EigenScore: OOD Detection using Posterior Covariance in Diffusion Models
Authors: Shirin Shoushtari
Yi Wang
Xiao Shi
Salman Asif
Ulugbek S. Kamilov
Venue: ICLR 2026 Poster (2026)
Keywords: OOD detection
diffusion models
uncertainty estimation
Primary area: generative models
TL;DR: We introduce EigenScore, a novel OOD detection method that quantifies distribution shift through the spectrum of the posterior covariance derived from the diffusion model.

Abstract: Out-of-distribution (OOD) detection is critical for the safe deployment of machine learning systems in safety-sensitive domains. Diffusion models have recently emerged as powerful generative models, capable of capturing complex data distributions through iterative denoising. Building on this progress, recent work has explored their potential for OOD detection.
We propose *EigenScore*, a new OOD detection method that leverages the eigenvalue spectrum of the posterior covariance induced by a diffusion model.
We argue that posterior covariance provides a consistent signal of distribution shift, leading to larger trace and leading eigenvalues on OOD inputs, yielding a clear spectral signature. We further provide analysis explicitly linking posterior covariance to distribution mismatch, establishing it as a reliable signal for OOD detection.
To ensure tractability, we adopt a Jacobian-free subspace iteration method to estimate the leading eigenvalues using only forward evaluations of the denoiser.
Empirically, EigenScore achieves state-of-the-art  performance, with up to 2% AUROC improvement over the best baseline. Notably, it remains robust in near-OOD settings such as CIFAR-10 vs CIFAR-100, where existing diffusion-based methods often fail.

EIGENSCORE: OOD DETECTION USING POSTERIOR COVARIANCE IN DIFFUSION MODELS
Shirin Shoushtari
Yi Wang Washington University in St. Louis y wang1 @wust1.es
Washington University in St. Louis shir phirrineee www.w.ullee...
Xiao Shi
M. Salman Asif
University of Wisdonnin-Madison xiao. shiswisc.. edu
University of California, Riverside sasif@ucr edu
Ulugbek S. Kamilov
University of Wisconsin-Madiss kamilov samilov@wisc.er edu
ABSTRACT
cut-of-distribution (OOD) detection is critical for the safe deployment of machine learning systems in safety-sensithe domains. Diffusion models have recently emerged powerful generative models, capable of capturing complex data dis- tributions through iterative denoising. Building on this progress, recent work has explored their potential for OOD detection. We propose EigenScore, detection method that leverages the eigenvalue spectrum of the posterior covariance induced by a diffusion model. We argue that posterior covariance provides a con- sistent signal of distribution shift, leading to larger trace and leading cigenvalues on OOD inputs, yielding clear spectral signature. We further provide analysis explicitly linking posterior covariance to distribution mismatch, establishing it as a reliable signal for OOD detection. To ensure tractability, we adopt a Jacobian-free free subspace iteration method to estimate the leading eigenvalues using only forward evaluations of the denoiser. Empirically, EigenScore achieves state-of-the- per- formance, with up to 2% AUROC improvement over the best baseline. Notably, it remains robust in near-OOD settings such as CIFAR-10 vs CIFAR-100, where existing diffusion-based methods often fail.
OOD
INTRODUCTION
Most machine learning systems that test data matches the training distribution, but distribution shift or -itt---diittiitto (OOD) data can severely degrade performance in safety-critical domains such medical imaging and autonomous driving (Yang et al., 2024). OOD inputs may stem from sensor noise, semantic differences, or acquisition changes, leading to unreliable predictions (Zhang al., 2023). To address this, many OOD detection methods have been proposed, ranging from supervised approaches that require labeled OOD data to unsupervised approaches that rely only on in-distribution (InD) training data (Graham et al., 2023).
Existing OOD detection methods can be broadly categorized into four families: (i) uncertainty- based methods, which rely on signals such as softmax confidence (Hendrycks Gimpel, 2017), ensemble variance (Choi et al., 2018), or Bayesian inference (Wang & Aitchison, 2021: Charpentier et al., 2020) to identify anomalous inputs; (ii) distance-based methods (Regmi et al., 2024), which compare test embeddings to in-distribution features, commonly via Mahalanobis distance (Colombo et al., 2022; Lee et al., 2018); (iii) density-based methods (Huang et al., 2022), including flow- and energy-based models (Kumar et al., 2023; Liu et al., 2020), which attempt to estimate likelihoods but have been shown to assign spuriously high likelihoods to OOD data (Nalisnick et al., 9a): and (iv) representation-leamiig methods (Wang al., 2022), including self-supervised and contrastive

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techniques (Seifi al., 2024; Hendrycks et al., 2019; Tack et al., 2020), which improve robustness by explicitly shaping feature spaces (Also see reviews in Koh et. al (2021); Salehi et al. (2022)).
Diffusion models (DMs) (Ho et al., 2020; Song et al., 2020) have emerged as ofate-of-the-ar generative models, achieving high-quality samples across diverse domains. Their success has spurred both architectural advances (Vahdat et al., 2021; Dhariwal & Nichol, 2021; Rombach et al., 2022; Karras et al., 2022) and applications beyond generation, such as imaging inverse problems and medical tasks (Chung et al., 2023; Adib al., 2023) (see also recent reviews (Daras al., 2024; Croitoru et al., 2023; Li et al., 2023; Kazerouni et al., 2023)). Crucially, diffusion models are especially relevant for OOD detection because their iterative denoising process does not simply produce samples, but also provides access to score functions that explicitly characterize the data distribution. Early work exploited this property through likelihood- or reconstruction-bast
scores (Graham al., 2023; Gao et al., 2023; He et al., 2025). More recent studies have explored structural aspects of the diffusion trajectory, such as score geometry and intermediate representations (Heng et al., 2024; Graham et al., 2023; Liu et al., 2023; Choi et al., 2023). These developments highlight both the promise of diffusion-basee OOD methods and the need for principled approaches that move beyond heuristic scoring rules.
Building on recent work in diffusion- based OOD detection, we introduce EigenScore, an unsu- pervised, deature-based framework for identifying distribution shift. Unlike reconstruction-bee based methods that measure input-outuut similarity (Graham et al., 2023) or trajectory-basse -base methods that analyze diffusion-patt geometry (Heng et al., 2024), EigenScore leverages the covariance structure of the denoising process to capture uncertainty signals. In particular, DDPM-OOD (Graham et al., 2023) detects distribution shift by measuring reconstruction fidelity, using perceptual or pixel-space discrepancies between the input and its denoised reconstruction as an OOD score. While effective in some regimes, this strategy implicitly assumes that distribution shift manifests primarily through degraded reconstruction quality. In contrast, EigenScore does not depend on reconstruction error magnitude or perceptual similarity; instead, it probes the posterior uncertainty of the denoiser by analyzing the eigenvalue spectrum of the posterior covariance. As shown in Section 3.1, even when OOD inputs admit visually plausible reconstructions as is common in near- OOD settings the posterior covariance inflates in structured directions, yielding consistent and theoretically grounded signal of distribution shift.
Recent work by Kamkari et al. (2024) provides a geometric explanation of the likelihood -ased OOD detection paradox by analyzing the local geometry of generative models. Their approach studies the Jacobian of the generative mapping and shows that likelihood scores conflate radial distance to the data manifold with tangential volume effects, leading to misleading OOD behavior. While their method also involves spectral quantities through singular values of Jacobian-related matrix, the motivation and signal differ fundamentally from ours. EigenScore does not operate on likelihoods or generative mappings; instead, it analyzes the posterior covariance of the diffusion denoiser, capturing predictive uncertainty rather than volume distortion. As result, EigenScore yields stable uncertainty-bist based signal even regimes where likelihood geometry is known fail, such as near-OOD settings.
By explicitly linking posterior covariance, estimated from the denoiser's Jacobian, to distribution mismatch, EigenScore provides an interpretable uncertainty-based signal while remaining practical at scale through a Jacobian-free empirical results demonstrate that applying in-distribution diffusion model to OOD samples leads to systematic inflation of posterior covariance, enabling stable and discriminative OOD detection. Our contributions are threefold:
cigenvalue estimation algorithm. Our theoretical analysis and
We introduce EigenScore, unsupervised, Srature-based framework for OOD detection in diffusion models. FigenScore leverages the posterior covariance of the denoising process to characterize distribution shift.
We provide supporting analysis establishing a direct connection between demoising uncertainty from posterior covariance and distribution mismatch, thereby explaining why EigenScore reliably separates InD from OOD samples.
We conduct extensive experiments standard OOD benchmarks (CIFAR-10 10 (C10), CIFAR-100 (C100), SVHN, CelebA, Imngeet),Net showing that EigenScore achieves average sfate-of-the-art performance and remains notably robust in challenging near-OOD scenarios.
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2 BACKGROUND
Diffusion Models. Diffusion models learn to generate samples by simulating gradual denoising process. During training, clean sample p(x) is perturbed by Gaussian noise across timesteps
t= 1,. ... .T, producing noisy states e, through the forward Markov chain p( (x) = o(
which allows for direct sampling via 2; - x + 2, where z (N o0,
The reverse process is approximated by a denoising network Da( 1), trained to predict either the clean signal or the injected noise. standard training objective mean squared error (MSE):
CMSE(Do) = Le,e, x Da (2 t)
(I)
Once trained, the model generates new samples by iteratively denoising from Gaussian noise at f = T back to f = 0. Importantly, Tweedie's 's formula (Robbins, 1956; Miyasawa, 1961) connects Gaussian denoising with score estimation, linking the posterior mean to the gradient of the log-density as
D2(2)) = Ep | æ = x + of log p(zz),
(2)
where Dp(x)) denotes an MMSE estimator trained on samples from distribution p. Here, the gradient is with respect to Ve and p(æ,) denotes the marginal distribution noisy image
p(ana) p( (z)p( / Go,, (2re 2 (
where Gz, denotes the Gaussian density function with standard deviation o, 2 0 (Vincent, 2011; Raphan & Simoncelli, 2011). This relationship implies that denoising does more than produce samples-it provides access to score functions and posterior statistics of the underlying distribution. In the context of OOD detection, this observation motivates our use of the denoiser's covariance structure as a principled signal of distribution shift. Diffusion models admit several formulations (c.g.,
variance-exploding, and SDE-based), but all share the key property of learning the score function V., log p(æx)) to guide denoising (Ho et al., 2020; Song et al., 2020; Song & Ermon, 2019; Yang et al., 2023).
Unsupervised OOD detection. Unsupervised OOD detection aims to determine whether given sample a originates from the same distribution as the training data, using only unlabeled InD samples p(æ). The goal is to learn detector that assigns an OOD score to each input, where higher scores indicate a greater likelihood that æ was drawn from a different distribution, such as the OOD density g(z) (Graham et al., 2023; Heng al., 2024).
, Tn .
Likelihood- methods. These methods use generative models including VAEs, flows, diffiusion models to estimate sample likelihoods, under the assumption that OOD data should receive lower likelihoods (Salimans et al., 2017; Kingma & Dhariwal, 2018; Morningstar et al., 2021: Ding et al., 2025). However, it has been shown that generative models often assign high likelihoods to OOD inputs (Choi et al., 2018; Nalisnick et al., 2019a; Kirichenko et al., 2020). To mitigate this, refined scores have been proposed, including likelihood ratios (Ren et al., 2019), compression corrections (Serrà et al., 2019), WAIC ensembles (Choi et al., 2018), and typicality tests (Nalisnick al.. 2019b). Diffusion-bassd variants further extend this idea by analyzing statistics across the denoising trajectory (Heng et al., 2024; Livernoche et al., 2024).
Reconstruction-basee methods. Another line of work assumes that InD samples reconstruct well, whereas OOD samples do not. Early examples include autoencoders (Zhou & Paffenroth, 2017) and GANs (Schlegl et al., 2019). More recently, diffusion models have been exploited for their strong reconstruction fidelity, leading to perceptual quality scores (Graham et al., 2023), projection regret (Choi et al., 2023), and masked inpainting like LMD) (Liu et al., 2023). OOD sample detection can alse be via subspace reconstruction of features or gradients, using PCA (Guan et al., 2023), kerrll PCA (Fang et al., 2024), or gradient projections such as GradOrth (Behpour et al., 2023).
beature-based methods. These approaches distinguish InD from OOD by leveraging learned representations, such as Mahalanobis distance in latent space (Denouden et al., 2018), unsupervised contrastive features (Hendrycks et al., 2019; Bergman & Hoshen, 2020; Tack et al., 2020), encoder features from invertible models (Ahmadian & Lindsten, 2021). Pretrained feature extractors have also proven effective (Xiao et al., 2020).
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CLEATUO BaD) CIPARIOO (OOD))
CIFARIO SVII (OCI)
Denetty
NLL (bits/dim) 240 60
Σ 1)1
160
100
18hea(ar, 913
550
Σ Ae(zss)
4
Figure 1:
We compure negative log-likelihood (NLL), score nOrm E, (eolooasss score derivative norm
VE, |1 (wr near OOD task for C10 (InD) vs. C100, NLL and score-based metrics fail separate distributions, showing substantial overlap. Bottom row: for C10 (InD) vs. SVHN (OOD), the ordering of metrics inverts-scon score and derivative norms assign lower values to OOD than InD, making thresholds unreliable. In both settings, our eigenvalue-bes basse metric achieves clear separation and consistently assigns higher scores to OOD samples.
(wi, 1) and the eigenvalue sum (ours) E.P Ng(()s) as OOD detection statistics. Top row:
Complementing these approaches, introduce new perspective based posterior covariance in diffusion models, which provides a principled feature for quantifying distribution shift.
3 DIFFUSION MODEL FOR OUT-OF-DISTRIBUTT DETECTION
EigenScore novel OOD detection method that exploits covariance structure of the denoising diffusion process. Our key insight is that when a diffusion model trained on InD data is applied to OOD inputs, the variance of its denoising predictions inflates, leaving a characteristic signature the eigenvalue spectrum of the score Jacobian.
To motivate EigenScore, we first revisit why commonly used ciftusion-based OOD met- rics-likelihood, score norm, and score derivatives- are unreliable (Sec. 3.1). We then show, both theoretically and empirically, that posterior covariance offers consistent marker of distribution shift (Sec. 3.2), before formalizing EigenScore and its efficient computation (Sec. 3.3).
3.1 WHY LIKELIHOOD AND SCORE DYNAMICS ARE INSUFFICIENT
Since diffusion models trained via variational lower bound (ELBO), likelihood-batt scores such as negative log-libe ligeiioel (NLL) are natural candidates for OOD detection. However, likelihood does not necessarily align with semantic structure: diffusion models often emphasize low-level statistics while ignoring higher-level semantics, making NLL unreliable (Nalisnick et al., 2019b; Serrà et al., 2019). Empirically, NLL can even assign higher likelihoods OOD samples than Inl ones (Heng et al., 2024). As shown in Fig. 1, NLL is not a reliable metric for separating InD from OOD samples.
Beyond likelihood, diffusion-basse based OOD metrics have also used the score function c0 t) and its temporal derivative Öcco( as statistics (Heng et al., 2024). Their norms provide some empirical separation. but they remain unstable. In near-OOD settings (C10 vs. C100), the distributions overlap substantially 1, top row). some settings (C10 vs. SVHN), the ordering can invert, with OOD samples receiving lower scores than InD (Fig. 1, bottom row). These limitations motivate shifting from scalarized scores toward a covariance-basee based perspective, where the structure of denoiser variability itself provides a more principled signal of distribution shift
3.2 UNCERTAINTY AS SIGNAL OF DISTRIBUTION SHIFT
We formalize why denoising uncertainty yields principled OOD signal. Let p(x: denote InD and (xx) an OOD distribution. Under Gaussian corruption with variance of, the KL divergence admits the score-based representation (Song al., 2021; Kadkhodaie al., 2024; Shoushtari al., 2025;
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CalAA 340)
OOD InD
A2
Aq
Aa
output
Uncertainty map
Eigenvatuus
Figure 2: Denoised outputs (left), corresponding uncertainty maps (first principle component) (middle), and violin plots the three largest eigenvalues for CelebA dataset (right). Top: clean CelebA image and its noisy variants for varying t. Middle: InD model (trained on CelebAj applied CelebA inputs. Bottom: OOD model (trained on C100) applied to same inputs. models yield sharp reconstructions and localized uncertainty with smaller leading eigenvalues, whereas OOD models produce blurrier outputs, diffuse uncertainty, and inflated eigewalles-hhghhighhin the eigenvalue spectrum indicator of distribution shift.
Algorithm 1 OOD Detection with EigenScore Train/Validation (left) and Test (right)
Train/Validatien Select time-steps, aggrega- tion, and compute Z-score stats
Test: Compute EigenScore
Require: Trained DM Dy. test set Nest. number of eigenvalues K, number of repetitions J,
Require: Trained DM Dp. train set Xyain K
number of eigenvalues, 1 number of repeti- tion, Lirain =
{ ot)) Liest = []
for Must do
1: for æ E Xmain do
2: Compute M(x) using T and agg"
2: Compute M(x) via Eq. (10)
ze <<(2) (x) for 1 ..... T
3: Append M(z) to Luain
5x
4: end for
So(x)
Σ Et=1 -(=12()) (x)
5: Compute
across Luain 6: Use validation tune T and aggregation
5: Append So(z) to Lyeat
end for
method (mcan/median/none))
7: return Liest
b OOD scores for all test
7: return (I" agg", {Me, o())
samples
Heng et al., 2024)
T (R /" B.n.. [ logg(w:) V log (xx) [|2] or dt.
(4)
where 2e is generated by the forward diffusion applied to æ. Building on prior analyses of KL divergence in diffusion processes (Shoushtari et al., 2025; Heng et al., 2024), restate the divergence in terms of denoising error (a derivation is given App. A.1 for completeness).
Proposition Let Pt. and qu denote the noisy marginals of InD and OOD distributions generated by the forward diffusion process (Eq, (3)). For MMSE denoisers Dy(xt) Exp[ar and (a:))) Eg zz).
(KI(pp 11 - [MSE(D,, -MSEED, 3
where MSE(D,..) t) B[[[ (p(x))]1 noise level t.
This proposition, adapted from earlier derivations, shows that KI. divergence and thus distribution shift- can be viewed as the accumulation of excess denoising error incurred when contrasting the optimal MMSE denoiser under with that under p. Thus, we have MSE(D,; q. > MSE(D,, ; for each t and the denoising error single InD denoiser D, is larger expectation on OOD inputs, yielding practical detection signal without access to q.
For the MMSE denoiser Dp, the mean mean-squared error admits law-of-tottll-varinnc decomposition (proof in App. A.3):
MSE(D,, p. t) = El| - (x, 11 = E,, tr (tr(Covp(z xx)))]-
(5)
Thus, denoising error equals the total posterior variance the trace of the conditional covari- ance-averaged noisy observations at noise level f. Intuitively, Cov, [x æx] quantifies the

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spread of plausible clean signals consistent with zt. When æ; is informative (e.g.. InD low noise), this spread is small and the MSE remains low; when Dt lies off-manifold, the spread inflates and the MSE increases. In this sense, denoising error corresponds directly to the model's predictive uncertainty, providing a principled basis for OOD detection and motivating our focus on the spectral structure of posterior covariance in Section 3.3.
While Eq.4 relates distribution shift to differences score norms, such metrics do not preserve ordering: depending on the dataset and noise scale, OOD samples may appear either larger or smaller than InD ones. By contrast, the MSE-based formulation in Prop. guarantees that OOD denoising error systematically larger in expectation than InD error. This ensures consistent separation with preserved ordering, whereas score-norm methods such as DiffPath (Heng et al., 2024) capture only relative differences without indicating direction.
3.3 EIGENVALUE- BASED UNCERTAINTY ES ESTIMATION
Section 3.2 established that denoising enror equals the total posterior variance and that this variance inflates under distribution shift. We now make this connection operational by expressing posterior covariance through the Jacobian of the MMSE denoiser and analyzing its eigen-spectrum. For the MMSE denoiser (xc) [.) the posterior covariance admits the Miyasawa identity (Miya-
sawa, 1961):
/ = of (I + v² log p(x:)) = of =f ((te),
(6)
so that
MSE(D,, = Ex, [tr (Cov, æx)) of [tr (VD,(
(7)
Thus, the Jacobian trace measures how much posterior uncertainty remains after observing 2;: small traces indicate concentrated belief's (InD), while large traces signal inflated uncertainty (OOD).
Derivations are provided in Appendix (A.2).
Writing (x,) /ov,|=/2,] = (xDD) the covariance is symmetric positive definiee and admits the digen-decompositionn Σx = U, diag(Ai, Xr M,,, with nonnegative eigen- values Xi. Since tr(E.) = Σx Xt. the MSE corresponds to the sum of eigenvalues- the total uncertainty across all principal directions noise level t:
MSE(Dp. p. t) = EEz, [tr
= Ez, [tr (E, = En,
Σ (am)) R=
(8)
For InD samples, the spectrum is compact with smaller leading eigenvalues, reflecting structured denoising aligned with the training data. For OOD samples, uncertainty spreads across multiple eigen- directions, inflating both the spectrum and the trace. This spectral inflation provides quantitative signal of distribution shift. Figure 2 illustrates this effect: OOD samples exhibit consistently higher uncertainty, confirming the link between spectral inflation and distribution mismatch.
3.4 EIGENSCORE AS AN OOD METRIC
While Prop.1 links distribution shift to excess denoising error, it requires two denoisers, whereas in practice we have only single InD model. Combining Prop. I with Eq. implies that OOD inputs exhibit inflated posterior covariance, which we can read from its eigenvalues. For each input æ and diffusion step t, we compute
K Σ (x)), 2; = oiz, N),,I),
(x)
the sum of the top-K eigenvalues across I noise realizations. These are aggregated (e.g.. by mean, median, or full set) into mx()), yielding the EigenScore feature vector
M(x)
(m))
(10)
We normalize each coordinate using Z-scores with statistics (14, o1) computed training set 2, (x) (m, - o.) The final OOD score the sum of normalized coordinates, with the number of timesteps T tuned for efficiency validation data.

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Direct Jacobian evaluation is costly. Following (Manor & Michaeli, 2024). estimate the top K eigenvalues using Jacobian-Fie free subspace iteration. The method approximates Jacobian- vector products with finite differences of the denoiser vt = +D( D(æ, cv)) //cc, where < 1 is the linear approximation constant, v the current principle component, and o+ in the next principle component. We then orthogonalizes directions via QR. After few iterations, eigenvalues are obtained as
(xx) = of ||D(a= + co** D(xt - cl2
(11)
2c Full pseudo-code is given in App. B.1.
3.5 EIGENSCORE vs. MSE: CAPTURING STRUCTURE INSTEAD OF COLLAPSE
Eq. (4) and Prop. show that distribution shift manifests as excess denoising error, measurable through the posterior covariance. However, scalar metrics such MSE score norms collapse this uncertainty into a single number, discarding how variance is distributed across directions. At high noise levels, this collapse is particularly problematic: many small eigenvalues are dominated by isotropic noise, obscuring class-specific structure. The following lemma formalizes this effect.
Lemma 1.
Let p(ax.)= p * (00 of denotes the noisy marginal in Eq. (3) and let (x.)
of (I o log p(g)) from Eq. (6). As ot 00, 2| p(x;)) - 0 uniformly compact sets. Hence
Ex) of + o(o))
so all eigenvalues satisfy X{ of + o(of)-
Lemma implies that the spectrum flattens under heavy Gaussian smoothing: all directions approach the same variance of. so low-variance components lose discriminative information. Consequently, MSE and score norms aggregate mostly isotropic noise, rather than meaningful structure (proof in App. A.4). To retain the informative structure, we focus on the dominant modes. Ky Fan's theorem guarantees that the top-K eigenvalues capture the maximal variance among all K-dimensional projections:
Proposition 2 (Ky Fan's theorem (Fan, 1950)). Let Ex())) 2 have eigenvalues X 2 ... 2 X with eigenvectors forming U, [uf,... uxx]. For any K E ( ,
tr (VTE((Z;)V)
Σ A
max
VEReX VVV=Ik
and maximizer is V* [*= u'xl-
By retaining only the top-K eigenvalues, EigenScore preserves the most informative uncertainty directions while discarding noise-dominated components. This explains its consistent advantage over MSE and score-based metries. The choice of K not dictated by theory, but reflects practical trade-off between capturing discriminative information and computational efficiency (see App. A.5 for proof). Reconstruction- be interpreted as relying on scalarized denoising error, either in pixel space perceptual feature space. As with MSE, these approaches collapse uncertainty into a single value, discarding how variance is distributed across directions. EigenScore differs fundamentally by retaining the dominant spectral modes of posterior covariance, which remain discriminative even when reconstruction error alone becomes unreliable. We further discuss the implications of using learned denoisers, whose Jacobians need not be strictly SPD. and their relation to spectral truncation in Appendix B.2.
diffusion methods such as DDPM-OOT (Graham et al., 2023) can
4 EXPERIMENTS
We now evaluate the effectiveness of EigenScore for OOD detection. Specifically, we benchmark EigenScore across suite of pairwise OOD detection tasks and compare its performance against state-of-the-ar baselines.
Datasets. We evaluate OOD detection on standard image benchmarks commonly used with diffusion models: C10 (Krizhevsky, 2009), SVHN (Netzer et al., 2011), 1), C100 (Krizhevsky, 2009), and CelebA (Liu et al., 2015a;b). For near-OOD tasks (Yang et al., 2022), we additionally include TinyImageNet (Le & Yang, 2015). Further details can be found in Appendix (B.5)
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Table 1: Main OOD detection results (AUROC). Comparison of EigenScore with likelihood based, reconstruction- based, and diffusion--asse baselines across multiple InD-OO1 dataset pairings (CelebA, C10, C100, SVHN). Best and second best are highlighted. Note that EigenScore achieves the best average perfor- mance and is either best second best in most settings.
InD
CelebA
C10
C100
SVHN
Avg
OOD
C10 C100 SVIIN C100 SVII CelebA
C10
CelebA
SVII
C10 C100
CelebA
DoS
0.630 0.615 0.808
0.504
0.752
0.456
0.491
0.520
0.777
0.911
0.904
0.956
0.693
TT
0.676 0.655 0.773
0.558
0.714
0,469
0,538
0.464
0.648
0.957
0.961
0.994
0,701
WAIC
0.589 0.569 0.793
0.476
0.760
0.469
0.502
0.530
0.782
0.978
0.974
0.955 0.698
Diffusion-based
NLL
0.507 0.671 0.753
0.558
0.545
0.599
0.480
0.484
0.481
0.635
0.660
0.636
0.584
IC
0,510 0.673 0.755
0.552
0.540
0,583
0.460
0.466
0.469
0,625
0.653
0.625
0.576
DDPM-OOD
0.922 0.928 0.992
0.618
0.944
0.642
0.462
0.496
0.870
0.963
0.972
0.996
0.817
LMD
0.886 0.848 0.950
0.601
0.821
0.834
0.569
0.595
0.748
0.780
0.749
0.872
0.771
DifIPath (CelebA)
1.000 1.000 0.964
0.554
0.729
0.885
0.475
0.887
0.724
0.919
0.941
0.328
0.784
DiftPahV2 (CelebA)
1.000 0.995 0.969
0.535
0.812
0.862
0.483
0.513
0.724
0.969
0.975
0.883
0,810
EigenScore (Ours)
0.965 0.944 0.888
0.880
0.810
0.873
0.642
0.427
0.661
0.992 0.982 0.994
0.838
Baselines. We compare method against several generative baselines for OOD detection, in- cluding Improved CD (Du et al., 2021), DoS (Morningstar et al., 2021), TT (Nalisnick et al., 2019b), WAIC (Choi al., 2018), NLL,
Table 2: Near-OOD detection results (AUROC). We evaluate on semantically related datasets, including C10 vs. C100 and Tinylm ageNet, which porticularly challenging due to shared low-level statistics berween InD and OOD samples. The best and second best methods are highlighted. EigenScore achieves the best aver- age performance across both tasks, with a clear margin over prior dilfusion-buser upproaches.
IC. DDPM-OOD (Graham et al., 2023), LMD (Liu al., 2023). DiffPath V2 (Abdi et al., 2025), and DiffPath (Heng et al., 2024). Details regarding the baselines can be found in Appendix (B.4).
C10 vs.
C100 vs.
Avg
OOD
CTO0
ImageNet
CTO
Tiny ImageNet
DIPM-OOD
0.618
0.570
0.462
0.457
0.527
LMD
0.601
0.592
0,569
0.558
0.580
DiffPath
0.554
0.993
0.475
0,995
0.754
EigenScore
0.884
0.973
0.652
0.888
0.849
4.1
MAIN RESULTS
Table reports AUROC results across all dataset pairs. EigenScore achieves the highest average performance across all settings and is best second hest nearly every dataset pair. Its advantage is most pronounced in the challenging near-OOD regime (CIFAR-10 vs. CIFAR-100), where likelihood- based scores and trajectory metrics often fail. For example, EigenScore improves AUROC by up to 2% over the best diffusion- bbased baseline, consistent with our theoretical claim that retaining leading eigenvalues preserves discriminative structure.
On the near-OOD task (Yang et al., 2022) (C10 vs. C100/TinylmaeNNet), EigenScore continues to deliver strong separation, achieving higher average AUROC than other diffusion-based metrics in Table 2. In particular, while baselines such as NL..,, IC, and WAIC (C10 vs. C100) struggle to distinguish the closely related distributions, EigenScore consistently maintains reliable performance, showing its robustness even under challenging near-OOD settings. Notably, EigenScore outperforms DDPM-OOD, where reconstruction quality remains high but posterior uncertainty inflates.
4.2 ABLATIONS
Number of timesteps. The parameter T determines how many points along the diffusion trajectory contribute to the score. Larger T includes more noise levels but increases computation and even- tually saturates, since high noise merely lifts all eigenvalues uniformly (Lemma 1) without adding discriminative power. As shown in Table 3, even small budget (e.g., T=5) achieves nearly the same AUROC as larger T, with only marginal improvements from denser schedules. This confirms that a compact subset of timesteps captures most of the useful information, balancing accuracy and efficiency.
8

---

Table 3: Ablation on timesteps '. repetitions and number of eigenvalues K. AUROC performance of EigenScore across InD OOD dataset pairs.
CelebA vs.
C10 vs.
C100 vs.
SVHN vs.
Timesteps C10 C100 SVHN C100 SVHN CelebA
Avg
C10
CelehA SVHN
C10 C100 CelehA
0.964 0.943 0.893 0.869 0.778
0.866
0.635
0.448
0.648
0.990 0.977 0.995
0.834
7
0,963 0,942 0.878
0.848
0,703
0.862
0.619
0,500
0.603
0.988 0.975
0.996
0.823
10
0,952 0,931 0.850 0.819
0.627
0.867
0,578
0.589
0.521
0.987 0.975 0.998 0.808
CelebA vs.
C10
C100
SVHN
Repetitions
Avg
C10 C100 SVHN C100
SVHN CelebA
C10
CelchA
SVHN
C10
C100
CalebA
0.959 0,940 0.885 0.863
0.773
0.857
0.635
0.455
0.649
0.990
0.977
0.995
0.832
10
0.962 0,942 0.889 0.867
0.776
0.864
0.635
0,450
0.650
0.990 0.978
0.995
0,833
15
0.962 0.942 0.891 0,869
0.778
0.866
0.635
0,447
0.649
0.990
0.977
0,994
0.833
20
0.964 0.943 0.893 0.869
0.778
0.866
0.635
0.448
0.648
0.990
0.977 0.995
0.834
CelebA vs.
C10
C100 vs.
SVHN
Eigenvalues K
Avg
C10 CT00 SVIN C100
SVIIN CelebA
C10
CelebA
SVHN
CTO
CTO0 CelebA
1
0.968 0.950 0.945 0.871 0.803
0.830
0.639
0.432
0.713
0.983 0,966 0.983
0.840
2
0.967 0.946 0,919 0,872 0.793
0.852
0.636
0,439
0.679
0,987 0.972
0.991
0.838
3
0.964 0.943 0.893 0.869 0.778
0.866
0.635
0.448
0.648
0,990 0.977 0.995 0.834
Table 4: Comparison of MSE vs. EigenScore. Average AUROC across all InD-OOD dataset pairs. EigenScore consistently outperforms MSE by leveraging spectral structure of the uncertainty.
CelebA
C10 vs,
C100
SVIIN
Method
Avg
C10 C100 SVHN C100
SVHN CelebA C10
CelebA SVHN C10 C100 CelebA
MSE
0.804 0.783 0,220
0.629
0.184
0.841
0.552
0.675
0.147
0.994
0.994
0.652 0,977
1,000
EigenScore
0.964 0.943 0.893
0.869
0.778
0.866
0.635
0.448
0.648
0.990
0.995 0.834
Number of repetitions. The parameter sets how many noise draws are averaged per timestep. Larger reduces variance in the estimated eigenvalues and stabilizes OOD scores, but also increases run-
AUROC
time. As shown in Table 3, small values (e.g.. /=5) are already sufficient, with only marginal gains be- yond I=15. The results are reported with timesteps t E {100, 150, 200, 250, 300), mean aggregation, and K=3 eigenvalues.
0
200
400
300
Number of eigenvalues. The parameter K deter- mines how many leading eigenvalues are aggregated at each timestep. Table 3 shows that K= achieves the best average performance, though in some set- tings K=3 yields slightly higher AUROC. This in- dicates that the bulk of discriminative information resides in the first few modes, consistent with Lemma 1. The results are reported with the same timestep schedule, mean aggregation, and /=20 repetitions.
Figure 3: Ablation on eigenvalue informativeness across I. Performance declines with increasing noise, consistent with Lem. 1, while X retains the strongest OOD signal compared to xa and Xa. supporting Prop.
EigenScore vs. MSE. EigenScore consistently outperforms MSE by retaining the spectral structure of posterior covariance rather than collapsing uncertainty into a single scalar. As shown in Table this leads to higher AUROC across diverse dataset pairs. All experiments here use 20 repetitions, timesteps / E (100, 150, 200, 250, 300), and mean aggregation.
To further examine Lemma 1 and Prop. 2, we analyze the informativeness of individual eigenvalues across noise levels. Lemma predicts that at larger t, eigenvalues converge toward oū diminishing their discriminative value. Figure 3 confirms this: AUROC with λ decreases gradually t increases, while Aa and Ag deteriorate more sharply. Prop. further implies that the leading eigenvalues capture the most informative variance. Consistently, ê1 provides the strongest OOD signal. followed by X and then Ag. These results highlight why focusing on dominant modes, as done in EigenScore, yields more stable and informative detection than scalarized measures such MSE

---

5 CONCLUSION
We introduced EigenScore, a principled OOD detection method for diffusion models that leverages the spectral structure of denoising uncertainty. By linking KL divergence excess denoising error and showing that posterior covariance inflation consistently signals distribution shift, EigenScore offers both theoretical justification and strong empirical performance. Across diverse benchmarks, EigenScore consistently outperformed likelihood- robust gains near- hear-OOD settings where traditional approaches fail. Ablation studies further con- firmed that most discriminative information lies in the leading eigenvalues at moderate noise levels, validating our spectral perspective.
and score-norm methods, with particularly
LIMITATIONS
Despite its strengths, our approach has several limitations. First, FigenScore only leverages a subset of the information available in diffusion models: but discard eigenvector structure, which may contain additional discriminative cues. Moreover, we compute features at limited set of timesteps rather than across the full diffusion trajectory, potentially overlooking temporal dynamics of uncertainty. Second, our framework focuses on the magnitude of eigenvalues, but does not explicitly exploit their rate of change across eigenvalues, which itself may differ systematically between InD and OOD inputs and could serve as a detection signal.
the eigenvalues of the posterior covariance
REPRODUCIBILITY STATEMENT
We have taken several steps to ensure the reproducibility of our results. All datasets used in our experiments (CIFAR-10, CIFAR-100, SVHN, CelebA, l'nylmmgeNNet publicly available. We provide detailed descriptions of our training and evaluation protocols, including the diffusion model architecture, noise schedules, hyperparameters, and aggregation strategies. Experimental results averaged across multiple random seeds, and we report the effect of varying key parameters (number of timesteps, repetitions, and eigenvalues) in Section 4.2. To further facilitate reproducibility, we have released the full anonymous source code and scripts for running experiments in supplement B.5.
LLMs USAGE STATEMENT
During the preparation of this manuscript, we made limited use of large language models (LLMs), specifically OpenAl's ChatGPT, to assist with language refinement and organization some sections. All technical content, equations, derivations, and experimental design were developed entirely by the authors. The LLM was not used for ideation methods, data analysis, or generation results.
ACKNOWLEDGMENTS
This paper is partially based on work supported by the NSF under CAREER Awards CCF-2043134, 2046293 and Grants CCF-2504613, 2504614.
10

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A PROOFS
A. CONNECTING KL DIVERGENCE WITH DENOISING ERROR
Eq. (4) is inspired from the work of Song et al. (2021) [Theorem and Shoushtari al. (2025) [Theorem IJ, where KL divergence between two distribution is stated in terms of fisher divergence:
(p (1))
Bu,w,
[ |og V log (
or dt.
(12)
By using the Tweedie's formula from Eq. (2)
Dy(2r)) E 120 [0, xt + of pogp(
and replacing the score functions for distributions and with their corresponding MMSE estimators, we have:
DKL (1
|| [x]p]]]]]]]]]]] xer] B, [ac] (22222)) dt
||
E2 [2]| [|]] Ex ||
Epl 2e 1 j0- dt, (13)
where in the second line, we used the following decomposition:
ae ||ar
F. (a: [ma]
Ear,ar
||
Ep [a [m PEp |ac [ar]are]
By (ax [o:
Ex,
||
12 xx [1:
| - E, [ac (x 2
22,,, me
(xx Ep (a: mx))
EEp [2> [x]]]
E. (xee))
Emn,,
|| Ep [2 ae
[|mp|]]] - E. (ar ac: 10:
where in the last equality, we used the fact that
Ex,, (xc E [ac] [x)e) = 0, where a x~ p(x) and or ~ p(æx).
By replacing the result of this equation into Eq. (13)
Ez,z, [|B||=/,,]] Bg[ax/12)
[ [ Ex zz
||: Eg[ (2] 112]
Ex,,
| Eppl 2
MSE(D, t) MSE(D,.. t).
A.2 CONNECTING MSE OF DENOISING TO COVARIANCE
We assume that MMSE estimator D, is an optimal denoiser trained on images sampled from data distribution plx). Consequently, the MSE for this operator is defined
MSE(D,. t) = Ex,,, || Ea 1x = Ex Ez, tr(x Ex(xx/=:))((= [x(x)/2))) a
Ex tr [0vv][x/211]
where the first equality follows from the definition of MSE using the MMSE estimator D,(x;) = Ep az: [x],), and the second equality applies the law of total expectation along with the identity /013 ir( (((weT) T). the last equality, we used the fact that the inner expectation corresponds to the conditional covariance matrix Cov ( æz). whose trace gives the expected squared error. From Theorem of Manor & Michaeli (2024), we have
K
Covp[z/<xe] oZVD,((x)) ~ Σ (z)
(14)
15

---

where DD, is the Jacobian, assumed to be positive semi-definite and symmetric. We approximate the Jacobian matrix using the top K eigenvalues λ1. ...... Ax. nd their corresponding eigenvectors u1 u which is low-rank approximation of the full covariance matrix based on its spectral decomposition of symmetric positive semi-definite matrices. By retaining only the top K < terms in this expansion, obtain rank-K approximation of the covariance, capturing most of its variance while reducing dimensionality and computation. The trace of this approximate covariance is then given by
(r(Covplæaa))
(Z)))
(15)
1=1
since each term u & is rank-one projection matrix with unit trace. This expression allows to estimate the total variance (or the MSE) using only the top eigenvalues, under the assumption that the remaining eigenvalues contribute negligibly. Consequently, we have:
MSE(Dp, = EE tr (ovpl=/<:) = oz
X (2)).
(16)
A.3 MIYASAWA RELATIONSHIP
The connection between MMSE estimation under Gaussian noise and the score function was first es- tablished in Miyasawa (1961) and later generalized in Raphan Simoncelli (2011). For completeness, we provide derivation here.
Let 2t = x + 2, denote a noisy observation of æ ~ p(=),, where = ~ (N00, r) and define p(a; (z) as the Gaussian likelihood. The marginal distribution of y is given by:
p(x.)= = f p(x) p( dæ.
To derive the score function Va, logp(w;)) differentiate the log-marginal using the identity Vh(x,) = h(x,) V log hxx;):
/ 1 p(ar)p(ar(100) Vn, log p( (a)) p((x))
Va, log log)(
p(an(ar)) Vn, log pan (x))a
VV,, logp(x;,22) (x) ax).
which can be interpreted as a chain rule applied to score functions rather than densities.
Next, compute the Hessian 4 loz)) Differentiating again:
2 log p(æ;)) = / (()) (Vx, (ogp(al)x;,) logp(0; T + V2 logp((z)(2)) dæ.
Using Bayes' rule:
log mx)) log logp(ar, log log log logg(
and differentiating with respect to y, we obtain:
Vae (ogp(x/<)) Væ, log - V p(x;).
Substituting this into the previous expression yields:
2 pog)( / p(alax.) ((V,, (ogp(x;<<) - V logp((;)) logp(x)(())
ee log log((xx)(r)) (z)) da
E [(V., logg(()(<)) - logg((:t) logp(w, T
xe + 22 log,(ar, (a:) xx]]
Cov VV, log log/(x)/))) æ + V2 log 1(x,(()))) xx]]
16

---

Now, since p( (2) is Gaussian with variance of I, we have:
log ((g = 20 01 x1 + const,
V x; log ((x)(()((2)
(xt x)
We log (ogp(z;(())
I.
of
Plugging into the previous results gives the Miyasawa identities:
log)( (Ele [es] - x+)) 2 logp(æ.)
(17)
1
Covlæ [i] o
I.
(18)
Rearranging, we recover both the posterior mean and posterior covariance in terms of the score and its Hessian:
E[æ æx] = æt + of l(æ:))
(19)
Cov(æ æ = of (I + V² logp(x;))
(20)
Finally, the optimal denoising error is given by the expected trace of the posterior covariance:
E
||
E(x xx ]]]]] E [tr Cov|æ æx]].
A.4 PROOF OF LEMMA
Lemma 1 Let p(x;) = 11 denotes the noisy marginal distribution in Eq. (3) and (x( = of (I+ log from Miyasawa (Eq. (6)). As oe + cc, 2 logp(<;))] + uniformly compact sets. Hence
Ex(x;) = 00,1 + o
so all eigenvalues satisfy X₁ (z()== of o
Proof. Let Go, denotes the Gaussian density function with standard deviation o, 0. Then, we have
p(a,)) = p(a))/=)p(z)))) / Go, (Ete - x))(xx)
(21)
Differentiation under the integral yields
(22) I
(z. =- 02 Go(z),
Gz)
(2x 0²
Gz.
(22)
Hence
Ve,p(x:)= = (a: x) p(x) dæ,
V2 (2,p((;) =
V'G x) p(x) dx.
(23)
Fix a compact set K dR. Using Schwarz and that p has finite second moment, one obtains bounds of the form
Ci
VV'((>))l
C2
Yo K,
(24)
p(a.)
az
p(xx)
of
for constants C1, C2 independent of oz. Consequently.
p(xx) = V2p(x;)
Vp(x,) V
(25)
p(x:)
p(x:)²
17

---

satisfies
| logg(t))) C2 C
(
(26)
of
uniformly on K. Thus v + -0as as dt + x..
Plugging into
Ex(x)) of + of v logp(x;))
(27)
gives
E((x,) = of + o(o)),
(28)
and hence (xx)) of + o(o?) for all k.
11
This implies that at large noise, the posterior covariance becomes asymptotically isotropic; the spectrum flattens flattens and "small" eigenvalues are lifted to oî. Their contribution is therefore indistin- guishable from isotropic noise, which explains why including late timesteps or many tail eigenvalues does not improve OOD discrimination.
A.5 PROOF OF KY FAN'S THEOREM
Prop. Let Ex(e)) have eigenvalues M ... 2 X with eigenvectors forming U. [uf...., 3 For any K E (1,...,n)
K
tr (VIE,(I;)V)
= Σ AR-
max V VTV=Ig
k=1
and a maximizer is V = =[uf ux]]
Proof. Let V E R=nK with V = Ix. Write (x = U,AUU, with U orthogonal. Set Q == V Raxk Since U, is orthogonal, Q Q VTU,U.V Ix, so the columns Q are orthonormal.
We have
tr ((V" tr(QQ = Σ X 11:11
i=1
where of denotes the i-th row of Q (so |1::112 1 2 0). Because QTQ Ik.
Σ = tr( = K, and
11:1 < 1 for all i (each row is a subvector of a unit vector).
Thus the objective is linear functional of the nonnegative weights = 11:1 subject Σ. wy K and S w; < Since the eigenvalues are ordered X ... X², the sum Σ, Ao is maximized by assigning wy = for I,... K and wy 0 otherwise (rearrangemeettgreen) argument), which yields
(v(V Σ X
max vTV=Ig
This maximum attained by taking Q = [e....., j.i., V UU |uf. u J. the matrix of the top-K eigenvectors.
B ADDITIONAL DETAILS
B.1 COMPUTATIONS OF EIGENVALUES
To compute the leading eigenvalues of the posterior covariance efficiently, we follow the Jacobian-fre subspace iteration method introduced by Manor & Michaeli (2024). The algorithm approximates Jacobian-vecee vector products using finite differences of the denoiser output, followed by QR orthogo- nalization stabilize directions. Iteratively refining these directions yields approximate posterior
18

---

Algorithm 2 Efficient computation of posterior principal components (Manor & Michacli, 2024)
Require: N (Number of PCs), K (number of iterations), D(-) (MSE-optimal denoiser), (noisy input), of (noise variance), < 1 (linear approx. constant)
1: Initialize (8) Jam1 (0
2: for < to K do
3: for i to N do
(4)
Diy + +yy+cuk D(y cok-1))
4: k
2c
5:
end for
QR DECOMPOSITION ( (10) .... 1 11
6:
Q, +
7:
(1)
uk
(N)
Q
8:
end for
9: (1))
t K
10:
( - 1
D + +uk
- cok-)))
2c
principal components and their corresponding eigenvalues. This approach avoids the costly explicit Jacobian calculation while still capturing the dominant spectral structure of the denoising uncertainty, Algorithm reports the calculations of eigenvalues.
B.2 ON THE SPD ASSUMPTION FOR LEARNED DENOISERS
Our theoretical analysis relies on properties of the optimal MMSE denoiser, whose Jacobian corre- sponds to the posterior covariance and is therefore symmetric positive semi-definite (SPD). In practice, however, diffusion models employ learned denoisers that only approximate the true score function and not guaranteed to satisfy this property exactly. As result, the Jacobian of a learned denoiser may exhibit deviations from idcal covariance structure, including spurious or weakly negative directions.
This observation provides additional insight into the empirical behavior of different uncertainty- metrics. Scalarized measures such as MSE or trace-based scores aggregate contributions from all -igen-directions, making them more sensitive approximation errors the learned Jacobian. In contrast, EigenScore focuses on the dominant eigenvalues, which are more likely to correspond to meaningful uncertainty directions that persist under model approximation. From this perspective, top- K spectral truncation can be viewed as a robust estimator that better approximates the behavior of the true posterior covariance, even when the learned denoiser deviates from the ideal MMSE solution.
B.3 PARAMETERS FOR THE REPORTED PERFORMANCE
Our method maintains strong and stable performance even without hyperparameter adjustments, demonstrating that EigenScore is not sensitive the choice of T, J, K and generalizes well across datasets. The fixed setup uses k = 3. T {100, 150, 200, 250}. I = 20, and all aggregation.
B.4 BASELINES
We compare our method against several generative baselines for OOD detection, including Improved CD (Du et al., 2021), DoS (Morningstar et al., 2021), TT (Nalisnick et al., 2019b), WAIC (Choi al., 2018), NLL, IC. DDPM-OOD (Graham al., 2023), LMD (Liu et al., 2023). DiffPath (Heng et al., 2024). We use the official repositories for each method, along with diffusion models trained under the EDM framework (Karras al., 2022) and the Glow model (Kingma & Dhariwal, 2018). For the Glow-based baselines (DoS, TT, WAIC), we follow (Morningstar et al., 2021). In DoS, we extract three statistics-the log-likelihook latent log-probability, and Jacobian log-determinant- fit Kernel Density Estimators (KDE) on the training data for each, summing across statistics to obtain the final score. For TT and WAIC, we adopt the sample-wise versions from (Morningstar et al., 2021): TT measures the deviation of individual sample's 's likelihood from the training set average, while WAIC is computed from five independently trained models using the mean and variance of their log-likelihoods. For diffusion- based baselines, compute NI.I using the official implementation
19

---

EigenScore from OpenAl's improved diffusion Github repository' and derive IC by combining this NLL with PNG compression. Other methods (DDPM-OOD LMD', DiffPath*)) are implemented using their official repositories.
B.5 DATASET AND MODELS
We follow the official train/test splits when training diffusion models. For validation, randomly sample 500 in-distribution (InD) and 500 OOD images; for testing, we sample disjoint set 500 Inl and 500 OOD images. All images resized to 32 32. All diffusion models are trained using the EDM framework of Karras et al. (2022). The code can be found here
B.6 ADDITIONAL. EXPERIMENTS
B.6.1 COMPARISON WITH IMAGENET-IK
We have compared our method using ImageNet dataset InD and SVHN and Texture dataser as OOD, Table report the results.
Table 5: ImageNet- 1K dataset detection results (AUROC), We evaluate related datasets, including SVHN and Texture OOD samples. Note the competitive performance of EigenScore over prior diffusion-based approaches.
InD
ImageNet-IKK vs.
Avg
OOD
SVHN
Texture
DDPM-OOD
0.208
0.570
0.389
LMD
0.901
0.486
0.693
DiffPath
0.753
0.699
0.726
DifiPath V2 (Abdi et al., 2025)
0.547
0.860
0.703
0.755
0.709
0.732
B.6.2 COMPARISON OF CIO AND C100 WITH LSUN, ISUN, AND TEXTURES
We compare our method using CIFAR-10 and CIFAR-100 as InD datasets and LSUN, iSUN, and Textures OOD datasets. Table 6 reports the results.
Table 6: C10 and C200 datasets detection results vs. LSUN, iSUN, and Textures. Note the competitive performance of EigenScore DiffPathh
InD
C10 VS.
C100 vs.
Avg
OOD
LSUN
iSUN
Textures
LSUN
iSUN
Textures
DifiPath
0.618
0.911
0.780 0.756
0.609
0.869
0.761
0.758
EigenScore
0.977
0.974
0.877
0.879
0.627
0.848
B.6.3 ABLATION STUDIES ON PARAMETER C OF THE POWER ITERATION
We perform ablation study on the parameter used in the power-iteration step when computing the top-1 eigenvalue. We evaluate multiple values of while keeping all other settings fixed: we use the z-score detection method, mean aggregation across steps, and timesteps {100, 150, 200, 250, 300). The experiment conducted on CIFAR-10 (InD) versus CIFAR-100 (OOD). As reported Table
htcreeer/gghhb..ccooooo//////oee ffusion
Ywjjhu3am9y3_
20

---

the results show that EigenScore is stable across wide range of values, indicating that the method not sensitive to this parameter and that our default choice provides strong and reliable performance.
Table 7: Ablating Parameter
c
0.005
0.01
0.05
0.1
0.5
AUROC
0.846
0.844
0.850
0.847
0.866
0.875
B.6.4 SENSITIVITY OF EIGENSCORE DIFFUSION TIMESTEP
Figure 6 shows the OOD detection performance for the CIFAR- 100 vs. CelebA pair across all diffusion timesteps. The results reveal that this pair exhibits peak separability at later timesteps (approximately 400-500), where the posterior covariance inflation between the two datasets becomes most pronounced. Earlier timesteps (100-300), used in the ablation table, do not reach this peak region, explaining the lower performance observed there. This behavior reflects dataset-specific sensitivity to the diffusion trajectory rather than instability, and EigenScore consistently improves once the anisotropic covariance modes dominate.
OOD Detaction Performance Across Timesteps (ID) (OOD)
AUROC
0.51
0.24
Figure 4: Sensitivity of EigenScore to diffusion timestep for CIFAR-100 vs. CelebA. We plot OOD detection performance across diffusion timesteps. This pair exhibits peak separability at later timesteps (approximately 400-500), where posterior covariance inflation between ID and OOD samples is most pronounced. Earlier timesteps (100-300), used the ablation study, do not reach this peak region, explaining the lower performance observed there.
21

---

Dimnersion
6.0000
0.0003
6.0004
0.0002
0.0002
0.0001
0.0037
0.0000
0.0009
0.00%
Dimension
Dimersion 5
Dimersion
6,0003
0.000100
WAN
6.0002
0. 0.002059
0.0001
0.0000
0.002000
-5000
-2500
2520
5000
50000
100000 150000 200002
-100000 -200000
Xio Hll.
Nissele His
Looseles Eily
Figure 5: DiffPaat feature distributions for SVHN (ID) CelebA (OOD). Each of the six DiffPath dimensions shows substantial overlap between Il) and OOD histograms, indicating that the metric provides weak separability for this pair.
CelebA vs. C10
CalebA va. C100
CelebA SVHN
MUROC
C1000 ve. SVHN
C100 ve. C10
C100 CelebA
AUROO
Timestace
Figure 6: AUROC as a function of diffusion timestep (100-700) for the largest three eigenvalues (K 1, 2, 3) across 6 ID/OOD pairs. The plots illustrate how OOD separability evolves along the diffusion trajectory and highlight the timestep regions where additional eigenvalues improve or saturate performance.
B.6.5 TIME/NFE COMPARISON
To contextualize computational cost, Table reports the number of function evaluations (NFE) and wall-clock time for key baselines and our method per image.
Table 8: Time/NFE comparisons
Method
DiffPath
LMD
DDPM-OOD
EigenScore
NFE
10
10000
350
300
Time (8)
0.1s
1.8s
0.53s
1.9s
B.6.6 RIGH-REEOLLIION OOD DETECTION EXPERIMENTS (256x256)
We evaluate EigenScore on high-resolution and large-scale image datasets. We conduct additional OOD detection experiments using 256 DDPM (Baranchuk et al., 2022). We evaluate OOD)
http:s////githhbb.om///eeeeete resea rch /ttp//ddggeeeettttt...
22

---

performance AFHQ (Choi al., 2020) and Microscopy (CHAMMI) (Chen et al., 2023). are reported in Table 9.
Table 9: OOD detection results on high-resolution 256x256 models. In-distribution dataset is FFHQ-
256. OOD datasets are AFHQ and Microscopy (CHAMMI).
Method
AFHQ
Microscopy
Avg
LMD
0.485
0.552
0.5185
DiffPath
0.593
0.998
0.7955
EigenScore
0.532
0.899
0.7155
23


Reviews:
[{"review_id": "qZ0gWIi1i1", "reviewer": "Reviewer_DEm7", "summary": "The paper proposes an unsupervised OOD detector for diffusion models that uses the leading eigenvalues of the posterior covariance. \nEigenScore leverages the covariance structure of the denoising process to capture uncertainty signals, which is theoretically grounded and interpretable.\nEmpirically, EigenScore achieves state-of-the-art performance", "questions": "see weakness", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "1. Generally clear and readable; \n\n2. figures and tables are informative; \n\n3. Codes are publicly available.\n\n4. Investigate OOD detection for Duffusion models is timely.\n\n5. the use of Jacobian-free eigenvalue estimation algorithm is new to me.", "weaknesses": "1. lack of experiments on large-scale dataset e.g. ImageNet-1K\n2. when c10 and c100 as ID data, SVHN LSUN iSUN Textures Places365 are standard OOD dataset in the literature of OOD detection. but the corresponding results are missing\n3. lack of time computation analysis since QR requires O(n^3) computation complexity\n4. this paper only introduce 1 baseline published on/after 2024. more advanced baseline should be included.\n5. given a ID dataset, the optimizied hyper-parameter setting are not identical for OOD datasets.\n6. The paper  mostly restates these connections and asserts that “inflation” will occur OOD; but formal conditions ensuring consistent ordering for individual samples (beyond in-expectation statements) are not established.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "h4kw9gYHZY", "reviewer": "Reviewer_sUXt", "summary": "The authors propose EIGENSCORE, an unsupervised framework for out-of-distribution (OOD) detection using uncertainty estimates derived from the posterior covariance of diffusion models during denoising. They highlight the limitations of likelihood and score-based OOD methods and establish a connection between the KL divergence of in-distribution (InD) and OOD data and the MSE of the optimal denoiser. Since denoising error (MSE) is higher for OOD inputs, they express this MSE as the trace of the conditional (posterior) covariance, which is approximated via the Jacobian of the denoiser using the Miyasawa identity. Assuming a symmetric positive semi-definite Jacobian, they perform an eigenvalue decomposition and use the inflation of its trace (the sum of eigenvalues) as the OOD metric for several timesteps. They further show that top-K eigenvalues provide stronger OOD discrimination than using all eigenvalues. EIGENSCORE achieves competitive AUROC results across multiple datasets (CIFAR-10, CIFAR-100, CelebA, SVHN) and shows promising performance on near-OOD detection, supported by ablations over key hyperparameters.", "questions": "Please see weaknesses above.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper clearly lists the limitations of existing diffusion-based OOD frameworks with evidence in near-OOD scenarios as well. and highlighted theoretically why the proposed method maintains consistent ordering in tasks. \n\n\nThe paper first lists the existing work, their limitations with evidence, an overview of their method, and a logical flow of the framework supported by theory and empirical evidence.\n\nThe paper combines existing theoretical concepts and intuition about covariance and eigenvalues, and applies it for diffusion posteriors.\n\n\nThe method performs better compared to different SOTA OOD detection framework paradigms. and shows promising results in a near-OOD scenario, an improvement compared to the previous diffusion-based SOTA, DiffPath.", "weaknesses": "It seems, the theory, proposition 1, and equations 5-8, which directly talk about MSE instead of top-K, predict good performance, but in Table 4, MSE performs poorly in some tasks (worse than random). This is a disconnect between prop 1 and equations 5-8, and lemma 1 and Proposition 2. Lemma 1 attempts to justify use of top-k but difference in performance on MSE (supported by prop 1) and top-K (supported by lemma 1) seems big. Can you please clarify?\n\n\nThe assumption that covariance or jacobian of diffusion model is symmetric positive semi-definite might be strong. Can you please clarify?\n\n\nThe method involves explicit training of diffusion model on InD data for OOD detection, whereas previous SOTA, DiffPath, uses diffusion model trained on different/generic dataset (ImageNet or CelebA). \n\n\nComparing table 1 (main result) and table 3 (ablation), for C100 vs CelebA, there is a difference in hyperparameter setting (timesteps are different). C100 vs CelebA in table 3 performs poorly compared to the same pair in table 1. Table 1 for this pair uses high noise levels (450, 500) and table 3 is for (100-300). For the same pair, MSE performs better compared to EIGENSCORE in table 4, which for higher noise values, according to lemma 1, should perform badly. More analysis on this could help. \n\n\nC100 vs CelebA in table 1 uses higher timesteps (450, 500) as compared to other input pairs (100-300), is there any hyperparameter sensitivity? Does higher noise help better performance in table 1?\n\n\nWhy does MSE fail so catastrophically (AUROC < 0.5) when theory predicts it should work? Can you please provide an explanation that reconciles Proposition 1 with Table 4?\n\n\nCan  you please mention the diffusion models used?\n\n\nDiffPath gets 0.328 AUROC, which is way less than their original paper, any insights would be great.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1GP5yn6zcY", "reviewer": "Reviewer_54fH", "summary": "This paper introduces EigenScore, a diffusion-based method for out-of-distribution detection. The core idea is that when a denoiser trained on in-distribution data is evaluated on OOD inputs, its denoising error increases. The authors leverage the fact that the MMSE denoising error corresponds to the expected total posterior variance of the clean image, and that this posterior variance can be expressed through the denoiser’s Jacobian via Miyasawa’s identity. Since the trace of the covariance equals the sum of its eigenvalues, inflation of the leading eigenvalues provides a practical surrogate for detecting excess uncertainty induced by distribution shift. Based on this intuition, EigenScore estimates the top K eigenvalues at selected low-to-medium noise levels and aggregates them into an OOD score. Experiments on standard benchmarks demonstrate improved behavior over norm-based diffusion OOD metrics, with consistently strong performance and particularly clear gains in near-OOD scenarios.", "questions": "1. Hyperparameters such as K, T, I, and the finite-difference step size c appear to be tuned specifically for each ID–OOD setting. Could you report results using a single fixed configuration across all experiments to better evaluate generalization when the nature of the OOD data is not known in advance.\n\n2. Could you include runtime and memory comparisons with classifier-based or other diffusion baselines.\n\n3. The finite-difference step size c influences Jacobian estimation quality and numerical stability, yet its impact is not analyzed. Can you comment on how sensitive EigenScore is to this parameter.\n\n4. Could you provide a denser analysis over diffusion timesteps to better understand which parts of the trajectory contribute most to OOD separation. For example, reporting AUROC as a function of t on a finer noise grid, and including a heatmap over (t × K) to visualize where additional eigenvalues help versus hurt. Evaluating whether restricting the score to the most informative t values improves robustness would also clarify the design choices for T.\n\n5. How does EigenScore perform on more complex and high-resolution OOD benchmarks, for example ImageNet vs Places365 or SUN (OpenOOD).", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **Novelty and strong theoretical motivation**.\n\n     The paper presents a new OOD detection metric based on the spectral properties of the diffusion denoiser’s Jacobian. The theoretical path, from excess denoising error to posterior variance and then to dominant eigenvalues, is well established and provides a compelling motivation for the method.\n\n- **Clear and well-structured presentation**.\n\n     The paper is easy to follow, with a logical progression from the theoretical insights to the algorithmic design and experimental validation.\n\n- **Consistent and competitive empirical performance**.\n\n     The method performs reliably across a variety of OOD detection scenarios, showing steady improvements over diffusion norm-based baselines. These results suggest that the proposed spectral metric captures a stable indicator of distribution shift.", "weaknesses": "- **Sensitivity to multiple hyperparameters.**\n\n    The method depends on several tunable parameters (K, T, I, aggregation rule, finite-difference step size c) and performance appears fragile when deviating from tuned configurations.\n\n- **Tuning strategy may limit real-world applicability.**\n\n    Hyperparameters are selected using ID–OOD validation splits, which assumes prior knowledge of the exact OOD distribution being evaluated. This setup does not reflect realistic deployment scenarios where the nature of the shift is unknown. Demonstrating strong performance with a single default configuration that does not rely on ID–OOD validation would improve the method’s practical usability.\n\n- **Computational efficiency not addressed.**\n\n    Repeated denoiser evaluations and eigenvalue estimation introduce substantial computational overhead, yet there is no runtime or throughput comparison with alternative detectors.\n\n- **Limited evaluation scope.**\n\n    The experiments use relatively small and low-resolution datasets. The effectiveness of EigenScore on larger-scale OOD tasks, for example ImageNet-based settings, remains unverified.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ooh1UEQjfG", "reviewer": "Reviewer_s9qH", "summary": "This work proposes to adopt the posterior covariance in diffusion models for out-of-distribution (OOD) detection. To be specific, through theoretical and empirical results, earlier diffusion-based scores, like likelihood and score dynamics, are shown to be not reliable enough. Then, in the proposed EigenScore, during the reverse diffusion process, the aggregated eigenvalues of the intermediate diffused results are leveraged to capture the differences between OOD and InD (in-distribution) as a detection score. Experiments verify the effectiveness of EigenScore in some small-scale low-resolution images.", "questions": "Questions correspond to the three weaknesses above. I will raise my rating if all the concerns are well addressed.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 1, "presentation": 3, "contribution": 3, "strengths": "1.\tExploring the posterior covariance in the reverse diffusion process is novel and interesting, which contributes new insights into diffusion-based OOD detection.\n2.\tExtensive theoretical results are provided, which is appreciated.\n3.\tTo enhance the computational complexity, acceleration techniques are introduced into the proposed EigenScore", "weaknesses": "1.\tAbout the validation part of EigenScore  \n\nThe proposed EigenScore requires a validation set, which further requires OOD samples to tune some hyper-parameters such as the selected time steps. In this sense, the selection of OOD samples directly affects the evaluation fairness. That is, if the OOD samples in the validation set are exactly from the same dataset in the test phase, then such a setup is not a fair one for comparisons, since EigenScore has “saw” OOD samples during validation. The authors are suggested to specify how the OOD samples in the validation set are collected. Those OOD samples for validation should be from an additional dataset that is different from those in the test phase.\n\n2.\tExperiments are not sufficient enough  \n\nThe included datasets in experiments are all small-scale low-resolution image datasets, such as the 32x32 CIFAR10 and CIFAR100. Nevertheless, note that the powerful generation ability of diffusion models is mainly reflected on large-scale high-resolution images. Besides, in existing OOD detection methods, experiments on the large-scale ImageNet-1K dataset are essential. Considering the two aspects, it is highly recommended for the authors to supplement results on the large-scale 224x224 ImageNet-1K dataset in order to better demonstrate the spectrum differences between InD and OOD from diffusion models, which could provide an even comprehensive evaluation and further strengthen this work.\n\n3.\tSome reconstruction-based OOD detection methods that do not rely on generative models should be cited and discussed for an enriched literature review. For example, in [a-c], proper subspaces are identified via (kernel) PCA and reconstructions are executed in features and gradients to distinguish OOD from InD. It would be much appreciated to also provide comparisons with [a-c].\n\n[a] Revisit pca-based technique for out-of-distribution detection. ICCV 2023.  \n[b] Kernel PCA for out-of-distribution detection. NeurIPS 2024.  \n[c] GradOrth: A Simple yet Efficient Out-of-Distribution Detection with Orthogonal Projection of Gradients. NeurIPS 2023.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)