283-311 310, 2024.
Richard Jordan, David Kinderlehrer, and Felix Otto. The variational formulation of the fokker planck equation. SIAM journal on mathematical analysis, 29(1):1 17, 1998.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion-basel generative models. Advances in Neural Information Processing Systems, 1126565-265777 2022.
Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. Advances in
Newral Information Processing Systems, 34:21696- 94:21696-21707, 2021.
Holden Lee, Jianfeng Lu, and Yixin Tan. Convergence for score-based generative modeling with polynomial complexity. Advances in Neural Information Processing Systems, 35:22870-223 -27822, 2022.
Holden Lee, Jianfeng Lu, and Yixin Tan. Convergence of score-based generative modeling for general data distributions. In International Conference on Algorithmic Learning Theory, pp. 946-985. PMLR. 2023.
Tengyuan Liang. How well generative adversarial networks learn distributions. Journal of Machine Learning Research, 121 2021.
Tengyuan Liang. Distributional shrinkage i: Universal denoisers in multi-dimensions. arXiv preprint arXiv:2511.09500/ 2025a.
Tengyuan Liang. Distributional shrinkage ii: Optimal transport denoisers with higher-order scores. arXiv preprinl or vxXi::251.099000 2025b.
Frederike Lübeck, Charlotte Bunne, Gabriele Gut, Jacobo Sarabia del Castillo, Lucas Pelkmans, and David Alvarez-Melis. Neural unbalanced optimal transport via cycle-consistent semi-couplings. arXie preprint arXiv:2209.1550 15621, 2022.
Dimitra Maoutsa, Sebastian Reich, and Manfred Opper. Interacting particle solutions of fokker planck equations through loileett-oggdenssiie estimation. Entropy, 22(8):802, 2020.
Andrea Montanari. Sampling, diffusions, and stochastic localization. or Xiu preprint or Xiv:22300 10690, 2023.
Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. Tn International Conference on Machine Learning, pp. 8162 8171. PMLR, 2021.
James R. Norri.. Markov chains. Cambridge university press, 1998.
Kazusato Oko, Shunta Akiyama, and Taiji Suzuki. Diffusion models are minimax optimal distribution estimalors. In International Conference on Machine Learning, pp. 26517 7-5822 PMLR, 2023.
George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshmi- narayanan. Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research, 22(57):1 64, 2021.
Herbert E. Robbins. An empirical bages opproach statistics, pp. 388 394. Springer New York, 1992.
Saced Saremi and Aapo Hyvärinen. Neural empirical bayes. Journal of Machine Learning Research, 20(181): 1-23, 2019.
23

---

Published in Transactions on Machine Learning Research (01/2026)
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference Machine Learning, pp- 2256 2265. PMLR, 2015.
Yang Song and Stefano Ermon. Generative modeling by estimaling gradients of the data distribution. Advances in Newral Information Processing Systems, 32, 2019.
Yang Song and Stefano Ermon. Improved techniques for training score- based generative models. Advances Neural Information Processing Systems, 33:12438 12448, 2020.
Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishck Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXi::2011.13456, 2020.
Yang Song, Conor Durkan, lain Murray, and Stefano Ermon. Maximum likelihood training of score-based diffusion models. Aduances in Newral Information Processing Systems, 34:1415-1428, 2021.
Roman Vershynin. High-dimennsionll probabilily: on introduction with applicalions in dala science. Cam- bridge University Press, 2018.
A Proofs
The Appendix is organized as follows.
Scction A.1 gathers the technical preparations for denoising and localization, where prove the results in
Section 2 We derive the optimal transport denoising map and explain how the curvature function appears
from a localization perspective.
Section A.2 presents analyses of the forward and backward processes, where we prove the results in Section 3 We demonstrate how the curvature function influences backward denoising via a sensitivity analysis under the Wasserstein metric. We also study the diffuse-then-denoise process, showing that adding noise then denoising can be beneficial in the presence of negative curvature.
We extend beyond the log- concave setting by defining and analyzing multi-scale complexity that captures an average notion of curvature across all time scales. This multi-scale complexity plays a key role in the sensitivily analysis of the diffusion model under the Wasserstein metric.
Section A.3 provides proofs for the results in Section 4
A.1 Proofs for Section
Proof of Proposition By Lemma 10.1.2 in Ambrosio et al. (008
1
" (t i) (G(++)))
where (G(4))) is the strong subdifferential. Recall that ((v) = + For a given v with density == == p- 24, by Lemma 10.4.1 Ambrosio et al. (2008), we know any E(x) E (v; in ag(v) admits the representation
8g V
{(a)
Vf(z) + B v log p(x), for v-a.e x E Rdd
8p
Take Metn we finish the proof.
Proof of Proposition This result follows from Tweedie's formula (Robbins
1992).
The conditional distri-
bution of Y y given X admits the following density: (x) ay P(Y x
L X)/o) where
Φ Rdd + IR is the density of d-dimensional standard normal. This implies
1 Φ2 (v X)pr(v(X) and Vpr(u)
1
Vpr(y
El(y X (xx((x)X))
(16)
24

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Published in Transactions on Machine Learning Research (01/2026)
Therefore, we obtain
Vpy (v) (2) 02 Blyy X)
pY (x)
1
02 lv E y1
(y(
where we used the fact that (X) 1 = E(h(X) Y y] for any integrable function h with respect to X. This completes the proof.
Proof of Proposition Taking the derivative of 16 with respect to y, we have
p = -
o2 La + os (v X) X) x) p
and (2pr(y) [B la + X) - x) (o>))r(()X)) xx] This gives
Verrly) (Y(
Ia gt (y -X)(u -x) PY (
E
σ²
pr(y)
Ia
El(y X) - x) Y = yy
0²
0
Combined with the result Epr(g) pr(u)
32 (Ely - X y)) from Proposition
we obtain
(err(y) Vpr(y) Vor(u)) T
V logpy(U)
DY(U)
py(U) (v(
gz
la +
σ El(y (y - X) Y = yl
1
04 (Ely X u) (Ely X yl)
1
1
0² la
πt Couly x y)
With Covly X = yl = Cou = v| = و² Coulo - X Y we complete the proof.
Proof of Proposition
Using the integration by parts, we have
/ br| (V log Pe (x))du(()) = Σ / (x)
d
8²
log (x)dde
Da. (1)
log logdd
Dri
V log (x)) (
A.2 Proofs for Section 3
Proof of Theorem
7 Take any two μ, E P)(X), we have by convexity of E(-)
E(p) - E(v) 2
(Vlogre, to i)du = (V (t)) i - dd
E(v) {(K) 2 / PVoogPanth i)dμ.
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Therefore, adding up these two inequalities, have
/ (Vlogp Vlogpo(t?); t2 i)dp - 0.
(17)
Now denote t = ti we know (P o # E I1(();)) v)) where para, pn are defined in
(2(mmtmettt)) p(tn) =:T 1n) / 1| (|r--g1 y11
inf
$ / [[ - [iin -1 Pwg (1-17Vf (t) n log Pv. due)
= / || - -tl/'dus 20 (i- (i Vf dftt
+ 2n (V (Vlogmm- Vlog (t), - -iddut
+ +0(1))
S (1 2nd) 14) 0
Here we use 17 and the fact (Vf(x) Vf((), x - v) 2 A|I2 - yll?, for all x,y- Rearrange and then let + 0, we prove the theorem.
Proof of Theorem First, for any v (X) define two quantities
:: |/ (m):))))))) W2(u,
E = (t% - 1)/l1% ill<a())))))
Then th, -i+ cg. For any (X),,, we have
W2264" # Me, Me, v) / |bb" (i cE) b
= / [b <x())) b (10
Define an auxiliary function ge(c) = [bb + cE(x)) - (x) (x) we can verify that
gE(E)
(0( = 0, lim
We also know that
/ 2(Vbb (E)))) b"))) + <E(x)) - -b bz)dd)))
ge(()
b
I [bbu <<())) - ba. (x) (ddo(x) +e. / [=
gE(E) sup (b) / [((=)))Ppp((s).
Notice f [(((=))))pp((() = 1, and divide both sides by e, we obtain
d gE(E)
(c(C)c ge(e)
I sup (bb''
de
Integrate this inequality and recall lim 00 gele) 0, we get.
ge(E)
S S sup [V)
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Published in Transactions on Machine Learning Research (01/2026)
which implies
ge(e) I sup (x)) 2 [I(()))))((())= - sup |>>b* W2
Recall the definition of ge(c), have shown for any PX(X),
W (14 Mr, v) ge(e) I sup (>>* W
(18)
Therefore, we have for any V E (X)
Wb a # pr: by (3( v)
S supl Vb** (x) 1.
W24,, v)
To further bound the right-hand side, for small enough, we notice
Vb" = La + n(V2 + 8 log log (x)) I (1 + nnk 7k 3-)
We complete the proof by taking the - 0 limit.
Proof of Corollary First note
Vb)*(()) = Ia + n (V2
122 (
= (1
nnn - 3)
b = (1 + n(k - 8 f = (b**) (1 + 7( -3-
Define for convenience, Mr-1 : b Mc: v-n = " v, then in the proof of Theorem
188
already proved
(2 v)
W (f f&v-n) sup [[ (1+ (1+ 13)
W P.9 Pr, b v)
(2 V v
Thus taking the reciprocal, then taking the limit inferior as - 0, we obtain
(22 # Pe. b) W2( v) > 2(x -)
inf 7-0
W2(me,
Theorem 2
already proved
1 WW2b' (b*." pe, b4 ())))WW(mup,
W)(m,, (2()ev
lim sup
n- 17
Thus, the equality holds.
Proof of Theorem Note by definition of the diffuse-then-denoiie step, we have
w (2((b
u.n Mon # # (1
||bl o (x) x1/2d
/ (1)))())) 7u (z)) 3- V-'VV-''PVppp Pp, (F dd
/ ()) (z))
V log + - =11
|| </V5(> n 1 log pM xz) + (/ 0(12) - =11ddp
=(17),
27

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where the second to the third line is applying the change of variable formula
log Pas (x))
= log det (ugp:(p>)-oocccd(111--1111111111
= -aappp/)----mmrx110000"""""//010
and the third to the fourth line uses the fact
= V [UP
The proof is completed.
Proof of Corollary Without loss of generality, assume W2(H4 v) 1. By Theorem we know Wg(Hkm 1 nk + 0(7²) s exp(-1) + 0(7)))
Using Theorem
W2((b> ( ) (bx") 02 " (uK))
1 <1+ - 8 (kn) + 0(122,
W2(pKn VKn)
S <xpp(n(1 - B (31((xn)) + 0(n²)).
W2 W2((b # (b*"))UkK)) S exp(-n(K 1) n + 0(1²))
Now recall Theorem
8
we have
W2(M(K-Dm (b*))**K)) = We(m(K-1)n: (b PR-l =
and thus
W2(M(x-1)m (b*""*
xVk))) & W2(H<( (bk* ( (bH4"))
S exxp(-n/ 1K - 1) 1 + O(1²))
Repeat the same argument, know
W2(P(K-2)) (bit o "x") #kk)) s exp(-n(K - -2) - 1 (ka)+ + 0(12))
Chaining this bound recursively, we have
K
(2(1,, (b(x]] ((k) u) = W22(1, (b/ (aKK) S exp
-118-1
Skn + 0(13)
k=
A.3 Proofs for Section
Proof of Proposition
Recall the definition
sr(1 () -
f (u)ddu
dō
82
(1) In the log-concave case, know s1) 0, therefore
sx(1 8) - J (u)du s- 8) sr(u)ldu
(6,r)
20
d8
82
8²
which shows the non-decreasing shape of m,r)..). (2) In the non-log-concave case, we have 8, (1(1) 0, and
thus
(
10 <0,
d8
28

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Published in Transactions on Machine Learning Research (01/2026)
and we claim that (by taking the derivative w.r.t 6)
L s,(u)du 1-6
00
8 - 5,- -)
is non-decreasing in 8. Therefore, 8 + dm,( either crosses zero (in non-dccreasing way) stays negative for d E [0, 1]. Therefore, (6, is either (i) non-increasing in 5 E (0, 11, or (ii) U-shaped in d (0. 1], namely first non-increasing then non-decreasing.
dó
Proof of Theorem As in the proof of Theorem
2 for any v E M(j,, M), define two quantities
e: || jt - (m) = W2(2,
{ = (ty - -i)/ - i1]
Then tx = =i+ eg. and as v the corresponding & satisfies
3 μe, we know + 0. Now note additionally if we assume v E M(H, M), then
& E Th. (M) :00)= {s {: / |E1|7dmm =1, sup =EDom(us)
{{{
|E(x)|1 M }
Claim that for any sequence of v E P5(X): v $ le that attains the limit superior, we have
W (b p4, b"" (3,,))))
limsup
(2(
" m: vEM(pr.M)ae
(x){( (ddu [((x)))()aaa()
sup (M)
[[{())]] (2)
To derive this claim, notice that
Wbb n.7 # Pe, v)= S / | (1 - b" ill
/ [|b>>( ||IbI(o++ (x())) (x)
2 ( (I + +o (1)
The last step requires some justification. (ddd we can verify that
Define an auxiliary function g(e) = I [I +
g'(0) 0,
(0) / (w)
(e) = 2 / (x+
+ / ( ( ((x) & g(x) & (b"* - b*)())))
By the Taylor's Theorem, there exists ē E [0, c), such that
g(e) ())e2
Notice (²) = +0 due the fact that [| < M uniformly bounded and that b has bounded derivatives, establish the claim.
29

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Now we proceed to control the term
| (=)(()))))) - [[b
Now let's study || (z)lop; recall s(t) = V 8- (1 21)
[V)lo 114 p/1 - (1) Coolr(t)Xx = 21)llop
= (1 + 11 I--- + I- - Cou|r X0) X; = =1]lop
= =1+ I-e 21
I-c x Le(()) (2)))
Here we recall the conditional covariance (localization function) defined before
(v( = | ylllop
Continue, we have
/ (N)))))))))))))))))() - / 11(())))dmm(())
s 24 [1(()))(a)((0)) - ---= /
|((()))Pdm(())) -
+ 2n M(( ((7))
We analyze the last term: first, we define the region
Rs = e Las s(()
S 0}
and bound the integral depending on the region,
f
Lice ((T) ( )1
JERRS
Tr(o 201) )
ZERS
11(())11d44((x)) + [ J
$ (1 5)
|1E()))12
V
+ o JEERS
(Z)))) ( -11))))(3)())
Recalling & E (M),
(1 5) / [k(x)))dmd)))))
/
JOERS (LrCe) -11- (1))))(((()((d))(())
(1 (1- 5) [(()))))m(())+++ +
- M [1((r))12(x((()) /18T00////p)))))
/ [sens (Lree) (5)T) --(11 - 6))
= -I(()))Pauul (1--)) M 1. 2 sn(o(2)dz))
where the last step uses Proposition 1 and the Fubini's theorem: for nen-negative random variable Z > 0,
if [ < co, then fe (z C))z(=)dz= Je P(Z 2 z)dz for C> 0: set Z Int (2))) and C = 1 - 6. Put
things together, for any 8 [0,1]
[ (x) (d) - (E(x)))aa S ((E()))(2d)1(
sup SET,, (M)
< 2n
2n
M h1 (6, r(t))] + +0(1)
T-e
30


Reviews:
[{"review_id": "UvpkQJMfO5", "reviewer": "Reviewer_AQYj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a sensitivity analysis of diffusion-based generative models, examining their stability under the Wasserstein-2 ($W_2$) metric. Consider a standard forward diffusion process that transforms a data distribution $\\mu$ into a noised distribution $\\mu_{K\\eta}$ at time $K\\eta$, the author investigates the behavior of a backward denoising process initiated from a perturbed distribution $\\nu \\approx \\mu_{K\\eta}$. This analysis tracks the evolution of the $W_2$ distance between the distributions of this perturbed chain ($\\nu_{k\\eta}$) and the true backward chain ($\\mu_{k\\eta}$) at each step $k$.\n\nThe primary contribution is demonstrating that the final error between the generated distribution $\\nu_0$ and the target data distribution $\\mu$ (i.e., $W_2(\\mu, \\nu_0)$) is fundamentally governed by the \"multi-scale complexity,\" a quantity defined by the average-case curvature of the log-density of the intermediate distributions, $\\nabla^2 \\log p_{k\\eta}(x)$, under the distribution $p_{k\\eta}$. The authors also compare this complexity across different signal-to-noise ratios, illustrating their findings with several specific examples.\n\nStrength: \n\n1. The paper provides a novel stability analysis of the backward denoising process using the Wasserstein-2 metric. This approach extends the existing theoretical framework for diffusion models, capturing more detailed behavior than previous analyses based on KL divergence or TV distance. A key benefit is that this $W_2$ analysis goes beyond worst-case scenarios and provides tight characterizations of the model's stability.\n\n2. The findings of this paper could have practical value. The connection between \"multi-scale complexity\" (i.e., average-case curvature) and stability at different time steps may offer a useful guide for designing and scheduling the noise parameters in diffusion models.\n\nWeakness:\n\n1. The paper's conceptual descriptions are at times overly stylized, which can hinder a clear understanding.\n\n2. The focus on the \"average-case behavior\" of curvature is not an entirely novel concept. This idea has been explored in prior literature on diffusion model analysis, and its connection to stability has been previously discussed, even though they stems from a different setup (discretization analysis).", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main theoretical claim is justified by mathematically rigorous analysis.", "audience_yn": "Yes", "audience_explanation": "The theoretical analysis of diffusion models is a highly active and popular topic within the machine learning theory community, which forms a key part of TMLR's audience. This paper contributes several interesting observations and a novel analytical framework to this specific area. The findings will likely be of significant interest to researchers working on the foundations of generative models.", "requested_changes": "1. Clarify conceptual sentences in the Introduction. Some of the framing is ambiguous and not convincing. Here are two examples\n\n(1)The paper states: \"Consequently, $d(\\mu_0, \\bar{\\nu}_0) \\le d(\\mu_K, \\nu)$ is solely determined by the contraction of the forward diffusion chain alone.\" This is confusing, as the bound $d(\\mu_0, \\bar{\\nu}_0) \\le d(\\mu_K, \\nu)$ in KL/TV analysis typically arises from applying the data-processing inequality to the backward chain (i.e., the denoising steps). It is unclear what is meant by this being \"solely determined by the contraction of the forward diffusion chain.\"\n\n(2)The paper states: “Therefore, justifying the diffusion model as a time reversal Markov chain aiming to recover the past from the future is an effort in vain. In contrast, we propose to study the diffusion models as a sensitivity analysis.” This is also a potentially confusing claim, as current theoretical studies often analyze the denoising process starting from a large time T (where the forward process is near-Gaussian), and the 'time reversal' perspective remains important in those studies. And those studies also provided some stability analysis of the backward process, though it's under a different metric. \n\nThe authors should revise these and similar sentences for clarity to avoid misunderstanding the setup and the paper's positioning. These  do not affect the overall evaluation, but it is important to make the paper clearer to the readers.\n\n2.Add discussion on the related work regarding average-case curvature. The paper should give credit to previous papers that discuss the \"average case curvature,\"  (though they mainly aim to give upper bound)  such as https://arxiv.org/abs/2211.01916 and https://arxiv.org/abs/2308.03686. Moreover, the later work has improved the bound to $E \\|\\nabla^2 \\log p_t(x)\\|_F^2 \\le d$, whereas this paper appears to use a naive bound.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "OhFNBTSWka", "reviewer": "Reviewer_r25w", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper provides a rigorous theoretical framework for analyzing the difficulty of denoising diffusion models by connecting them to Wasserstein-2 (W2) Optimal Transport (OT) and information geometry. The paper's main claim is that the difficulty of the reverse process is governed by the local curvature of the smoothed data distribution $p_t$ at different signal-to-noise ratio (SNR) scales.\n\nThe paper’s core contributions are the following:\n\n**1. Score Function as W2-Gradient Flow:**\nThe paper first proves that the denoising score function, $\\nabla \\log p_t(x)$, is the velocity field of the reverse diffusion process, which can be framed as a W2-gradient flow of the KL divergence (known as the Benamou-Brenier formulation).\n\n**2. Curvature (Hessian) Governs Localization Uncertainty:** \nThe \"difficulty\" of a single denoising step—defined as the localization uncertainty (i.e., the conditional variance $\\text{Var}(x_{t-1} | x_t)$)—is proven to be governed by the local curvature, specifically the Hessian of the log-density, $\\nabla^2 \\log p_t(x)$.\n\n*Negative Curvature (Log-Concave Regions):* $\\nabla^2 \\log p_t \\prec 0$. These regions are stable, \"contractive,\" and easy to denoise (low conditional variance).\n\n*Positive Curvature (Non-Log-Concave Regions):* $\\nabla^2 \\log p_t \\succ 0$. These regions are \"expansive,\" unstable, and hard to denoise (high conditional variance).\n\n**3. Multi-Scale Curvature Complexity:** \nThe paper introduces a new metric that quantifies the total generative difficulty. This metric is an average-case measure that integrates the positive (expansive) part of the curvature across all SNR scales. This metric is crucial for identifying the \"bottleneck SNR\"—the specific noise scale $t$ where the average positive curvature is highest, representing the most difficult part of the generative process.\n\nKey Illustrations\n\nThe authors provide extensive illustrations of their result using Gaussian Mixture Models (GMMs):\n\n*Curvature Analysis:* \n\nThe paper shows that the geometry of the denoising task changes dramatically with the noise scale $t$:\n\nSpecifically they show that within modes, the high-probability regions within each individual Gaussian mode are log-concave (negative curvature). These regions are \"easy\" to denoise. However, the low-probability \"ridge\" or \"trench\" between the modes is non-log-concave (high positive curvature). This region is \"hard\" to denoise, as the model must learn to \"split\" the noise and \"pull\" data points back towards their respective modes, creating an \"expansive\" flow.\n\nThey then show that the \"Multi-Scale Curvature Complexity\" metric successfully identifies the most difficult part of the generative process, and this sits between the low noise as well as high noise regimes, specifically the \"bottleneck\" occurs at the specific intermediate noise level where the noise is just enough to \"lift\" the low-probability ridge, causing the modes to merge. At this scale, the unstable positive-curvature region is at its most prominent and widespread before being completely washed out by noise.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper’s claims are accurate and they provide detailed proofs. I have skimmed these and the proofs seem correct to me.", "audience_yn": "Yes", "audience_explanation": "This paper focuses on theoretically understanding the complexity of the denoising process of a diffusion model. The paper sheds light on the difficulty of denoising at various SNR scales, and I feel this is of particular interest to members of the machine learning community at this point.", "requested_changes": "I feel the paper would be significantly strengthened by improving clarity of the presentation. It would be nice to have, for instance, a clear section defining notation and perhaps in the appendix a section at least stating several properties and theorems that have been just stated in the body.", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Y4RBo3Pnmb", "reviewer": "Reviewer_sMiA", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper studies the denoising process, that is, the reverse process in the diffusion model. In particular, the paper studies the Wasserstein contraction that is characterized by the curvature complexity of a smoothed data distribution as a specific signal-to-noise ratio scale. The multi-scale curvature complexity determines the difficulty of the denoising chain. The multi-scale complexity quantifies a notion of average-case curvature instead of the worst-case scenario. It depends on the integrated tail function, measuring the relative mass of locations with positive curvature versus those with negative curvature. Several examples are given.", "claims_and_evidence_yn": "Yes", "claims_explanation": "This is a theoretical paper. The proof seems to be rigorous. The paper is relatively well-written.", "audience_yn": "Yes", "audience_explanation": "I believe so. I think there has been a lot of interest in generative models such as a diffusion model in recent years. Various convergence guarantees have been obtained in the literature, mostly in TV and KL divergence, and some papers deal with Wasserstein convergence. But the papers that I am aware of that deal with Wasserstein convergence usually need to assume some very strong assumptions, such as strongly-log-concave data distributions. This paper introduces some new ideas to use a notion of average-case curvature instead of the worst-case scenario to measure the contraction of the reverse process. This might lead to some future research in this field.", "requested_changes": "(1) On page 2, you wrote that most of the current theoretical literature (Lee et al., 2023; Chen et al., 2022;2023) treat this curvature as a nuisance by, for example, leveraging a form of data-processing inequality and only focusing on non-expansion metrics $d$... for $d\\in\\{d_{TV},d_{KL}\\}$. In your paper, you did not cite any theoretical works in Wasserstein convergence. You should cite and discuss the following papers. \n\nGao et. al. (2025). Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models. Journal of Machine Learning Research.\n\nBruno et al. (2025). On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates. Transactions on Machine Learning Research.\n\nBoth papers dealt with log-concave data distributions, and obtained contractions in Wasserstein distance.\nThere is another recent paper, \n\nSilveri and Ocello (2025) Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance (ICML 2025), \n\nthat goes beyond the log-concave case. The field is developing very fast, and there might be other papers that derived contractions in Wasserstein distance. As a result, your sentence on page 7 that the current theory falls short in addressing the behavior of the backward denoising process under the Wasserstein metric is not completely accurate.\n\n(2) After equation (5), you should introduce $\\beta$ and explain the role it plays.\n\n(3) The assumption in your Corollary 1 seems super strong. When you assume $\\nabla^{2}f(x)=\\kappa\\cdot I_{d}$ and $\\nabla^{2}\\log p_{\\mu_{t}}(x)=-\\zeta\\cdot I_{d}$, did you assume this holds for every $x$? If so, you are assuming the data distribution is Gaussian right?\n\n(4) In the three papers I mentioned previously, sometimes, strong-log-concavity is only assumed for the data distribution, and then\nthe strong-log-concavity at any time $t$ can be derived as a result. In your results, it seems that you directly assume strong-log-concavity\nat any time. At least, you should provide some discussions on that. For example, in your Corollary 2, you assumed strong-log-concavity coefficient $\\zeta_{k\\eta}$, but it is not clear how $\\zeta_{k\\eta}$ depends on $k\\eta$.\n\n(5) Your main results, such as Theorem 2 and Theorem 4, seem to be asymptotic in nature, i.e. requiring $\\eta\\rightarrow 0$. \nIt would be nice if you can extend that to non-asymptotic results, which are more relevant in deriving iteration complexity.\nThe current results are presented in an asymptotic way, which is one of the weaknesses of the paper.\n\n(6) In Definition 6, please make $\\{$ and $\\}$ bigger. \n\n(7) In Section 5.2, Beyond Log-Concavity, you presented a few special examples. I am wondering whether it is possible\nto derive some general results beyond log-concavity. For example, instead of particular examples, how about\na general distribution that satisfies dissipativity condition or strong-log-concavity outside a compact domain?\nThat will significantly strengthen the contributions of the paper.\n\n(8) Please move $+$ at the end of the first line in equation (13) to the beginning of the second line.", "broader_impact_concerns": "N/A.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)