Optimal Transport-Induced Samples against Out-of-Distribution Overconfidence
Authors: Keke Tang
Ziyong Du
Xiaofei Wang
Weilong Peng
Peican Zhu
Zhihong Tian
Venue: ICLR 2026 Poster (2026)
Keywords: out-of-distribution
overconfidence
optimal transport
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: Deep neural networks (DNNs) often produce overconfident predictions on out-of-distribution (OOD) inputs, undermining their reliability in open-world environments. Singularities in semi-discrete optimal transport (OT) mark regions of semantic ambiguity, where classifiers are particularly prone to unwarranted high-confidence predictions. Motivated by this observation, we propose a principled framework to mitigate OOD overconfidence by leveraging the geometry of OT-induced singular boundaries. Specifically, we formulate an OT problem between a continuous base distribution and the latent embeddings of training data, and identify the resulting singular boundaries. By sampling near these boundaries, we construct a class of OOD inputs, termed optimal transport-induced OOD samples (OTIS), which are geometrically grounded and inherently semantically ambiguous. During training, a confidence suppression loss is applied to OTIS to guide the model toward more calibrated predictions in structurally uncertain regions. Extensive experiments show that our method significantly alleviates OOD overconfidence and outperforms state-of-the-art methods.

Published as a conference paper at ICLR 2026
OPTIMAL TRANSPORT-INDUCED SAMPLES AGAINST DUT-OF-DISTRIBUTION OVERCONFIDENCE
Keke Tang',, Ziyong Du' Xiaofei Wang2.3r Weilong Peng', Peican Zhu ZZihong Tian 'Guangzhou University, "SmartMore Corporation Univeesiity of Science and Technology of China, "Northwestern Polytechnical University 'Guangdong Key Laboratory of Industrial Control System Security tangbohutbhh@gall com, duxiaoshuaicst@@ge wxf9545 1044mmmall..mm westccedu... cn, wlpengetju. edu, er ericcan@nwwpr edu. tianzhihong@gzzuu edu.cn
com
ABSTRACT
Deep neural networks (DNNs) often produce overconfident predictions on out- of-distribution (OOD) inputs, undermining their reliability in open-world envi- ronments. Singularities in semi-discrete optimal transport (OT) mark regions of semantic ambiguity, where classifiers are particularly prone to unwarranted high- confidence predictions. Motivated by this observation, we propose a principled framework mitigate OOD overconfidence by leveraging the geometry of OT- induced singular boundaries. Specifically, we formulate an OT problem between a continuous base distribution and the latent embeddings of training data, and identify the resulting singular boundaries. By sampling near these boundaries, we construct class of OOD inputs, termed optimal transport-induced OOD samples (OTIS), which are geometrically grounded and inherently semantically ambigu- ous. During training, a confidence suppression loss is applied to OTIS to guide the model toward more calibrated predictions in structurally uncertain regions. Extensive experiments show that our method significantly alleviates OOD over- confidence and outperforms state-of-the-art methods.
INTRODUCTION
Deep neural networks (DNNs) have achieved remarkable success in classification problems (He et al. 2015 Tang et al. 20256). However, they are typically trained and evaluated under the closed- world assumption that all test inputs are drawn from the same distribution as the training data (Ben-) dale & Boult 2015). In open-world scenarios, this assumption often breaks down, as inputs from previously unseen classes or domains naturally arise. When confronted with such dittrf-disttibutionn (OOD) inputs, DNNs tend to produce confident predictions despite lacking relevant experience. This overconfidence has been extensively documented (Nguyen et al. 2015 Goodfellow et al. 2015)) and poses serious risks in safety-critical applications. Mitigating such overconfidence therefore fundamental step toward reliable deployment of DNN systems in open-world environments.
To address OOD overconfidence, widely adopted strategy is perform OOD detection test time, where confidence-basee scoring functions are used to separate in in-distribution (ID) and OOD inputs (Hendrycks & Gimpel| 2017, Liang et al. (2018 Tang et al. 2023a b 2024 2025a Zhao al,
2025 Yang et al.] 2025] Fang et al. [024|] 2025a blc). However, such post-hoc filtering merely avoids high-confidence mistakes without fundamentally altering the model's tendency produce overconfident predictions on unfamiliar inputs. As more proactive alternative, recent work has explored exposing the model to proxy OOD samples during training, encouraging it to produce low-confidence predictions
inputs outside the training distribution. These proxy samples typically constructed using external datasets (Hendrycks et al.) (2188). input corruption (Hein et al.] 2019), class mixing Tang et al. 2021), ) latent outlier synthesis (Du al.) (002). Despite their empirical success, these heuristically designed samples often lack theoretical grounding and fail to target semantically ambiguous regions where overconfidence is most likely to occur.
"Equal contribution 1 Corresponding authors

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Published as a conference paper at ICLR 2026
(a)
(b)
(c)
OT Mapping
Sampling
Singular Boundary
Neither Gat
Nor Fox!
Continuous Base Dist.
Discrete Target Dist.
Semantic Ambiguity Set
Figure 1: Given a remi-discrete optimal transport (OT) map from (a) a continuous base distribution to (b) discrete target distribution over images, the singular boundaries in the source domain mapped to (c) the semantic ambiguity set in image space, which typically contains images with features from multiple classes.
promising direction overcome these limitations is exploit the geometric insights offered by optimal transport (OT) (Villani et al. 2008; Santambrogio, 2015 Mérigot 201T), which provides a principled framework for understanding data structure and distri outional uncertainty. In the semi- discrete OT setting, a continuous source distribution (e.g.. Gaussian noise) is mapped onto a discrete target measure via the gradient of a convex potential. This potential induces a partition of the source domain into convex regions, each assigned to discrete target point. The interfaces between adja- cent regions correspond to non-differentiable loci the potential, known as transport singularities, where the direction of transport changes abruptly (Figalli) 2010 An et al. (2200). These singular boundaries mark structurally unstable zones the transport map and often coincide with seman- tically ambiguous regions in classification models, where decision behavior becomes less reliable (see Fig. [ As such, these regions provide a theoretically grounded signal for identifying where overconfidence may arise, offering compelling basis for model regularization.
Building on this insight, we propose principled framework for mitigating OOD overconfidence by exploiting the geometric singularities arising in semi-discrete optimal transport. Our approach first encodes ID samples into compact latent space via an autoencoder. A semi-discrete OT problem is then solved between a continuous base distribution, c.g., a Gaussian uniform distribution, and the discrete set of latent embeddings, inducing convex partition of the latent space. Among the adja- cent regions, those with large angular deviations in transport direction are identified as structurally unstable zones. To construct ambiguous inputs, interpolate between the centroids of neighbor- ing regions associated with these zones, and decode the resulting latent vectors back to the input space. The generated samples, termed optimal transport-induced OOD samples (OTIS), are used during training with a confidence suppression loss, guiding the model to produce low-confidence predictions structurally ambiguous regions. Extensive experiments across multiple architectures and diverse ID/OOD settings demonstrate that our method effectively reduces OOD overconfidence without sacrificing IDD accuracy, consistently outperforming state-of-the-art approaches.
Overall, our contribution is summarized as follows:
We establish a theoretical link between geometric singularities in semi-discrete optimal transport and the emergence of overconfident predictions on OOD inputs.
We develop a novel OOD overconfidence mitigation framework that regularizes model confidence using semantically ambiguous samples derived from OT-induced singularities.
We demonstrate through experiments across multiple architectures and ID/OOD settings that our approach outperforms state-of-the- methods in mitigating OOD overconfidence.
PROBLEM FORMULATION AND MOTIVATION
2.1 OOD OVERCONFIDENCE
Preliminaries OOD Overconfidence Issue. We consider standard multi-class classification task with label space {1, 2,..,... K}. .)........ Let A denote the space of all candidate inputs, and let I C A be the in-distribution (ID) subset. The set of oft-off-istributee
(OOD) inputs is given by A I. A classifier of A + {1...., ., K} } is trained on I, but may produce overconfident predictions on inputs from A I, compromising reliability in open-world deployments.
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Mitigating Overconfidence via Suppressing Predictions Proxy OOD Samples. A widely adopted strategy to mitigate OOD overconfidence is to expose the model to set of proxy OOD samples O - A \ I during training. These samples are typically constructed by injecting noise, applying input corruptions (Hein et al. 2019). performing class mixing (Tang et al. (021), or sam- pling from unrelated datasets (Hendrycks al. 2018). The objective encourage the model to produce low-confidence predictions on unfamiliar inputs. The corresponding regularization loss is often formulated as
(1) where s (x)) denotes a confidence score (e.g., the maximum softmax probability). However, exist- ing methods typically construct O with heuristic rules and don't 't explicitly target structurally unstable or semantically ambiguous regions where overconfident predictions are most likely to occur.
praxy = Ex
2.2 SEMI-DISCRETE OPTIMAL TRANSPORT
Semi-discrete optimal transport (OT) models the alignment between continuous source measure and a discrete target distribution. It naturally induces geometric partition of the source domain, providing structured view of how samples are assigned to semantic prototypes.
Preliminaries on Semi-Discrete Optimal Transport. Let fs be an absolutely continuous proba- bility measure supported on a convex domain f2 S Rd, such as a Gaussian or uniform distribution. Let the target domain be a discrete set {y1,.... 8n } S Rd equipped with probability measure E w; where W; 2 0 and Σ, 10 =
The semi-discrede optimal transport problem seeks a measurable map T Ω - I2 that pushes onto v/ i... T μ and minimizes the expected transport cost:
1 T(2) =||² du((),
where: 2 E Ω denotes a source point.
According to Brenier's 's theorem (Brenier 1991), if μ is absolutely continuous, the optimal transport map T exists and given by the gradient of a convex function:
T(:) (2), ux(2)
max (v,, =) + h.),
where h = } are scalar offsets. The function un is the upper envelope of affine functions, forming piecewise-linear convex surface over Ω. To ensure identifiability, a normalization condition such as Σ, 142 = 0 is typically imposed.
Partition Boundaries Induced by Optimal Transport The convex structure of un induces partition of the continuous source domain Q into convex Laguerre cells, such that Ω = U, W, where
W: = {= E Ω (ui, 2) + h4 2 (u). + h2, i)
(4)
These cells form power diagram over Q, generalizing Voronoi diagrams by incorporating the off- sets {ha}. Each adjacent pair (W,, W3) shares boundary hyperplane:
S f= S (a.j, =) bij =], aij yj, bij h: h1j
(5)
These transport-induced boundaries delineate semantic transitions and provide the foundation for constructing structurally ambiguous samples.
2.3 OT SINGULARITIES AS SOURCES OF SEMANTICALLY AMBIGUOUS OOD SAMPLES
According to the regularity theory of semi- mimi-discrete optimal transport (Figalli 2010 Chen & Figalli, 2017], the Brenier potential 4p becomes non-differentiable on a singular set Σ C Q when the target measure is multimodal or disconnected. The transport map T(z) Vun(z) (z) is discontinuous across Σ, leading to abrupt changes in transport direction.
Although Σ has measure zero, its neighborhood contains inputs near multiple transport boundaries, where assignments are unstable and semantic alignment ambiguous. These regions often corre- spond to high-confidence mispredictions and indicate heightened robustness risk.
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Published as a conference paper at ICLR 2026
Upper Envelape Uh
Singular Boundary
B)
Enc
up
Mj y/u
OT map
2j
Dec
Dec
W
w
Input
Latent feature
Mapping
Synthesized Feature
OTIS
Overview of our framework for generating OTIS. Input images are encoded space, where semi-discrete optimal transport (OT) map establishes power-diagram partition with its convex potential visualized as the upper envelope. Singularity boundaries identified from this OT map guide the generation of interpolated latent features, which are decoded to synthesize OTIS.
Figure 2:
latent
Therefore, by setting the OT target distribution to the support training images, we can construct transport-induced singularities and map from their vicinity back the input space, yielding semanti- cally ambiguous samples near class transitions. These samples serve as proxies for OOD uncertainty and offer principled basis for training-time regularization mitigate OOD overconfidence.
METHOD
This section presents our framework for alleviating OOD overconfidence by exploiting the structural properties of OT-induced boundaries introduced in Sec. 2.2 The proposed pipeline first encodes input samples into compact latent space, then constructs optimal wraspprt-inducedd OOD samples (OTIS) based on the geometry of the OT partition, and finally mitigates overconfident predictions by training the classifier with these ambiguity-aware samples. Please refer to Fig. 2 for demonstration.
3.1 LATENT REPRESENTATION FOR OT
To facilitate optimal transport modeling a compact geometry, we construct a latent representation space )' C IRd using an autoencoder. Each ID sample x E I is encoded by a neural encoder Enc into a latent vector:
y = Enc(x).
(6)
The resulting y lies in the latent space D. which is designed to be geometrically structured and compact. To enable reconstruction and downstream training, a decoder Dec maps latent vectors back to the input domain:
Dec(y).
We denote by {2:) I set of ID training samples, and let their latent embeddings {y: Enc(x;) ) serve as support points for the discrete target measure y = Σn wyS,, in the semi- discrete optimal transport formulation. Modeling in latent space improves regularity and tractability compared to operating in the original input domain.
3.2 CONSTRUCTION OF OTIS
Partition Estimation via Potential Optimization. Building on the semi-discrete OT formulation
introduced Sec. 2.2 we aim to construct the transport-induced partition in the latent space by estimating the yı-ollme of each Laguerre cell. These volumes characterize how the source measure distributes mass across regions associated with the support points {
We estimate the y-volume of each Laguerre cell (h) using Monte Carlo strategy. Specifically, a and assign each point to the region that maximizes the potential: (2z) = arg max; (y). 27) + hx). The proportion of points assigned to each region yields empirical estimate of its p-volume: ((W)(h)) #tJ (x,))((
we sample M latent points {

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Published as a conference paper at ICLR 2026
To compute the optimal offsets h, minimize the discrepancy between estimated μt- v-volumes and a target measure / E, wydy. using uniform weights wA 1/n:
min
{(h) Σ (a(W(h))) - wx) s.t. Σ h4 -.
(8)
After convergence, each region pair (ye, Uj) defines a boundary S.j. and we collect S = Uies Sij as candidate boundary set.
Identification of Singular Boundaries. We operationalize the concept of transport singularities by assigming a geometric score to each OT-induced boundary and selecting those most likely to exhibit discontinuous unstable behavior. Given the candidate boundary set S constructed from adjacent support points in the OT partition, we quantify the angular deviation across each boundary Siy as
( (y6, Mi) [Tall |1
score(S.g) arccos
This score reflects the change in transport direction between adjacent cells. Larger values suggest sharper directional shifts and greater likelihood of singularity. We sort all boundaries by score and retain fixed top-ranked proportion to form the singular boundary set S' S S, which is used to guide ambiguity-an aware sample generation in the latent space.
Synthesis of Optimal Transport-induced OOD Samples (OTIS). For each selected boundary Sig E S', we estimate the mass centers é and ég of the corresponding Laguerre cells W; and W, via Monte Carlo approximation. Specifically, each center is computed as:
Σ SWE.
Zh: with 2 nr, LE (i,}}.
(10)
# W.)
We then sample latent point 2 ~ μ and compute inverse-distance interpolation weights:
1/1l= è,1 X 1/|| è4 + 1/| àgll'
b =
(11)
To ensure continuity near transport-induced boundaries, we adopt a smoothed transport extension T(-), defined as:
(12)
This extension provides softened approximation of the discrete OT map in structurally ambiguous regions, reducing aliasing artifacts and enhancing the semantic coherence of generated features.
Finally, decode ý via å Dec(ÿ), and include it in the proxy OOD set Oor- These samples, termed optimal transport-induces OOD samples (OTIS), are used during training to suppress over- confident predictions near OT-induced singularities.
3.3 TRAINING WITH OTIS FOR MITIGATING OOD OVERCONFIDENCE
To mitigate OOD overconfidence, we incorporate OTIS, è E Oor, which serves as proxy OOD inputs generated near OT-induced singular boundaries, into the training of DNNs. We regularize model predictions on these inputs using a confidence suppression loss following (Tang et al. 2021):
Sxup (t)
log Vi($),
(13)
where Vi(it) denotes the softmax probability for category i. This loss encourages the model to distribute its confidence evenly across all classes, thereby enforcing averaged uncertainty and sup- pressing overconfident predictions on ambiguous inputs.
During training, each batch consists 50% ID samples supervised with cross-entropy loss and 50%
OTIS guided by the suppression loss in Eqn. 13
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Comparison of eight methods in mitigating OOD overconfidence. We report test error (TE), mean maximum confidence ID (ID MMC 1) and OOD (OOD MMC 4) inputs. All values are in percent (%). Note: OE and CCUd leverage auxiliary datasets, while others do not.
Table I:
ID
metric
0OD
CEDA ACET CCUs CODES
vOs
Ours
OE
CCUd
with auxiliary dataset
TE
CIFAR-IB
5.79
8.55
6.87
5.86
7.73
5.81
7.52
6.80
5.55
ID MMC.
CIFAR-TO
97.98
96.06
96.84
97.50
93.46
97.14
95.46
96.80
97.30
SVHN
84.22
71.62
82.16
72.48
72.35
73.16
13.18
55.82
76.52
CIFAR-I00 100
85.98
80.18
82.36
75.95
74.69
81.04
64.79
80.94
81.49
CIFAR
LSUN_CC
77.18
60.70
65.15
55.93
51.64
70.5
30.36
68.61
79.49
Textures_C
86.28
72.47
78.58
59.01
61.32
78.5
48.75
67.66
74.35
OOD MMC
Noise
85.55
60.51
64.74
50.61
43.70
51.38
16.18
10.13
74.95
Uniform
87.45
10.04
10.00
10.00
11.13
80.65
10.00
26.20
10.00
Adv. Noise
98.90
43.04
10.00
10.00
37.66
95.56
26.42
91.15
10.00
Adv. Samples
95,45
69.44
26.27
84.20
92.37
95.85
57.71
88.86
98.00
TE
CIFAR-I00
23.03
30.96
25.95
50.66
25.68
23.56
27.73
25.63
23.83
ID MMC
TO0
82.96
80.36
83.75
56.43
87.76
85.41
83.90
84.02
82.01
SVHN
48.27
63.03
62.85
65.49
66.11
58.76
9.30
52.57
58.62
CIFAR-II 100
CIFAR-10
56.34
62.65
65.36
45.70
68.96
62.29
61.88
65.01
56.34
LSUN.CE CR
55.01
51.34
50.35
51.82
41.85
58.02
40.64
50.11
56.98
Textures.C C
57.42
66.69
66.26
58.30
63.77
61.96
$5.89
60.66
56.14
OOD MMC
Noise
92.17
82.99
78.94
68.00
80.15
92.92
71.61
1.07
70.59
Uniform
74.76
1.09
1.00
1.00
1.10
72.44
1.00
3.25
7.38
Adv. Noise
86.47
23.58
1.00
1.00
27.85
77.58
8.94
67.36
46.52
Adv. Samples
92.86
19.35
2.25
51.16
87.88
84.01
13.71
47.75
63.88
TE ID MMC
SVHN
3.68
4.66
3.50
8,42
4.29
3.51
4.83
3.53
3.17
SVHN
98.41
97.17
97.78
90.64
93.35
97.99
96.25
97.67
98.50
CIFAR-10
75.15
73.70
62.54
46.92
61.09
71.39
61.37
62.48
66.36
100
75.36
74.01
63.42
44.81
54.09
72.5
56.88
62.44
66.12
SYHN
LSUN CR
66.75
66.44
50.98
49.99
56.11
64.39
48.85
54.52
75.41
Textures. C
74.22
73.03
50.30
51.55
22.77
71.58
51.20
45.76
56.06
OOD MMC
Naise
98.41
97.17
97.78
97.64
93.35
97.99
92.42
97.67
98.00
Uniform
75.43
10.21
10.00
10.00
10.25
71.32
10.00
10.08
10.00
Adv. Noise
93.75
80.21
10.00
10.00
25.59
91.99
24.53
21.88
10.00
Adv. Samples
89.05
84.83
71,20
82.33
68. 68.05
88.13
66.81
42.23
99.90
TE
MNIST
0.67
0.82
0.72
1.28
0.66
0.68
1.00
0.83
0,61
ID MMC
MNIST
99.47
99.32
99.27
98.41
99.65
99.43
98.98
99.30
99.18
FMNIST
58.80
55.62
40.76
32.09
57.29
62.21
20.05
40.83
34.88
EMNIST
82.18
82.40
79.12
78.34
86.02
81.85
50.01
79.87
80.66
MNIST
CIFAR
41.41
29.35
12.80
10.01
25.98
39.85
15.48
13.76
10.02
Kylberg
26.56
10.42
10.00
10.00
10.09
19.48
10.00
10.17
10.00
OOD MMC
Noise
10.95
10.62
10.01
10.30
10.22
11.25
10.00
10.07
10.91
Uniform
27.20
10.25
10.00
10.00
10.01
19.56
10.00
10.13
10.00
Adv. Noise
95.08
75.67
10.00
10.00
65.47
19.33
10.00
25.50
10.00
Adv. Samples
89.09
88.01
59.03
10.27
93.77
87.75
10.00
76.37
10.00
TE
FMNIST
6.91
6.10
7.52
11.51
7.25
5.98
7.38
6.46
4.84
ID MMC
FMNIST
97.01
97.48
94.49
88.53
97.40
97.31
95.64
98.02
98.32
MNIST
77.89
75.83
80.80
48.84
89.39
76.46
44.63
75.86
70.20
FMINIST
EMNIST
81.32
77.58
85.56
33.64
90.43
77.28
47.93
78.64
67.16
GrayCIFAR
90.73
$6.79
32.01
11.37
38.54
82.86
11.14
16.63
45.66
Kylberg
94.25
17.30
15.25
10.93
17.11
81.92
10.35
18.43
59.11
OOD MMC
Noise
92.37
35.67
11.27
12.76
13.32
64.16
10.00
10.02
52.77
Uniform
85.78
10.03
10.00
10.00
10.31
66.53
10.00
32.88
10.00
Adv. Noise
99.76
16.98
10.00
10.00
19.64
98.69
19.39
97.57
10.00
Adv. Samples
89.64
67.10
50.20
98.60
51.22
83.71
15.09
40.81
99.98
4 EXPERIMENTS
4.1 EXPERIMENTAL SETUP
Implementation. For latent representation, we train an autoencoder using a five-layer convo- lutional encoder with 512 channels paired with symmetric five-layer transposed convolutional decoder for small images (28 x 28 or 32 x 32), resulting in a latent dimensionality of 256. For ImageNet (224+ x 224), symmetric VGG-16 architecture is adopted as the autoencoder, yielding a latent dimensionality 1024. All models are trained for 200 epochs using the Adam optimizer with learning rate of 0.0001. The semi-discrete OT problem is solved following (An al. 2020

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Confeiinne))
Conficnneet)
Cor carfo)
Corfedmon)))
Corriderio7]]
Coeridence
"Cont dwecess)
Figure 3: Histograms of maximum confidence scores on OOD inputs before and after applying our method. Results are shown for ResNet-18 18 trained on CIFAR-10 (left) and CIFAR-100 (right).
We select the top 10% of boundaries with the largest singular scores for OTIS generation and set the weight X; according to Eqn. 11 The overall training procedure for confidence suppression fol-
lows the default configuration of CCU (Meinke & Hein 2020). All experiments are implemented in PyTorch (Paszke et al. 2019) and conducted on workstation with NVIDIA RTX 4090 GPUs.
ID/OOD Configurations. 100 (Krizhevsky et al. 2009), SVHN (Netzer et al., 201T), MNIST, and FashionMNIST (FM- NIST) (Xiao et al., (017) as ID datasets. Each can also serve as an OOD dataset when ex- cluded from training, enabling diverse ID/OOD configurations. Additional OOD datasets include LSUN_CK (a classromm subset of LSUN (Yu et al. al.5 (015 Textures_C (cropped from Textures (Ash- worth & M.Lellann 1985)), EMNIST (Cohen et al. 2017), GrayCIFAR (grayscale CIFAR- 10), and Kylberg (Kylberg) 2011 We also include Noise and Uniform samples following the procedure in (Meinke & Hein 2020 Furthermore, we evaluate on Adversarial Noise (obtained by maximiz- ing classifier confidence near random inputs) and Adversarial Samples (constructed near ID inputs but off the data manifold) (Meinke & Hein 2020). For high-resolution evaluation, we use Ima- geNet (Deng et al., 2209) as the ID dataset, and consider OpenImage-O (Kuznetsova et al.) 2020), iNaturalist (Van Horn et al., 2018), SUN (Xiao al.) 2010)), Places (Zhou al. 2016), and Tex- tures (Ashworth & McLellan 1985) as OOD benchmarks.
For low-resolution benchmarks, we use CIFAR-10. CIFAR-
Baselines. On low-resolution datasets, we compare our method with five approaches that do not rely on auxiliary OOD data: CEDA and ACET (Hein et al. 2019). CCUs (Meinke & Hein 2020) (a noise-based variant of CCU), CODES (Tagg et al., 2021 and VOS (Du et 2022). We also include two methods that utilize external datasets: OE (Hendrycks et al., variant of CCU trained with the 80 Million Tiny Images dataset (Torralba et al. 2008). excluding all samples overlapping with CIFAR- following (Meinke & Hein 2020). For high-resolution experiments on ImageNet, we compare against CEDA, ACET, CODES, and VOS as baselines.
2018), and CCUd,
Setup and Evaluation Metrics. We use LeNet for MNIST and ResNet-18 18 for CIFAR-10. CIFAR- 100, SVHN, and FMNIST. For high high-resolution datasets, ResNet-50 is employed. All models are evaluated on their respective ID test sets to report test error (TE), and on both ID and OOD datasets to compute mean maximum confidence (MMC), following the protocol of (Meinke & Hein 2020). To further assess the model's ability to suppress overconfident predictions unfamiliar inputs, also report the false positive rate at 95% true positive rate (FPR95), where confidence serves as the scoring function distinguish between ID and OOD samples.
4.2
MAIN RESULTS
Results on Low-resolution Benchmarks. Tab. reboorss the performance of our method and prior approaches across five datasets. Our method consistently outperforms all basclines in mitigating OOD overconfidence, achieving the lowest OOD MMC in nearly all settings while maintaining strong ID performance. On 10 and CIFAR-100, we obtain significantly lower OOD MMCs (e.g.. 13. 18% on SVHN and 9.30% on C0FA than all competitors, including OE and CCUd which rely on large auxiliary datasets. Meanwhile, test errors and ID MMC remain competitive, in- dicating no sacrifice in ID accuracy. On MNIST and FMNIST, our approach achieves near-optimal suppression (e.g., 10.00% Noise and Uniform) and outperforms prior methods lenging shifts such as GrayCIFAR and EMNIST. For adversarial shifts, our method again delivers superior performance (e.g., 8.94% CIFAR-100 and 15.09% FMNIST), without relying on any
on chal-
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Table 2: Comparison of five methods in mitigating OOD overeonfidence for ResNet-50 on Ima-
geNet. We report test error (TE), mean maximum confidence (MMC; ID T. OOD J) and FPR95 L.
ID
OOD
Method
ImageNet
OpenImage-O
INaturalist
SUN
Places
Textures
TF
MMC
MMC
FPR95
MMC
FPR95
MMC
FPR95
MMC
FPR95
MMC
FPR95
23.87
79.70
43.84
66.84
35.84
52.77
46.15
68.58
48.33
71.57
46.31
66.13
CEDA
24.72
79.87
44.37
67.50
34.76
50.67
46.25
68.36
47.77
70.83
47.18
66.33
ACET
25.15
78.91
43.04
67.76
35.31
54.39
46.18
71.02
47.17
72.40
46.49
68.33
CODES
27.71
81.39
34,25
65.26
32.87
56.38
39.36
70.23
46.70
68.10
45.80
62.62
VOS
24.73
82.60
37.84
67.67
39.04
52.12
39.26
66.42
51.02
70.72
47.62
63.58
Ours
26.43
88,93
34.96
63.96
31.01
49.16
38,80
68.17
45.78
68.08
38.81
66.50
Original
OTII
(6)
1b)
(d)
Figure 4: Visualizations of original inputs and corresponding OTIS from (a) CIFAR-10 10 and (b) Im-
ageNet. t-SNE plots of inputs and corresponding OTIS from (c) FMNIST and (d) CIFAR-10.
(2)
Figure 5: OOD confidence of ResNet-18 trained on (a) CIFAR-10 10 and (b) FMNIST under differ- ent sampling strategies. Boundary-based methods use top-&% singularities or random boundaries (RanB). L-Inter and I-Inter denote latent and image-level interpolation without boundary guidance.
handcrafted augmentations or external supervision. Overall, our method consistently achieves the best overall trade-off between accuracy and confidence calibration across diverse OOD conditions.
Visualization of Confidence Distributions. Fig. 3 shows histograms of maximum softmax con- fidence scores on OOD inputs (SVHN and LSUN CR) for ResNet- 18 models trained on CIFAR- 10 and C1FAR-100. Compared to the baseline, our method substantially reduces the prevalence of high- confidence predictions, shifting the distributions toward lower confidence values. This illustrates the effectiveness of our approach in alleviating OOD overconfidence.
Qualitative Analysis of OTIS. As shown in Fig. OTIS instances exhibit semantically ambiguous yet visually coherent content, often obscuring discriminative features and reflecting structural uncer- tainty aligned with our objective. The t-SNE (Maaten & Hinton, 2008) visualizations further reveal that OTIS instances cluster in inter-class transition zones rather than collapsing onto any single class manifold. This validates their role as structurally grounded inputs for suppressing overconfident predictions near decision boundaries.
ImageNet Results. Tab. 2 presents results ImageNet with ResNet-50. Our method achieves the highest ID MMC (88. (8%.93%) and significantly reduces OOD confidence, attaining the lowest OOD MMC on most benchmarks. It also achieves the best FPR95 on key datasets such iNaturalist 16%) and Openlmage-O (63.96%). Although the test error (26.43%) is slightly higher than some baselines, the overall performance demonstrates favorable trade-off between ID accuracy and OOD calibration, outperforming CEDA, ACET, CODES, and VOS in most cases.
4.3 ABLATION STUDIES AND OTHER ANALYSTS
Effectiveness of High-Singularity Boundaries. We investigate whether sampling near boundaries with higher singularity scores improves OOD regularization. As shown in Fig. selecting the top
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aoppanaaeet 201
107 ABCC- 31
Figure 6: ROC curves for OOD detection on four ID -00 settings using MSP, ODIN, and ReAct, with and without applying our training framework.
10% of boundaries consistently achieves the lowest OOD MMC across multiple datasets, indicating that these regions are most effective in capturing structural ambiguity and eliciting overconfidence. Performance gradually degrades when selecting lower-ranked boundaries, but remains superior to random boundary selection (RanB), confirming the benefit of singularity-aware sampling.
Singularity-Barn -Bsed Sampling vs. Interpolation Baselines. We compare boundary-based sampling with interpolation strategies that do not leverage OT geometry. As shown in Fig. []] both latent- space (L-Inter) and input-space (I-Inter) interpolation yield substantially higher OOD MMC than boundary-basee methods. This suggests that naive mixing fails to produce semantically ambiguous inputs effective for regularizing overconfidence. contrast, OT-induced singular boundaries offer geometrically grounded, uncertain regions that serve as more informative training signals.
Performance in Improving OOD Detection. To assess whether our confidence regularization en- hances standard OOD detection, we apply MSP (Hendrycks & Gimpel 2017), ODIN (Liang et al.) 2018), and ReAct (Sun et al., 202T), with and without training the classifier using our OTiS-based suppression loss. As shown in Fig. 6 models trained with our framework (MSP+Ours, ODIN+Ours, ReAct+Ours) consistently exhibit ROC curves that dominate their respective baselines, indicating improved true positive rates across range of false positive thresholds in all test cases. The im- provement is especially pronounced under challenging shifts such as CIFAR-10->Textu results confirm that suppressing overconfidence in structurally ambiguous regions during training yields more calibrated models and enables more effective OOD detection at inference.
C. These
Why OTIS Instead of Other Heuristic Boundary Sam- ples? Tab. 3 reports Fréchet inception distance (FID) to CIFAR-10 and the mean maximum confidence (MMC) of a ResNet- 18 classifier trained on CIFAR-10 evaluated on the CIFAR-10 test set and on the boundary samples used by CEDA, ACET, CODES, VOS, OE, and OTIS. Among these sample sets, OTIS achieves the smallest FID while still inducing high MMC, meaning that OTIS samples stay closest to the CIFAR-10 ID manifold and predicted with high confidence. Although ACET at- tains the highest MMC, the Adv. Noise it uses lies much farther from the ID manifold, as reflected by its much larger FID. Overall, OTIS focuses on near-distribution, high-confidence regions that previous schemes under-cover, clarifying the benefit of OT-based con- struction over heuristic boundary generation.
Table 3: FID CIFAR-10 (ID) and MMC (%) for boundary samples (OOD) used by different OOD overconfidence
mitigation methods, with CIFAR-10 re-
sults shown for reference.
Dataset
FID
MMC
CIFAR-10
0.00
97.98
CEDA
7.25
87.39
ACET
7.49
99.98
CODES
3.18
88.84
VOS
5.28
88.51
OE
5.73
84.70
OTIS
2.45
91.29
Confidence Calibration ID Data. Tab. reports expected calibration eтror (ECE) on ID test sets for the base classifier and models trained with OOD overconfidence mitigation meth- ods. OTIS achieves the best ECE on CIFAR-10 and FMNIST and matches the best baseline on MNIST. On CIFAR-100 and ImageNet datasets, several methods worsen ECE relative the base model, whereas OTIS stays close to the baseline. These results indicate that OTIS reduces overcon- fidence OOD inputs without degrading, and sometimes improving, ID calibration.
Table 4: ECE (x 10 10-1101-00000 on ID test sets for the base classifier and models trained with OOD overconfidence mitigation methods.
Dataset
CEDA
ACET
CCUs CODEs VOS Ours
OE CCUd
CI
3.77
4.61
3.73
3.81
4.26
2.95
1.88
3.65
2.91
CIFAR-100
6.08
11.34
9.70
12.88
11.45
8.97
6.91
9.69
6.16
SVHN
2.10
1.83
1.30
2.82
2.74
1.51
1.66
1.28
1.67
MNIST
0.21
0.14
0.16
0.37
0.31
0.24
0.14
0.34
0.30
FMNIST
4.00
3.61
4.12
4.53
4.65
3.39
3.26
4.50
3.32
ImageNet
2.05
3.07
2.64
4.75
4.26
2.71
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Table 5:
Comparison of OTIS variants for mitigating OOD overconfidence with CIFAR-10 10
as ID. Columns (Gaussian-5, Uniform-3/7/5) follow the format base distribution number of en- coder/decoder layers. We report ID MMC (1), TE (1). OOD MMC (L). and FPR95 (L).
Gaussian-S 5
Uniform-3 3
Uniform-7
Uniform-5 5 (Ours)
Dataset
ID
TF
ID MMC
TF
IT MMC
TE
ID MMC
TE
CIFAR-I0
96,48
7.31
96.00
7.60
92.31
8.55
95.46
7.52
OOD MMC
FPR95
OOD MMC
FPR95
OOD MMC
FPR95
OOD MMC
FPR95
SVHN
17.77
1.54
24.95
3.11
14.27
1.23
13.18
1.21
CIFAR- 100
68.81
64.78
67.74
65.34
59.83
68.63
64,79
63.38
LSUN .R
39.33
15.83
34.47
19.27
34.72
22.77
30.36
10.95
Textures_C
$7.77
45.64
56.88
46.22
$4.57
50.07
48.75
36.70
Noise
24.23
0.00
28.70
0.23
21.59
4.54
16.18
Uniform
10.00
0.00
10.00
0.00
10.04
0.00
10.00
Adv. Noise
27.91
3.43
44.51
3.00
31.37
1.09
26.42
Adv. Samples
59.96
76.78
60.21
77.55
59.63
78.64
57.71
74.71
Effect of Base Distribution. Tab. compares Gaussian base distribution (Gaussian-5) and the
uniform base distribution (Uniform-3, Ours) under the same 5-layer symmetric encoder-decodes Both yield similar CIFAR-10 test error and ID MMC, while Uniform-5 consistently achieves better OOD metrics (lower OOD MMC and FPR95). This suggests that OTIS is insensitive the exact base distribution and that a uniform base is a strong default choice in practice.
Effect of Autoencoder Depth. Tab. 5 further compares OTIS with different AE depths under uniform base (Uniform-3, Uniform-5, Uniform-7). CIFAR- test error and ID MMC remain close across depths, whereas Uniform-5 generally offers the best OOD trade-off, with the lowest near- lowest OOD MMC and FPR95 on most datasets. This indicates that OTIS is robust to the AE depth and that a 5-layer symmetric encoder meoder-decoder decoder provides good balance between computational cost and the effectiveness in alleviating OOD overconfidence.
5 RELATED WORK
Handling OOD Overconfidence. Test-time detection methods estimate prediction confidence via softmax responses (Hendrycks Gimpel 2017), temperature scaling (Liang et al. (2018), energy scores (Liu et al. 2020), or feature-space distances (Lee et al.] (018b). Though widely used, these post hoc techniques do not resolve the model's inherent overconfidence on unfamiliar inputs. A proactive alternative introduces proxy OOD inputs during training. Some methods use generative models to estimate ID boundaries and penalize confidence in low-density areas (Lee et al.] 2018a Meinke & Hein| 2020), though these often require careful tuning partial OOD access (Li & Vasconcelos] 2020). Others employ unrelated datasets (Hendrycks et al.] 2018), or apply input transformations e.g., noise, permutation, mixing (Hein et al., 2019: Maaten & Hinton 2008). or sample from low-likelihood regions (Du et al., (002).. These methods typically rely on heuristics and lack principled ways to characterize semantic ambiguity or boundary instability. We instead exploit semi-discrete OT geometry to identify transport-induce singularities, enabling grounded OOD example construction for confidence regularization.
Optimal Transport (OT). OT offers a principled framework for comparing probability measures via cost-optimal mappings (Villani et al., 22008; Santambrogio 2015; Mérigot 2011). It has been broadly applied in domain adaptation (Courty et al. 2017; Damodaran et al.) 2018), generative modeling (Genevay et al. 2018 Tolstikhin et al., (018), and representation learning (Cherian & Acron 2020; Chen et al. 2022). In contrast, are the first to exploit structural singularities in semi-discrete OT for OOD robustness, introducing OT as a novel tool for confidence calibration.
CONCLUSION
In this paper, we have presented a theoretical perspective on OOD overconfidence by linking to singularities in semi-discrete optimal transport, which highlight structurally ambiguous regions where classifiers tend to be overconfident. These regions provide natural targets for confidence regularization. Extensive experiments across various architectures and ID/OOD settings show that our method reduces overconfidence without sacrificing ID accuracy. We hope this work motivates further studies robustness beyond OOD challenges.
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ACKNOWLEDGEMENTS
This work was supported in part by the National Natural Science Foundation of China (62472117, 62572400, U2436208, 62372129), the Guangdong Basic and Applied Basic Research Founda- tion (2025A1515010157, 2024A1515012066 (2025A0300137, 2024B0101010000 the CCF-NetEase ThunderFire Innovation Research Funding (CCF-Netease 202514), and the Project of Guangdong Key Laboratory of Industrial Control System Security (2024B1212020000;
the Science and Technology Projects Guangzhou
ETHICS STATEMENT
This work adheres to the ICLR Code of Ethics. It does not involve human subjects, sensitive user data, or potentially harmful applications. Our method is designed to improve the robustness and trustworthiness of deep neural networks against out-of-distribution inputs, and does promote enable misuse. All datasets used are publicly available and commonly adopted in the literature, and have ensured no bias or discriminatory behavior is introduced during training evaluation.
REPRODUCIBILITY STATEMENT
To facilitate reproducibility, we provide key ayper-parameters training settings, and evaluation pro- tocols in Sec. 4 The complete source code and reproduction instructions will be released publicly upon acceptance of this paper.
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A APPENDIX
A.1 MORE QUANTITATIVE RESULTS
Comparison on Mitigating OOD Overconfidence Measured by FPR95. Tab. 6 reports the FPR95 across broad set of ID and OOD dataset pairs. Our method consistently achieves com- petitive or superior performance across most settings. Compared to baseline models, which often yield FPR95 above 60%, our approach substantially reduces overconfidence, particularly on near- OOD and corrupted variants. Among methods without auxiliary data, our method significantly outperforms CEDA, ACET, CODEs, and VOS on most datasets, especially on Adversarial Noise and Adversarial Samples. For instance, CIFAR-10 with Adversarial Noise, our method achieves 0.66% FPR95, far below VOS (94.12%) and CODEs (19.55%). Compared to methods using aux- iliary datasets, such as OE and CCUd, method is competitive while remaining free of extra supervision. These results confirm that our OT-based framework effectively identifies structurally ambiguous regions and suppresses overconfident predictions diverse range of OOD inputs.
Comparison on Mitigating OOD Overconfidence Measured by AUROC. To further evaluate the model's s ability suppress overconfident predictions, we report the area Under the receiver operating characteristic curve (AUROC), where we use the confidence as a threshold for the detection problem (ID vs. OOD). As shown in Tab. 7 method achieves consistently high AUROC scores across various OOD types. It outperforms all baselines on CIFAR-I0 with SVHN and Noise as OOD (99.79% and 99.97 99.97%) and maintains strong performance on challenging shifts such as Adversarial Noise (100.00% on CIFAR- 100). On simpler datasets like MNIST and FMNIST, it achieves near- perfect AUROC under multiple shifts. These results demonstrate the effectiveness of our method in mitigating OOD overconfidence across diverse conditions.
Robustness to Common Corruptions. We treat CIFAR-10 and CIFAR-100 as ID data and their corrupted counterparts CIFAR-10-C and 10FAR-100-C as OOD inputs, modeling more realistic, corruption- based distribution shifts. Tab. fabbBBaa summarizes the results on these corruption benchmarks. On CIFAR- OTIS achieves the lowest OOD MMC and FPR95 and the highest AUROC among all methods. On CIFAR- 100-C, OTIS again attains the lowest MMC and FPR95, while its AUROC remains competitive with the strongest baseline CCUd. These results show that OTIS effectively mitigates overconfidence and improves OOD detection under challenging corruption-nnuuced induced shifts.
A.2 MORE VISUALIZATION RESULTS
Visualization of Confidence Distributions. We visualize the distribution of maximum softmax confidence on various OOD inputs before and after applying our method. As shown in Fig, 7 stan- dard models exhibit long tail of overeonfident predictions even on clearly OOD samples. Our method effectively reduces high-confidence spikes and flattens the overall distribution. Fig. high- lights the robustness of our approach across different types of synthetic OOD inputs, including random noise and adversarial variants. In all settings, confidence mass is shifted away from the high end, indicating more calibrated predictions. Finally, Fig. 9 demonstrates that our method remains effective under large-scale settings. Despite the difficulty of the ImageNet task, approach duces confidence saturation on OOD datasets such as OpenImage-On and SUN, suggesting improved generalization to unseen distributions.
Visualization of Transport-Indueee OOD Samples (OTIS). We visualize OTIS instances gen- erated from different datasets in Figs. 12 Across all domains, OTIS instances exhibit seman-
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Table 6: Comparison of eight methods for mitigating OOD overconfidence, measured by FPR95
(%, 1). Note: OE and CCUd leverage auxiliary datasets, while the others do not.
ID
OOD
Baseline
CEDA ACET
CCUs
CODEs
VOS
Ours
OE
CCUd
with auxiliary dataset
v
v
SVHN
59.51
55.78
69.99
21.20
72.23
47.5
1.21
31.87
8.24
CIFAR-I00
64.99
70.69
69.66
40.18
74.32
62.93
63.38
66.90
61.18
CIFAR-IN 10
LSUN_CR
45.27
35.11
39.40
25.27
35.25
42.33
10.95
45.67
12.02
Textures_C
66.49
57.46
63.76
25.32
51.54
57.77
36.70
50.55
20.05
Noise
59.49
17.80
25.27
32.75
18.06
11.00
0.00
0.00
0.01
Uniform
68.59
0,00
0.00
0.00
0.00
62.95
0.00
0.39
0,00
Adv. Noise
97.46
31.94
0.00
0.00
19.55
94.12
0.66
84,61
0.00
Adv. Samples
94.28
92.01
76.84
100.00
91.74
92.81
74.71
86.64
99.83
SVHN
71.39
86.46
79.01
17.90
82.08
75.07
8.81
66.86
10.00
CIFAR-10
79.49
86.02
83.51
12.92
82.42
79.76
77.38
82,43
10.07
CIFAR 100
LSUN_G _R
78.60
72.86
64.81
9.93
45.45
76.48
55.82
64.31
9.01
Textures C
80.11
89.29
84.52
83.16
75.30
78.74
75.57
75.62
74.47
Noise
99.61
99.75
99.87
33.23
97.83
99.49
28.20
0.00
0.55
Uniform
97.95
0.00
0.00
98.78
0.00
0.00
0.00
Adv. Noise
99.73
21.48
0.00
0.00
27.31
99.34
7.41
84.69
0.00
Adv. Samples
92.89
92.57
85.65
100.00
90.60
88.84
86.09
82.74
99.00
CIFAR-IO
30.45
45.61
18.12
16.47
7.68
27.09
11.45
19.85
8.27
CIFAR-I00
30.61
46.32
19.84
17.45
17.64
28.89
13.29
21.00
16.51
LSUN ASUN_CR
17.50
31.89
10.01
29.24
49.03
15.75
11.04
10.87
10.00
SVHN
Textures.C
28.62
46.15
12.61
10.90
8.63
26.79
41.17
11,40
0.00
Noise
99.54
95.38
96.19
97.26
99.95
96.23
90.12
95.48
96.28
Uniform
28.91
0.00
0.00
0.00
0.00
26.65
0.00
0.00
0.00
Adv. Noise
77.50
60.74
0.00
11.47
74.55
7.98
4.60
0.00
Adv. Samples
66.55
67.06
46.26
99.67
39.60
63.15
59.07
23.14
99.00
FMNIST
1.44
1.56
0.48
6.78
0.80
2.48
0.36
0.85
0.26
EMNIST
22.15
21.56
19.88
40.95
19.70
21.78
15.38
21.60
35.10
GrayCIFAR
0.03
0.00
0.00
0.00
0.00
0.04
0.30
0.00
0.00
MNIST
Kylberg
0.00
0.00
0.00
0.00
0.00
Noise
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Uniform
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Adv. Noise
0.04
0,00
0.00
0.00
0.00
0.00
0.00
0.00
0,00
Adv. Samples
4.06
0.43
0.00
0.00
1.44
0.99
0.00
0.00
0.17
MNIST
59.07
48.92
35.12
57.83
79.43
51.54
53.38
40.78
$2.88
EMNIST
66.61
52.37
43.71
42,97
81.04
53.36
41.62
47.97
43.94
FMNIST
GrayCIFAR
86.65
22.53
16.56
11.90
7.61
66.91
7.46
0.47
7.10
Kylberg
92.75
1.07
6.03
10.85
0.31
65.2
2.56
0.16
0,00
Noise
92.99
0.02
21.32
2.42
0.00
16.77
0.00
0.00
0.01
Uniform
79.85
26.54
0.00
0,02
0.00
Adv. Noise
99.92
1.30
0.00
0.00
0.91
99.26
0.00
95.78
0.00
Ady. Samples
82.28
28.65
17.19
100.00
16.65
64.45
38.15
12.51
98.50
tically coherent yet structurally ambiguous characteristics that differ from typical ID patterns. On ImageNet (Fig. 10), the generated instances contain mixed blurred attributes resembling multi- ple classes (e.g., dog-coo-ooo cat or ship-truck), often with disrupted contours and overlaid textures. On CIFAR-10 (Fig. I. OTIS instances present hybrid features and indistinct boundaries, lacking the sharp discriminative traits of clean samples. On FMNIST (Fig. 12), shape distortions and pattern mixing are especially prominent, producing examples that lie between known fashion categories. These results demonstrate that our OT-based sampling process successfully identifies regions of structural uncertainty and generates visually plausible OOD inputs for suppressing overconfidence.
A.3 STATEMENT ON L.LM USAGE
LLMs used solely writing- assistannee tool to polish the language of this manuscript (c.g., grammar, phrasing, and clarity). They were not involved research idcation, experiment design, data analysis, or substantive content generation.
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Published as a conference paper at ICLR 2026
Comparison of eight methods for mitigating OOD overconfidence, measured by AUROC (%, 1). Note: OE and CCUd leverage auxiliary datasets, while the others do not.
Table 7:
ID
0OD
Baseline
CEDA
ACET
CCUs
CODEs
VOS
Ours
CCUd
with auxiliary dataset
v
SVHN
91.29
92.56
89.00
88.75
86.67
92.56
99.79
95.27
85.42
CIFAR-100
88.35
85,69
85,64
85.83
83.77
85.76
87.19
86.15
82.38
CIFAR-IR 10
LSUN_CR
93.80
95.42
94.93
90.66
94.57
92.94
98.26
94.12 12
98.34
Textures_C C
88.50
90.26
88.31
94.52
90.10
88.32
93.01
90.04
97.29
Noise
93.60
96,40
95.47
98.38
97.71
98. 14
99.97
99. .6
97.22
Uniform
90.46
100,00
100.00
100.00
100.00
91.61
100.00
99.84
100,00
Adv. Noise
56.38
85,36
100.00
100.00
93.11
92.96
95.78
98.77
100.00
Adv. Samples
54.77
57.31
66.79
63.40
65.68
59.26
67.27
63.11
62.79
SVHN
84.15
82.14
86,35
86.10
89.82
90.6
98.04
93.37
97.37
CIFAR-I0
78.61
72.33
74,20
67.54
76.23
77.83
74.05
74.61
80.25
CIFAR 100
LSUN_CO CR
80.38
81,45
85.16
86.08
89.83
81.89
87.28
85.55
82.32
Textures
77.74
68.02
72.09
86.24
78.45
78.09
79.00
75.45
76.45
Noise
53.61
52.09
60.34
53.55
54.27
61.93
100.00
56.60
Uniform
69.77
100.00
100.00
99.99
100.00
75.63
100.00
99.99
100.00
Adv. Noise
52.33
91.12
100.00
100.00
90.83
69.57
100.00
73.15 15
100.00
Adv. Samples
60.26
74.23
84.75
82.88
81.20
84.06
85.36
89.11
85.59
CIFAR-I0
91.68
92.26
96.26
98.26
98.45
92.44
89.83
95.94
99.94
CIFAR-
91.82
92.09
96.01
97.96
97.64
91.91
98.18
95.66
99.92
LSUN_CE
96.04
94.87
97.80
95.15
92.43
95.75
95.47
97.59
99.90
SVHN
Textures C
92.93
90.84
97.36
99.24
98.33
92.52
97.56
97.68
99.94
Noise
84.09
87.07
88.07
96.18
91.83
87.35
95.72
99.99
95.07
Uniform
94.22
100.00
100.00
100.00
100.00
93.19
100.00
100.00
100.00
Adv. Noise
50.25
76.98
100.00
100.00
95.62
85.05
92.43
98.64
100.00
Adv. Samples
61.44
76.69
81.43
84.84
89.10
74.44
86.68
92.06
92.79
FMNIST
99.34
99.19
99.69
99.25
99.46
99.11
99.77
99.50
99.61
EMNIST
94.74
94.74
95.31
94.05
95.06
94.9
96.55
94.84
96.15
MNIST
GrayCIFAR
99.83
99.94
100.00
99.99
99.95
99.85
99.87
99.99
99,99
Kylberg
99.99
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
Noise
100.00
100.00
100.00
99.99
100.00
100.00
100.00
100.00
99.99
Uniform
100.00
100,00
100.00
100.00
100.00
100.00
100.00
100.00
100,00
Adv. Noise
97.55
99.28
100.00
100.00
99.57
100.00
100.00
98.91
100,00
Adv. Samples
97.80
98.32
99.75
100.00
97.98
98.08
100.00
98,88
99,99
MNIST
90.72
94.16
95.58
85.08
79.23
93.16
97.87
95.10
97.26
EMNIST
87.70
92.71
94.02
88.12
78.44
92.37
92.96
93.40
96.11
FMNIST
GrayCIFAR
83.89
97.23
97.89
99.00
98.97
92.25
98.44
99.90
98.24
Kylberg
79.93
99.78
99.18
99.30
99.92
92.97
98.74
99.92
99.08
Noise
86.65
99.80
96.46
97.94
100.00
97.01
97.96
100.00
96.98
Uniform
90.90
100.00
100.00
99.49
100.00
96.54
100.00
99.98
Adv. Noise
57.87
99.80
100.00
99.94
99.86
67.08
98.87
83.45
99.99
Adv. Samples
81.54
96,29
97.57
82.72
96.85
86.76
93.84
97.74
87.60
Table 8:
Comparison of different methods for mitigating OOD overconfidence under common corruptions, treating CIFAR-10/100 as ID data and
OOD inputs. We report
OOD MMC (J), FPR95 (4), and AUROC (f). all in percent (%)
CIFAR-I0-C
CIFAR- CIFAC-100-C
Method
MMC
FPR95
AUROC
MMC
FPR95
AUROC
Base
93.21
83.30
70.43
65.61
84.06
68.92
CEDA
86.81
80.92
70.66
69.85
89.30
63.17
ACET
90.81
85.00
66.16
71.80
87.40
65.62
CCUs
85.42
80.96
66.16
66.19
79.78
65.94
CODES
84.44
84,80
68.00
74,96
85,80
67.04
vos
90.89
82.38
66.07
70.90
84.72
67.67
Ours
70.78
66.16
76.63
59.22
73.82
72.72
OE 87.75
79.86
70.15
71.18
85.20
66.16
CCUd
83,82
68.79
73.99
63.27
78.16
79.20
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Published as a conference paper at ICLR 2026
LSUN_CR
CIFAR-10
SVHN
Conndende(N))
Coriidence(1))
Confience(%)
Corfiienda((5)
CIFAR-100
Textures
(antidinaa/5)
Coriidence(1))
Confdance(
confidenca((s)
EMNIST
MNIST
FMNIST
Confienno(
GrayCIFAR
Coniidence(()
Confidens(()
Kylbrr
Confidien0(55)
Contidence/N)
Comtiencce
Confidence(%)
Confidence())
EMNIST
FMNIST
MNIST
ConfidenceN)
Confidence(N)
Confidence(
Confidence(55)
GrayCIFAR
Kylerr
Confidecce)))
Confdenioe
Confidenc()))
Confidanca
Histograms of maximum softmax confidence on OOD samples from common datasets. Each model is trained on SVHN, FMNIST, or MNIST using ResNet- 18, and tested on datasets such as CIFAR- 100, L.SN,, and EMNIST.
Figure 7:
17

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Published as a conference paper at ICLR 2026
Uniform
Note
CIFAR-10
(o)
Adv Noise
Comfidennee(1)
Confidence((5)
Adt Samples
Canfdence)(19)
Confidence(S)
Conficeence(S)
Confidence(1))
Confidence(5)
Unifor
Noise
CIFAR-100 100
(ontianca(5))
Adv Notse
Conficiono(())
Confidence(51)
AdySSmples
Canidanca(S4)
(onfiance(33)
Conficiono(%5
Confidence(5)
Confidenc((5))
Uniform
Noise
SVHN
Conhdanca(33)
Noise
Conficence(%
Confidence(55)
Samples
Confidenc((5))
Conndance/54)
Conficenco(5)
Comficncc((5)
Confidence(s))
Uniform
Notsp
FMNNST
Contdencc//4)
Confidncc((5)
Comfdence((5)
Confience((5))
Advivoise
Confidence(S)
Confiencc((5)
Confienner(5))
Comfdence)(9)
Figure 8: Histograms of confidence on synthetic OOD inputs. Each model trained on CIFAR-10, CIFAR-100, SVHN, or FMNIST using ResNet-18. 18, and evaluated Uniform, Noise, Adversarial Noise, and Adversarial Samples.
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Published as a conference paper at ICLR 2026
Operimege-O
iNsturslist
ImagaNet
Conlidence(5)
Conlidecc,1%)
Confidanca(
Conndenda
SUN
Placss
Corfidence(S)
Confidence
Confidenc(())
Contidricce(5)
Figure 9: Histograms of maximum confidence scores on large-scale OOD datasets. A ResNet-50 is trained on ImageNet and evaluated on OpenImage-O. O, iNaturalist, SUN, and Places.
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Published as a conference paper at ICLR 2026
Figure 10: Visualizations of OTIS generated from ImageNet.
20

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Published as a conference paper at ICLR 2026
Figure 11: Visualizations of OTIS generated from CIFAR-10.
Figure 12: Visualizations of OTIS generated from FMNIST.
21


Reviews:
[{"review_id": "vKb3vBCsI7", "reviewer": "Reviewer_rQDu", "summary": "This paper addresses the general issue of deep neural networks producing overconfident predictions on out-of-distribution (OOD) inputs. The authors propose to generate specific proxy OOD samples. The core idea is to identify regions of semantic\nambiguity in the latent space of an AE by leveraging the geometry of semi-discrete Optimal Transport (OT). They solve an OT problem between a base distribution and the latent embeddings of training data, in order to identify singular boundaries. Then, they generate OTIS by interpolating latent features near these boundaries of interest. The paper claims this geometrically grounded approach significantly reduces OOD overconfidence compared to SOTA methods.", "questions": "The work introduces an interesting idea for generating targeted samples to reduce OOD confidence and shows promising results on the OOD MMC metric, however it has some important weaknesses\nin my opinion. The theoretical justification based on OT appears too complex for the actual mechanism employed and lacks comparison to natural alternatives (k-NN) and the absence of standard calibration metrics (ECE). The paper’s heavy reliance on dense OT and writing makes it difficult to understand the core mechanism, which is essentially an advanced latent-space mixup. Concerns about scalability/stability and potential negative impacts on robustness further weaken the submission. I am open to reconsidering my score if the authors can provide a convincing rebuttal that addresses these weaknesses, namely : \n\n- clarify the necessity of the OT framework\n- clarify the impact on ECE\n- clarify the complexity implications\n- clarify the standing on corruption/domain shift", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- the conceptual link drawn between singularities in semidiscrete OT and semantic ambiguity is interesting. Targeting these specific regions for confidence suppression is potentially more founded than using heuristics like noise or generic outlier datasets \n\n- the method for generating OTIS is detailed and the sample creation between latent concepts should provide a more effective signal for training than random interpolations or generic augmentations (e.g. Figure 5) \n\n- strong performance in experiments", "weaknesses": "- The paper heavily relies on the theory of semi-discrete OT (Brenier potential, Laguerre cells) in order to define a notion of neighborhood (i.e., adjacency cell boundaries). However, the core measure of ambiguity is simply the angle between neighbor latent vectors, which is independent of the OT framework. Hence, the paper doesn’t clearly explain/justify  the additional complexity : why the OT-neighborhood concept is necessary over a much simpler and standard k-NN approach.\n\n- The primary goal is to mitigate overconfidence, (i.e, lead the model toward better calibrated predictions). While the work shows ID MMC remains high, this doesn’t eliminate  the possibility that the method makes the model uncalibrated on correct ID predictions due to the nature of the loss and the generated OoD inputs. It would be crucial in my opinion to report the ECE score on the ID samples as it may reduce overconfidence on ID inputs, which would open the discussion a little bit.\n\n- I might have missed some things, but there are some major complexity drawbacks potentially. Solving a semi-discrete OT problem for the entire training set (n=50k for CIFAR, n=1.2M for ImageNet) is certainly computationally demanding. Do the authors rely on solving OT on mini-batches of target points {$y_i$} ? If so, the geometric partition (Laguerre cells, boundaries) may be unstable and change with each batch. In this case, the paper should provide a theoretical error bound for this approximated domain/boundary relative to the true partition (over all the dataset for instance), using for instance Hausdorff distance. This may depend on the number of points, the dimension of the latent space and also probably involve some topological assumption [1] on the representativeness of the mini-batch. If not, the paper must provide a clear analysis of the computational complexity and demonstrate the feasibility and runtime of their OT algorithm on large-scale datasets. Not discussing this aspect is detrimental for the readers.\n\n- The method trains the model to reduce confidence on inputs deviating from the clean ID manifold, using semantic ambiguity. This might teach the model to also reduce confidence on inputs suffering from diverse domain shift such as common corruptions (e.g., CIFAR-10-C dataset). As modern architectures are well known to be overconfident on such inputs, it would be interesting to see MMC and ECE score on corruption benchmarks.\n\n[1] Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. Geometric and topological inference. Vol. 57. Cambridge University Press, 2018.", "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": "FVcfB7hXUL", "reviewer": "Reviewer_u5rM", "summary": "This work proposes a method to reduce a classifier's confidence on out-of-distribution (OOD) samples. The high-level idea is to construct a set of samples on the semantic boundary, and then train the model to have low confidence on this set of samples. The authors propose to use semi-discrete optimal transport (OT) to construct the boundary. Specifically, the authors sample a set of ID samples, use an encoder to get their latent features, and get the boundary for each pair of samples using Eqn. (9). The authors test their method on some vision datasets.", "questions": "1. How is the encoder in Eqn. (6) trained?\n2. If the encoder is not well trained, how will it affect the performance of your method?\n3. How to choose $x_1,\\cdots,x_n$ in line 194?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The writing is fine. It is not hard to understand the paper\n2. The description of the OT method is clear", "weaknesses": "1. Motivation is not very clear. It is a natural idea to lower the model's confidence on boundary samples in order to lower its confidence on OOD samples. However, it is not clear to me why using the specific method proposed in this work. There seems to be a bunch of possible ways to construct the boundary samples, and the authors do not elaborate on why the specific method proposed here is more efficient. While the formulation looks reasonable, it looks quite complicated and I am not really sure what this extra complication buys us. \n2. Some important details are unclear. I might have missed it, but the two important details I fail to find in the paper are (a) how the encoder in Eqn. (6) should be trained, and (b) how the $n$ training samples after Eqn. (7) should be drawn. I think these two details are particularly important to the success of the propose method, so some extra elaboration is needed.\n3. Regarding the experiments, from Table 1 I notice that the proposed method always has a high test error, so it is not clear to me if the decrease in the confidence on OOD samples actually comes from this trade-off. I also think that a brief description of the compared baselines is necessary.", "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": "4z5lHkzsFD", "reviewer": "Reviewer_QHM5", "summary": "This work presents Optimal Transport-induced OOD samples (OITS), a novel method for synthesizing OOD samples from ID labeled datasets to regularize the training process (tackling the overconfidence problem). OITS leverages ideas from the Optimal Transport (OT) literature, constructing OOD samples via latent embeddings near the singular boundaries, which are often aligned with the model's decision boundary. Overall, the paper provides detailed theoretical derivations and comprehensive empirical analysis validating the proposed method.", "questions": "In line 157—Could the authors provide some explanations on why \"these regions often correspond to high-confidence mispredictions\"? Or, are there any links between the Optimal Transport problem and the training process of a NN? Any intuition would be helpful.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "To the best of my knowledge, the motivating idea of this work is novel. By incorporating the semi-discrete OT problem into DNN training, the authors successfully capture a picture depicting the generalization of NNs.\n\nThe paper is clearly written, and I find the theoretical explanation in Section 2 is accurate and easy to follow for a wider audience. Empirical results also demonstrate the effectiveness of the proposed method, with several helpful visualizations.", "weaknesses": "As shown in Tables 1 and 2, the proposed method suffers from a relatively high test error, indicating that some accuracy is sacrificed for robustness against OOD samples. Nevertheless, the proposed method still has on-par test error compared to other baselines, and the OOD robustness (measured by OOD MMC) is significant.", "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": "1eF84CSDjj", "reviewer": "Reviewer_mC83", "summary": "This paper proposes OTIS (Optimal Transport–Induced Samples), a training‑time regularization framework to mitigate overconfident predictions on OOD inputs. The core idea is to (1) encode training images into a latent space, (2) solve a semi‑discrete OT problem between a continuous base distribution and the discrete set of latent embeddings, which induces a power‑diagram (Laguerre) partition; (3) identify singular boundaries in the OT map as regions of semantic ambiguity; (4) synthesize proxy OOD samples via interpolation across those boundaries and decoding back to image space; and (5) apply confidence suppression loss (uniform‑target cross‑entropy) on the synthesized samples + standard cross‑entropy on ID data during training. \n\nThe paper presents improvements in OOD overconfidence metrics (MMC, FPR@95, AUROC) across multiple canonical datasets in the empirical results, and shows that training with OTIS strengthens standard test‑time detectors.", "questions": "1. For Table 3, how were “Adversarial Noise/Samples” generated (step sizes, budgets, initialization)? Some reported gains are striking—for example, CIFAR‑10 with adversarial noise FPR95 **0.66%** vs VOS **94.12%** (Table 3). Is there any intuition for such striking differences?\n2. How sensitive are results to the encoder/decoder architecture and reconstruction quality (e.g., using a WAE vs. the chosen AE)?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. Presentation of the pipeline is easy to follow and fairly well justified: Partition estimation, scoring boundaries (Eq. 9), OTIS synthesis, and training are easy to follow (Fig. 2, Sec. 3).\n2. The background knowledge for the semi‑discrete OT setup is carefully crafted with sufficient support. The Induced power diagram is also visually appealing and elaborative, giving good clarity to the geometric motivation on the selection of method. From the description, readers can have a clear picture on latent‑space structure and appreciate singular boundaries as a credible proxy for ambiguity.\n3. The quality for experiments is good. Benchmarks span small and large scale (CIFARs, SVHN, MNIST/FM‑NIST, ImageNet), with multiple OOD types (near‑OOD, synthetic corruptions, adversarial noise/samples).\n4. On many conditions the method outperforms baseline methods in the experiment setup. The margin is relatively clear, signifying the work's potential to be further applied in later researches.\n5. Training with OTIS consistently boosts MSP/ODIN/ReAct ROC curves (Fig. 6). \n6. Boundary ranking ablation and comparisons to latent/input interpolations are helpful (Fig. 5); histograms and OTIS visualizations support the “ambiguous but plausible” claim (Figs. 7–12).\n7. The idea of leveraging OT‑induced boundaries to synthesize targeted, ambiguous training examples seems broadly applicable beyond the specific detectors and datasets evaluated", "weaknesses": "1. The paper states a theoretical link (singularities <-> overconfidence), but provides no formal guarantee that the proposed angular deviation score identifies regions of high miscalibration. The result is a strong intuition with empirical support rather than a theorem (Sec. 2.2–2.3; Eq. 9).\n2. Eq. 11 defines inverse‑distance weights $\\lambda_i$; the implementation then weight is fixed to 0.5 (line 319). This is a non‑trivial discrepancy. Even if the hard-coded weight seems reasonable, proper explicit justification (and potential ablations) should be applied to amend for the discrepancy. This weakens the credibility of the results produced on more generic setups.\n3. The number of Monte Carlo samples used to estimate cell volumes (M), base distribution choice (Gaussian vs. uniform) in each experiment, and convergence criteria for offset optimization are not fully specified in the main text; only high‑level statements are provided (Sec. 3.2).\n4. While Table 2 shows improved OOD MMC on ImageNet, ID MMC rises substantially (to **88.93**), which may reflect over‑smoothing/confidence effects; a calibration analysis (ECE/Brier/NLL) would clarify the trade‑off.", "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)