Cross-fluctuation phase transitions reveal sampling dynamics in diffusion models
Authors: Sai Niranjan Ramachandran
Manish Krishan Lal
Suvrit Sra
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion Models
Fluctuation theory
Phase Transitions
Sampling
Markov Chains
Interpretability.
Primary area: deep_learning
TL;DR: We explain how discrete phase transitions govern the emergence of structure in diffusion models through the idea of cross-fluctuations from statistical physics.

Abstract: We analyse how the sampling dynamics of distributions evolve in score-based diffusion models using \emph{cross-fluctuations}, a centered-moment statistic from statistical physics. Specifically, we show that starting from an unbiased isotropic normal distribution, samples undergo sharp, discrete transitions, eventually forming distinct events of a desired distribution while progressively revealing finer structure. As this process is reversible, these transitions also occur in reverse, where intermediate states progressively merge, tracing a path back to the initial distribution.  We demonstrate that these transitions can be detected as discontinuities in $n^{\text{th}}$-order cross-fluctuations. For variance-preserving SDEs, we derive a closed-form for these cross-fluctuations that is efficiently computable for the reverse trajectory. %, thus tying cross-fluctuation dynamics to event formation within the desired distribution. 
We find that detecting these transitions directly boosts sampling efficiency, accelerates class-conditional and rare-class generation, and improves two zero-shot tasks--image classification and style transfer--without expensive grid search or retraining. We also show that this viewpoint unifies classical coupling and mixing from finite Markov chains with continuous dynamics while extending to stochastic SDEs and non Markovian samplers. Our framework therefore bridges discrete Markov chain theory, phase analysis, and modern generative modeling.

Cross-fucctution phase transitions reveal sampling dynamics in diffusion models
Sai Niranjan Ramachandran" +t
Manish Krishan Lal*t
Suvrit Sra*t
Abstract
We analyse how the sampling dynamics of distributions evolve in score-based diffusion models using cross-fluctuations, tistical physics. Specifically, we show that starting from an unbiased isotropic normal distribution, samples undergo sharp, discrete transitions, eventually form- ing distinct events of a desired distribution while progressively revealing finer structure. As this process is reversible, these transitions also occur reverse, where intermediate states progressively merge, tracing path back to the initial distribution. We demonstrate that these transitions can detected as disconti- SDEs, we derive
centered-moment statistic from sta-
nuities in -oddrr cross-fluctuatios a closed-fore form for these cross- cross-fluctuations that is efficiently computable for the reverse trajectory. We find that detecting these transitions directly boosts sampling efficiency. accelerates class conditionii and rare-class generation, and improves zero-shot tasks lasks-image classification and style transfer-wiid without expensive grid search or retraining. We also show that this viewpoint unifies classical coupling and mixing from finite Markov chains with continuous dynamics while extending to stochastic SDEs and non Markovian samplers. Our framework therefore bridges discrete Markov chain theory, phase analysis, and modern generative modeling.
For
Introduction
Diffusion models are cornerstone of generative systems. They learn to reverse process that gradually erases structure from data, transforming it into featureless (isotropic) noise. During inference, the model starts with this noise and progressively refines it, adding structure step by step until realistic data samples emerge. [Ho et al., 2020, Sohl-Dickstein et al., 2015]. These models can generate lifelike images, 3D scenes, audio, and even molecular structures [Rombach al., 2022, Blattmann et al., 2023, Chen et al., 2022b, Corso et al., 2022, 2023, Karnewar et al., 2023, Kynkäänniemi et al., 2024]. They also excel in complex tasks like protein design [Jumper et al., 2021] and zero-shot vision tasks [Li et al., 2023, Meng et al., 2021, Rombach et al., 2022].
Yet. the sampling process remains a black box. Each step blends thousands of values in ways that difficult to anticipate. Like other generative paradigms [Hutchinson and Mitchell, 2019, Hendrycks al., 2023, Rudin, 2019, Sharkey al., 2025], small tweaks in code hyperparameters can turn perfect sample into a failure or introduce unintended biases. This unpredictability underscores the need for clear understanding of how successful and failed outcomes diverge as the sampler operates.
We provide a clear framework by defining a user-specified goal (e.g., a class label, an aesthetic style, or a semantic attribute) as desirable evení, which is a set of outcomes that align with the user's objective. As the sampler evolves backward in time, iteratively generates distribution over possible outcomes at each step. Initially, paths corresponding different outcomes, whether desirable undesirable, may appear indistinguishable. However, we demonstrate that at certain critical intermediate steps, these paths undergo discrete phase transitions, where they become distinguishable they separate
School of Computation, Information and Technology. Technical University Munich, Germany Munich Center for Machine Learning (MCML)
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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into distinct regions of the outcome space. These transitions occur when the sampling process reveals structural differences between outcomes that align with the user's goal (desirable) and those that do not (undesirable), marked by a breakdown of path separability.
Main contributions
The main contributions of this paper are the following:
1. Framework and theory. We present rigorous framework using fluctuation theory from statistical physics [Chaikin et al., 1995, Kivelson et al., 2024, Landau and Lifshitz, 1980, Pathria and Beale, 2011] to identify and quantify discrete phase transitions in the diffusion process (Section 2.2, Appendices B.1.1 and B.3.6). This framework incorporates user-specified goals as desirable events and shows how their dynamics can be tracked (Section 3). Moreover, we establish a connection between ensemble based techniques in statistical mechanics as well as classical maxxing-coupling results for Markov chains [Aldous and Fill, 2002, Levin and Peres, 2017, Saloff-Coste, 1997], using generalised notion of phase transitions, thereby providing unified perspective on probability flows (Section B.1.3.2 and Appendices B.1.4 and B.3.6).
2. Practical toolkit. We develop practical toolkit with clear diagnostic criteria for identifying and leveraging these discrete transitions (Section 3, Appendices and C). The proposed Algorithm systematically identifies discrete transitions in cross cross- fluctuations to characterise the sampling dynamics of desired outcomes. This enables precise intervention to enhance the generation and likelihood of desirable samples, thereby improving overall model performance.
3. Illustrative applications. We illustrate the power of the proposed framework and toolkit through several upplications: accelerated sampling (Section 4.1), conditional generation (Section 4.2), rare-class coverage (Section 4.3), and improved zero-shot performance in classification (Sec- tion 4.4) and style transfer (Section 4.5). Remarkably, simple fluctuation driven tweak can improve the performance baseline methods.
The remainder of this paper is organised as follows: background (Section 2); methodology (Section 3); experiments (Section 4); theoretical foundations (Appendix B); implementation details and additional experiments (Appendix C); and limitations and related-work context (Appendix D-E).
2
Background
We first review prriancc-preserving diffusion models and the fluctuation -hituattionttheorticc -thooretic statistics that let us detect emergent phases. For detailed notation used throughout the paper, see Appendix A.
2.1 Diffusion models in continuous and discrete time
Let Xo Po be sample from distribution with support & C Rd. The forward variance-preserving stochastic differential equation (SDE) of Song et al. [2021] is
dx, --==--pp ((t) x, dt / B(1) dw.,
E [0,
(2.1)
where (() 0 is the noise schedule and w, standard Wiener process. For linear schedules and sufficiently large T we have pr =s A/ I). Removing stochasticity yields the probability-flow ODE (PF-ODE) with identical marginals,
dx: dt 3() 3(t) Vx log (x1)
(2.2)
neural network sg (x, I) learns an approximation of the score; sampling integrates the reverse ODE from é 7' back to 0. In discrete time, the Denoising Diffusion Probabilistic Model (DDPM) update of Ho et al. [2020] is
B, x, v Be Et. E, (O.
1=
(2.3)
Both the continuous and the discrete processes admit the closed-form marginal
(x xo) = .N(((() xo. (1 (())))
J(() = exp( (6
(2.4)
which separates signal from noise cleanly, where Jt) is known the signal-attenuation factor.

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2.2 Measuring statistical distinctiveness
The central insight of our work is the observation that given coarse categories of data e-g. object classes, tracking their statistical distincttvvness throughout the diffusion process suffices to pinpoint the emergence of structure during generation.
Figure 1:
addition noise causes distinct categories data to "merge" through the forward diffusion process as statistical properties progressively converge to those the standard normal distribution.
As Figure shows, the statistical properties of data show well-defined convergence on noise addition and conversely show divergence as the sampling process progresses. We formalise this notion through the use of fluctuation theory adapted from physics.
2.2.1 Basics of fluctuation theory
Fluctuation theory supplies statistics that reveal when trajectories become (inydistinguishable operational signature of a phase transition. We present a formal framework for analyzing these fluctuations in vector-valued state spaces, which generalises the concepts from statistical physics and provides a direct connection to modern tools like Centered Kernel Alignment (CKA) (Kornblith et al. [2019]).
Fluctuations. Let (2,F,P) be a probability space. The state of the system is described by vector-valued random variable p Ω + Rd. The fundamental object for our analysis is the nhth ooder fluctuation, random tensor representing the nh centered moment at single point w E Q:
F(n)
(p(w) E[lp).
(2.5)
This is a tensor of rank n. For instance, for 2, this gives F3 (p(w) -(W)-))))))
which is d x matrix. Its expectation, [[ is the familiar covariance matrix of p. We assume these random fluctuation tensors (and their expectations) reside suitable Hilbert spaces H., (for rank-n tensors) equipped with an inner product (: .H.. For matrices (n 2), Me is typically the space of d x matrices with the Frobenius inner product, (A, B) = Tr( B).
Joint dynamics and conditional expectations. We consider two disjoint events, Ωn and Ω2. While samples are drawn independently from each, the events themselves are coupled: they originate from the same data manifold and evolve under the same stochastic dynamics. This coupling justifies the comparison of their respective conditional expectations of the fluctuation tensor:
E, [F(()) B[ سا E 24] and I2 ( E(F(() w Ω2). For = 2, these are the conditional covariance matrices for each event.
Normalised and unnormalised cross-fluctuations these conditional expectations.
We now redefine our key statistics based on
1. Unnormalised cross-fuctuation (G): This measures the alignment of the conditional expected fluctuation tensors of the two events.
G(1) (Q,, Q2) (E, [,
(2.6)
2. Within event fluctuation (F): This measures the magnitude (or self-alignment) of the conditional expected fluctuation tensor for a single event.
(22m) (Ω.) = (Ea(F)()) (F(")))) n. || [ (
(2.7)
Note the 2n in the symbol F is convention.

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3. Normalised cross-fluctuttion (M)) This is the magnitude of the cosine similarity between the conditional expected fluctuation tensors.
|Gp (d (S1. 922)
[E, (F)") || (1) H., [F 11
Bg
[F Draen
(2,,
(2.8)
V ((2m))
(22) (( (S2)
((2m)
(3))
The normalized cross-fluctuation, MM [0, 1}, measures the structural similarity between the events' n-th order moments. value of M = indicates that the events' statistical shapes have aligned, condition define "merge." Conversely, a value significantly less than 1 implies distinguishability. For the important special case of n = 2, My quantifies the similarity between the conditional covariance matrices (Ex) of the two events and is connected to CKA.
3 A practical toolkit for diffusion models
We now present the theoretical construction for our methodology. The main practical outcome is Algorithm 1, an algorithm whose applications are demonstrated in Sections 4.1 4.5.
Embedding in diffusion time. Let pi be the marginal at step = 0...., n of a forward diffusion
process (For simplicity, we focus the PF-ODE case*), with Po Paesired- where Pdesired represents a constructed distribution that encodes properties of interest, and p.. = (O, Choose two disjoint events Q 0, Z2,0 supp(pa) and propagate them i,, As the forward trajectory converges to white noise, the regions coincide at n in theory, but due to hyper-contractivity and numerical effects, they often appear to merge much earlier. We thus redefine the operator M capture this discrete transition (Appendix B.3.6),
( (2,, 1,
d( ((F)2")(() (S1,)), (0I.),FF ((22)((222)) otherwise,
(I22,))
M (1)
(3.1)
where -) is a distance metric which for our case is the & distance between the traces of (()) (.) and is fixed threshold. The earliest index
===min M M 11)
generalises the coupling time of Markov chain [Aldous and Fill, 2002, Levin and Peres, 2017] (Appendix B.1.4). The operator undergoes discrete phase transition dictated by Conceptually, the smooth distance function has been replaced by a "relu" like non-linearity to capture numerical effects. Further details on this kind of phase transition and how it differs from the traditional notion of phase-transitiin can be found in (Appendix B.3.6).
We justify our choice of the forward trajectory in Appendix 1.2. establishing that the detected transitions also oceur in the sampling trajectory of trained diffusion model. Crucially, this approach allows us to rely solely on the computationally efficient empirical forward process. We tailor Paesired and (9,,o, S22.)) to illuminate specific phenomena. Algorithm implements these steps. Note that, when Po has compact support (e.g., images), the PF-ODE ensures spatial continuity of ( [Coddington and Levinson, 1955, Thm 5.2]; any jump thus indicates genuine phase transition (for SDEs see Appendix B.2.3). A single forward Monte- Carlo sweep yields unbiased estimates of all terms in M((") see Appendix B.2.5.
'A detailed treatment for the PF-ODE appears in Appendix B.2.2, and the SDE case is discussed in Appendix B.2.3.
'Details on the topological equivalence of metrie space on within- event fluctuations and the absolute distance of normalised cross- foss-fluctuation from can be found at Section B.1.3.1

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Algorithm 1 Compute (M (1) and the first merger step 8*
Require: paa, steps n,, forward sampler PF, task objective L, fluctuation F
metric threshold
1: Prlcaired DEFINEDESIRED (Paara L)
2: (LI.o, (2,0) < PICKEVENTS(Ptisiiie L)
3: M [[]
P dynamic array for Ma " (i)
4: i" +n
merge no later than white noise
5: for i to do
6: if > O then
(Di,i, < FORWARDSTTE((II.-......
8:
else
9:
(0,,1,92,1) < (S,,,,,2..0) end if
10:
+ ((F(") (11,), (S22.))
(1) M ( (2,1
8
12:
8 S
13:
append MD to
14:
if M (i) and = then
15:
b. first step where events merge
16:
end if
17:
end for
18: return M,
4 Demonstrations of our methodology
We next present case studies that show how our machinery can diagnose and subsequently improve the behaviour of diffusion samplers on practical tasks. Our aim is to illustrate simple recipes for working with user-defined events that need not require a deep dive into the underlying theory but at the same time showcase the ufility our framework.
4.1 Warm-up: Predicting convergence of the data distribution
In this case study, we abuse notation and let the sequence (p Pr represent the marginals under the PF-ODE for data distribution Po- Assume that the support of the data distribution, Do spp) essentially disjoint from the support the Gaussian endpoint, D, supp(p.) = sup I.)). In high dimensions, this is realistic because D, Vd Sdd1 and therefore has asymptotically negligible volume [Vershynin, 2018
We now show the setup of Algorithm in this setting. Let Pdrsired the desired distribution be defined over the union of all supports _ such that,
Po(Z)
Z Do:
Pdesired ( ())
(4.1)
0, otherwise,
where Paesired, Po are the probability masses associated with Pdesired: Po respectively. is easy to see that the PF-ODE on PO Pdesired leads to the set of marginals that we term augmented process, aug = { defined as,
P((Z).
7 ZCD;
P((Z) 0, otherwise,
(4.2)
where P. (resp. P.) the probability measure associated with p; (resp. P.). Operationally, aug coincides with the original PF-ODE but with an explicitly enlarged state space.
'Note that, D,, becomes null set for d - oo, so overlap with any finite-volume data manifold is vanishingly unlikely. Concentration inequalities showing rapid decay function of increasing can be found, e.g.. in [Vershynin, 2018J. Formally, assume that 2 Da D... then Po(D"), P.. (D")
'The above construction designed to measure how probability mass transfers away from any marginal p: towards another marginal where j > i. In our work we only concern ourselves with the case where n so probe convergence to stationarity.
5

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Set l2 = Du and Q220 Dr. Because 92 is already the support of the stationary distribution, 92 = 22.00 for all > 0. Detecting the smallest index for which M(()) p,00y:0g (1) ~ therefore amounts to deducing the onset of convergence to the stationary distribution, this is in practice equivalent to multivariate normality test on p;. We adopt the D' D'Agostino-Pearson omnibus test [D' Agostino, 1971, D3 Agostino and Pearson, 1973], the p-value threshold is the conventional 0.05. We evaluate MNIST [Deng, 2012], CIFAR-10 [Krizhevsky et al., 2009], and compressed INT-8 variant of ImageNet [Deng et al., 2009, Ryu, 2024], using the popular DDPM noise schedule [Ho et al., 2020]. Results appear in Figure in Appendix C.2.
Acceleration via early stopping. Let i" denote the smallest index flagged as Gaussian by the test. Starting the reverse sampler from = i", rather than the conventional n, preserves visual quality while saving n i" steps, summarised in Table across datasets. By convention, throughout our work a (4) sign denotes that lower values are better and vice-versa for an (1). For ImageNet we employ the state-of-the-art DiT-XL/2 model at 512 x 512 resolution [Peebles and Xie, 2022]; open-source checkpoints are used for the other datasets. comparable observation was reported by Raya and Ambrogioni [2024], who derive i* analytically under stronger assumptions on the data. Intriguingly, an unrelated theorem on Brownian equilibrium time predicts i* surprisingly well for data in Appendix B.3.1; clarifying this connection is left to future work, but the theorem already provides a useful heuristic.
Model / Dataset
FID(L)
Steps(L)
(FLOPs(
DiT-XL/2 (ImageNet, full)
3.42 = 0.21
250
4100
DiT-XL/2 (ImageNet, ours)
3.37 0.31
175
2870
DDPM (MNIST, full)
2.27 0.19
1000
2000
DDPM (MNIST, ours)
2.29 0.17
1200
DDPM (CIFAR-10) 10, full)
3.62 = 0.35
500
6000
DDPM (CIFAR-IO. 10, ours)
3.47 0.34
300
3600
Table 1: Acceleration achieved by stopping reverse diffusion at 1 = i* instead of n (DDPM schedule). FID scores are averaged over three runs with 95% confidence intervals.
Thus, starting the reverse sampler from the observed convergence time yields the same perceptual quality while reducing wall-clock cost often by thousands of gigaflops (GFLOPs).
Visualizing the merger cascade. Before detailing our next set applications, it instructive to visualise mergers occurring over disjoint partition of events. Figure 2 presents the "merger cascade," a temporal hierarchy visualizing how disjoint events merge over diffusion time t. Events start as distinct leaves at t 0. As time progresses upwards, branches join at a black dot-a discrete merger event (Appendix B.3.6) the precise moment their fluctuation tensors become indistinguishable. This dendrogram provides intuitive map of how structural similarities are lost, motivating the targeted interventions that follow.
Visualizing the Marger Cascude: Temporal Herrechy Merger
Diffuston
2.6
Figure Visualizing the merger cascade. This temporal hierarchy illustrates how distinct, mutually disjoint events merge as the diffusion process evolves. Time flows upward. At = 0, events are distinct leaves. they diffuse, pairs whose fluctuation tensors become indistinguishable undergo a discrete merger event (black dot), and their branches merge. This cascade continues until all discriminating information has vanished into single cluster.

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4.2 Class-conditional generation
We now study the behaviour of Algorithm when Puesired is just the data split into K mutually disjoint classes {R1.0..... .,. 92,o0 such that - S4y0 = Q where is the sample space of the data distribution and 94,0 7 52,0 = Ø whenever le & L. Throughout we write U2 for the image of 2x.o under the forward diffusion map at (discrete) time A formal algorithm summarizing this modification is present in Algorithm 4.
second-order statistic via centred kernel alignment. Higher-order fluctuations for vector states become unwieldy, so we fix 2, leveraging covariance matrices to efficiently capture fourth-order moments. This captures non-linear feature information while maintaining second-order structure (Appendix B.3.3). For completeness, we also demonstrate experiment with higher-order fluctuations, simplified using Isserlis / 's theorem [Isserlis, 1918, Wick, 1950] (Appendix B.3.2). Let us for the case define,
k Cov(pl Sk.c) E Iddxd
(4.3)
The normalised second-order cooss-fluctuation M (Slg,t, St,,e) is then the centred kernel alignment (CKA) between Ex,t and Ee,t [Cortes et al., 2012, Cristianini al., 2001, Kornblith al., 2019]:
Tr Me,))
CKAre(t) = Ex.)) Tr (ET, EE))
(4.4)
where We convert the above definition (4.4) into a discrete phase transition indicator (3.1) detecting whether the events have "merged / not merged" by the rule
Tr(E,, L(),) , F. = T> D..)) , (Q,,)) = Tr(IT,
(CKALE(t), if (F((F (Rt,,) (P (R2z)) e;
M (2) (1)
(4.5)
11
otherwise,
Because the forward SDE is isotropic, Σs., contracts in the radial direction only; its spectrum therefore collapses towards scaled identity. Hyper-contractivityy implies that the merger of two classes is governed by the largest eigenvalues (See Appendix B.3.3 for further details)
A (t) = Amax (E,,e).
X (1) = >max (Ee,);
so we may replace the trace metric by d( (Amax (t), Xmax (t) < In practice, we fix
max; A Amax (0)
E
400
and use absolute eigenvalue differences for This criterion yields clear merger times across datasets; trajectories are plotted in Appendix C.3.
Practical payoff:
class-wise Interval guidance. The state-of-the-art guidance algorithm, Interval guidance (IG) [Kynkäänniemi et al., 2024] argues that class conditioning should be applied only for sub-interval t (tstnn tead) for optimal results. The reference implementation runs an expensive grid search as detailed in Appendix C.3. Our fluctuation analysis supplies both bounds for free:
Sstart i", the convergence index from Section 4.1;
fend = (t
M (2) ke (t) < for some l + k).
Using these class-specific windows, we match or improve IG performance on ImageNet, CIFAR- 10, and MNIST datasets. Our method is capable of cutting the combinatorial search cost by orders of magnitude. We present the results in Table 2. We utilise the metrics Frechet Inception Distance (FID) Heusel al. [2017]. Precision, Recall, and their more suitable variants for generative models such as Density and Coverage Nacem et al. [2020]. (We also show results for the StableDiffusion Rombach al. [2022] model using Imagenet and the fine-grained classification dataset OxfordllTTPets in Appendix C.3 along with further results.)
Monitoring when class supports merge lets us place guidance exactly where it helps, eliminating costly grid searches and boosting class-conditional quality with negligible overhead.
7

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ModeV/Dataset
FID (J)
Precision (1)
Recall (1)
Density (1)
Coverage (1)
D/T-XL/2 (Imagenet, IG Baseline)
3.22 + 0.16
0.78 = 0.01
0.23 + 0.05
0.83 = 0.01
0.35 = 0.02
D.T-XL22 (Imagenet, IG Ours)
2.86 1. 0.15
0.83 1 0.02
0.26 1 0.04
0.85 1 0.01
0.39 1 0.02
DDPM (MNIST, I, IG Baseline)
2.15 1 0.06
0.80 1 0.02
0.25 1 0.01
0.85 1 0.01
0.36 1 0.02
DDPM (MNIST, IG Ours)
1.99 1 0.11
0.85 1 0.01
0.28 1 0.03
0.89 1 0.02
0.40 1 0.01
DDPM (CIFARIO, IG Bascline)
3.32 + 0.25
0.77 + 0.01
0.19 0.14
0.81 = 0.03
0.32 + 0.02
DDPM (CIFARIO, IG Ours)
3.01 1 0.14
0.79 1 0.02
0.22 + 0.01
0.84 : 0.01
0.35 1 0.04
Table Class conditional generation
Model/Datasef
CLIP Similarity (
Precision (f)
Recall (
Density (1)
Coverage (*)
SD (NNaturalist, IG baseline)
0.21 + 0.03
0.70 0.01
0.15 0.00
0.75 0.01
0.27 0.02
(iNaturalist, IG Ours)
0.24 0.02
0.73 0.01
0.00
0.02
0.04
SD (NNaturalist, IG-ILVR Ours)
0.27 + 0.01
0.76 0.001
0.19 0.02
0.81 0.01
0.31 + 0.03
SD (CUB200, IG baseline)
0.24 + 0.05
0.75 0.00
0.14 0.01
4.79
0.30 4 0.02
SD (CUB200, IG Ours)
0,26 +: 0.01
0.78 + 0.05
0.17 = 0.01
0.82 4: 0.02
0.33 4: 0.01
SD (CUB200, [G-IIVR Ours)
0.27 1: 0.02
0.82 0.02
0.18 0.02
0.85 +: 0.01
0.31 + 0.02
Table 3: Rare class generation using StableDiffusion
4.3 Rare-class generation
We reuse the same modification of Algorithm as Section 4.2 but focus on tail classes-categories under-represented in the training data, and therefore poorly synthesised by off-the-shelf diffusion models. Concretely, we study the following datasets,
CUB-200 (200 bird species), and
. iNaaurrlist 2019 (fine-grained flora & fauna).
Both have been identified as failure cases for Stable Diffusion [Samuel et al., 2024]. For each species k, we denote the source event by 2g.0 and its forward image at timestep by DR
Merger-aware guidance window. Adopting the notation of Sections 4.1 1 and 4.2, let
feomy
mergee.k min .M(22 (SeE,t, St,,e) = for some + k).
The class-specific guidance interval is then (Kaan,ks fend,k) = (fcomv, l'merge.k) k),, that is, from global data convergence up to-but not beyond-the first time the target class becomes indistinguishable from any other class.
Interval guidance with corrupted exemplars. We observe that this setting as compared to naive interval guidance, slight tweak using corrupted exemplars can yield better results Inside this window, we upply Interval Guidance [Kynkäääniemi et al., 2024] and interpolate with a single exemplar Sup00 that is forward-noised in lock-step with the individual sample being generated. The conditioning term is effectively therefore the pair (Ee, =xef,r), making the scheme class-specific variant of ILVR [Choi et al., 2021] with intervals. No additional hyper-parameters are introduced; we keep the noise schedule and scaling constants from Section 4.2. formal version of the algorithm is presented in Algorithm 3, refer to Appendix C.4 for further details.
Table 3 summarises the results using standard metrics, where we compare three settings: (i) naïve Interval Guidance over the full horizon, (ii) merger-aware Interval Guidance in Section 4.2 and (iā) merger-aware IG corrupted exemplar (ours). Across metrics, the merger-aware schedules consistently outperform the baselines, with the exemplar variant giving the largest gains.
We conclude that as fluctuation-driien merger times generalise from mainstream classes to the tail classes without re-tuning, leveraging them yields a lightweight yet effective strategy for rare-class synthesis, mitigating class imbalance at essentially zero extra computational cost.
4.4 Zero-shot classification
Once again continue with setting of Algorithm 1 as is in Section 4.2: where each sample 2~ po belongs to exactly one class event 2g Given a trained class-conditional diffusion model fo with
We observed appreciable gain for the setup in Section 4.2, hence omit this tactic there.

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label index set A = f1,.... ... } the goal is to predict
k- arg Ph (z),
REA
where (2) estimates the posterior class probability. We build upon the procedure proposed by Li et al. [2023] for this task. Let =x(2,) e) E R denote the forward-noiser version of at step t, obtained with seed e ~ VV(0, 1). For that configuration, define the logit
sx(t, E) = - Il (z - =
k E A.
With N noise seeds ( and weights w(t), our importance-weighteed class score becomes
(x(2)= = Σ futaaat
Assop. A
N n= on ((,)))) w(1) [ 1 E
(x(1,))
exp(ss(ttE)) ELEA exp(se(t,e))
(4.6)
where (t, E) is the corresponding softmax probability. The choices uniform, inverse-SNR and truncated inverse-SNR correspond to three concrete w distributions, each summing to 1 its support. For the inverse-SNR strategies, the weights at each time are proportional to the inverse of the Signal to Noise Ratio (SNR) [Kingma et al., 2021] at that time, while uniform assigns constant weight. The lower bound is set to f = Ak for the uniform and inverse-SNF strategies. For the truncated case set & 20 for all k; (Appendix C.5) the upper bound
kstop, min M (2) (0x,e. Re,e)) for some é * k}
ensures that no timestep beyond the first class merger contributes to (4.6). A formal version of the above algorithm is present in Algorithm 4 along with further details in Appendix C.5. Using a Stable Diffusion backbone [Rombach et al., 2022], we obtain the accuracies in Table 2. Merger-aware weighting improves the uniform baseline of Li et al. [2023]. where fatart 0 and fstop, T for all classes k, where T is the horizon of the diffusion process; truncated inverse-SNE performs best.
Method
Image 54.96
CIFAR-I0
Oxford-IIIT Pets
SD, uniform (Li ct al.)
14.67
82.87:0.339
SD, uniform (ours)
57.91
85.1740.17
86.1740..26
SD, inverse-SNR
04.17
87.26
88.17
SD, trunc. inverse-SNR
8..48
88.38 1 0.43
89.15 - - 0.26
CLIP RN-50
50.41
75.42+0026
85.61#0.29
OpenCLIP VIT-H/14
76.91
96.87#0.59
90.37
Table 4: Zero-shot multi-class accuracy (%); = 95% CI over five runs.
That truncated inverse-SNE outperforms uniform weighting suggests the corrupted marginals remain most discriminative shortly before class supports merge. Appendix C.5.1 analyses this effect in detail via linear probes for binary classification between Imagenet classes.
Thus, merger times not only optimise guidance but also delimit the timesteps that matter for zero-shot recognition, yielding tangible gains at negligible cost.
4.5 Zero-shot style transfer
Let Po be source image distribution on X' C IRd and letp" TTyle be the same semantic content rendered in new artistic style. We assume
1. Bijectivity. Tayle X X' invertible on the supports, so p valid density and supp(p*)= (supp)pp())))
2. Structure preservation in the Fourier domain. For a density write q(E) =
Jx e xdx and define the frequency-domain norm ||p glls 1p @ (( We posit the regularity bound
[lpo 1pl-p"lls s 8, with 0 < 1
(4.7)

---

Fluctuation adaptation lemma for style transfer. Let p(x) x be the identity state operator for Po and let pa = po Taylo for p". Appendix B.3.4 proves that for every measurable f S X' and each moment order E 21, 3,4, 4}.
(S)) - (Tayle < C. 6,
(4.8)
with a constant Cn independent of Q. Hence the fluctuation trajectories of the source distribution po approximate those of the target-style distribution p* to O()) accuracy.
Choosing a start time for reverse denoising. For each semantic class
let
ma min :M) (S R4,)) for some & + ** }}
be its earliest merger time under the forward diffusion of po (see Section 4.2). By the lemma above, mA remains valid approximation for the target-style chain. Therefore, when we apply a target-style model (e.g. Stable Diffusion fine-tuned on van-Gogh paintings) we can kick-start reverse denoising at t =====m my rather than at the full horizon t = n, achieving style transfer in fewer steps. Thus Pdesired in Algorithm here is the source distribution while the diffusion model is actually trained on different target distribution. A formal version of the above algorithm is presented in Algorithm 5. We also note that the current procedure naturally extends inverse problems using diffusion models, following similar procedure under conditions analogous to (4.8). These conditions align with prior methodologies for inverse problems Song et al. [2022]. Exploring these extensions is left for future work
We follow the setup of [Meng et al., 2021] wherein an optimal time needs to be estimated till which noising must occur. Our baseline is the grid search technique used in [Meng et al., 2021] computed over an entire dataset. We present sample results for the Studio Ghibli Miyazaki and Ghibli [2014] and Van Gogh Roojen [2019] artistic styles in Table 5 using the PSNR and MSE metrics. See Appendix C.6 for experimental details, figures and empirical verification of the fluctuation adaptation lemma, and additional results for the Elden Ring Software and Entertainment [2022] and Arcane Production and Games [2021] styles.
Style
Ghibli
van-GGgh
Models/Metrics
PSNR (t)
MSE (L)
PSNR (f)
MSE (J)
SD Edit (OxfordllIPPees)
25.67 1.14
0.09 + 0.001
26.16 1 0.72
0.09 0.003
Ours (OxtordllI(Pcts)
28.71 + 0.86
0.03 + 0.005
28.65 + 0.49
0.03 + 0.002
SD Edit (AFHQv2)
27.12 1: 0.59
0.05 + 0.006
27.49 + 0.27
0.04 +: 0.004
Ours (AFHQ v2)
27.65 0.57
0.04 0.004
28.07 1 0.32
0.03 0.006
Table 5: Style Transfer results for Ghibli and van-Gogh styles
Thus, as under mild Fourier regularity, the merger schedule learned on a source dataset transfers directly any purely stylistic variant of that dataset, the result is true zero-shot style transfer procedure that inherits the efficiency gains of Section 4.1 I without additional tuning.
Conclusion
We introduced cross-fluctuttonn mergers: centred-moment statistic that fires precisely when two regions of a diffusion trajectory become indistinguishable. The concept links statistical-physic fluc- tuation theory with generative modelling and yields practical benefits: early stopping the forward process cuts a quarter to two-fifths of reverse steps; class-wise guidance windows raise conditional image quality without grid search; merger-aware weighting upgrades zero-shot classification, and the same signal powers zero-shot style transfer and rare-class generation- all without re-training or new knobs to tune. Our analysis assumes a variance- pressrvvngg noise schedule; extending to non-VP VP schedules (e.g.. EDM [Karras et al., 2022]), anisotropic diffusions, higher-order fluctuations for complex data, and different modalities are promising directions for future work. We expect our framework to inspire further developments in tighter theory, training-time objectives, and applications.
10

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Acknowledgments
We thank Xiang Cheng for his support in the conceptualisation of this project and for valuable discussions throughout its development. We are also grateful to Joshua Robinson for insightful early conversations and for suggesting experimental setups. We thank Sanket Kumar Tripathy for engaging discussions on the conceptual connections between our work and ideas from statistical physics. SNR, MKL, and SS acknowledge generous support from the Alexander von Humboldt Foundation.

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Contents
Introduction
2
Background
2.1
Diffusion models in continuous and discrete time
2.2
Measuring statistical distinctiveness
2.2.1
Basics of fluctuation theory
3
A practical toolkit for diffusion models
4
Demonstrations of our methodology
4.1 Warm-up: Predicting convergence of the data distribution
4.2 Class-conditional generation
4.3 Rare-class generation
8
4.4
Zero-shot classification
8
4.5
Zero-shot style transfer
5
Conclusion
10
Notation used throughout the paper
26
B Theoretical contributions
27
B.1 Theoretical foundations of the cross fluctuation framework
27
B.1.1
A concise primer on fluctuation theory with physical intuition
27
B.1.2 On the validity of fluctuation theory
28
B.1.3
Merger Times an efficient proxy for convergence in total variation
29
B.1.4
Equivalence in continuous and discrete state spaces
32
B.2
Application and analysis in diffusion models
33
B.2.1
From the empirical reverse process the learned sampler
33
B.2.2
Pull-back of data events along the PF-ODE
33
B.2.3
Stochastic-flow formulation for the SDE view
34
B.2.4
Fluctuations offer fine-probes into the geometry of data
35
B.2.5
Unbiased monte-carlo estimators for cross fluctuations
36
B.3
Framework connections, extensions, and interpretation
37
B.3.1
Mixing time of isotropic Gaussians under Brownian diffusion
37
B.3.2
Higher-order fluctuations proof of concept
38
B.3.3
Centred kernel alignment (CKA)
39
B.3.4
Structural regularity bounds fluctuations
40
B.3.5
Extending the framework to certain non Markovian samplers
40
B.3.6
Phases of diffusion model dynamics
C Experimental details and further results
44

---

C.1 Compute and reproducibility
44
C.2 Convergence of data
44
C.3 Class-conditional generation
45
C,4 Rare-class generation
51
C.5 Zero-shot classification
53
C.5.1 Binary classification via linear probes
53
C.6 Zero-shot style transfer
55
Limitations and future work
59
Related work
59
25

---

A Notation used throughout the paper
Table 6: Global symbols
Symbol
Domain / /type
Meaning
Xa po
_
Data sample from initial distribution pr
Piesirrd
density on R R
Desired distribution
X;
Forward-diffused variable time /
B(t), Bc
Rya/ (0, 1)
Continuous discrete noise schedule
(())
(0, 1]
Signal attenuation factor (2.4)
p
density on B
Marginal distribution x;
so (x,1)
R - R
Learned score network State operator (usually p(x) x) (Sec-
p()
map 52 +
tion 3)
(Q)
Rz0
n centred fluctuation moment on event ? (2.5)
(2t), ((le) (St:)
Rz0
2n within-event fluctuation moment on
Q (2.7)
(l (2), G(u)
(2,, 92)
Rz0 [0.1]
Unnormalised cross- cosss-fuucuatioon (2.6)
M (Q1, 02), PL (21, Q2)
Normalised cross-fluctuation (2.8)
Qk.o
event in Ω
Class-k source region (k ,,...,K)
(Section 4.2)
De.t
event S Rga
Image of S& after t forward steps
Ex.t
Covariance of Ωx.
A (t)
Maximum eigenvalue of Σx.
M2(()
[0.1]
Shorthand for M (2). Q2,1)
{O....,
First index where M² (1) = (merger) (Section 3)
wtt)
probability mass
Importance weight over timesteps (Sec- tion 4.4)
SNR(()
R20
a, (11 - az) (signal-to-noise)
Tr(+)
R
Matrix trace operator
Table 7: Time- and index speciffc symbols
Symbol
Type / range
Meaning
R20 Rz0
Continuous diffusion time (PF-ODE or SDE)
t
Generie continuous times used in flow composition s
T')
{O...., n) positive real integer
Discrete forward- forward-process process index (i data; i white noise) Continuous final time (SDE/ODE) or discrete horizon with schedule
{3,)T=
N
Chosen number of forward (or reverse) discrete steps in experi- ment (may be n= (
At
R>D
Integration step size in continuous numerical solvers
3,
sequence on {1, (I.....77 ,T
Discrete noise-schedule value at step / Continuous noise-schedule function Earliest discrete index where two events first merge (convergence index)
((t)
function [0, 7 R
4*
integer
teonv
Rz0
Samc as i" but expressed on the continuous time axis First time class & merges with any other class Lower bound of guidance / weighting window for class k Upper bound of the window for class Thermodynamic phase- transitionn times (unbiased speci- ation ccidde-connnsstionn
Imerge.k
R20
betart,k
Ez0 or int
Istopuk
same type above
bu-ss to
R20
(mix(E) (epi(E)
Exp (Appendix B.3.6)
e-mixing time of the VP- VP-SDE (Appendix B.1.4)
integer
E-coupling time for discrete Markov chains (Appendix B.1.4) First lattice-merger time between classes and with tolerance e (Appendix B.3.6)
dat (E)
Rzp
1]
Interpolation schedule value used in Algorithm 3 (rare-class genera- tion)
26

---

B Theoretical contributions
This appendix provides the theoretical support for the results presented in the main paper. The content structured in three parts: we first introduce the general framework, then apply it to diffusion models, and finally, we connect our work existing methods while also presenting new theoretical results and interpretations.
Appendix B. introduces the cross-fluctuation framework. It covers the necessary definitions, proves that merger times can serve as an efficient proxy for convergence in total variation, and shows that the method is applicable to both discrete and continuous state spaces.
Appendix B.2 applies this framework specifically diffusion models. This section justifies the use of the forward process for our analysis, details the dynamics of event structures under the model's SDE/ODE, confirms the consistency of the resulting merger events, and provides an efficient method for their estimation.
Appendix B.3 discusses connections to other methods and provides additional context. We show the relationship between our framework and Centered Kernel Alignment (CKA), prove the fluctuation adaptation lemma for style transfer, and conclude by framing our analysis of discrete phase transitions (what we term lattice transitions) within the context of statistical physics.
B.1 Theoretical foundations of the cross fluctuation framework
B.1.1 A concise primer fluctuation theory with physical intuition
Here provide concise overview of the nuccuattiontheorrtii framework that underpins the notation introduced in Section 2.2.
From observables to fluctuations. In statistical physics, system's properties are understood by measuring observables (e-g.. energy, momentum). For a distribution p, the expectation of an observable A is [A]. Deviations from this mean, or fluctuations, reveal the system's internal structure and correlations. Our framework generalises this by treating the state vector p(w) itself as the core observable.
The key object, the -order fluctuation (Eq. (2.5)),
(w) (p(w) - E(pl).
and its conditional expectation E, [F m) over event Ω2 directly correspond to the centered moments used to characterise complex distributions. For example, )= is the conditional covariance matrix, fundamental second-orde statistic.
Diffusion models as dynamical systems. The forward diffusion process, whether described by the SDE (Eq. (2) or the PF-ODE (Eq. (2.2)), defines dynamical system that evolves an initial data distribution Po into sequence of marginals {pr}. The PF-ODE,
dx;
di 3() -()) V log px.))(
describes a deterministic flow governed by vector field. Its associated continuity equation, 0,p. Vx (p&v = O, confirms that evolves via deterministic transport of probability mass. This deterministic evolution makes it perfect setting for applying our fluctuation analysis track how the moment structures of different subpopulations (events) evolve and merge over time [Biroli and Mézard, 2023, Raya and Ambrogioni, 2024].
Fluctuations and the significance of mergers. A cornerstone of statistical physics is the deep connection between a system's internal fluctuations and its large-scale correlational structure [Chaikin et al., 1995, Kivelson et al., 2024]. This principle, often formalised in Fluctuation iiituaation-issipationn Theorems, provides the physical intuition for our approach.
For our purposes, this connection justifies why monitoring cross-fluctuttions is meaningful. The normalised cross coss-fluctuation ( acts generalised correlation function between the moment
27

---

structures of two events. A sharp change in this correlation (i.e., merger where [M, + 1) signals that the two events have become statistically indistinguishable with respect to their n-th order structure. In physical systems, such loss of distinguishability is the hallmark of a phase transition, where the system undergoes a qualitative change. While we do not compute system-wide
thermodynamic quantities directly, detecting the merger via M serves as direct, practical probe for these critical points in the diffusion trajectory.
Working with multiple states.
In Eqs. (2.6) and (2.8) we defined
G(r) and Mp
allowing # and pa to be heterogeneous (i.e., P + p2). As the following example illustrates, different state-operator choices can yield qualitatively different cross- pross-fluctuation behaviour.
Example Let Ω the unit circle IR,, Consider two events: Qn the are the first quadrant (a: > 0,y > 0), and l2, the are in the second quadrant (x < 0, 0). Assume a uniform distribution on the circle. Let the state operator be the identity, p(a, y) = [z, yl T. The global mean
p) [0, 0]
For n = 2, we compute the conditional covariance matrices ΣΙ = = [F) and Σ₂ = 12 by evaluating standard integrals. This leads to the following conclusions,
The mean of S2 is [p]p]] 12/ 2/ T Its covariance matrix Mn will have a dominant eigenvector along the direction [1, -]],, indicating negative correlation (as y decreases when increases along the arc).
The mean of 922 is [p]p]] = 1-2 2/2 m/m]T. Its covariance matrix 2g will have dominant eigenvector along [1, 1] indicating positive correlation.
Since the dominant structures (covariance matrices) Σn and Eg are nearly orthogonal, their Frobenius inner product Tr(E; Σ2) will be close to zero. Consequently; their normalized cross-fluctuation M (91,, 122) will be close to 0, correctly identifying the events as structurally distinct.
This flexibility crucial in quantum mechanics, where one often considers distinct pure-state projectors
Pr (w)s) (vta)
on Hilbert space H. 8 Their flostflluttatioons underpin tasks such as quantum state tomography and discrimination of non-orthogonal states Nielsen and Chuang [2010], Peres [1995].
More generally, heterogeneous Pl. P2 detect alignment between two evolving state spaces. For example, if text and image embeddings are both driven by the same diffusion dynamics, choosing P1 and P2 to sample each modality reveals how given concept manifests across them. We leave such multimodal extensions for future work.
B.1.2 On the validity of fluctuation theory
The entire fluctuation framework rests the existence and properties of moments, which fundamentally linked to the derivatives of the characteristic function (CF). For random vector Rdd its characteristic function is the Fourier transform of its probability law:
(t) = E
it p]. E
The CF always exists and satisfies (t) < 1. If moments up to order exist, the CF is n-times differentiable at the origin. The derivatives of the CF generate the moment tensors. For example, the second moment tensor (a matrix) is given by the Hessian of the CF at the origin:
8pp
E Eppypa] 72 Ōt t=
'Mathematically, measurements project the density operator onto the eigenspace associated with the outcome; Nielsen and Chuang [2010] for details.
28

---

To obtain the centered moment tensors used in our framework, one differentiates the characteristic function the centered variable p E(p]. The conditional expectation of the nt noddorder fluctuation tensor, E, [F,)), is thus completely determined by the derivatives of the conditional characteristic function (t((2t))) at the origin. This connection is crucial for the moment- inequality used in our theoretical results (Theorem 5).
Foundational facts. The theory of characteristic functions is well-established [Lukacs, 1970]:
1. Uniqueness. If (x( -y (t) for all t, then X and Y have the same distribution.
2. Inversion. The distribution can be recovered from the CF. For instance, in one dimension:
1 π /- (e (xx(t) dt [Dudley, 2018].
CDF =
3. Convolution. For independent X, Y, the CF of their sum is the product of their CFs: (t)+y(t) (x(())
Theorem (Bochner's theorem for R4) A function ip: Rd + C is characteristic function of some random vector if and only if is positive-defnite continuous at the origin, and (p(0) = 1.
Proof. See Rudin [1962, Thm. 15.2] for the ID case, which generalises to TR.
Remark (Existence of the characteristic function) We assume the state operator p has well- defined characteristic function, which is very mild condition. It is automatically satisfied by most data distributions in machine learning and physics [Blanchard and Brüning, 2015, Bertini et al., 2002]. This assumption is more fundamental than the existence of a density, characteristic functions also exist for purely discrete or singular measures [Dudley, 2018, Rudin, 1962], giving the approach greater utility [Ansari et al., 2020, Sriperumbudur et al., 2010].
B.1.3 Merger Times as an efficient proxy for convergence in total variation
The justification using merger times as a proxy for convergence a two-step argument connecting our practical measurement to fundamental theory.
First, we validate our specific measurement technique, proving that the computationally efficient method of directly thresholding the distance between moment tensors is topologically equivalent to the intuitive alternative of monitoring the cross-fluctuation similarity (M
Second, we establish the core theoretical link. moment-TV inequality proves that this proximity between moments rigorously guarantees that the total variation distance between the underlying distributions also vanishes.
This validates our method: measuring tensor distance is a sound and efficient proxy for observing true distributional convergence.
B.1.3.1 Equivalence of distance and similarity based merger thresholds
In main methodology (Section 3), we define the discrete transition detector M (t) using threshold on the distance between conditional expected fluctuation tensors:
() LM (a R22)
|| (7) (Qi,)) E2( (A2,))ll e,
(B.1)
otherwise.
An alternative, and perhaps more direct, formulation would be to threshold the value of [M (r) itself:
IM (2,)
1- 1-MM (2,,))22)11 > 0,
M(n) P.alt
(B.2)
otherwise.
Here, we show that these two thresholding criteria are topologically equivalent, meaning they detect the same notion of convergence. The first formulation is often more computationally stable and efficient and hence is the prefered choice in our work.
29

---

Theorem 4 Let the conditional expected fluciuation tensors Bx(FF (9t) for events Q2 reside in Hilbert space Hn with norm || -lR... Define two metrics on the space of pairs of events (91,, S22):
1. The direct distance between their moment tensors: dx = P ()) (22)) 12 (F)) (S2)]l
2. The similarity-asse distance: dM((I), 02) = MM (12, 92))
If the mapping Ω + [ (()) is continuous with respect to suitable topology on the space events, then the metrics dy and dm are topologically equivalent in any region where [[ (l,,,,, is bounded away from zero.
Proof. To establish topological equivalence, we show that for any sequence of event pairs ) {( 222 k== convergence under metric dy is equivalent to convergence under metric dm- Let
AІ ( and Bg (2(FF() (S22,)) be the corresponding sequence of tensors in the
Hilbert space Hn.
Part 1: dy + 0 = dm + O. Assume that d1 (2, as - oc. This means
|| B& BR.
+ 0.
First, by the reverse triangle inequality, have:
|el - [ S || BAIl..-
Since the right-hand side goes zero, the norms converge to each other. As we are in region where the norms are bounded away from zero, if ||Ab|| converges to a limit > 0, then |Bx] must also converge to L.
Second, the inner product continuous function on Hr x Hn. This follows from the Cauchy- Schwarz inequality:
KAE, BB) (Bk. B&) ((Ax Bk. B&Y
< A| BRll 17..
As ||Ax B& 11 and || |l,, bounded, the right-hand side goes zero This shows
(Ax, Bx) B + O. Since ||As| + BBll, we have (Ak, Bx) + L
Now consider the limit of [
KAR, BBY
1
lim (M(1) (S2,,6, = lim R-000
1.
k-00
|| Nt.
BB Ptn
L.
Therefore, (m 222,k) (2=x)=1 =1 1 (Rr,k: 22,x)l -
Part 2: dm + 0
- O as ht + xc. This means [M (SI, 22,x)1 - Let Og be the angle between A₁ and Bg- Then cos Oyl (
df + O. Assume that
[M((") (01.k,522,11 which implies cos 0x + 1.
Consider the squared distance dz = || 1 Using the law of cosines in a Hilbert space:
||A - 2 = || + | (AA, Bx) = || + ||
2|| Ax B& cos 0k-
For the distance to converge zero, require not only that the angle between the vectors vanishes (i.c.. cos 0g + 1). but also that their norms converge to the same value. The physical process of distributional merging implies that not only the orientation of the moment structures but also their magnitudes become identical. We therefore assume that as events merge, if ||Ax|] converges, then | converges to the same limit L O. Under this assumption:
lim A k-000
B& l|
lim AI |& - 2|1 Ax B&| cos 0k)
k0
1,2 122 =1²)2=-2000000
= 0.
30

---

Thus, (101,,,22xx) = ||A, - B&ll + 0.
Since convergence under dy is equivalent convergence under dm, the two metrics induce the same topology.
B.1.3.2 Fluctuation moments bound total variation distance
now show that asymptotically utilizing fluctuation theory to understand mergers is equivalent to probing the similarity of probability distributions using the Total Variation Distance.
Theorem (Moment-TV inequality) Fix an integer n 2 2. Let P.g be probability densities on IR: that
(i) are of bounded variation;
(ii) admit centred moments (R) (K) ,ph for k 1....,
(iii) obey the moment proximity bound (2(k) (kk S M for k I,...,
(iv) satisfy the uniform first- and second-moment bound (7621), 1.p. S B;
(v) have matching tails:
lim R-400
(p 9) = O. |e|>R
Then
drv(p. 3 CM + B).
where may take C₂ = (1 + n!) (2 48) and co 0 is absolute constant.
bound. To connect the proposition's conditions to the proof, work with characteristic functions (CFs) of the centered random variables. Let [p(() and (t) denote these adjusted CFs. Since p. are absolutely continuous and of bounded variation, fp. fa are bounded by 1 and possess derivatives up to order n + at the origin. Write the order-n Taylor expansions with integral remainder
Proof. 1. Characteristic-ffnntiii
(it)* b!
(it) a+l
S (1- 8)" s) (1+1) jistX ds.
(t)
Pp
(K)
n!
Subtract the analogous expression for Sq. use hypothesis (iii), and take absolute values and the standard property of integrals:
1
14 n1+
(( (11-s))
(0x()) f((1)
M
s)" 1 ((ntt) Dg Dytt ds.
M
(n)!
1=1
Now, use the triangle inequality and note that the integral is simple weighting integral this gives, n-11 (n + 1)! (f(n+1) 1) + 1 1(x+1)).
(fa(t) - fo(1) <
M
(B.3)
k!
Bounding the (n+1)ss moments. By Jensen's inequality, m((+1) < (m!22) (n+1)/2 Insert this in (B.3) to get
< B(n+1)/2
(fo(t) - fa(t)) S a 1| + bn B (1 120+1)
(An)
where : R bn = (a+1y!
2
3. Esseen's 's smoothing inequality. For any T 0 (Ibragimov,, 1975, Thm. 1.5.4),
1
[p(1) f((1)
24 #f (Var(p) Var(g).
drv g) S
dt +
(B.4)
2T
31

---

Integral term: divide (A,) by [t] and integrate,
1
T fp fq MT n n+1
bn g(n+1)/2Tm++1
2
Variance term: hypothesis (iv) yields Var(p), Var(1) S B + M2, the second term in (B.4) is bounded by 48 (B + (²T)(
4. Choice T. Set T = 1. (A different 7 only rescales the constant.) The bounds become
48
drv(p, g) S [an + Ma]M + [an + bn) B(++1/22
(B + M²)
C, (M² 2 + B).
where the last line uses M < M and B(++1//2 < B for 2 1, and absorbs all numeric factors into C₁ (1 + n!) (2n 48) with a universal co-
Remark 6 Pf are sub-Gaussian (or sub-exponential) [Vershynin, 2018], all moments exist and (k) ((VB)((())))))) *): the conditions of Proposition are then automatically satisfied Rdd
satisfy Up
Generalisation multivariate distributions and moment tensors. The logic Theorem extends to multivariate distributions R As established in Appendix 1.1.2, the derivatives of the multivariate characteristic function (t) at the origin generate the moment tensors. A
multivariate version of Esseen's 's inequality bounds the TV distance by integral over the difference of characteristic functions, which is in turn bounded by the norm of the difference between moment tensors, [F (FP ]]|
B.1.4 Equivalence continuous and discrete state spaces
We recall (and slightly adapt) the terminology Aldous and Fill [2002], Levin and Peres [2017] so that it aligns with the notation used in the main text and provides a foundation for generalizing concepts to continuous processes.
Single chain and mixing time. Let S S1., be the marginal sequence of an ergodic Markov chain finite state space X. Its transition matrix is II [0, x|xxx and the (unique) stationary distribution satisfies IIS = S For two probability vectors p, q on X, the total-variation (TV) distance is
dTv(p.g)
-l((z)-o - d(>)) sup P(A) -(A))-
ACX
The -miiing time of the chain is the first time the distribution is E-close to stationarity:
(E) = min 2 0 drv(S,, Sax) {
(B.5)
Multiple chains and coupling time. Let A be finite index set. For every À E A, fix an initial event f2x,0 S X partitioning the state space. Run the same transition matrix on each event to obtain collection of conditional chains S {Sx. ) where Sx, is the law of the chain at time conditioned on starting in f2x,0- Because the dynamics are identical, all chains converge to the same stationary distribution S but their finite-time marginals differ. The coupling time is the first time all conditional chains become E-close to each other:
(e))(E) = min 2 0: max drv(SS,t, S {e)
(B.6)
a.BEA
Hence, tepl measures when all initial subpopulations have effectively merged in distribution.
Generalisation continuous state spaces. The total-variation distance ill-suited for comparing distributions on continuous state spaces like R&d, where it is often trivially I unless the distributions have overlapping singular parts (Bhattacharyya et al., 2024, Tao al., 2024]. Our fluctuation metric provides a natural generalisation.
32

---

As established in Section B.1.3.2, our cross-fluctuation statistic M is intimately linked to the similarity distributions via their moment structures. We can therefore define generalised coupling time using this measure of structural similarity:
(m) gen (=) = min 2 0 min (Da, lz,1)l 2 2 1
This definition measures the first time at which the least-related pair of events (a,, (5) has achieved structural similarity of at least 1-& E. directly generalises the discrete coupling time to continuous diffusion processes. For main application with 2, this provides a practical tool for tracking when the covariance structures of different event distributions have merged.
B.2 Application and analysis in diffusion models
B.2.1 From the empirical reverse process to the learned sampler
Our justification for analyzing the forward process hinges on its tight correspondence with the learned reverse sampler. This connection forged during training, where the diffusion model minimises the simplified ELBO, an objective that intuitively trains the model fe to predict and reverse the noise added during the forward process.
simplified ELBO:
Caimple Unif (o....T- (x(. - =112 No-Po: (0.1)
(B.7)
where xx(o 8) = ar Xo + v1 0,e pı (cf. Eq. (2.3)).
While practical training results in non-zero error Lsimple O, key theoretical result guarantees that the learned sampler's trajectory remains faithful to the true process.
Theorem (Chen al., 2022a, Thm. 1) Under standard regularity conditions, if the score error e, is bounded and sufficient number of reverse steps are taken, then for all steps i:
lppt"
IIlTV S E.
where ph the marginal of the true reverse process and Ps the marginal produced by the learned sampler fo-
The power of this result its implication: closeness in total variation guarantees that all statistics, including the cross-fluctuation metrics we use, also close. This validation allows to use the computationally efficient forward process to identify merger times (i"), confident that these critical points directly correspond to observable behaviors in well-trained reverse sampler.
B.2.2 Pull-back of data events along the PF-ODE
Our method tracks the evolution of distinct subpopulations, or events, through the forward diffusion process. For the deterministic probability-fior ODE (Eq. (2.2)), this evolution is governed by well-behaved flow map.
Let -.-- Rd + Rdd be the flow map of the PF-ODE. Because the drift field is globally Lipschitz under standard assumptions, T x is bijection for every S s <ts S T. The continuity equation,
O.p. + Vx (pe vz) 0,
implies that the time-t marginal p. is the push-forward of the time-s marginal p. under the flow map:
Pt = (#s- which means P2( = P. (B) for any measurable set B C Rdd
where P; is the probability measure associated with the density pt-
From source events time-t marginals. Fix two disjoint source events 91 9220 C supp(po). We define their images at later time by pulling them forward along the flow:
Pg (0-=x(94.0)
k E (1,2), I [0,
Since Φ a diffeomorphism, each Qx a well-defined measurable set. While the source events Dg.0 are disjoint, their images 2g.e may overlap the diffusion progresses.
33

---

Event probabilities along the flow. The mass of each event preserved over time. For any lo E {1.2}:
nature of the flow ensures that the probability
Ps (9 ( Po (S2.,0)
This confirms that (Ra are valid events with constant probability under their respective marginals Pt at every time t.
Monitoring mixing via cross- flassffucttations. Even when the supports of the events, Ru, and a2 begin to overlap, our cross- foass-fluctuttion statistic (22,, remains well-defined as compares the internal moment structures of the distributions conditioned on these events. value of | approaching one marks the moment the two event distributions become structurally indistinguishable i.e. the merger time merge.
B.2.3 Stochastic-flow formulation for the SDE view
Appendix B.2.2 defined 94 T (94.0) via the deterministic flow 4 of the PF-ODE (2.2). We now show that the same construction works pathwise for the stochastic forward SDD
dx.
1/2 (t) dt + v B(t) dw
whose solution map zg 2e depends on the Wiener path Sprobb
Kunita's 's stochastic flow of diffcomorphisms. Let (p :): IRd IRd, 0 T, denote the Kunita flow generated by (2.1) [Kunita, 1990, Ch. 4J. For every fixed the map + (w,) a C diffeomorphism and
Pe, (w,-) -) = (e,,,w,, pale(w,)))
0 OSs<truSR
Hence the pathwise inverse Po. (w, :) exists almost surely. Given disjoint data events 21,00 2220 C I2 set
(w)) (w 24.0),
A E (1,2).
The map - (()) closed set. law equals the push-forward po by the SDE:
(x) ) is Fe-measurable (Kunita's measurability theorem), so Ui.t is random
Pfx E A} = Ew P(Po, xo) E A}. To ~ po.
and we again write pe L(xx).
Annealed cross-fuuctuations. Fix n. Because Po (w, is C'l and Po has finite n-thh moments, the pathwise conditional expected fluctuation tensor E[F(") (Q,((w)) exists. To obtain deterministic statistics, we define the annealed conditional expected fluctuation tensor, which denote E by averaging its pathwise counterpart over all Wiener paths:
F., [E[F(") [ [w)]]
From this, the annealed normalised cross-fluctuation (n) is the cosine similarity between these deterministic annealed tensors:
E 1.1
(m)) (Sr,e, P2,e))
2, Ma.
Since expectation commutes with the inner products and norms, all fluctuation identities remain valid for these annealed quantities. Thus, all merger-time results proved for the deterministic PF-ODE hold for the SDE in expected sense, justifying the use of the same algorithms.
34

---

B.2.4 Fluctuations offer fine-probes into the geometry of data
In this section, we illustrate that the analysis of fluctuations on a diffusion process provides powerful probe into the intrinsic geometry of the data itself. Our main tool for this purpose is the following theorem,
Theorem (Exponential contraction of fluctuations and MST construction) Let p be a Lips- chitz state operator, such that the components of the fluctuation tensor function Fp are square integrable with respect to the invariant measure p. of the diffusion (i.e, F(e) LA(u, Ha)). Let the evolution be governed by a diffusion semigroup Pt. that is self adjoint on (2( and possesses spectral gap X>
Then, for any two initial event distributions Ps H4 with densities he, h2 with respect to μu, the distance between their conditional expected fiuctuation tensors decays exponentially:
[[E,[F(")()) (6) (t)ll,, S (1F())) || (L,.)H ||A, h2 hy]lep(a)-
Consequently, since the right-hand side converges to zero t 0c, for any fixed e > the graph on
events with edges between pairs (S2,, Q2)) satisfying [|E, [B,, (1)) - ( S becomes
(1))]l
complete graph for sufficiently large L. A complete graph on finte number of vertices always
admits a well- well-defined minimal spanning tree (MST).
P Proof. Before going into the details of the proof, we justify the reasonableness of the theorem's core assumptions in the context of a d dimensional Sariance-Preservinn (VP) SDE used in diffusion models.
Existence of an invariant measure Jr: The theorem assumes a stationary distribution μr, which is clearly satisfied. This distribution for our case is N(2,, Taxa)
2. Self adjointness of the semigroup P:: The theorem critically relies on the self adjointness of the semigroup operator P. on the Hilbert space L2 The VP-SDE process is reversible process with respect to its Gaussian invariant measure μ. A fundamental result in the theory of Markov processes is that reversibility of process with respect to a measure f is equivalent to its generator being self-adjoint adjointness of the associated semigroup [Bakry et al., 2014].
I. (1)), which in turn implies the self
3. Existence of a spectral gap À > O: This is the crucial assumption providing the exponential decay rate. The VP-SDE process a textbook example of system with a spectral gap.
4. Square integrability of the fluctuation tensor F: The proof requires the norm (F (H. be finite. The invariant measure μ is Gaussian, meaning its density decays extremely rapidly (exponentially in ||x]|3). If the state operator p is Lipschitz (a mild regularity condition), the components of the fluctuation tensor F(1)) will be polynomials in the state variables. Any polynomial function square integrable with respect a Gaussian measure, making this assumption easily satisfied.
In summary, the VP-SDE process is a prime example for which all assumptions of the theorem hold, making the conclusion of exponential convergence robust and directly applicable. We now head towards the main proof.
The proof relies on the consequences of the spectral gap property of the diffusion semigroup P.. Let = = denote the tensor-valued function the state space. The difference in conditional expectations at time t, for initial distributions μ4, My with densities h, hy with respect to the invariant measure μ. is vector in the Hilbert space H.,
:= = E; - - [t)) ( - -(
hx)=))
This is a Bochner integral of an H., -valued function against scalar function. The semigroup P. is assumed to be self adjoint on the Hilbert space (n)) of scalar functions. This property extends to the space of tensor-valued functions I.2 (1, H.)) We can thus move the action of the semigroup from the function F to the density difference (h; hz):
Di,(t) = / F(x) (Pi(h, - h₂)) du(r).
35

---

We bound the norm of this integral using the Cauchy-Schward inequality for Bochner integrals, followed by the standard Cauchy-Schwer Schwarz inquaaitt:
|| [ = [1F())]m, (Pi(h; -h))(a))
- (1 ( du(x))
1/2
(fICRiCh (1IPP(hh--)) (h)))))
1/2
- -1 -IP((h::-h))llaa -1)))
The existence of a spectral gap A > O is equivalent to the following contraction property on the space of mean-zero functions in (x)): for any g (m)) with J g du O, we have || <a(a) S e e-> e-Millgl Since h, and h, are probability densities, their difference g = h4 - hy has zero mean. Applying the contraction property yields:
[[)))(()] - -,,(F()],
[FFI 144 hy]lep(w)-
The term ((,, [l,, (111-h,) -1yl (P()) is constant for any fixed pair of events. The exponential term oc. Therefore, for any threshold > 0,
e-xr guarantees that the distance converges there exists a time such that for all f T, the distance less than e for all pairs (i.j), making the corresponding graph complete. An MST can always be constructed using standard algorithms such as Kruskal's 's [Kruskal, 1956].
weighted complete graph
This exponential convergence justifies why we expect mergers to occur and provides geometric interpretation of the process: initially distant event structures are pulled together until they collapse into single point in the space of moment tensors. The above geometric perspective the evolution of event structures has two key implications:
Hierarchical refinement. We can progressively refine our analysis by choosing finer partitions of the initial state space. Starting with broad categories (e-g., animals vs. vehicles), we can track their mergers, then move to finer sub-partitions (e.g.. cats vs. dogs) and track their subsequent mergers. This allows for probing the system's dynamics increasingly fine resolutions, all on the same underlying data distribution.
2. Connection manifold learning. Tracking the evolution of the graph Ge on events (where is conceptually similar to
edge weights are given by the distance || [|E,]]]]] E, algorithms that build neighborhood graphs learn low-dimensional embeddings. Methods like t-SNE [van der Maaten and Hinton, 2008] and UMAP [McInnes et al., 2018] also rely connecting nearby points (or neighborhoods) to reveal underlying manifold structure. Our framework can be seen as applying similar principle in the time domain, tracking how neighborhoods of events connect and merge as the diffusion process evolves. Exploring this connection in more detail is promising direction for future work.
B.2.5 Unbiased monte-carlo estimators for cross fluctuations
Theorem 9 (One-sweep unbiasedness) Let 2 2220 S S be disjoint events with probabilities P1,P2 > 0. Simulate once N ii.d. forward trajectories {x o from the VP process. Let z
1 (x)))
First, we form an unbiased estimator for the conditional expected fluctuation tensor itself:
N
B
Σ
k E (1,),
(B.8)
NPk
where Fx()(xx()) the random fluctuation tensor for sample time This estimator an average of tensors and is itself a tensor in Hn.
Nexi, we define plug-in in estimators for the unnormalised cross-fluctuation Glo) and the within-event fluctuation magnitude F p(22n) based on these tensor estimators:
G( (S 1221) ((E(())
/ B(T) H..
(B.9)
(2m)
(Ok,e) == [16()² Ii....
(B.10)
36

---

(inner product and squared norm) of the tensor estimator (n) Since É(m) is an average of i.i.d. random tensors, is consistent estimator for its mean by the Law of Large Numbers. By the The tensor estimator E(()) is unbiased for the true conditional expected fluctuation lensor (Ax,)))- (9x,0] Consequently, the plug-in ratio estimator for the normalised cross- prossffuctuation,
T (S1,1,
(E
E(m) Hr.
(1)
Ha
(2m) ((2n) (21,1) F, (D2,,)
E
1.
2,
consistent and asymptotically unbiased for the true value M (") (St,,t S22,1) E M (t)] - (Si,,
(0,,,,922)) P22,) + O(N-)).
Proof. The proof proceeds in three steps, starting with the unbiasedness of the core tensor estimator.
Step I:
Unbiasedness of the tensor estimator ((n) Ret
Fix l E [1,2]. By the linearity of expectation,
we can move the expectation inside the sum:
Ep
Bp
(x
k.C
NpR
For each term in the sum, we condition on the starting point x (0)
Ex 2 F
[x{"))
/
l (2o)Po (Eo) ( (xx) dro
po(zo) ( dar
Pk Suyo
Pk Ex (F(" ((k,))
Summing N identical terms gives Np, (F, (S4,e))- Ep
Dividing by the Npx prefactor proves that
(()) k.t = E&(F% (S4,,)]-
Step 2: Consistency of plug-in estimators. The estimators (6a) and (2n)
are continuous functions
continuous mapping theorem [Billingsley, 2012], G(n) and (2n)
therefore consistent estimators
for G( and Ta(2"), respectively.
Step 3: Bias of the ratio estimator: The estimator M (t) is smooth function of the components of the estimators D(n) and (((m)) Fach of these components an average of N i.i.d. random variables. By the multivariate Central Limit Theorem, the joint distribution of these averages converges to normal distribution. The delta method for ratios of random variables then applies directly. multivariate second-order Taylor expansion of the cosine similarity function around the true expected tensor values shows that the linear terms in the bias expansion vanish, and the bias of order O(N-1).
Remark 10 All computations rely on the same set of forward trajectories. The primary computa-
tional step is to compute and store the random fluctuation tensors {F(() (x] ())) for each sample and time step. from which all other quantities be derived.
B.3
Framework connections, extensions, and interpretation
B.3.1 Mixing time of isotropic Gaussians under Brownian diffusion
We quantify how long the forward VP-SDE (2.1) needs to forget an isotropic sub-Gaussian input and become e-close (in total variation) to its Gaussian limit. Write
J(t) = exp - 2 ((s)
37

---

the deterministic attenuation factor from (2.4).
Proposition 11 (Mixing time for sub-Gaussian data) Let E RR have ii.d. mean-zero, unit- variance components that are -²-sub- oe-sub-Gaussian: Pr([y:] > t) S 2e-2120" Evolve with the VP-SDE (2.1) and denote ((x,) Pt- For every EE (0, 1) the e-mixing time
(mix(E) == inf 0: drv (PO, =²}
satisfies
(+) 2
log {) and [(mii( = =(
Proof. Step 1: tails of the input. Sub-Gaussianity yields E| E|y = d and Var llyl S Cd (for C' = C(a)). Bernstein's 's inequality Vershynin [2018] gives lyll = d + O(vd) w.h.p.
Step second moment under the SDE. Conditioned on y. E X 1 (1y)l d(11 J(t))). so averaging produces B||x, 1| d + J())OO(vd)
3: bounding x² Pinsker's inequality Cover and Thomas [2006] gives dly S 2x² 2 For Gaussians
with equal means, x² (det Σ)- exp(1 ( ir(1 - -)-1) - 1. With = [() + J(I)
J(t)*
N) s d 1 J()) (1 (1+0
Step4: 4: solve for (t).. Setting the rhs to 2² and solving yields the claimed bound.
Closed form for linear schedule.
With 3(t) 30 + (3r Bo)ttTT,
- But ( -Bo) 12
J(t) exp
T
Taking E = e- in Theorem 11, log J (fmuxx) ~ log d + loo 2, so tmix solves quadratic. For the DDPM defaults (So, Br, T) (10 0.02, 1000):
mmix + 0.0995 1 = 5000 log( d/2)..
Data dimension
Pred. fmix
Obs. i"/T
3 32 x 32 (CIFAR-ID 10) 1 28 x 28 (MNIST)
0.602
0.60
0.543
0.60
4 32 x 32 (emageNet latents)
0.614
0.70
Table 8: Theoretical mixing index vs. empirical convergence index.
Table 8 shows theory versus measured convergence indices; the ((d-1) scaling persists real data.
Thus, it is possible to estimate the mixing time for sub-gaussian data analytically, in fact due to concentration bounds on high-dimensional sub-gaussians Vershynin [2018] it could be shown that the above time manifests physically a symmetry breaking transition, leading an estimate consistent with Raya and Ambrogioni [2024] for spherically symmetric distributions and in Biroli et al. [2024] for gaussian mixtures. Interestingly, ImageNet latent representations align closely with these theoretical estimates. We hypothesise that this occurs because the compressed space corresponds to the latent space of a Variational Autoencoder (VAE) trained to approximate Gaussian distribution, potentially making these latents effectively nau-Gausinnn. Verification of this hypothesis and related questions remains future work.
B.3.2 Higher-order fluctuations as a proof of concept
To make higher-order analysis computationally tractable, we transition from analyzing the full rank-n fluctuation tensor (m)) to analyzing the moments of a corresponding scalar-valued random variable. This simplification is equivalent to assuming the components of the state vector are i.i.d. and studying the dynamics of single component. This bypasses tensor computations and allows us to work with scalar moments.
38

---

Under this assumption, we can track the scalar moment identity:
p. (1)= (m) (0) + (1-
where n. (1) is the n-th centered moment for class time For the Gaussian limit, Isserlis' /Wick's theorem[Issen 1918, Wick, 1950] can be applied because the components are treated as jointly Gaussian. The theorem states that higher-order moments of a Gaussian are polynomials in the variance, and odd moments are zero. This rule lets us draw the higher-order generative diagrams in Figures 3-4. Compared with the second-order diagram, high-order curves fan out more widely at early times- evidence that non- non-linear features dominate, but they collapse sooner, consistent with the rapid J(t)" decay.
Isserlis/Wick's
forms oaa Canaeii Laggnn 304
Poitth Ooder Gereratio Dlaynon for WNIET
Figure 3: Fourth order generative diagram for CI- Figure 4: Fourth order generative diagram for FAR10. We show the emergence of classes using fourth-order correlations.
MNIST.We show the emergence of classes using fourth-order correlations.
B.3.3 Centred kernel alignment (CKA)
Kernel alignment measures how similarly two Gram matrices embed the same data. For K, L Raxx the uncentred score the cosine in "R [Cristianini et al., 2001]:
(K, L)F
A)K,
(), : Σ Kig
|K]| ||
Because A reacts to mean shifts, Cortes et al. [2012] introduced centred kernel alignment
Ac(K, L)) = AHKK,, HLH), H =- - 111
A, is invariant to any feature-space translation and, for linear kernels, to orthogonal mixing and isotropic scaling.
CKA for covariance kernels.
Section 4.2 sets K = Ei,t, L. = Ej.. For symmetric matrices
(K, 1/F
Σ
Nitt djst.my
(B.11)
|
Σ
X2 aa.
(B.12)
where } are the eigenvalues Σx, and mutatis mutandis for (. Under the VP flow they contract as Ai,t, (1)) + (1- (()?) [Biroli et al., 2024]. The largest eigenvalue dominates once J(t)² & (,0.2//5,00111 so
Ene) 1 < A ~ Aja.1-
Therefore tracking the top eigenvalues A A3,,1 gives an efficient proxy for merger of Qi and Ly.t in Section 4.2.
39

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B.3.4 Structural regularity bounds fluctuations
This section provides more rigorous justification for the Fluctuation adaptation lemma from Section 4.5, which posits that structurally similar distributions exhibit similar fluctuation dynamics. The premise is the Fourier regularity condition from Eq. (4.7), stating that the L distance between the characteristic functions (CFs) of the source and target-style distributions is small: ll -0 S
The challenge is to show that this integral bound the CFs implies a bound the fluctuation tensors, which are determined by pointwise derivatives of the CFs at the origin.
The bridge between these two concepts is provided by the Sobolev Embedding Theorem [Adams and Fournier, 2003]. This theorem connects function's average smoothness, measured by Sobolev norm H to its poinhwise smoothness, such as the boundedness of its derivatives. However, the Sobolev norm, | = [1 + (( dg, is stronger than the standard L norm due to the Sequeeccy-eeihhing factor (1 + |E| ) Therefore, a small L norm does not automatically guarantee small Sobolev norm. To bridge this gap, we must introduce a more explicit regularity assumption on the style transfer process itself:
The style transfer transformation Tayle is assumed to be sufficiently regular such that the difference characteristic functions, g = (E) (E), not only has a small L norm but also has rapidly decaying high-frequency content. Formally, this means there exists constant K such that |s и- < K|lø7I a for the required Sobolev order s. This assumption holds if, for instance, the style transfer primarily modifies low-to-mid frequency components of the distribution, common case for artistic stylisation that preserves core structures.
Assumption:
With this assumption, the argument is complete. The Sobolev Embedding Theorem now be directly applied:
(0)) S Cllglla S CK S CK8.
This step guarantees that the difference between each individual scalar component of the correspond- ing moment tensors is bounded by (68). This component-wiss bound is then extended to norm on J. is an element of (F(")] be the difference tensor.
the entire fluctuation tensor. The conditional expected fluctuation tensor, Eo,
the Hilbert space H. of rank-n tensors. Let A, [a,a(F ( --E The squared Frobenius norm of this tensor is the sum of the squared magnitudes of its components: | Σj (A)) Since we established that each component is bounded, [(Ar)j] I C3ő,
the norm of the full tensor is also bounded:
[a E.
[F) H4,
S C.,
Finally, since the normalised foossfluuctttioo M((n) is defined as the cosine of the angle between these tensors in the Hilbert space H.,, it is a continuous function of its tensor arguments [Conway, 1990]. Therefore, small perturbation in the input tensors, bounded by O(6). leads to corre- spondingly small change in the value of ( This ensures that the merger times are stable under the stylistic transformation, rigorously justifying the use of a schedule computed on the source distribution for the zero-shot style transfer task.
B.3.5 Extending the framework to certain non Markovian samplers
For a non Markovian latent chain the marginal ps depends on the entire future tail (pi+1, pi+2..... P.... Hence the pull-back S4,0 + S4gs well defined only if every conditional kernel beyond step is known. This hurdle disappears when the latent family belongs to a natural exponential family (NEF).
Tail statistic Markovisation. An NEF on IRd has densities po(a) (()) exp (00, T(x)) A(0)), with sufficient statistic T and log partition function A. For an independent sequence {x,), drawn from an NEF, define the tail statistic
G =
T(X.),
i 1, 1....,
The Pitman-Koopman Koopman -oormaan-Darmoiss theorem gives
40

---

Theorem 12 (Tail statistic Markov property Pitman, 1936)
(i) If (X:) are Lid. from an NEF, the conditional sequence {C(X; G.) firttorder Markov.
(ū) Conversely, if statistic sequence {Gi) makes {C(X; G.)) Markov for all n, then the marginals must form an NEE.
Thus the random vector G; captures all future information relevant at step
Injecting the tail statistic into fluctuations. Fix disjoint initial events Ω1,0, 92.0. Condition on G = g and apply the deterministic PF-ODE of Appendix B.2.2 inside the fibre {X; G. = g):
(g) := { E Bd (0, E Sk.u. E (1,2).
Because the conditioned kernels are Markov (Theorem 12), this construction mirrors the purely Markovian case. The pathwise cross cross-fluctuation ( (000,,()),,,,00 (22,(9)) is the cosine similarity between the conditional expected fluctuation tensors for a given g. Averaging over the law G yields the annealed cross- flostflucttation:
M (n) (S2,, S22,)) = / MM (11,)) d'' (g).
Every algebraic identity from the main framework now carries over to this tail-averaged counterpart. A merger is detected when [M 1. Star-DDPM Okhotin al. [2023] is recent work that uses a similar formulation to obtain non Markovian diffusion generative models.
Thus, whenever non Markovian diffusion admits mimee-dimensional tail statistic- property guaranteed for exponential-family latents conditinnnn on that statistic restores the Markov property and lets the fluctuation framework operate unchanged. Identifying broader classes of tail markovizable samplers is promising direction for future work.
B.3.6 Phases of diffusion model dynamics
We analyze diffusion model dynamics by distinguishing between two types of phase transitions. Our approach inspired by discretisation based analyses in physics [Kogut, 1983], but we treat the system's 's discrete nature. its time steps and finite precision as a fundamental property, not computational artifact. This leads us differentiate between:
(1) Thermodynamic. A discontinuity in the n-th classical derivative of a system- wide property, (T())), at some time to (assuming ( E C²)). Such transitions, studied in the continuum limit, are typically exclusive, meaning most one can occur for the whole system at given time [Biroli et al., 2024].
(2) Lattice For given step size > O, if there exists discontinuity threshold lase > such that for the n-th finite differences from the left (LD(") Φ) and right (RD("))) the below inequality holds, then the observable & undergoes a lattice transition step lo of order
(r) ((o)) RD(") (lo) 2 Qdisc-
Lattice transitions include thermodynamic ones in the limit 7 but are generally non-exclusive.
Our core observable for detecting these transitions is the absolute normalised cross-fluctuation, [ This quantity measures the cosine similarity of the conditional expected fluctuation tensors of two evolving events. merger signifies that these tensors have aligned, i.c., |MM[') + 1. Our
merger time it is precisely lattice transition point for this observable, defined by our chosen merger distance threshold (as in Eq. (3.1)).
Theorem 13 The merger event detected by M (a)
defined by our merger distance threshold in Eq. (3.1) constitutes a lattice transition at the merger time i".
Proof. Consider two initially disjoint events Ωr,, 22,0 C Ωg- Let i* denote the merger time given by Eq. (3.1). From the topological equivalence proved Theorem 4, there exists a constant 0 > 0 such that at the merger,
[M (Bi, S2,2
s 0.
41

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By selecting suitable merger distance threshold >> 0 in the metric d-.--.) used in Eq. (3.1), we
can ensure 1/2. Let our observable be (()) |M (Sq,s, 92,1).. It follows that $(i") 2 1 1-12 Due to the hypercontractivity of the underlying Brownian motion, the structural similarity between events monotonically increasing, hence (() is monotonically increasing This monotonicity implies that for discrete step size the rate of approach to the merger is bounded. For the n-th
order finite differences, this can shown to satisfy:
RD(") (o(**)
S 3 7
and LD (+(r)) 1 -
Φ(i 1T)
The key insight is that the jump into the merged state is finite, discrete event. Applying the reverse triangle inequality to the difference in the finite derivatives gives:
LTD. (o(i)) RD," (+(+))) 0,
which establishes the existence of lattice transition the merger time, by demonstrating non-zero jump in the finite differences, which can be thresholded by a chosen Edisc- Notice that as this relies on finite 7 and 0, this transition is not necessarily thermodynamic.
Thermcdynamic phases from prior works. For class-conditioned VP diffusion, Biroli et al. [2024] proved two thermodynamic boundaries tu-s (unbiased + speciation) and tg- (speciation + condensation):
unbiased [0, (u-)s) speciation (u--s:l--ed C condensation (fx-xe; T).
Relation of class conditional lattice mergers thermodynamic phases. For two classes k + define the centred cross cross- fluctuation Mte(()) ((4.5) in Section 4.2). Its e-merger time is
kg lat (e) = inf (t 2 0 MMR(()) 2 - e}, (0,
Lemma 14 (Merger times lie inside the speciation phase) For all + and e E (0,
bu
kt (lat( (E) Ls-
Proof. Unbiased phase. If < lu-ys then pa, = Pt,t. so Mue(!) 1; no upward crossing can occur.
Condensation phase. For & bs-se each covariance (4.3) satisfies Amax (En,)) S cef--tt-- [Biroli al., 2024, Prop. 4]. Using (B. Mue(t) ((t-b-(t)))))) hence Mue(t) 2 for all large t. No new crossing can start after tg-
Speciation phase. Because crossing cannot start before fu-s after 18- any merger time must lie in (tu-asi [c-.
The phase spectrum. Figure 5(a) details the spectrum of phases revealed by our lattice approach. For any choice of > O, the system can characterised by the number of merger events required to connect all events in partition. The plot visualises a sequence of distinct states:
Unrealizable thermodynamic phase possible merger at t - 00.
An unattainable limit which admits only a single
ii Unrealizable phase (0 < < 6min): Below the computational precision limit, a maximal number of K potential mergers exist, but they are not observable.
ii.. Realizable plateau (Emin S < (start): The system is observable and the threshold allows for maximum number of mergers possible.
iv. Staircase phase (Estart < < Cena): As increases, it becomes less sensitive the structural differences between events. The single macro-state of "maximum potential mergers" splits into spectrum of fine-grained states, visualised downward staircase. Each step is lattice transition where the number of distinct merger events required to connect the class graph decreases.
V. Saturated phase (e 2 : The threshold is so large that single overarching rule merges all events. The system becomes a singleton for discrimination purposes, losing all distinguishing power
42

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Quantifying Merger Events: Phase Spectrum VS. E
Merger Feasibility Frontiers: The - Trade-off
Feasible
Fradiin
Nomber of Ditinet Groups,
Secoeed Jepin
Mitreedyy Foodle fernie
Motively Foocle fern
Distaace Threshold.
Acaloesty Temble
Distance Throshold.
Merger Time,
a The phase spectrum of merger events
b Merger fcasibility frontiers
Figure 5: Conceptual visualisation of lattice transition dynamics. (a) The phase diagram illustrates how the number merger events changes with the threshold c. Adopting a lattice perspective reveals spectrum of distinct phases: an unrealizable thermodynamic phase at = 0 which admits single speciation transition, an unrealizable region below the system's precision limit, realizable plateau of maximum potential mergers, staircase of soquential transitions where the system "splits" into fine-gre ine-grained merger states, and final saturated state. (b) The feasibility frontiers visualise the trade-off between merger time t and threshold e. The relationship analogous to an uncertainty principle, where higher-order fluctuations merge faster, shifting their feasibility frontiers to the left.
Feasibility and uncertainty. Figure 5(b) visualises a fundamental trade-off in our framework. To precisely identify merger (small e), one must observe the system for a long time (large 1*). Conversely, to detect merger quickly (small f*), one must accept lenient threshold (large c). This relationship, which holds for any fluctuation tensor, is analogous to an uncertainty principle. For the specific, practical case where the state operator is the identity, p(x) z, the plot shows additional phenomenon: higher-order fluctuations (n) merge faster. This is because when analyzing the data vectors directly, high-order statistics capture fine-grained details that are more fragile and are erased more quickly by the diffusion process. The layered shading illustrates how the "feasible region" of detectable mergers expands of p.
moves to higher orders any given time for this specific choice
Thus, lattice transitions give fine-grained view inside the most relevant operational regimes of standard diffusion models, which are invisible to purely classical criteria.
43

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C Experimental details and further results
C.1 Compute and reproducibility
We conducted all experiments using a single Nvidia A100 GPU and provided sample code for reproducibility. Our implementation builds on open-source code from Hugging Face (diffusers library) and publicly available code from Kynkäänniemi et al. [2024], Li et al. [2023], Peebles and Xie [2022]. Our method plug-and-pla requiring simple hyperparameter adjustments for these techniques without any major code modifications. For chro-shot style transfer (Section 4.5), we used the Img2lmg transfer pipeline in diffusers (Meng et al. [2021], 1,2 Rombach al. [2022]), fixing the VP/DDPM schedule and adjusting the strength parameter.
Baseline experiments using grid search computationally intensive, typically requiring 24-48 hours of GPU time. Fluctuation computations (Section 3) were primarily constrained by covari- ance matrix calculations and eigendecompositions. These can be efficiently optimised using multi- processing, though we used single-process approach in this work. Our method is directly compatible with any standard implementation of the baselines. Preliminary small-scale experiments verified our theoretical framework but were excluded from the final paper.
C.2 Convergence of data
We present images visualizing the normality tests stated in Section 4.1
Normality Test P-values Timesteps for MNIST
6.00a-6.02
0.8
0.6
Pr00s
3.00
4.00
5.00
600
700
500
Titsosteps
a Normality test p- v-vuuess over time for MNIST
Normality Pvaluses Timesteps
alphu-0.05
0.8
0.6
Pealo
0.2
3.00
400
5.0
800
Timentage
b Normality test p-vuuues time for CIFAR10
44

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Normality Text P-walues Timestaps for Laberts
0.5
03
6.4
04
0.5
Timeshope
Normality test p-values over time for Imagenet
Figure 7: Normality test p-values for the DDPM schedule
C.3 Class-conditional generation
We compare two guidance schedules:
1. grid-search baseline that follows Interval Guidance (IG) [Kynkäänniemi et al., 2024] with a single dataset-level interval found by brute force";
2. our merger-aware schedule, in which each class k receives its own window (Istart,ks bend,k) derived from fluctuation theory (Sections 4.1 and 4.2).
Interval guidance baseline. Let 0 be the classifier-free guidance Ho and Salimans [2022b] (CFG) weight, and let 7 be the full diffusion horizon. During reverse sampling we switch CFG on only for (fend,es esta
Algorithm 2 Interval Guidance (class c)
Require: latent xoy NO I). CFG weight I,.... .. do
for
2: fead.e t fstart.e then 3: x - CF
4: else
5:
I, CFG
6:
7: end for
8:
return do
Hyper-parameters and Grid-search baseline. Following Peebles and Xie [2022] we set w for DiT-XI//2. Stable Diffusion requires stronger guidance; we use a fixed w E [3.5, 4.5] per dataset. For every dataset we sweep
lend,e E (0.17,0.2T, ... 0.87), Lstart, E {0.2T..... T}.
under the constraint fstart.c > tend.e- yielding 44 admissible pairs. On ImageNet the best pair is (0.8T, 0.27") (lowest FID); MNIST and CIFAR- 10 select (0.67, 0.17) and (0. 7T, 0.17),, respectively.
'In Kynkäänniemi al. [2024], it argued that a class-level or even sample-level search preferable. However, both these settings require at least 10 10 x the bascline compute, making them infeasible for us. Our primary objective is to demonstrate that leveraging finer hierarchical levels can compensate for compute limitations. Specifically, we reason that for class-level brute-force search, it is more practical to consider the transitions of fine-grained hierarchy derived from the representations of semi/self-suprrvised learning model Chen et al. [2020], Grill [2020], We leave such an extension to future work. We note that asymptotically, our method converges to the same output as perssalll--eeee grid search, as ultimately each data sample can be treated as a distinct singleton event Q2.0.
al. [2021]. A theoretical backing for the same is found in Theorem
45

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For Stable Diffusion we replace FID by CLIP similarity and obtain (0.8T, 0.17) for both ImageNet and Oxford-II11 Pet. One exhaustive sweep on DiT-XL/2 costs 4100 GFLOPs x 50 000 samples x44 configs z 9.0 PFL.Os; five repeats per pair multiply the cost five-fold. Results for Imagenet, MNIST and CIFAR are in Table in the main paper while that for Imagenet and Oxford-IIlTPets using Stable Diffusion is in Table 9. Note that for the case of Imagenet, identical intervals are used for both settings as the empirical forward process trajectory is independent of the model choice.
Merger-aware schedule (ours). For each class k. fttart,k i", fend, Emerge,k: where i" is the global convergence index (Section 4.1) and merge, is the first at which M Pr.,) = 1 for
some & (Section 4.2). No search required; windows differ automatically across classes.
Model/Dataset
CLIP Similarity (7)
Precision (7)
(e)
Density (1)
Coverage (7)
SD (Imagenet, baseline)
0.26 0.03
0.75 0.04
0.18
1.50::0.00 0.05
0.30 +: 0.03
(Imagenet, Ours)
0.31
0.02
0.01
SD (OxfordlITPPe, IG buseline)
0.28 0.02
0.79 1 0.03
0.21 0.03
0.84 +: 0.06
0.33 +: 0.05
SD (Oxtordil]TPer, IG Ours)
0.34 #: 0.03
0.81 + 0.01
0.26 + 0.04
0.89 +: 0.01
0.36 + 0.02
Table Class conditional generation using Stable Diffusion
Generative diagrams. Figure 13 plots the leading eigenvalues Amax (Ex,e) of the class covariances Σg.: for ImageNet, CIFAR-I0, and MNIST, highlighting merger points (proofss in Appendix B.3.3). Additional zoom-ins for ImageNet appear Figure 12. For long-tail datasets used in Section 4.3, analogous diagrams are given in Figures 14a and 14c. We also plot the subplots for the 10 classes for Imagenet and OxfordlIITPet, having the greatest magnitude of principal eigenvalues in Figure
SD output for prompt "Siamese Cat"
SD output for prompt "Golden Retriever"
b SD output for prompt "British Shorthair"
SD output for prompt "German Shepherd"
Figure 8: Stable-Diffusion samples for four
classes, generated with naive interval guidance (top of each column) versus our method (bottom).
Figure 11 compares ImageNet samples from our merger-aware IG with the grid-search baseline, using a guidance weight of 4.5 for visual clarity; Figure 8 does the same for Oxford-IIIT under Stable Diffusion. Our method yields crisper details and fewer artefacts artefacts-despie despite eliminating ss 9 PFLOPs
46

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of search for Imagenet. Thus, class-wise guidance windows obtained from cross cross- fluctuation mergers can match or exceed the quality an exhaustive dataset-level search, while slashing computational cost by orders of magnitude. Genurudave Diegrum Itagetnii class wilh larged Dkuccirass eigeensizee
Canerilee www.anterrooooooooooooo. diieed arguriri ergarre
Halee Com
X08
thar baaerty
2000
Thandegs
ImageNet top 10 classes
b Oxford-IIIT-Paa top 10 classes
Figure 9: Merger transitions are upper-bounded by those of the ten classes with the largest principal- cigenvalue magnitudes in their covariance matrices.
Merge prob. (2 0)
FID (J)
200
0.027
0.
100
0.013
2.86::0.15
80
0,010
0.12
67
0.009
0.11
20
0.002
1.181
Table 10: Effect of the MAE threshold (ImageNet).
Cumulative Probability
1.00
Cumulative Probability
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
0
20000
40000
0
200
400
600
Difference Between Eigenvalues
Difference Between Eigenvalues
a CDF of principal -igenvlal differences, ImageNet
e CDF of Oxford -llordlll---Pett
differences.
Margs Fretanilky be Difieenn Valane
Moepa Prokabikky Diffeeen Vilues
b Merge-probabiiiity vs. threshold, ImageNet
d perge-pbobbbillly threshold, Oxford-IIIT-Pel
Figure 10: -eeevaaleediiffeeecee distributions and their associated merge-probability curves for ImageNet and -ifford-IIT--et.....
To understand the sensitivity of the parameter we plot the distribution of the difference in eigenval- ues (CDF) and evolution of merge probabilities for different choices e (Figure 10). Our default choice 100 has merge probability 0.01 at t 0. We show corresponding FIDs in Table 10.
47

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Naive interval guidance for Imagenet
b Our optimised interval guidance for Imagenet
Naive interval guidance for Imagenet
d Our optimised interval guidance for Imagenet
Naive interval guidance for Imagenet
f Our optimised interval guidance for Imagenet
Naive interval guidance for Imagenet
h Our optimised interval guidance for Imagenet
Naive interval guidance for Imagenet
j Our optimised interval guidance for Imagenet
Figure 11: Visual Comparison of guidance for the Imagenet dataset [Deng et al., 2009, Ryu, 2024]. All samples were originally generated in 512 x 512 resolution.
48

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Oeporetion Diagner for 10 cissos Ixsegooe
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49
Figure 12: subplots for ImageNet (ten classes shown in two parallel columns).

---

3000
30000
660
200
405
425
a Generative diagram of MNIST
Generative diagram of ImageNet
264
304
500
2:0
431
250
7001
D Generative diagram of 10
d Generative diagram of Oxford-1I1 Pet
Figure 13: Merger- transittonnniiionn measure obtained from the intersection time of the principal eigenvalues of the class-ovvariance matrices.
Ocnordiiv Diagrers INgaade korpett aoaneeus
Diagrars QUe 200
106
Gode gadoo
540
300
20
220
940
200
Naturalist all classes
CUB-200 all classes
Generstive Diagress fer (Ntaandt lurpet masnnna dpendus
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Bouas roseber
Shar Seeed Altem
Dilbar
dieatted Jicentigiod
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122
300
Timarep
iiNatrrlist top 10 classes
(CU-200 top 10 classes
Figure 14: Aransittransition measures obtained from the intersection times of the principal eigen- values for iNaturalist and CUB-200 The bottom row shows that the classes with the ten largest eigenvalue magnitudes pounr-bound all other transitions.
50

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C.4 Rare-class generation
Interpolation- interval guidance. In Section 4.3 we observed that augmenting Algorithm with an interpolation correction- akiti to ILVR [Choi et al., 2021] gives the best fidelity for CUB- 200 and iNaturalist-2011 2019. The full procedure is listed in Algorithm 3; differs from standard Interval Guidance only in lines 5-7.
Algorithm 3 interpolation-basee interval guidance (class c)
Require:
Tr~~
latent to be denoised class-c exemplar
dg Pop with shape(ar) = shape(foo)
guidance weights CFGw, CFGu
interpolation schedule {n I where 0 7t $
1: for f = T 1,..., do
2:
if lamn. < < Lend,e then
guidance window
3:
4:
it FWD(
5:
3t
Th 24 + (1 m1)
6:
else
7:
Tt + CFGo
8:
end if
9:
end for
return x0
Note that in Algorithm 3, CFG,, applies classifier-free guidance with strength w; CFG, disables conditioning. FWD(ěu, t) generates the forward-noised version of the exemplar step f. The convex update in line nudges the current latent towards the exemplar's 's trajectory, counteracting class drift.
Choosing the interpolation schedule 7). Because guidance corrections most valuable late in the reverse chain, we derive 7) from the noise schedule 3, 7e s(3// max,<t Bu), < 8 < 1.
The scale factor is found by binary search over [10- 10 10-21: larger values degrade sharpness,
while smaller ones have negligible impact.
1.0
Cumulative Probability
1.00
Cumulative Probability
0.8
0.75
0.6
0.50
0.4
0.25
0.2
0.00
0.0
0
500
1000
1500
2000
0
200
400
600
Difference Between Eigenvalues
Difference Between Eigenvalue:
CDF of crincipal-eigenvalue differences (iNaturalist) Marpe Prmubiiyy by Dihuret Moatee
b CDF of principal-cigenvahee differences (CUB200) Hagga www.aluuyyy Scc DiDeee Vi3ke
122
d Merge--robabbiitt threshold (CUB200) Figure 15: Comparison of eigenvalue-difference distributions and merge conge-probability curves for iNaturalist vs. CUB200.
-Merge-probabilityy vs. threshold (iNaturalist)
51

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Experimental settings. For all runs, we fix e = 5 on iNaturalist and on CUB-200 when computing merger statistics. Figures 14a and 14c visualise per-class merger probabilities, with complementary CDF and eigenvalue trajectories in Figures 15a to 15d. Top-10 10 eigenvalue plots appear Figures 14b and 14d. Qualitative comparisons between Algorithm and naive Stable Diffusion, using identical random seeds, is given in Figures 16 and 17.
Original SD Image
Original SD Image
Original SD Image
Conditioning Image
Conditioning Image
Conditioning Image
Image
Image
New Image
Figure 16: Visual comparison of generation algorithms for stable diffusion for the iNaturalist class prompt "clouded leopard walking".
Original Image
Original Image
Original Image
Conditioning Image
Conditioning Image 2
Conditioning Image 3
New SD Image
h New SD Image
New SD Image
Figure 17: Visual comparison of generation algorithms for stable diffusion for the CUB-200 prompt
"pine warbler."
52

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C,5 Zero-shot classification
In Li al. [2023], a pretrained caas--conditioall diffusion network fo is repurposed as a classifier by averaging softmax logits over forward-diffused diffused replicas of the query image. We keep that spirit but (i) restrict the average the class-specify guidance window [tstart.), fatop.A] identified in Section 4.2, and (ii) attach an importance weight w(()) to every timestep I. Setting w(t) = 1/T and the bounds recovers the estimator of et al. [2023].
, 0, /stop.A
Throughout this section, A {1, [1....,A K} is the label set; En üid (0, Ta) noise seeds (n 1...., N); FWD,(x, E) denotes the forward map that produces the noisy latent žạ at step i; w(t) 20 probability mass the integer interval [Estart, A Astop.A]-
Algorithm 4 Zero-shot class probability px(=)
Require: query 2 2~ class label time window & S S tatop. weights (w((t) summing noise seeds (sn)m=]
1: px
2: for latart, ... A do
3: for n 1...., NN do
4: 2t + FWDr(zz,n)
5: 82 + -1 -I En, 2) -x
w(t)
exp(s>)
6:
Pa Pa +
N
ΣpEn exp(-11fo(21, En 11?
7: end for
8: end for
9: return ps
Let a₁ be the signal coefficient of the VP process (2.4). The signal-to-noise ratio is SNR(t)
af/(1 af). We consider three discrete weight laws: Uniform: w(t) 1/[fstop,> fstart, + 1). INverse-SNR: w(t) x SNR(()-1 with lstart, O. Truncated inverse-SNR: same as previous but restricted to 2 20 (empirically, very early steps degrade performance as the score model is not exact at these time scales [Chen et al., 2022a, Karras et al., 2022]). We fix N 250 for all of our experiments following et [2023].
C.5.1 Binary classification via linear probes
To understand why non-uniform weights help, we study binary accuracy along the forward chain with no diffusion model involved. Pick two ImageNet classes 1x] at random and sample min min(card(A), (r), (ard(())) 10000) number of images for each class. Ar each step 6, we extract the noisy embedding 24 (VP schedule) and train a linear MLP on 80% of the embeddings; the rest forms the test set. We repeat the experiment 20 times and report mean + s.d.
Figure 18 shows that test accuracy rises sharply and peaks near the merger time of À and μr, after which it is not defined since the embeddings corresponding to X, fu become practically indistinguishable Averaging the accuracies with the three weight laws yields the means Table 11; using the single best / (the merger time) achieves the highest score but is undefined for the multi-class case, however, the general trend for the other strategies that are valid for the multi-class setting remains the same.
Weighting strategy
Avg. binary accuracy
Uniform
0.72 + 0.03
Inverse-SNR
0.79 + 0.06
Trunc. inv. SNR
0.82 + 0.04
Merge-time probe
0.90 + 0.02
Table 11:
Binary accuracy for random ImageNet pairs.
Each forward step convolves the data with Gaussian kernel, progressively smoothing non-linear features. Just before the classes merge, the representation is simpler yet still separates the two manifolds, making linear decision boundaries easiest to leamn This explains why inverse-SNR weighting, which emphasises mid-to-late timesteps- beats uniform weighting, and provides an additional reason why further truncation gives a small extra gain. Thus, merger-aware weighting focuses the zero-shot estimator on the most discriminative region of the diffusion trajectory, narrowing the accuracy gap to dedicated representation learners such CLIP.
53

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54 Figure 18: Linear-probe probe accuracy through the forward diffusion process. Later timesteps hold greater discriminative information between the two classes.

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C.6 Zero-shot style transfer
In [Meng al., 2021] the parameter fstop. is estimated by a grid search over the interval 0.17, 0.27...................... ..... T for an entire dataset for each style which we follow for our baseline imple- mentation using the PSNR metric. istop. estimated in this way for both datasets were observed to lie in the range 0.37, 0.47, 0.5T across styles. For our implementation /stp,, is estimated class-wise as the smallest merger time of the class(es) to which 2 belongs.
Algorithm 5 Zero-shot Style Transfer
Require: query 2 po noise seed En, maximum time lstop,s 1: Ststop. <- FWDtaapp. En)
2: return 2" + BWDe (Ststop.s)
We use open-source fine-tuned stable diffusion models available on Hugging Face for all of our experiments. These models are trained on the artistic styles Studio Ghibli Miyazaki and Ghibli [2014], Van Gogh Roojen [2019] and the styles of the animation game Elden Ring Software and Entertainment [2022] and series Arcane Production and Games [2021] 10 Note that some samples from the OxfordlITTPet Dataset may be randomly censored by the automatic filter present in these models. We observed that this happens mostly for dog images, where some breeds have samples with their mouth open. This can be mitigated to extent by center cropping and realigning: however, preprocessing fails, we discard such images.Tablee 5 in Section 4.5 of the main paper has results for the Studio Ghibli and Van Gogh styles while Table 12 has results for the Elden Ring and Arcane styles.
Style
Elden Ring
Arcane
Models/Metrics
PSNR (f)
MSE (L)
PSNR (1)
MSE (J)
SD Edit (OxfordlITPets)
24.19 0.72
0.09 + 0.006
25.17 + 0.42
0.09 0.005
Ours (OxfordlIIPPets)
28. 28.14 0.69
0.03:00002
28.35 + 0.72
0.03 0.0006
SD Edit (AFHQv2)
26.08 + 0.37
0.06 + 0.003
26.49 + 0.34
0.05 + 0.004
Ours (AFHQ v2)
27.56 + 0.37
0.04 + 0.009
28.23 + 0.18
0.03 +: 0.001
Table 12: Style Transfer results for Elden Ring and Arcane styles
Visual results are in Figures 19 and 20 for the AFHQv2 and OxfordllITPet datasets, respectively. We also show visual proof of our assumption from Section 4.5 elaborated in Appendix B.3.4 through visual plots of the evolution of the fourier transforms through the forward process of images mainly differing only in style, in Figure 21. Here we gap of 0.17 between the two images. These plots empirically show that for images differing only in style but with the same essential structure, their fourier transforms are close and this fact extends their noised versions well due the nonlinear decay in Brownian Motion (Appendix B.3.3).
All referenced trademarks, copyrighted characters, and original artworks remain the property of their respective owners. Algorithm 5 was utilised for research purposes only and is not intended for commercial infringe upon the intellectual property rights of the original creators.
"Disclaimer:
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Original
Ghibli
Van Gogh
Elden Ring
Arcane
Figure 19: Zero Shot Transfer results on AFHQ v2
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Original
Ghibli
Van Gogh
Elden Ring
Arcane
Figure 20: Zero Shot Transfer results on OxfordllTTPet
57

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Original
Mide Bing
sidee Ring
Figure 21: Fourier transforms and decay in Fourier spectra are close for structurally similar data
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D Limitations and future work
Modelling assumptions. Our analysis is built three structural premises: (i) vartance-preservinng
(VP) noise schedule, (ii) isotropic Brownian perturbations, and (iii) the Fourier-regulariyy bound of (4.7). These hold for the canonical SDE/ODE pairs [Song al., 2021, Ho et al., 2020J, but have not yet been extended to non-VP schedules (e.g. EDM [Karras et al., 2022]) or to anisotropic diffusions. Relaxing any of the three assumptions remains open.
Moment truncation. Empirically, the first few centered moments suffice to expose the phase boundaries, yet diffieult data (heavy-tailed, multimodal) may demand higher orders. Reliable estimation of tensor moments beyond order four is computationally expensive and lacks sharp concentration inequalities.
Memory overhead.
Computing covariances Σx.r E d for many classes incurs O(Kd) mem- ory. Random projections, sketching, or block-diagonal approximations could mitigate this without degrading detection power.
Modalities beyond 2-D images. All experiments target natural images. How merger-based guid- ance interacts with 3-D diffusion, video or audio, or with text diffusion unexplored.
Training-time usage. We use cross-fluctuations post hoc. Injecting merger times into the loss- as a curriculum on noise levels or as regulariser enforeing class separation-man -ay accelerate or stabilise training; we leave this to future work.
E
Related work
Statistical physics of diffusion models Thermodynamic transitions diffusion models have been studied in prior work, such as Raya and Ambrogioni [2024], which identifies a mixing time transition akin to ours in Section 4.1, but for a narrow class of initial distributions. This is similar to the Curie-Weit transition in ferromagnetic systems Kivelson et al. [2024], with analytical estimate. We provide an alternative derivation of this dependency for a broader class of sub-Gaussian data in Appendix B.3.1, but approach in Section 4.1 is more general, relying solely the assumption that the supports of the data and the isotropic Gaussian are essentially disjoint. Biroli and Mézard [2023], Biroli et al. [2024] extend Raya and Ambrogioni [2024] by modeling data Gaussian mixture and show three distinct phases in a glass-conditiona setup similar to Section 4.2. Our framework further builds these ideas in Appendix B.3.6, where showcase general framework relying on discrete lattice transitions more common in quantum systems.
Fast sampling for diffusion models. DDIM [Song et al., 2020]. IDDPM [Nichol and Dhariwal, 2021], DPM-Solver [Lu et al., 2022], and EDM [Karras et al., 2022] reduce the reverse step count; early-exit criteria such
are orthogonal and in principle could be combined with any of them. We leave such extensions to future work. As discussed earlier, the occurrence of this criterion has also been demonstrated by [Raya and Ambrogioni, 2024, Biroli and Mézard, 2023, Biroli al., 2024] through a different analysis based on symmetry breaking of a potential function; we show that framework extends such considerations in Appendix B.3.6.
Theoretical lenses on diffusion. Hyper-contractivit [Saloff- 1994, Chen et al., 2022a]. mixing- mixing-time bounds [Levin and Peres, 2017], and classical coupling [Aldous and Fill, 2002] traces back to Markov-chain theory. We recast these ideas as cross coss-fluctuation mergers, bridging discrete and continuous settings in Appendix 1.1.4
Conditional guidance. Score distillation [Poole et al., 2022], ILVR [Choi et al., 2021], classifier- free guidance [Ho and Salimans, 2022a], and Interval Guidance (IG) [Kynkäänniemi et al., 2024] dominate conditional generation. Our merger-aware IG trims the interval search from per-sample to per-class with no extra Syper-parameterr with details in Section 4.2 and Appendix C.3
Zero-shot classification using diffusion networks. Li et al. [2023] showed diffusion backbones encode class information. Importance weighting by intermediate times with cutoff at merger times
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boosts their zero-shot accuracy at equal compute (Section 4.4). We also demonstrate that near the merger times, the diffusion process demonstrates near-perfect diseriminability Appendix C.5.1.
Rare-class synthesis. Long-tail tail generation typically relies on fine-tuning [Bansal et al., 2023, Samuel et al., 2024]. Our fluctuation-basel guidance suning-free and alleviates the mode dropping at inference time by utilizing optimised IG based guidance (Section 4.3, Appendix C.4)
Zero-shot style transfer using diffusion models The intriguing property of style transfer by learning trained VP-SDE only on the target style after utilizing the forward process to corrupt inputs from the source style was initially shown Meng al. [2021] and has become standard practice across setups Rombach et al. [2022]. To our knowledge, we are the first to provide a theoretical foundation for this approach in terms Fourier regularity that manifests observed transitions (Section 4.5, Appendix B.3.4) while showing that setting the noising parameter based merger times can boost fidelity (Section 4.5, A.5.eppe C.6)
60


Reviews:
[{"review_id": "IJiZG6AwAV", "reviewer": "Reviewer_Sxsz", "summary": "This paper presents a novel framework for understanding the sampling dynamics of diffusion models using cross-fluctuation, a concept from statistical physics. The authors propose a useful operator to capture the discrete transition during the sampling of diffusion models and demonstrate the effectiveness of the proposed method on five applications, including early stopped sampling, CFG sampling, rare-class generation, classification, and style transfer.", "questions": "I don't have further questions.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper is well-organized, and the application of the concept of cross-fluctuation on diffusion models is novel. \n- The proposed method provides a practical criterion, which can be computed with only the forward process, to capture the discrete phase transition in the sampling dynamics of diffusion models. \n- Such a criterion is shown to be practically useful in various scenarios. \n\nWeaknesses:\n- The most crucial hyperparameter, the $\\epsilon$ in Equation (3.1), requires more detailed illustration. The choice of $\\epsilon$ determines the performance of the proposed method in all the included application scenarios. However, the criterion for choosing a proper $\\epsilon$ seems to lack discussion, except for class-conditional sampling, where a criterion based on the largest eigenvalue is proposed. If a proper $\\epsilon$ is sensitive to the dataset or task, it will still take a source-extensive sweep search to find the optimal one.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for acknowledging the novelty of our cross-fluctuation framework and its practical efficiency across multiple applications. To address the remark regarding the selection of $\\epsilon$, we consolidate and expand our sensitivity analysis as follows:\n\n(1) we recall our rule for selecting $\\epsilon$ without costly grid search (Appendix C.3);\n(2) we reiterate how this rule governs the probability of full dataset merging at each step (Appendix C.3);\n(3) we introduce a new phase diagram illustrating how varying $\\epsilon$ leads to distinct qualitative behaviors, spanning the spectrum from $\\epsilon = 0$ (infinite precision) to $\\epsilon \\to \\infty$ (zero precision);\n(4) we further extend the analysis by showing how admissible $\\epsilon$ ranges shift under higher-order fluctuations beyond the second-order (covariance) case.\n\n### **(1) Principled selection of $\\epsilon$.**\n\n**Selection criterion**:  \nInstead of running an expensive grid search, we use a simple rule of thumb: set $\\epsilon = \\frac{\\lambda_{\\max}}{400}$, where $\\lambda_{\\max}$ is the largest eigenvalue magnitude among class covariance matrices. We arrived at this choice by inspecting our datasets and plotting the CDF of pairwise **differences** between the top eigenvalues of unperturbed class covariances. We then chose $\\epsilon$ so that $P(\\mathrm{difference} \\leq \\epsilon)$ is small typically around $10^{-2}$. This helps avoid trivial mergers between classes at $t = 0$. Figure 7a in Appendix C.3 shows a representative CDF for ImageNet that illustrates this behavior.\n\n\n[Difference between eigenvalues for Imagenet classes](https://postimg.cc/qznfZp35)\n\n\n**Empirical validation**: (please also see Appendix C.3, Table 9). We demonstrate the relationship between $\\epsilon$ values, merger probabilities (probabilities of two random classes being merged at $t = 0$ with this $\\epsilon$), and the associated FID. \n\n| $\\epsilon$ | Merge probability (t=0) | FID (↓) |\n|---|-------------------------|---------|\n| 200 | 0.027 | 3.06±0.19 |\n| 100 | 0.013 | 2.86±0.15 |\n| 80 | 0.010 | 2.92±0.17 |\n| 67 | 0.009 | 2.90±0.11 |\n| 20 | 0.002 | 2.88±0.14 |\n\n\n### **2) Merger probability analysis.**\n\nFigures 7b,7d, and 12 b, 12d (Appendix C.3) present curves showing how the fraction of merged classes evolves over time $t$ for different values of $\\epsilon$ across our choice of datasets. These curves demonstrate that our criterion yields stable merger dynamics with merger probabilities at $t=0$ being of the order $10^{-2}$. We also present a sample curve for the Imagenet case (from Figure 7b, Appendix C.3)\n\n(**Click for plot**) [Merger probability](https://postimg.cc/0z0pkjTs)\n\n\n### **(3) Phase structure of $\\epsilon$ selection.**\n\nTo further substantiate the choice of \\$\\epsilon\\$, we have carried out additional theoretical analyses to explore the phase structure underlying the merger dynamics:\n\n(**Click for plot**) [Merger phase diagram](https://postimg.cc/3kQMG8tN)\n\nThe plot above reveals the fundamental phase structure governing $\\epsilon$ selection across four distinct operational regions. The optimal region enables detection of the maximum number of distinguishable mergers ($1 < K \\leq N-1$ for $N$ classes), while regions with $\\epsilon$ values too low or too high either are impossible due to a lack of numerical precision or fail to detect class distinctions. Our $\\lambda_{\\max}/400$ criterion operates within this optimal detection region as the merger probability for standard datasets is of the order $10^{-2}$ with this criterion.\n\n### **(4) Merger feasibility frontiers - dependency on order of fluctuation.**\n\n(**Click for plot**) [Merger feasibility frontiers](https://postimg.cc/Mctggyqf)\n\nThis analysis demonstrates the theoretical trade-off between merger time $t^{\\star}$ and $\\epsilon$ across different fluctuation orders ($n = 2,3,4,5$). Higher-order fluctuations expand the feasible parameter space, allowing merger detection across broader $\\epsilon$ ranges at lower timesteps while collapsing faster due to non-linear decay induced by the diffusion process. The plot confirms that second-order fluctuations provide stable discrimination for practical applications.  \n\n\n### **Summary**\n\nWe hope the above discussion addresses the reviewer's concerns, and we would be happy to engage in any discussion that may be helpful in further clarification. **We will incorporate the reviewer suggestions fully in the revised manuscript**."}, {"review_id": "FHaJ9Tqi4g", "reviewer": "Reviewer_Z71p", "summary": "This paper introduces a novel framework for controlling sampling in diffusion models through the lens of fluctuation theory. The central insight is that during forward diffusion, different classes (data events) exhibit distinct fluctuation behaviors, which eventually converge (or \"merge\") as the process approaches pure noise. The authors define a normalized cross-fluctuation metric   to quantify distinguishability between events and use it to detect phase transitions — points at which classes become statistically indistinguishable. \nThey apply this framework to: accelerated sampling via early stopping, improved class-conditional guidance, enhanced rare-class generation, zero-shot classification using fluctuation signatures.", "questions": "Sec 4.2: to what extent are the results sensitive to the threshold epsilon? It would have been interesting to give the results for various epsilons.\n\nSec 4.3: similar question: I understand that the criterion is whether the rare class has merged with any other class. Would the results be sensitive to a change of this criterion, e.g. the merging of the rare class with m classes with m>1? \n\nSec 4.4: would this method generalize to hierarchical or fine-grained classification settings (e.g., sub-breeds of dogs, rather than mnist classes)? How scalable is the covariance-based comparison in high-dimensional spaces?", "limitations": "The authors did not really discuss the limitations of this work.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n- originality: The use of fluctuation metrics as a lens to analyze diffusion dynamics is novel. Rather than relying on learned metrics or external classifiers, the method uses intrinsic model statistics.\n- applicability: The paper provides a practical algorithm (Algorithm 1). The method is evaluated across diverse settings: class-conditional sampling, rare class recovery, long-horizon compositional prompts, and even zero-shot classification.\n- significance: This approach has wide implications; among which acceleration of diffusion, improvement of rare-class generation, guiding conditioning more precisely, and reveal structure without training. \n- clarity (mostly): The paper is well-organized and mostly clear. That said, some parts, especially the explanation of how fluctuation metrics translate to guidance windows — require careful reading or visual aids.\n\nWeaknesses\n\n- clarity: The concept of fluctuations, merger times, and their link to class indistinguishability is dense and may be hard to parse without prior familiarity. The paper would benefit from intuitive visualizations or simplified examples early on.\n- Metrics/ interpretation of the numerical results for some applications: While precision/recall/coverage are well-adapted for generative models, the way numerical results illustrate the claims of the authors is not always clear. For instance, it is not clear how they indicate a greater rare-class coverage. the paper could be clearer about how this, especially since it doesn’t directly report class-wise sample counts.\n- Uncertainty in Merging Behavior: The notion that class k's merger time is based on indistinguishability from any other class (rather than all classes) is conservative but may underestimate useful guidance intervals. Discussion of this tradeoff is absent.", "was_revised": true, "final_justification": "kept my score", "rebuttal": "We thank the reviewer for acknowledging the originality and significance of our work, and for pointing out where the exposition and clarity should be improved. We respond to the points raised below, with the key aim of suggesting what we are doing to improve the clarity and accessibility of the work.\n\n### Addressing concerns on clarity\n\nWe acknowledge that the mathematical treatment of fluctuation theory may be dense. To provide an intuitive understanding, we present a visualization capturing what we are measuring: a hierarchy of mergers that reveals how members of a dataset partition relate to each other.\n\n(**Click for plot**) [Merger visualization](https://postimg.cc/N9hRsXdp)\n\nThis temporal hierarchy illustrates how distinct, mutually disjoint events merge as the diffusion process evolves, with merger points indicating when discriminating information vanishes.\n\n## Response to specific questions about §§4.2, 4.3, and 4.4 \n\n### **Section 4.2: Sensitivity to epsilon threshold**\n\n**Selection criterion**: We establish $\\epsilon = \\frac{\\lambda_{max}}{400}$, where $\\lambda_{max}$ represents the largest principal eigenvalue magnitude across class covariance matrices. This provides a principled, data-driven selection criterion that adapts automatically to dataset characteristics.\n\n**Empirical validation**: As detailed in Appendix C.3, Table 9 demonstrates the relationship between $\\epsilon$ values, merger probabilities (probabilities of two random classes being merged at $t = 0$ with this $\\epsilon$), and associated FID performance:\n\n| $\\epsilon$ | Merge probability (t=0) | FID (↓) |\n|---|-------------------------|---------|\n| 200 | 0.027 | 3.06±0.19 |\n| 100 | 0.013 | 2.86±0.15 |\n| 80 | 0.010 | 2.92±0.17 |\n| 67 | 0.009 | 2.90±0.11 |\n| 20 | 0.002 | 2.88±0.14 |\n\n**Merger probability analysis**: Figures 7,12 (Appendix C.3) present curves showing how the fraction of merged classes evolves over time t for different ε values across our choice of datasets. These curves demonstrate that our criterion yields stable merger dynamics with merger probabilities at $t=0$ being of the order $10^{-2}$. We also present a sample curve for the Imagenet case (from Figure 7, Appendix C.3)\n\n(**Click for plot**) [Merger probability curves](https://postimg.cc/0z0pkjTs)\n\n### **Additional theoretical analysis**\n\n\n**Quantifying merger events - Phase spectrum analysis**\n\n(**Click for plot**) [Merger phase diagram](https://postimg.cc/3kQMG8tN)\n\nThis plot reveals the fundamental phase structure governing $\\epsilon$ selection across four distinct operational regions. The optimal region enables detection of the maximum number of distinguishable mergers ($1 < K \\leq N-1$ for $N$ classes), while regions with $\\epsilon$ values too low or too high either are impossible due to a lack of numerical precision or fail to detect class distinctions. Our $\\lambda_{max}/400$ criterion operates within this optimal detection region as the merger probability for standard datasets is of the order $10^{-2}$ with this criterion.\n\n **Merger feasibility frontiers - Theoretical trade-offs** \n\n(**Click for plot**) [Merger feasibility frontiers](https://postimg.cc/Mctggyqf)\n\nThis analysis demonstrates the theoretical trade-off between merger time $t^{\\star}$ and $\\epsilon$ across different fluctuation orders ($n = 2,3,4,5$). Higher-order fluctuations expand the feasible parameter space, allowing merger detection across broader $\\epsilon$ ranges at lower timesteps while collapsing faster due to non-linear decay induced by the diffusion process. The plot confirms that second-order fluctuations provide stable discrimination for practical applications.  \n\n### **Section 4.3: Merger criterion sensitivity**\n\nWe appreciate the reviewer’s suggestion to explore the sensitivity of our merger criterion—namely, changing from merging as soon as any single class becomes indistinguishable to requiring merger with $m>1$ classes. Our conservative choice (merging upon indistinguishability with any other class) identifies the earliest point at which statistical separation is lost, and thus defines the most appropriate guidance‐window boundary. While one could certainly extend the windows by insisting on merger across multiple classes, doing so carries the risk of operating beyond the region in which class distinctions remain meaningful.\n\nThe reviewer asks about sensitivity to changing the merger criterion from merging with any single class to merging with $m>1$ classes. Our conservative approach (merger with any other class) captures the earliest point of statistical indistinguishability, which represents the optimal guidance window boundary. Relaxing this criterion to require merger with multiple classes would extend guidance windows, but risks operating beyond meaningful class distinction.\n\nWe provide empirical validation for this design choice in Appendix C.6, where Table 11 and Figure 15 demonstrate that linear separability between randomly chosen ImageNet class pairs increases monotonically up to the merger time. This confirms that the binary merger criterion identifies the point of maximum statistical distinguishability. We attach a sample plot from Figure 15 for reference.\n\n(**Click for plot**) [Linear separability](https://postimg.cc/2qqQhZ2N)\n\n### **Section 4.4: Hierarchical classification and scalability**\n\n**Generalization to higher hierarchies**: Our mathematical framework assumes only a partition of distinct events in the data space, enabling generalization to finer hierarchies. We refer the reviewer to Theorem 11 in Appendix B.5, which formalizes that partitions can be refined to greater detail down to singletons with well-defined mergers.\n\n**Scalability**: The covariance-based comparison scales as $O(Kd^{2})$ for $K$ classes and $d$ dimensions. For high-dimensional spaces, this can be mitigated through random projections or block diagonal approximations without degrading detection power, as discussed in Appendix D.\n\n### **Metrics interpretation and rare-class coverage**\n\n**Standardness of metrics**: We employ precision, recall, density, and coverage as standard metrics for rare class synthesis, consistent with established evaluation protocols for long-tailed datasets. CLIP similarity provides additional validation for semantic quality.\n\n**Rare-class coverage**: The numerical results demonstrate improved rare-class coverage through higher recall and coverage metrics. Our method ensures rare classes receive appropriate guidance windows before merging with dominant classes, preventing their collapse during generation.\n\n### **Limitations**\n\nWe acknowledge several limitations of our approach. Memory overhead scales as $O(Kd^{2})$ for computing class covariances, though this can be mitigated through approximation techniques. The method has been validated primarily on 2D images and the standard VP schedule, and extension to other modalities and different noise schedules requires further investigation.\n\n### **Summary**\nTo summarize: our criterion provides principled parameter selection with demonstrated stability across the tested range. The phase structure analysis reveals the theoretical foundations underlying this robustness, while our mathematical framework naturally extends to hierarchical classifications. These contributions establish a theoretically grounded and practically reliable approach for controlling diffusion sampling dynamics.\n\nWe trust that the new experiments and analyses effectively address the reviewer’s concerns regarding parameter sensitivity, criterion robustness, and scalability. **We sincerely appreciate the insightful suggestions and will incorporate them fully in the revised manuscript**. Should any questions or additional concerns remain, we would be delighted to discuss and resolve them. Thank you very much for your time and thoughtful review."}, {"review_id": "DoNHUVWFIx", "reviewer": "Reviewer_fbWE", "summary": "The paper analyzes sampling dynamics in score-based diffusion models through cross-fluctuations, an nth-order central moment that measures when two sets of trajectories become indistinguishable under the forward noise process. It shows the connection between phase transitions with the trajectory  discontinuities of this statistic. For variance-preserving SDEs, the authors derive a closed-form expression that can be evaluated without additional simulations. These theoretical results lead to several methods that can detect merger points and safely skip the earliest reverse steps, yielding faster sampling and other guidance tricks without retraining.", "questions": "1. The Main Contributions section states that this work establishes \"a connection between ensemble based techniques in statistical mechanics as well as classical mixing-coupling results for Markov chains, using a generalized notion of phase transitions\". Could you provide a high-level explanation of this result, and comment on how novel or significant it is within the statistics literature?\n2. Typo: line 292 reads \"methodlologies\" but should be \"methodologies\".", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper introduces a fresh mathematical viewpoint for studying diffusion models by bringing tools from fluctuation theory into the analysis.\n2. The study is thorough, moving from formal results to practical tests, and the theory-driven algorithm shows promising empirical gains.\n\nWeaknesses:\nMy main concern is presentation. To me, the key contribution is the new theoretical lens, while the experiments mostly serve as illustrations. The paper does not present clearly which theoretical results are formally proved and how to read them intuitively. It would help to make it explicit which parts are provable theory and which parts are theory-inspired heuristics.", "was_revised": true, "final_justification": "I thank the authors for their detailed response that address all my questions. I will maintain my positive score.", "rebuttal": "We are sincerely grateful to the reviewer for highlighting our work’s “fresh mathematical viewpoint” and for offering such constructive suggestions to improve the clarity of our presentation.  We have addressed the reviewer's comments point by point below.\n\n\n### **Mathematical context and distinguishing formal proofs from heuristics**\n\n**Mathematical foundation**: Cross-fluctuations are nth-order centered moment statistics computed over trajectory ensembles within event sets (partitions of the data space). The framework measures structural similarity between events through their moment interactions. We track the nromalized cross-fluctuation $\\mathcal{M}^{(n)}_{\\rho}(\\Omega_1, \\Omega_2) \\in [0,1]$ with values near $1$ indicating statistical indistinguishability.\n\n**Formally proven results**:\n- **Theorem 12** (Appendix B.7): Moment-TV inequality establishing that fluctuation moment proximity bounds total variation distance between probability distributions under regularity conditions.\n- **Theorem 10** (Appendix B.5): Topological equivalence between fluctuation-based distance d_M and Frobenius distance d_F on expected fluctuation tensors.  \n- **Theorem 11** (Appendix B.5): Diffusion time geometry theorem proving that cross-fluctuation merger sequences determine the metric structure of evolving marginals, with extensions to hierarchical partitions.\n\n**Theoretically inspired heuristics**:\n- The criterion for $\\epsilon$ parameter selection (validated through extensive experimentation in Appendix C rather than theoretical derivation).\n- Algorithm 1 implementation details for practical merger detection.\n- Application-specific guidance window calculations (detailed in Appendix C).\n\n### **More details on the theoretically inspired selection of $\\epsilon$**\n\n#### Quantifying merger events - Phase spectrum analysis\n\n(**Click for plot**) [Merger phase diagram](https://postimg.cc/3kQMG8tN))\n\nThis plot reveals the fundamental phase structure governing $\\epsilon$ selection across four distinct operational regions. The optimal region enables detection of the maximum number of distinguishable mergers ($1 < K \\leq N-1$ for $N$ classes), while regions with $\\epsilon$ values too low or too high either are impossible due to a lack of numerical precision or fail to detect class distinctions. Our $\\lambda_{max}/400$ criterion operates within this optimal detection region as the merger probability for standard datasets is of the order $10^{-2}$ with this criterion.\n\n\n#### Merger feasibility frontiers - Theoretical trade-offs  \n\n(**Click for plot**) [Merger feasibility frontiers](https://postimg.cc/Mctggyqf))\n\nThis analysis demonstrates the theoretical trade-off between merger time $t^{\\star}$ and $\\epsilon$ across different fluctuation orders ($n = 2,3,4,5$). Higher-order fluctuations expand the feasible parameter space, allowing merger detection across broader $\\epsilon$ ranges at lower timesteps while collapsing faster due to non-linear decay induced by the diffusion process. The plot confirms that second-order fluctuations provide stable discrimination for practical applications.  \n\n\n### **Connection to statistical mechanics and Markov chains**\n\n**Mathematical framework**: Our approach addresses fundamental limitations in extending coupling theory to continuous state spaces. In discrete Markov chain theory, the coupling time $t_{cpl}(\\epsilon) = \\min\\{t \\geq 0 : \\max_{\\alpha,\\beta} d_{TV}(S_{\\alpha,t}, S_{\\beta,t}) \\leq  \\epsilon\\}$ measures when chains ($S_{\\alpha,t},S_{\\beta,t}$) starting from different initial distributions become indistinguishable. However, total variation distance lacks discriminative power in continuous spaces, where it often equals $1$ for distinct absolutely continuous distributions.\n\n**Generalized coupling framework**: We establish a moment-based generalization through our cross-fluctuation statistic. As detailed in Appendix B.6, we may define the generalized coupling time $t_{gen}(ε) = \\min\\{t \\geq 0 : \\max_{\\alpha,\\beta} \\mathcal{M}^{(n)}_\\rho(\\Omega_{\\alpha,t}, \\Omega_{\\beta,t}) \\geq  1-\\epsilon\\}$, which coincides with classical coupling time for discrete state spaces while providing meaningful discrimination in continuous settings.\n\n**Theoretical bridge**: Theorem 12, Appendix B.7 establishes that moment proximity bounds total variation distance under regularity conditions. This enables practical merger detection in high-dimensional continuous spaces where direct TV computation is intractable.\n\n**Statistical physics connection**: The discrete phase transitions we identify in cross-fluctuations connect to ensemble theory in statistical mechanics (Appendix B.14). We distinguish thermodynamic transitions (discontinuities in classical derivatives) from lattice transitions (finite-difference discontinuities with threshold $\\tau$), where our merger events represent lattice transitions in the moment space.\n\n**Significance**: This work provides a computational framework for extending coupling concepts to modern generative models operating in high-dimensional continuous spaces, where traditional TV-based approaches face fundamental mathematical and computational challenges.\n\n### **Organizational improvements**\n\nWe acknowledge the presentation concerns and will implement several organizational improvements based on reviewer feedback. These include restructuring the main results into a dedicated theorem section, consolidating theoretical foundations in reorganized appendices, and providing clearer delineation between proven results and practical heuristics. These changes will enhance accessibility while maintaining mathematical rigor.\n\n### **Other Improvements**\n\nWe acknowledge the typo on line 292 (\"methodlologies\" $\\rightarrow$ \"methodologies\") and will correct this in the final version, along with additional proofreads to catch other related typos.\n\n### **Summary**\n\nOur work establishes theoretical connections between discrete Markov chain coupling concepts and continuous diffusion dynamics through fluctuation-based phase transitions. The formal theorems in Appendix B provide mathematical foundations that enable practical merger detection in high-dimensional continuous spaces. While parameter selection, such as $\\epsilon$, represents an empirically validated approach rather than a theoretical derivation, the underlying mathematical framework provides principled grounding for controlling diffusion sampling dynamics across multiple applications. \n\nWe sincerely thank the reviewer for their thoughtful and constructive feedback. **We are confident that the reviewer’s suggestions can be seamlessly integrated into the revised script**."}, {"review_id": "zbP1KiChBM", "reviewer": "Reviewer_yAPh", "summary": "The paper introduced a study of score-based models using cross-fluctuations. The aim of the paper is to show that in the sampling process there are several transitions between distinct states. The paper illustrates that this states can be detected with the cross-fluctuation formalism and be used to speed-up the sampling and improve zero-shot use-cases.", "questions": "See Strengths and Weaknesses.", "limitations": "See Strengths and Weaknesses.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is not really accessible to all the community. Even if the result is a purely mathematical one, we still have to remember that we are submitting the work to share the knowledge to most of the community. Perhaps, before diving into the mathematical formalism, I always recommend to give an highly level overview on the method and intuition behind the discovery. Due to the high-density of mathematical formalism without broader intuition, the paper is really hard to read. Half-way through the paper, I lost completely the scope of the paper. \n\nSaid so, I don't really understand what a \"transition\" is in this use-case. What does actually measure that there is a phase transition? Phase transition between what and what? In Physics, phase-transitions can be first or second order with their properties. Here, the use of transition between phase is used since the beginning but I haven't seen any definition on the states and/or transitions. In the algorithm, are you sampling with two score-based models simultaneously and measure the cross-fluctuations between the two? Is it the code available? \n\nI believe that this kind of approach is novel, I've never seen these concepts applied to score-based models. Experimental results show improvements in FID and inference time. However, I'd ask the authors to give a higher-level intuition in the main paper and part of the mathematical definition to the appendix. At the current state, I see a high-potential paper with lack in interpretability for most of the research community.", "was_revised": true, "final_justification": "See the answer to authors rebuttal.", "rebuttal": "We would like to express our sincere gratitude to the reviewer for their thoughtful and constructive feedback. We address these points carefully, one by one, below.\n\n---\n\n### **High-level intuitive overview**\n\nBefore addressing the specific technical points, we begin with the core intuition:\n\n**Core insight**: In diffusion sampling, structured data (e.g., class-separated) gradually transitions into noise. This process unfolds through **discrete \"merger events\"** where formerly distinguishable groups suddenly become statistically indistinguishable. Think of this as watching ice cubes melt: separate cubes soften gradually, then suddenly merge into uniform water.\n\n**What we measure**: We detect **cross-fluctuations**—the moment when the statistical \"fingerprints\" of different data groups converge. These moments mark **optimal intervention points** for early stopping, guidance, and other sampling strategies.\n\n**Practical value**: Instead of relying on expensive grid searches, our method automatically pinpoints the most effective windows for intervention based on these fluctuations.\n\n---\n\n### **Defining phase transitions in our context**\n\nWe acknowledge that the terminology around \"phases\" and \"transitions\" needs clarification. In our framework:\n\n**Phase transition**: A discrete point during forward diffusion where two previously distinguishable data clusters become statistically indistinguishable. These are detected as sharp discontinuities in the cross-fluctuation statistic.\n\n**States**: Each \"phase\" corresponds to a statistical regime:\n\n- **Structured phase**: Classes remain distinguishable, retaining unique moment structures  \n- **Merged phase**: Classes become statistically identical  \n- **Transition point**: A discrete timestep at which the regime switch occurs\n\n#### Relation to physical systems\n\nWhile not the main focus, these phase transitions are **emergent phenomena**, driven by finite precision in numerical computation. Classical theory predicts a single mixing event as $t \\to \\infty$, but for $d$-dimensional Sub-Gaussian data, this occurs practically at time scale $\\mathcal{O}(\\log d)$ (see Proposition 15, Appendix B).  \nOur framework uncovers **intermediate, computationally accessible transitions**—moments where merging occurs before the theoretical limit, enabling real-world intervention strategies.\n\n(**Click for plot**) [Merger phase diagram](https://postimg.cc/3kQMG8tN)\n\nThis plot reveals the **phase structure governing the choice of $\\epsilon$**, showing four distinct operational regions. The **optimal region** allows detection of the largest number of mergers ($1 < K \\leq N-1$ for $N$ classes).  \nValues of $\\epsilon$ that are too small fall below numerical resolution; values too large cause early overmerging. Our chosen threshold of $\\lambda_{\\max}/400$ lies within this optimal region, where the chance of premature merging at $t=0$ is typically $\\mathcal{O}(10^{-2})$ for standard datasets.\n\nThis emergent behavior can be interpreted, in physics terms, as a **Renormalization Flow** induced by discretization. While a full treatment is outside our current scope, Lemma 17 (Appendix B.14) provides a comparison between the **classical single-phase prediction** and the **multiple discrete regimes** observed under finite precision. We prove that a single theoretical phase **splits** into a rich sequence of detectable merger events.\n\n---\n\n### **Visualization of the merger process**\n\nTo support intuition, we include a visual hierarchy of the merger process:\n\n(**Click for plot**) [Merger visualization](https://postimg.cc/N9hRsXdp)\n\nThis diagram shows how initially disjoint events merge over time during diffusion. Merger points indicate when discriminative information vanishes. The resulting tree structure reveals the **natural statistical hierarchy** among events in the dataset.\n\n---\n\n### **Algorithm clarification**\n\nThe reviewer raised a concern regarding simultaneous sampling with two models. This reflects a common misunderstanding. Our approach:\n\n- **Uses a single forward diffusion model**  \n- Tracks how **different data subsets** (e.g., classes) evolve under this process  \n- Requires no separate models or retraining\n\n**Cross-fluctuation computation** compares the evolved states of different data subsets. When their statistics converge (i.e., similarity approaches 1), we label the groups as merged.\n\n**Practical implementation**: Algorithm 1 only uses **forward passes** through the training data. It involves no reverse sampling, retraining, or hyperparameter search.\n\nTheoretical support for this setup is given in Theorem 9 (Appendix B.4), under standard regularity conditions.\n\n---\n\n### **Methodology details**\n\n- **Measurement process**:  \n  We define cross-fluctuations $\\mathcal{M}^{(n)}_{\\rho}(\\Omega_1, \\Omega_2) \\in [0,1]$, where values near 1 indicate statistical indistinguishability.\n\n- **Detection criterion**:  \n  We declare a **merger transition** when the expected moment structures of two groups converge within a threshold $\\epsilon$. Sharp rises in $\\mathcal{M}^{(n)}_{\\rho}$ toward 1 mark transition points. These transitions serve as **precise markers** for optimal interventions like early stopping or guided sampling—**without** any grid search.\n\n- **Computational efficiency**:  \n  Our method is highly efficient. It relies only on **forward diffusion** and **covariance estimation**. The latter can be scaled using block-diagonal approximations or random projections (Appendix D), without compromising detection quality.\n\n---\n\n### **Addressing remarks on accessibility**\n\nWe agree that the current presentation could better serve a broader audience. Based on the reviewer's feedback, we will implement the following changes:\n\n- **Structural reorganization**:  \n  We will lead with intuitive overviews and visualizations before introducing mathematical details.\n\n- **Concrete examples**:  \n  Section 2.2 will be expanded to include simple, visualizable use cases that show cross-fluctuation dynamics in action.\n\n- **Appendix relocation**:  \n  Technical derivations will be moved to appendices to declutter the main text while preserving theoretical completeness.\n\n- **Practical emphasis**:  \n  We will highlight the method as a plug-in toolkit (Section 3), usable without deep theoretical background.\n\n---\n\n### **Experimental validation**\n\n**Performance**:  \nAs noted by the reviewer, our experiments show significant improvements in FID and inference efficiency. Our approach yields **orders-of-magnitude gains** over grid search-based methods.\n\n**Generality**:  \nWe validate across **five use cases**: early stopping, class conditioning, rare class generation, classification, and style transfer. This underscores the broad applicability of our phase transition framework.\n\n---\n\n### **Summary**\n\nWe sincerely thank the reviewer for their constructive feedback. We fully agree that presenting our framework more intuitively will broaden its impact without compromising rigor.  \n**We are confident that these improvements will significantly enhance clarity, accessibility, and practical utility of the revised submission.**"}]

Author rebuttal:

Meta-review: * Summery

The authors introduce tools from statistical physics to analyse the sampling dynamics of diffusion models and their underlying dynamics.  
Using fluctuation theory and fluctuation dispersion style tools, they can separate and distinguish between different regions of events and define the transition between such regions as a phase transition. The method is then explored, both through theoretical arguments and insights, as well as empirical experiments. All of the above is leveraged to improve accelerated sampling through early stopping, enhancing class-conditional guidance, and class generation, and performing zero-shot classification.

* Strengths and reasons for acceptance 

The idea is novel and, in a sense, builds upon the original diffusion models presented in "Deep Unsupervised Learning using Nonequilibrium Thermodynamics." Since this kind of fluctuation theory is what ties equilibrium and [non-equilibrium processes](https://www.pnas.org/doi/10.1073/pnas.1918386117), the paper is very well written and puts a major effort into being self-contained. While theoretical in nature, the authors also provide a practical algorithm for implementation and demonstrate it across several tasks related to training and using diffusion models. In addition to the above, this is one of the first methods to examine the model data independently in this manner, without requiring additional training or external classifiers.
With all of the above, I would recommend this paper for acceptance into the conference. 

* Weaknesses 
The paper is quite dense, and discerning the originality of ideas can be challenging; more specifically, the clarity of the core concepts is difficult to grasp. The biggest point of weakness and issue with the paper we should be critical of is the hyperparameters, and specifically the $\epsilon$ parameter. I would have been happier with a bit more insight into it and guidance on how to choose it.
Decision: Accept (poster)