Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Authors: Muskan Dosi
Chiranjeev Chiranjeev
Kartik Thakral
Mayank Vatsa
Richa Singh
Venue: ICML 2025 poster (2025)
Keywords: Diffusion Model
vMF Distribution
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: Exploring diffusion to generate class-aware hyperspherical geometry by leveraging angular uncertainty.

Abstract: Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce \textbf{HyperSphereDiff} to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold.

Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Muskan Dosi' Chiranjeev Chiranjeev Kartik Thakral
Mayank Vatsa Richa Singh
Abstract
noise, coupled with its isotropic nature, simplifies both theo- retical formulations and practical implementations, making it a default choice for most diffusion frameworks (Kingma et al., 2021). However, this assumption inherently biases the data to Fuclidean spaces (Song et al., 2021; Dhariwal & Nichol, 2021), limiting the model's ability to account for in- trinsic geometric structures non-Euclidean domains (Lui, 2012; Scott al., 2021; De Bortoli et al., 2022), such as hy- perspherical or manifold-constrainee data (Bronstein et al., 2017; Rezende et al., 2020). In such settings, Gaussian- based diffusion may fail to fully capture the directional relationships and nuanced variations inherent to the data, as shown in Figure (()) (top row) where the forward process of adding Gaussian noise leads to distortion in the angular geometry of the classes during sampling. It motivates a need for rethinking the noise distribution in the diffusion processes.
Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Eu- clidean spaces. However, many real-world prob- lems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to subop- timal generative performance. To address this lim- itation, we introduce HyperSphereDiff" to align hyperspherical structures with directional noise, preserving class geometry and effectively captur- ing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic ge- ometry of hyperspherical data, resulting in more accurate and geometry-awe aware generative models. We evaluate framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underly- ing hyperspherical manifold. Resources are avail- able at: Link.
Beyond the limitations of geometry, Gaussian noise also overlooks the flexibility required to model varying uncer- tainty levels across different samples. Ambiguous or noisy data points appear across different timesteps in Figure (a)) (top row). These points require a representation where un- certainty is explicitly modeled. However, the isotropic as- sumption of Gaussian noise treats all directions equally, failing to capture this uncertainty effectively. Researchers have explored alternative noise distributions better suited to specific data geometries and uncertainty modeling (Xu et al., 2023). One promising direction is the von Mises- Fisher (vMF) distribution (Mardia & Jupp, 2000), which is naturally defined on hyperspheres and parameterized by concentration parameter that controls directional uncer- tainty (Hasnat et al., 2017). For effective handling of un- certainty using vMF noise in diffusion models, we aim to align the generative process with the underlying geometry of the data, thus enabling more accurate modeling of direc- tional relationships. This approach builds on recent efforts to integrate geometric priors into generative models (Falorsi et al., 2018), expanding the scope of diffusion techniques to hyperspherical data.
1. Introduction
Diffusion models have revolutionized generative model- ing, achieving remarkable success in diverse modalities, including generation of images (Ho et al., 2020; Dhariwal & Nichol, 2021), audio (Kong et al., 2021), 3D scenes and 3D structures (Bautista et al., 2022; Shue et al., 2023). These models operate by progressively adding Gaussian noise to the data a forward process, followed by a reverse process that learns recover the original data from the corrupted versions (Sohl-Diee Dicksttin et al., 2015). The use of Gaussian
Department of Computer Science and Engineering, Indian In- stitute of Technology Jodhpur, India. Correspondence to: Muskan Dosi <dosi. I @ ac.i?>.
1.1. Research Contributions
We propose HyperSphereDifr (Diffusion with Hyper- Spheres), leveraging the von Mises-Fisher (vMF) distribu- tion to preserve hyperspherical class geometry by introduc-
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Geassine Based Difisioon Process
oMF AMI--Bard Diffasion Procees
Kapzatain
(4)
Figure 1. (a) Illustrating class geometry preservation in non- Euclidean hy hyperspherical spaces. The top row shows Gaussian diffusson, which fails capture structural relations, whereas the bottom row demonstrates von Mises-Fisher (yMF)-based diffusion, effectively modeling directional uncertainty. Red and green arrows indicate forward and backward diffusion process, respectively. (b) Comparison of the 3D sphere: Gaussian (top) vs. vMF (hottom). Gaussian distorts class boundaries, while vMF maintains the original geometry, preserving angular regions.
ing directional uncertainty into the diffusion process. It is visually illustrated through Figure ((a) (bottom row). Here, geometry precisely captures angular relationships defining class boundaries, whereas uncertainty reflects stochastic variation governed by the vMF concentration parameter. Figure 1(b) provides empirical visualization demonstrat- ing that vMF-driven diffusion preserves hyper-conical class geometry, while Gaussian- baased diffusion distorts it. By embedding vMF-based noise into both forward and reverse processes, the manifold-aware framework maintains global structure and local directional fidelity. It ensures directional uncertainty respects hyperspherical structures throughout the diffusion process. The key contributions of this research are:
distribution. We introduce two metrics, Hypercone Coverage Ratio (HCR) and Hypercone Difficulty Skew (HDS), for assessing geometry preservation.
Theoretical Foundations and Empirical Validation. Along with detailed theoretical foundations, through extensive experiments on four object datasets and two face datasets, we demonstrate improved alignment with hyperspherical geometry. We observed enhanced ro- bustness to varying uncertainty levels and superior per- formance class-conditional generation tasks.
1.2. Related Work
Denoising diffusion models can generate diverse unseen im- age samples by training on existing datasets (Ho et al., 2020; Song et al., 2021; Dhariwal & Nichol, 2021). First intro- duced by (Sohl-Dickstein et al., 2015), they added Gaussian noise in forward process and learned to reverse it for ple generation. Advances in score-based modeling (Song & Ermon, 2019; Pidstrigach, 2022), efficient sampling (Wat- son et al., 2022), and discrete diffusion (Austin et al., 2021) have further improved their capabilities. However, their reliance on Gaussian noise limits them to Euclidean spaces, restricting their ability to model directional relationships in oon-Euclidean domains (Bronstein et al., 2017). While manifold-aware and sphere-based extensions (Rezende et al., 2020) attempt to address this, they remain limited in han- dling anisotropic and angular uncertainties.
1. Class-Specific Hypercone Formalism. We introduce hypercone representation that preserves intra-class angular concentration on the hypersphere while en- forcing inter-class separation. This formalism captures class structure by focusing on directional relationships rather than purely Euclidean distances.
2. Based Forward and Reverse Processes. Gaus- sian noise is replaced with vMF noise to maintain angu- lar consistency. This ensures generated samples remain the hypersphere, effectively guiding them toward class-specific hypercones during the reverse process.
3. Diverse and Hard Sample Generation. contrast to Gaussian- based methods, the VMF-driven approach mitigates simplicity bias, producing a broader sample
The von Hisse--Fisher (yMF) distribution, commonly used for hyperspherical data, has demonstrated effectiveness in
2

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
face recognition (Hasnat al., 2017; Deng et al., 2019), outlier detection and generation (Du et al., 2022; Ming et al., 2023; Du et al., 2023) and representation learning tasks (Davidson et al., 2018). It has also been employed in generative modeling through hyperspherical GANs (David- son et al., 2018) and spherical VAES (Falorsi et al., 2018). However, the integration of vMF noise into diffusion mod- els remains relatively unexplored. Previous works explor- ing non-Gaussian noise in diffusion processes (Wang al., 2022; Xu et al., 2023) lack a systematic approach to leverage angular distributions like vMF.
distribution Pdata (Xo) to a standard Gaussian distribution (0,T)) as t+ T.
The uncertainty in the forward process is introduced through the isotropic Gaussian noise ~ NN I). At each time step, the variance of the noise component - (t increases as at decreases, representing a controlled diffusion of informa- tion. This uncertainty is modeled symmetrically across all dimensions of the data, resulting in a spherically symmet- ric probability density function. Formally, the marginal distribution at time L can be expressed as:
(xxx)) / q(xx) (x) (ro))oo))) P (Xo) dxo.
In addition to Gaussian and vMF noise, alternative noise types such as Laplacian noise (Dwork & et al., 2006), Stu- dent's t-noise (Matsubara & Imai, 2021), and blue noise (Huang et al., 2024) have been explored for robustness and outlier handling. Flow-based transformations have also been used in generative tasks (Kingma& & Dhariwal, 2018) 8), while dopain-specific noises in molecular generation (Luo et al., 2021) focus on aligning noise with data properties. Our work adds to this growing body of literature by intro- ducing a novel angular noise mechanism based on vMF distributions, enabling the diffusion process model class- specific and directional uncertainty in hyperspherical do- mains. Progressive Distillation (Salimans & Ho, 2022) uses an angular DDIM update in Euclidean space but ignores hyperspherical constraints. EDM (Karras et al., 2022) en- sures variance preservation via dataset-dependent scaling but lacks directional or geometric alignment. In contrast, HyperSphereDiff operates directly on the hypersphere, pre- serving both variance and manifold consistency without extrinsic normalization.
which represents the corrupted data distribution as a Gaus- sian mixture where each component is centered at vārx0 and has variance 1 - c₁ The reverse process is designed to approximate p(xo) xx) and recover the original data by progressively denoising the sample x,.
3. Revisiting Hyperspherical Data Geometry
The use of the Gaussian noise simplifies theoretical analysis and computational implementation by assuming data resides in the flat, unbounded Fuclidean space Rd. However, this approach is fundamentally sub-optimal when dealing with non- -on-Euulidean data geometries, such as hyperspheres Sd-1 or other manifolds.
operrtts
Theorem 3.1 (Gaussian Noise and Spherical Structure). Let (O, be an isotropic Gaussian in Rd. Then:
(a) The radial density follows:
2. Preliminary
P([z|] = r) x , r
In Gaussian-based diffusion models, the forward process defined as a Markov chain that incrementally corrupts the data by introducing isotropic Gaussian noise. Let xo ~ Paua (xa) represent the original data. The sequence of noisy samples x, is generated according to:
(b) As d 00: 00:
|Z| Vd
- 1
(c) For data x E Ed-1 the noised vector x + oz does not preserve angular relationships as a > xc.
VārXo + vI-a a.e,
Proof. (a) Follows from the Jacobian of spherical coordi- nates, (b) By the law of large numbers, ||z|i// + 1 in
where, a E (0, is a variance schedule controlling the relative weight of the signal and noise at time and (O, I) is isotropic Gaussian.
probability, and (c) As a - oc, the contribution of x be- comes negligible.
The geometry of the corrupted data distribution in time step t is defined by the weighted combination of the original data Xg and the noise component e. The resulting distribu- tion can be expressed as:
Gaussian noise works well in flat unbounded spaces due to its isotropic nature. However, many real-world datasets reside on curved spaces, such as hyperspheres or other Rie- mannian manifolds. In these cases, Gaussian noise disrupts the intrinsic geometry by introducing perturbations that ex- tend beyond the manifold (Bronstein et al., 2017; Huang
xxx/xx)= = N(xx V Vo, (1 I).
which defines a trajectory in the Euclidean space where the corrupted data progressively transitions from the original
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
al., 2022) (detailed proof in Appendix A). This misalign- ment motivates the exploration of alternative noise processes designed for data with non-Euclidean geometries.
On the hypersphere g4-1, classes naturally align within hypercones defined by angular constraints, and the vMF distribution inherently models these structures. Such distri- butions also relate to class separation on hyperspheres (refer Appendix B). Unlike Gaussian noise, vMF noise respects the manifold's geometry by modulating its focus through the concentration parameter K. When K = 0. vMF is isotropic, uniformly spanning the hypersphere, while larger x values focus the distribution within a hypercone around the mean direction μ.M This adaptability enables vMF-based diffu- sion to effectively learn and represent class-wise structures, ensuring a geometrically consistent generative modeling framework.
Facial data embeddings, often derived through deep neural networks, are typically normalized to lie on a hypersphere gd-1 C R4, reflecting their natural angular variability due to factors like pose. expression, and illumination changes (Majumdar al., 2017: Wang et al., 2018; Deng et al., 2019). This normalization ensures that the similarity be- tween two embeddings is determined by their angular re- lationship rather than their magnitude, aligning with the cosine similarity metric frequently used in recognition tasks. For two embeddings e1, C2 E Sd-1, their similarity can be expressed as cos( ej e2 where 0 = arccos(e) [e] 62) is the geodesic distance on the hypersphere. This structure inherently aligns facial embeddings with hyperspherical ge- ometry, where angular deviations are the primary measure of variability. In high-dimensional hyperspherical spaces, facial classes can be modeled as distinct regions, often vi- sualized as hypercones emanating from the origin. Each class Ck Sd-1 is centered around a mean direction vector H4 E gd-1, with intra-class variability characterized by angular deviations. The vMF distribution provides natural framework for modeling such hypercones due to its con- centration parameter Kg, which controls the spread of the distribution around 14g- The probability density function is given as:
4. Modeling Angular Uncertainty in Diffusion
To effectively model the uncertainty inherent in hyperspher- ical data, we employ a forward diffusion process driven by noise sampled from the vMF distribution. The forward process introduces angular-based uncertainty such that the data progressively transitions from structured, -lass-specific noise to a uniform distribution over the hypersphere gd-1, Initially, the noise is highly concentrated (x is large), pre- serving the class structure within narrow hypercones. As the process advances, decreases to zero, injecting isotropic noise, and the data diffuses uniformly over the hypersphere.
4.1. Angular Noise Injection
k (d/2))1 exp(ssu T x),
(x) = (2 d/2 (()2))1(k)
In the proposed HyperSphereDifff. the forward process adds noise to the data representation using angular interpolation. At each time step L, the data representation obtained in a latent space through an encoder is denoted as Z, which is up- dated as z, = cos )2 1 + sin(6,) (6,)v, where 2;-1 E $d-1 is the data representation at the previous time step, v is a unit vector sampled uniformly from gd-1, and 0, is time- dependent angle that increases monotonically from 0 to T 7/2. The angular interpolation ensures that the injected noise aligns with the hyperspherical geometry. The parameter 6g acts as a scheduler, gradually increasing to control the extent of deviation from the previous representation. Corre- spondingly, this forward step mimics the vMF distribution using Ka scheduler. The decrease in Ke defines the level of angular uncertainty introduced at each step.
where, ((k) is the modified Bessel function of the first kind. To formalize this relationship, we introduce the following lemma:
Lemma 3.2 (yMF Hypercone Representation), Let C2 C Sd-1 be a class hypercone centered at Pt with angular radius Og. The data follows a vMF distribution with concen- tration Kj. Then:
(a) The probability mass within the class hypercone is:
P(x E Cx)= = P(((x, S 0x) 2
By employing k as a scheduler, our forward diffusion pro- cess progressively incorporates angular uncertainty, tran- sitioning from class- structured perturbations to isotropic noise (Figure 1). This ensures that the initial steps of the corruption process retain class information, as larger x val- ues concentrate the data around class-specife hypercones. Over time, as + 0. the noise diffuses the data to uni- form hypersphere, reflecting maximum uncertainty. This process inherently respects the geometric properties of hy- perspherical data, with angular deviations tied to the data
(b) For any desired coverage probability 1 1-t E (0, 1), choosing:
1
1
Kk 1- cos
log
c Ux
ensures that P(x& E 21
Note:
As Oz + O, the required concentration Kss + 00, reflecting the scenario of a single-direction hypercone.
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
distribution rather than being purely random.
The concentration parameter Ki plays a crucial role in shap- ing this evolution. It starts small for isotropic diffusion and then grows during the reverse process, increasingly favoring class-specific directions. As Rg increases, the vMF sam- pling becomes more concentrated around the class means, reflecting the growing certainty in class membership while maintaining geometric consistency.
4.2. Backward Step: Hypersphere to Hypercone
In generative modeling with hyperspherical data, the reverse process of HyperrSherrDiiff transforms noisy samples into structured data aligned with class distributions. This pro- cess progressively refines noisy points toward class-specific hypercones using score-based methods. The reverse formu- lation leverages vMF sampling, ensuring angular relation- ships between classes are retained. The reverse process is defined as:
The combination of score-based updates and vMF sampling ensures smooth transition from noise structured data. The process gradually refines points until they converge near their respective class means Hec for a class effectively recovering true class representations while preserving the underlying hyperspherical geometry. This mechanism pro- vides mathematically principled approach to generating class-consistent samples on the hypersphere, maintaining both local structure and global class relationships through- out the diffusion process.
- VMF (2, 7/s Vz, log fzz; M4y (e)), K4).
The complete algorithm A and experiments are based on this reverse step and MSE loss as defined in standard Gaus- sian diffusion. Alternatively, the reverse step with angular updates using vMFF-based stochastic denoising is defined as:
2t-1 vMF (1( [I(cos(0))2 +
Algorithm 1 HyperSphereDiff Training: vMF Diffusion with Hypercone Preservation
Require: Data samples (x:). class labels {v:}. diffusion
V 2 log (Zzi Me) -V=, log [(=:; pe]]
(0.)
Kit.
steps T
Require: Angular schedule {0.)7 Learning rate 7
I: Initialize score network parameters 0
To further align optimization with hyperspherical geometry, the reverse denoising step is refined using an angular-basse based loss function alternative to the MSE loss. Cosine Loss:
2: while not converged do
3: Sample batch (x, y) from dataset
4: Sample time step Uniform
Encourages angular alignment between the score function and noise direction.
Ri f cot(0,)
Set concentration
6: Sample ~ Uniform(S"-1)
7:
II(cco((,)) + (i:() Forward process
Lc 1 - E Ze log [" pe) cx]
8:
similar to ~ vMF(x, Ki)
9:
Vz, log Score (Z,, t.y)
Geodesic Loss: Penalizes angular deviations.
10: L+ I log Vz, log p(zz [x)]2
Lg E arccoss
|V=, s, log f(z;; f(z;:e)lllll Pe)ll )
Vi log / (e) Tee
11: Update 0 using gradient of
12:
end while
13: return Trained score network parameters 0
where, 2, represents the current noisy sample, f(-) is the vMF distribution modeling class hypercones, and Vz, log f()) is the score function providing gradient in- formation. Here, interpolation using cos(6:) and sin(O:) maintains angular relationships, while the normalized score function preserves directional consistency. (Refer Appendix C) The comparative analysis of various loss functions and angular based denoising is provided in Appendix (Refer Appendix H.2).
4.3. Adaptive Class-Dependent Concentration
We consider data zo on the unit hypersphere and forward diffusion process as before with vMF transitions. In the reverse process (o(zrr) = po (zo 21,3) v) IT Ze, v). we guide samples into learned truncated regions on the hypersphere. We employ two networks: a direction predictor D& that estimates class-specific directions m = Dø t, y). and an angle predictor Co. that estimates class angular radii Og Co(Ze, I, y). These networks learn to form dynamic hypercones Ct,y = {z: L(z, m,) < 0g} that capture class-specific regions at each step. The reverse process uses truncated vMF:
Po (z,-1
The process evolves from isotropic noise class- class-specific structures through the interplay of two key components. The score function Vz, log (2: M4; Ke) points towards the class means Me. with the step size Th controlling the update strength. Simultaneously, vMF sampling adds controlled stochasticity, ensuring updates remain consistent with hy- perspherical geometry while progressively removing noise.
(o (21,9) TvMF(2o; mo No (o(21, Cr.))
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Algorithm 2 HyperSphereDiiff Testing: Sampling from vMF Diffusion with Class Guidance
Table I. Performance comparison Gaussian and HyperSphereD- if' across six datasets using FID (lower is better), HCR (lower is better), and HDS (lower values indicates harder samples). The vMF model demonstrates superior capability in generating chal- lenging samples with better FID and HDS scores.
Require: Class label y, diffusion steps T. trained score network 0
Require: Angular schedule {0, ) step sizes M
1: Sample 2т Uniform(S*-)
Dataset
FID
HCR
HDS
2: for t = T to do
Gaussian
vMF
Gaussian
vMF
Gaussian
vMF
3: fig < (ot(
> Get concentration
MNIST
1,55
1.86
0.17
0.14
0.76
0.52
4: i > then
CIFAR-10
3.45
3.52
0.23
0.2
0.72
0.48
CUB-200
9.11
0.19
0.13
0.81
0.41
5:
2n log ScoreNet tz)
Cars-196
9.09
7.87
0.19
0.17
0.85
6:
m: + Zt + 17 2z. log
CelebA
9.31
9.29
0.42
0.22
0.77
0.59
7:
m, hmmmlmmmm /|yl
Project to hypersphere
D-L.ORD
11.38
9.27
0.46
0.21
0.91
0.62
8:
2 vMF(mm,,e))
9: end if
4.4. Beyond Geometry and Uncertainty: Retaining Image Information
10: end for
11: return Final sample Z1
In our face generation process, we introduce two forms of uncertainty to capture both magnitude and directional com- ponents in the data representation motivated by (Chiranjeev et al., 2024). Specifically, we interpret image embed- ding having magnitude ||x|| (which relates to overall intensity scale) and a direction x/ sx|l Sd (which captures structural and class-related geometry). We employ a Gaussian distribution to model the magnitude uncertainty and a vMF distribution to model the directional uncertainty. This hybrid approach preserves crucial class geometry on the hypersphere, while still accommodating image-specific variability (e.g. changes in lighting intensity) through Gaussian noise. The detailed forward and reverse process is provided in Appendix F.
TYMF(Z = C(,,C) exp(R 2) 1 C),
We maintain adaptive concentration through (z y) Emax (3 [0,, Zize, m,))). where 3 is scaling factor, Kmaz is maximum concentration allowed and (-)) is sig- moid function, ensuring stronger concentration within pre- dicted regions. The normalization constant Z(K, C) for the truncated vMF can be computed as:
Z(K, C) = exp(xx ds (z)
where, dSd-1 is the hyperspherical measure. This integral has closed form in terms of the incomplete gamma func- tion when C is a hypercone. The networks Do and Cg are trained jointly with the score network to minimize the objective:
5. Experiments
We evaluate HyperSphereDifff on four object datasets (CIFAR-10 (Krizhevsky et al., 2009), MNIST (Deng, 2012), CUB-200 (Wah et al., 2011) and Cars-196 (Krause et al., 2013)), and two face datasets (CelebA (Liu et al., 2015) and D-LORD (Manchanda et al., 2023)). The architecture and training details are provided in Appendix H.
C(o, v-) BE. [lm. 1 X(0y- 0,,3
where, my and By are empirical class statistics computed from training data, and À balances the direction and angle losses. This ensures that the predicted geometry aligns with the true class structure while maintaining the flexibility of learned truncation regions.
Retaining Geometry: We define two metrics evaluate whether the generated samples preserve the class structure and maintain distributional properties of the original data. Hypercone Coverage Ratio (HCR): The HCR quanti- fies the percentage of generated samples outside the class distribution 's hypercone. For each class k, we compute:
The combination of learned directional prediction, adaptive concentration, and explicit truncation provides a powerful mechanism for class-conditional generation. At each step r, the process maintains dynamic balance between score- based updates and geometric constraints:
k
N₂ [cos (z, (4)) > 0
HCR
Σ
K
A=1
ElZz-1 1 v] m, 7/e Vz, log (zz phg k.) -1), Cr.Y
A lower HCR indicates better preservation of the class struc- ture, while a higher HCR suggests out-of-class or unrealistic samples.
This ensures samples remain within class-approprias re- gions while benefiting from the score function's gradient information. Detailed proofs of forward and reverse model- ing are shown Appendix D.
Hypercone Difficulty Skew (HDS); The HDS measures whether the model generates easy samples by analyzing how
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
CUB-200
Calabok
Figure 2. Generated samples of birds (left), faces (middle) and (right) using the proposed vMF based diffusion model trained on CUB-200, CelebA and Cars-195 196 dataset, showcasing the preservation of key attributes and diversity.
ing that the vMF generates samples that follow the original class structure, thus preserving angular relations. Similarly, the Hypercone Difficulty Skew (HDS) value with the vMF model in all six datasets is around 0.5, reflecting its ability to generate generalized samples across the hypercone rather than generating simple samples. Gaussian-based generated samples have HDS values around 0.6, showing that a higher proportion of samples are generated from the innermost hypercone demonstrating simplicity bias. For datasets like D-LORD and CUB-200, the reduction in HCR and HDS is particularly pronounced, showcasing the efficiency of the vMF model modeling angular variation and producing diverse samples.
Cluss Maan 6J
Conestration Uoepel diteersse
Figure 3. Real-world surveillance samples generated using class- dependent adaptive concentration s. through proposed diffusion model after training on D-LORD dataset.
they are distributed across sub-cones of increasing angular deviation. high HDS indicates the model generates easy samples, while a low HDS suggests a balanced distribution across difficulty levels (refer Appendix H.1). For each class k, we compute the fraction of samples in each sub-cone and compute HDS:
We evaluate the quality of generated samples using our method across six diverse datasets. These include natural images (CelebA, CIFAR-IO), fine-grained object categories (CUB-200, Cars- 196), and structured dig- its (MNIST). Figure 2 presents examples of generated faces and birds from models trained on CelebA, Cars- 196, and CUB-200. The images exhibit realistic structural coherence and diversity, resembling their respective distributions. We also compute the Fréchet Inception Distance (FID) (Byna- gari, 2019), achieving competitive scores across all datasets (Table 1). In particular, CelebA, D-LORD, and 196s-196 show high performance, indicating that the model captures both global structure and fine- greeggraied details. This demon- strates that HyperSphereDDiff generates high-quality samples with significant variation while preserving class structure, aligning with our theoretical insights (See Appendix H).
Generation Quality:
No
Σ
[0
00-1 < cos 'z: 14)) S 0m]
Pk
N
K
M
1
HDS
WmPk
m
K
1= m=1
The HCR and HDS metrics offer thorough assessment of generative models angular space. HCR measures class consistency and assesses how well intra-class variations are preserved, while HDS identifies whether the model favors generating easier samples. An ideal generative model should maintain low HCR while achieving HDS that aligns with the natural difficulty distribution of the original data.
We evaluate the hypercone con- straint by analyzing generated samples from both inner and outer hypercones, shown in Appendix H. Samples in the inner hypercone retain class-consistent features, while those in the outer hypercone exhibit noisy generations, validating the effectiveness of our metric in capturing generation qual- ity. Furthermore, for the real-world D-LORD (Manchanda
Hypercone Generation:
Table highlights the comparison between the Gaussian and vMF models terms of HCR and HDS in six datasets. The Hypercone Coverage Ratio (HCR) consistently decreases for the vFF model compared to the Gaussian model, indicat-
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Gaussian
Scatter Regression: Gaussian vMF Diffusion
Similerity to Meat
0.9
0.7
Variations
0.6
Pose Illumination Expressions Quality
Goussian
oMF
200
800
Sample
(a)
(b)
Figure 4. (a) Comparison of samples generated using Gaussian and vMF diffusion models, highlighting improved variation in pose, illumination, expression, and quality with vME (b) Scatter plot with fitted regression shows cosine similarity to the class mean, where HyperSphereDiff generates more challenging samples with broader similarity distribution compared to Gaussian.
400
Geussie
vMii-Uleed
Table 2. FID scores different noise configurations across Celeb- A and D-LORD datasets. The hybrid Gaussian + Spherical method achieves the best performance.
350
300
Dataset Celeb-A 9.31
Gaussian
Gaussian + Spherical Spherical
Trequercy
9.29
9.31
D-LORD 11.38
9.27
10.94
iff compared to the Gaussian model. This result highlights the ability of our approach to generate data that better aligns with the angular structure of real- world surveillance images, effectively capturing the underlying geometry.
1.0
Cosine Similarity to Class Mean (u_c)
Generating Hard Samples: We observed that the vMF- based approach consistently produced samples with varied cosine similarity to the class mean, indicating a higher de- gree of generalized sample generation compared to the Gaus- sian baseline, as shown in Figure 4. This is also supported by qualitative results, where vMF samples exhibit greater variations in pose, illumination and expression, closely re- sembling real-world challenges. Quantitatively, this ob- servation is validated using HDS metrie (Table 1), which measures the proportion of samples concentrated in the smaller hypercones. The vMF model achieved lower HDS values, confirming its effectiveness in generating harder, more diverse samples that can better represent challenging scenarios in real-world datasets. Further, Figure shows HyperSphereDiif yields broader coverage (mean cosine sim- ilarity=0.722, aad=0.137), reflecting balanced difficulty rep- resentation compared to Gaussian diffusion (mean=0.900. std=0.048).
Figure 5. Gaussian based diffusion generates samples tightly clus- tered near the class mean (high similarity, low variance), favoring easier cases. In contrast, HyperrShereDDiff produces diverse spread (lower similarity, higher variance), ensuring better coverage across difficulty levels.
al., 2023) dataset, we sample images at varying kappa values shown in Figure concentrating the samples around the mean vector to reflect distance- wiie variations in the surveillance imagery. These experiments highlight how dif- fusion with hypercone constraints ensures both intra- class consistency and controlled diversity.
We evaluate the performance of our proposed method on the real-world D-LORD (Man- chanda al., 2023) surveillance dataset, which consists of sequential frame images exhibiting angular relationships in the feature space. We generated samples of 1000 synthetic subjects using both manssiannbbsed diffusion and our vMF- based hyperspherical diffusion. Training the ArcFace model on the generated data leads to 5.0% improvement on the real test set identification accuracy with the HyperSphereID-
Face Recognition Results:
Ablation Study: Table presents FID scores comparison on Celeb-A and D-LORD datasets under three noise strategies: Gaussian, Spherical (HyperSpherrliif), and a hybrid of
8

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Foints
VNF
Cas
Oet
Gee
Cae
100
0.0
0.0
1.0
1.0
1.0
0.5
0.5
-10
2.0
-1.0
0.0
0.0
1.0 -1.0
-1.0
1.0
Figure 6. Feature representation of the 10-class MNIST dataset generated using Gaussian based diffusion (left) and vMF- diffusion (right). The vMF-based sampling aligns generated sample features within class-specific 3D hypercones, while Gaussian based sampling results in scattered features outside the class-hypercones.
(a) Expression
10 classes of the MNIST dataset. The figure highlights that the vMF-based reverse sampling effectively converges sam- ples within class-specific hypercones, capturing the angular geometry of the data. In contrast, Gaussian-based reverse sampling produces samples that converge within Euclidean space, failing to adhere to the hyperspherical structure.
Source
Interpolatin usins
base) Detuadion
Interpoaatinn using VIP Deluaiio
Interpolation Results:
Figure shows interpolations using Gaussian- diffusion (top row) which often produce unnatural or inconsistent transitions, especially with large attribute shifts. In contrast, our HyperSphereDiff (bottom row) angular interpolation on a hypersphere yields smoother, preeeeviig transitions across poses and expressions.
(b) Pose
Sounce
Source
using Ceusdim Defuaioo
Indepolation based
6. Conclusion and Future Work
Figure 7. Comparison between interpolation using Gaussian-based diffusion based diffusion generating images between variants of the same subject: (a) Expression, (b) Pose.
In this work, we have challenged the ubiquitous Gaussian assumption in diffusion models by introducing novel hy- perspherical framework HyperSphereDiff that leverages vMF distributions for generative modeling on angular mani- folds. By incorporating hyperspherical geometry and class- dependent uncertainty, our approach preserves angular struc- while producing diverse, semantically rich samples. Ex- tensive experiments facial and complex object datasets demonstrate its effectiveness in fine-grained tasks where angular relationships are critical. This work opens new avenues for manifold-constraing generative modeling, ad- vancing geometry-aware diffusion techniques. Future re- search will focus on developing adaptive k-based sched- ulers, adopting hierarchical hypercone partitioning for finer class variations, and extending the framework to conditional generation tasks, such as pose-invarian face synthesis.
both. The hybrid model yields the lowest FID scores across both datasets (9.29 for Celeb-A and 9.27 for D-LORD), suggesting that combining magnitude- based Gaussian noise with direction-aware spherical noise results in better gener- ation quality. Purely Gaussian or purely spherical models alone are less effective, particularly for D-LORD, where Gaussian diffusion performs poorly (FID 11.38) and the spherical-only model underperforms compared to the hybrid. This demonstrates the advantage of modeling dual uncer- tainty by capturing both intensity variation and directional structure to enhance the realism and class consistency.
Feature Representation: Figure illustrates the feature representations of conditional samples generated from the

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Impact Statement
Deng, J., Guo, J., Xue, N., and Zafeiriou, S. Arcface: Additive angular margin loss for deep face recognition. In Proceedings of the Conference Computer Vision and Pattern Recognition, pp. 4690-4699, 2019.
This work introduces a novel hyperspherical diffusion frame- work leveraging von Mises- (vMF) distributions to enhance the modeling of high-dimensional angular data in Machine Learning. By improving the fidelity and inter- pretability of generative models, the proposed method has applications in computer vision, line-grained classification, and surveillance. However, like other generative techniques, carries potential ethical risks, including misuse in deep- fakes, privacy concerns, and bias amplification. To mitigate these risks, we emphasize responsible use and transparency, particularly in sensitive domains (Mittal et al., 2024). Over- all, this research advances generative modeling for hyper- spherical data while promoting a deeper understanding of geometric structures in Machine Learning.
Deng, L. The mnist database of handwritten digit images for machine learning rescarch [best of the web]. IEEE signal processing magazine, 29(6):141-142) 2012.
Dhariwal, P. and Nichol, Diffusion models beat gans on image synthesis. In Advances in neural information processing systems, volume 34, pp. 8780-8794, 2021.
Du, X., Gozum, G., Ming, Y., and Li, Y. Siren: Shaping representations for detecting out disttri-itiittiionn objects. In Advances in neural information processing systems, volume 35, pp. 20434-20449, 2022.
Acknowledgements
Du, X., Sun, Y., Zhu, J., and Li, Y. Dream the impossible: Outlier imagination with diffusion models. In Advances in Neural Information Processing Systems, volume 36, pp. 60878-60901, 2023.
The authors acknowledge the support of IndiaAl and Meta through the Srijan: Centre of Excellence for Generative Al.
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
A. Gaussian Noise Distorts Angular Relationship
key challenge in hyperspherical data modeling is preserving the angular relationships that define class structure, especially when noise is introduced. In many generative and transformation- based approaches. Gaussian noise commonly used to perturb data points. However, in non-Euclidean spaces like the hypersphere, such noise can significantly distort the underlying geometric structure. Unlike structured perturbations that respect the manifold's constraints, isotropic Gaussian noise introduces deviations that shift points off the sphere and disrupt their relative angles. The following lemma formally establishes how Gaussian noise fails to maintain angular class structure, particularly in high- nigh-dimensional spaces where its effects become more pronounced.
Lemma A.1 (Gaussian Noise Distorts Angular Class Structure). Let {x:} be N points in S (grouped into classes) such that the angular distances (or equivalently, dot products) between points reftect some inter-class and intra-class relationships (e.g. points in the same class are close in angle, points different classes have larger angles). Define the perturbed points
X;
Eis E, - N o I',,,
i=
where, E, are ii.. isotropic Gaussians in Rdd Then in general, the inner products
X2 and x xj
are not preserved; i.e. with high probability, the angles among the perturbed points (X1,.... .x.. are significantly different from those among the original points (XI,...,XN) especially as d grows and/or a >> 0 large.
Proof (Sketch):
i) No longer on the hypersphere. Since ||ei|l & O almost surely, the perturbed points X; will not lie on gd-1, so angles measured in Rd w.r.f. the origin are already changed. More precisely.
|5&,11 ||x e1 = + ||e/1? + 2xx Es).
which (with probability 1) is not equal to 1. Thus, any "spherical" relationships are broken immediately.
ii) Inner products become random. Consider the inner product
x (x: (x) (x, + cj) xTx, + x, x, + x55 - c.TT.
Since 6;, 6, are Gaussians with mean 0, each cross-lerm is random variable whose distribution depends o² and d.
iii) High dimension amplifies distortion. In high d with o² > O, we rypically have [|e:]] ze Vdo, so the energy in the noise vectors can overshadow the original norm (||x|l 1). Hence ||x;||| = Vdo, dominating any small angular differences that originally existed among the }x:}. Even for moderate d, if a is large enough, X; and &j become nearly orthogonal or randomly oriented (depending on the sign and correlation among the noise). Thus, the relative angles among the perturbed points often bear little resemblance to the original class structure.
iv) Conclusion. Because isotropic Gaussian noise in Rdd shifts points off the hypersphere and injects random directions scale it fails to preserve the original spherical relationships (both inter-class angles and intra-class distributions). In fact. for large d or sufficiently large a, the new angles are effectively random, destroying the class separation that was originally encoded in angles on Sd-1
B. Class Separation using vMF
fundamental challenge in generative modeling on hyperspheres is ensuring class separability while preserving the underlying geometric structure. The von Mises-Fisher (yMF) distribution provides natural way to model directional data while maintaining angular coherence. Unlike Gaussian noise, which distorts class boundaries by introducing random
13

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
perturbations in Euclidean space, vMF-based modeling enforces directional consistency by concentrating samples around mean direction with tunable spread controlled by the concentration parameter
The following lemma establishes probabilistic bound on class separability when data points from different classes are modeled using vMF distributions. It quantifies how the probability of misclassification depends on g and the angular separation f between class centers. This result provides a theoretical foundation for setting to achieve a desired classification accuracy and demonstrates that stronger concentration (higher x) exponentially improves separation, reinforcing the effectiveness of vMF-based diffusion in preserving hyperspherical structure.
Lemma B.1 (Class Separation with von Mises-Fisher Distributions). Consider two classes C1 and C2 on the unit hypersphere gd-1, with mean directions Pl, M2 E gd-1 separated by angle = arccos ( pz2). Suppose data points in each class follow von Mises-Fisher distributions with the same concentration parameter is > 0:
p(x 16)
i= 1,2
where, Ca(R) is the normalizing constant. Then:
(a) The probability of misclassification Pa (classifying point from class I as belonging to class 2 or vice versa) is bounded above by:
Pe S exp(-x(1 - cos
(b) For any desired error rate 0, setting the concentration parameter
1
K 2 1 cos 0
log
guarantees that Pe c.
Proof. For point x drawn from class 1, i.c., x vMF(H1 R),, the probability density at angle &i from μ is:
p(o) = Ck( exp exp(x cos σ)
2. Misclassification occurs when x is closer to f4 than to f41- Byhyperspherical geometry, this happens when the angle & between x and br exceeds 0/2.
3. Therefore, the misclassification probability is:
{ p(0) sin (
P.
J0/2
where, sin sin"-?(d)dde) is the surface element on $d-1
4. For @ 2 0/2, have cos o S cos cos(0) (0/2), thus:
P. [" Ca(k) Ca(s))xx((xcospp sin sin 2nnd2 2
8/2
< Ca(k) exp(1 cos((/2) 1" sin d- (
J0/2
I expp(x(11>) cos
5. For part (b), solving exp(-x(1 - cos ()) < yields the required bound
Implications: (1) The bound tightens exponentially with (2) Larger angular separation 0 allows for smaller k. (3) For fixed e, required v. scales inversely with class separation.
14

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
C. Variational Bound
In typical hyperspherical diffusion, the reverse process attempts to invert the forward noising so that data points return precisely to their original directions. By contrast, certain tasks (e.g., flass-conditionll generation) may demand less rigid constraint: as long the final sample lies within small hypercone around class-specific prototype. the objective is satisfied. We formalize this idea by modifying the standard diffusion variational bound to allow for hypercone-coonttriined convergence in the reverse process.
C.1. Forward Process the Hypersphere
We assume data Zo gd-1 sampled from some distribution g(z)). The forward noising process gradually transforms Zn into approximately uniform distribution on the hypersphere by injecting von Mises-Fisher noise:
q(2 22-1) vMF Ki), I 1=.,...,?
where, (MF k) denotes a vMF distribution on gd-1 with mean direction μ and concentration parameter 2 0. For large t, fig + causing the distribution q(zr zo) to approach uniform on gd-1
Hence, the complete forward chain for (Zo:r) is:
T
g(zur)) = ((zo) II
g(zz 2;-1) t=1
(3)
Our goal is then to reverse this process, recovering Zy from a noisy Zy.
C.2. Class-Specific Hypercones and the Reverse Process
Hypercone Definition. We consider C' distinct classes, each identified by a direction μe E gd-1 and an angular radius
0g 2 0. The class hypercone Co(Oc) is then defined as:
(..) = (ZES L(z,pe) S 0e}
(4)
Thus, each class corresponds to all directions hypersphere within angular distance tg of Me-
We introduce a class-conditiona reverse process that moves from to Zg in T steps:
Po y) (zo Zi, y) II po(z: Zet1s 4),
(5)
where, y (1...., C} is the class label. For each intermediate t 2 1, we let
Po (2 Zz: y)
vMF (mm v), kz),
(6)
where, mo y) is (learned) unit vector on the hypersphere and R; is (possibly deterministic or learned) concentration.
Class Hypercone at O. Instead of insisting that do exactly match the original data direction, only require that An lie in the appropriate class hypercone (0y). Hence, can define:
pz( Z1, y) (truncated vMF with support in C,,)))),
so that zo & (02)) has zero probability. Equivalently, one may define a suitable parametric distribution that is sharply peaked around My but has finite support within angle 0g-
C.3. Variational Bound with Hypercone Constraint
Let Zo belong to class y. We seek to maximize po(zo Zo y) (the likelihood of reconstructing zu within the correct hypercone). standard approach introduces the forward chain as variational distribution and applies Jensen's inequality.
15

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
C.4. Evidence Lower Bound (ELBO) Derivation
We start from:
log (Zo 1/) log / po y) dz
g(2).T Zo. v)
log
po Zo:r y) dz
(8)
q(Z1.r
Zu. y)
where, ((Z1.T 2o: v) - II a(zz 2.-1, but note that practice g(zz 1;- usually coincides with q(zz (z-11)
the forward noising class-agnoster Applying Jensen's inequality to the expression inside
(zo.T y)
Po (zo.T 1)
log (zo (v) = log E (y(zzr//z0w) g(21.r Zu: y)
By T (RO.))
log
(9)
(Z1:T Zo: y)
Hence define the negative ELBO:
(()) = E Ee(xoor mo.z)) log gZ Zo: 1)
so that
log pa 13) - C(0)
(10)
po (z 1)
C.5. Decomposition
By writing out q(a1:T
20, y) and po (Z0:r) y) explicitly, we get:
Zoo
II
g(z Zz-1, y), 1=1
Po (zo:T (/) = po Z1 v) II po Zet1, 9)
Therefore,
log (Z1rT Zu. v)
log
Z1. v)
log g(zz 22-1.8) log pe 2141,8))
(11)
po (go.T) y)
Taking expectation under ((Zo.r 20, v) yields:
((0) = EngRo. (Ro.9) [- log Pe (zu Z1, y)
Ba m0.9)
[og q(z 2z-1.8) - log Po 22-1. v)
(12)
(ecconstruction hypercone)
between forward vMF and
Hypercone Constraint.
The key difference from standard hyperspherical diffusion is that
Po 2o
Z1, y) is constrained to Cy (6g),
ie. we ensure that Zo stays within angular distance 0, of the class mean My. Hence the "reconstruction term" - log Po (Zo 21, 2) does not drive the model to single point Hy, but rather to the full hypercone (o,)),, Mathematically. this can be implemented by truncated vMF distribution or any parametric distribution that has zero probability outside Z(z, My) >> 0y.
C.6. Resulting Objective
Combining the above, we arrive at:
Po v) 2 L(()) Eyglar (oow) po 21, v)
Σ (logpo po 2 v) log 2 w))
(13)
each (z, 2 y) is a vMF(ma, Ri)) and each q(z, 2;-1, y) is vMF(2,11, fig) then each KL term has a known closed form. The final f term effectively enforces that Zg remains in the class hypercone. Thus, the chain converges to a distribution localized around Py, with an angular radius fy. rather than collapsing single direction.
16

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
D. Uncertainty Modelling
Lemma D.1 (Bquivalence of Angular Interpolation and vMF Diffusion), Let be E gd-1 be generated by either:
1. Angular interpolation: z, = ccos(f))4,.)) sin(0,)v, where v Uniform(St-1)
2. vMF sampling: 2, ~ (M-
For K₁ = cot(6:). these processes generate equivalent distributions over the hypersphere.
Proof. Under angular interpolation, 0, = 0 gives 2, = Zz-1 (perfect preservation), while 8, = = gives 2, = (uniform noise). For vMF with A; = col(O.), these correspond to Ki - oo (perfect concentration) K; - (uniform distribution) respectively. The equivalence follows from the conditional density:
p(zzlz--1) x exp(cot(O))zz.
which matches the vMF density [(2;; 2t,1 me) xx exp(Rez(_112) when Ki = cot.
Implications: This equivalence offers geometrie (angular interpolation) and probabilistic (vMF) views of the forward process, with K₁ cot) ensuring compatibility with Smooth progression 0 n/2 matches Kig 0
Lemma D.2 (Concentration of vME Reverse Process into Class Hypercone). Suppose have discrete-tine reverse Markov chain {Zr 1i-r C Cdd- defined by
2 ~ vMF( [T( + me Vn, log s(ze pc)); mit)
(14)
where:
1. [Txx)= = x/1 |x|| is the projection onto the unit hypersphere Sd-1
2. Vx, log (21 Pt) is the gradient (score function) of density [Z2 pa) that is sharply peaked around the class mean
Me E gd-1, In particular, this gradient points largely in the direction of p Ze (where u = pec whenever z₁ is
not too close to pe.
3. K4 (the vMF concentration) increases over time, and + 0 at suitable rate (e.g. Tikt + oo but - 0 as + 0).
Define the class hypercone of half-angle a > -0 around Hec by:
Ca = {uESt-I u Tue 2 cos(a)}
Then under mild smoothness assumptions on and the above monotonicity/decay rates for Ka and qu the chain converges (in distribution) into the hypercone C. (p.) as 1. - Specifically, for any a >
lim [[ E Co = t-0
Equivalently, the angle between 2, and pe goes to zero with high probability.
Sketch of Proof. We outline the main arguments:
1. Directional Gradient Alignment. By assumption, V x, log ['; pa) points generally toward μt, for z, not close to fec- Hence the update z, T) Vz, log f rotates 2, closer in angle ph. After projection 11(.) to the hypersphere, this remains unit vector closer μe than
2. Sharp Concentration under vMF As K4 + 0o, vMF(m, Ki) puts most of its mass around m, with the variance of the angular distribution shrinking like 1/k,. Thus, if the mean direction II [z + m) Va, f) is already within a of Me then with high probability the new sample 2:-1 remains near pcc
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
3. Iteration and Convergence. Given that ve - O, the movement per step in the hypersphere's tangent space shrinks, preventing large excursions away from Me. Meanwhile, Hy (the concentration) grows that each step's sample is drawn from an increasingly peaked distribution. Iterating backward from & = ' down to t = 0, the probability of Ze being outside any cone (pt,) diminishes at each step. By the last steps near t = 0, 2, concentrates with high probability in the chosen hypercone around P.c
Thus, in the limit -> O, we conclude that 2, converges in distribution directions arbitrarily close to Me i.e. within any desired half-angle Equivalently, the angle ((zt; Me) goes to zero with high probability, implying lime- 2 Me I almost surely.
Algorithm 3 Snneercone-Cosstrainee Sampling with Learned Truncation
Require: Class label y. diffusion steps T
Require: Angular schedule {0: Yē=1 step sizes {m )T=1
1: Sample zr ~ Uniform(S" (S
2: for = T to do
3:
m, - (z( t,
Predict direction b Predict angle
4:
0g - Co t,
5:
Ci - fz: (zz m²) S By)
6:
di (zi; mi)
7:
Kt K (310, dcl)
8:
ift > then
9:
ZZ, log f t- ScoreNets (Z,, t,
10:
u, < 2, + 1/p Va, log S
11:
u, - uz/ ///[
Zt-1 Ky, R4, Cty.)
13:
end if
14: end for
15:
return Final sample
E. Brownian Motion on Hypersphere
Recent advances in continuous-time score-based generative models (Song et al., 2021; Karras et al., 2022) suggest viewing the forward noising process as SDE, whose time-reversal recovers the data distribution. When data reside on hypersphere gd-1, an analogous approach involves constructing a Brownian motion restricted to gd-1, then reversing it to produce samples from the original (or a conditional) distribution. Concretely, let z(t) E gd-1 evolve for E [0, T] such that follows an intrinsic Brownian motion on the hypersphere. In the simplest form,
dz(t) = (2)2(t) P Pa(t) dw(t),
(15)
where, (() 2 0 is noise scale (or diffusion coefficient) and P Pz(() projects IR increments dw(() onto the tangent space at (t) E gd-1
Spherical Forward Process and Hyperspherical Score Estimation. By choosing (t) such that the marginal distribution of (T)) approaches the uniform measure on 1 we obtain a continuous analog of spherical diffusion. In practice, one can incorporate small drift term b(2, t) to ensure that ((x())) is nearly uniform, similar to the variance- or variance-explodingg schedules in Euclidean SDEs (Song et al., 2021). Concurrently, we learn a score network se (z, that approximates V. log (z), using a spherical variant of score matching (Vincent, 2011).
Reverse-Timi SDE and Generative Sampling. Once so trained, we generate new samples by solving the reverse-time SDE. Formally, time reversal of the forward process (15) yields
(z) [-o² =1) so (z, t)] dt 02 P Pz(c) dw(l),
18

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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
where, / [T. 0] and dw()) is again Brownian noise on the tangent space, and we solve backward from Z(T) Uniform(S* down to (()). Intuitively, the term (t) so so acts drift that guides samples toward the data manifold the hypersphere.
Comparison to MMF Diffusion. In discrete-time spherical diffusion using vMF noise, each forward step is vMF (2 He), while the reverse step estimates vMF(- RR) with a learned center. Spherical Brownian motion (15) can be viewed as the continuous limit of many small vMF perturbations. Inversely, discretizing (E) via Euler-Maruyama method yields a small-angle vMF reverse step that remains tangent to Sd-1
Lemma E.1. Discrete-Continuont Correspondence Let qae (244224 Ze) be vMF transition with concentration kÂt. As At 0 with KYt (1/Att), the process converges weakly the solution of the spherical Brownian motion SDE (15).
Handling Class Hypercones and Adaptive Freezing. Finally, respect class geometry or hypercone constraints, one may introduce a drift term that becomes very large (or reflection boundary) whenever z(t) moves outside the class-constrained cone. Alternatiely, if each class y has known center My and angular tolerance Oy. the reverse SDE (E) can incorporate penalty encouraging (2. 1yy) S Oy. effectively "locking" constraining to sub-manifold like a hypercone on the sphere) trajectories in the appropriate region of ga-1, Such mechanisms ensure the final sample remains class-consistent while leveraging the flexibility and elegance of spherical Brownian motion the underlying diffusion.
F. Beyond Geometry and Uncertainty: Retaining Image Information
For the experiment with two diffusion process, the forward (noising) step at time t can be viewed as:
0t 04-1 + aee d, vMF KR)
X; = aq d:
where, a' is small Gaussian perturbation (e.g. ((0, 1)) controlling how the magnitude spreads, and de is sampled from a vMF distribution centered at d,-1 with concentration K₁. As t increases, Kr may decrease (making directions more diffuse), while a can enlarge the variance of ag, allowing the embedding norm to fluctuate more widely.
Reverse (Denoising) Process. We define the reverse chain to invert both magnitude and direction back to the original class-consistent configuration. To invert this process, we learn parameters 0 that predict a suitable Gaussian mean for the magnitude and a suitable mean direction for the vMF. Formally, define
x Muo de), )²)
d_- vMF(mo dr), Kzz
x م di-1
Here, (-) is neural network output that infers the ideal norm given the current 0r and de, while m is another network output that estimates the best directional center for the vMF.
This design provides powerful way to incorporate both global image information and class geometry: we learn how to denoise both magnitude (via the Gaussian) and direction (via vMF), thus recovering the class-relevant structure on the hypersphere encoded in the angular direction while preserving essential image information encoded in a. This dual- uncertainty approach ensures that face embeddings can vary naturally in intensity or brightness, yet remain consistent with class geometry, capturing both how bright an image is and which identity it belongs to.
Theorem F.1. Dual-Uncertainty Preservation Let Xo Rdd with do | and dn xo/||x0||- Under the forward process:
Q4 = a1-1 + 0
E (((,0 dt vMF Ki) x; at
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
The following holds for all & 0:
B(a au] =
Var( ao) Eld d0] = II Aa(k:)
where, As(k) La/2(()//a//-11/ (R) is the ratio of modified Bessel functions. Moreover p(xi|xo) pdd//dn) with d, E gd-1 almost surely.
G. Adaptive Class-based Hypercone Learning
Class Hypercone Setup. Each class E {I....,C') is associated with a direction My E gdd- and an angular radius fy 2 0. Hence, the class hypercone is defined by:
(o,))) = {z gd-1 1z,, mby) S 0g}
(16)
Forward (Noising) Process. We consider forward chain of length T, where each step injects vMF noise:
((1:3 20) I 2-1:; where | 21-1 vMF(21-1, Kit)
(17)
Here, K₂ is a (potentially decreasing) schedule that pushes z, toward uniform on gd-1 as grows. Adaptive Reverse with Learned Concentration. Instead of using fixed reverse schedule, let
(2 y) gd-1 x {1, (1....,C) C}
(18)
be a learned function of the current state E gd-1 and the class We define the reverse model as
T
(o v) = p(zr) II
po 21 Ze, v). where po pz 21 vMF mo U), (o(Z),)) v)
(19)
Here, p(2r) is uniform on the hypersphere (i.e. the limiting vMF with zero concentration), and m ( v) E gd-1 is a learned mean direction. The key distinction is that No (zt, v) can grow large once is in the correct hypercone, effectively "freezing" further denoising.
Example of a Freezing Mechanism. Let
a(z,,)) v) Aze, and Ro(Ze, v) Kima (3 alze; w)l)
(20)
where, o--) is monotonic squashing function (e.g. sigmoid), Kmax is large positive constant, and 3 > 0 controls slope. If alz, y) S fg. then rig ko(z,,8) y) saturates near Kmax. locking 2 into the hypercone Cy(Oy) by making the vMF distribution highly concentrated.
H. Experimental Details
For feature extraction, we employ different architectures based on the domain: facial representations obtained using pre-trained ArcFace (Deng et al., 2019) model with iResNet50 (Duta al., 2021) backbone, while object categories use CNN-based feature extractor, both producing hyperspherical embeddings. The latent space is configured with dimensions 32 x 32 x 3, balancing detail and computational efficiency. Our diffuusin process uses an angular parameter 0 that progresses from to 722 across diffusion steps, with the concentration parameter к derived as cot(f), naturally transitioning from high concentration (N >> at = 0) to a uniform distribution. For ctass-conditional generation, we utilize a context dimension of 512 encode class information. The class specific hypercone constraints are implemented through adaptive
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Hypooeee Corragy
Maraan anwore sampli surade Bie Hight HCA Angulaa datariaa
Heporos Eiilic Nooe
extippe postppooe for clse
Koup sellheghe
Lever detter perentzation slight demplity Bla
Semplio under with anglic
timpple under with angli X Seceple under arrlee
Where sip- p. Prporoee ingley
Figure 8. Geometric interpretation of proposed metries HCR and HDS.
values, ranging from 0 to class-specific maximum determined by the class angular radius 0.. which is predicted by UNet architecture. The maximum follows Kmaz log(1/e)/(11 cos(0.)), ensuring generated samples respect class-specific geometric structure. For training, we use the Adam optimizer with learning rate of le-4 and batch size of 128, with the model trained for 100K iterations on a single NVIDIA A A100 GPU. For comparison with Gaussian based diffusion, we used Variance preserving variant of diffusion.
H.1. Analysis of metrics
The Hypercone Coverage Ratio (HCR) and Hypercone Difficulty Skew (HDS) together offer a comprehensive framework for evaluating generative models in terms of their alignment with the original class distribution and their ability to handle sample difficulty.
The HCR primarily assesses whether the generative model preserves the class structure by examining the percentage of generated samples that fall outside the class's hypercone. A lower HCR suggests that the model is generating samples that remain within the expected angular region of the class distribution, indicating that it faithfully reproduces the structure of the original class, On the other hand, higher HCR implies that the model is producing out-of oftlas- or unrealistic samples that deviate significantly from the class's ' expected region. This could be a sign of overfitting or a failure to learn the true distribution of the class, leading to unrealistic or poorly sampled outputs.
The HDS, in contrast, focuses on the model's ability balance difficulty across the generated samples. By dividing the class's hypercone into multiple sub-cones based on increasing angular thresholds, HDS captures how well the model distributes its generated samples across regions of varying difficulty. The model is expected generate mix of easy (close to the class mean) and hard (further from the mean) samples, reflecting the full diversity of the class. A high HDS indicates that the model primarily generates easy samples clustered in the innermost sub-cones. This could suggest a bias towards overfitting simpler patterns. This bias is undesirable because fails to capture the full range of complexity in the class distribution. Conversely, a low HDS suggests that the model is distributing its samples more evenly across different difficulty levels, which is indicative more robust generative process that captures both simple and complex variations in the class.
Together, these two metrics help to paint a fuller picture of generative model's performance. A well-balanced generative model should ideally have a low HCR, reflecting good preservation of class boundaries, and moderate HDS, indicating that it generates a variety of sample difficulties, capturing the full complexity of the class distribution. A high HCR combined with a low HDS might indicate that the model is overly focused on easy samples, while high HDS with a low HCR could suggest that the model is struggling to maintain class consistency. Thus, an optimal model should strike a balance, maintaining a good coverage of the class's 's angular space without overfitting to simpler, casier samples.
H.2. Results
Reverse Denoising Comparison The results presented in Tables to 5 provide a comparative analysis of Euclidean versus angular-basee based reverse denoising strategies across three datasets: 10, MNIST, and D-LORD. For datasets, we observe that angular-bases based methods, particularly those using cosine and geodesic formulations, consistently offer
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
improvements comparable performance across key metries. Specifically, angular approaches yield lower FID scores, indicating better sample quality. For example, FID improves from 3.52 to 3.28 CIFAR-10 and from 1.86 1.77 on MNIST. In terms of Hypercone Coverage Ratio (HCR), angular losses preserve class structure similarly or better than MSE-based approaches, while Hypercone Difficulty Skew (HDS) values suggest that angular methods produce a more balanced range of sample difficulty. These improvements highlight that angular loss functions better align with the intrinsic hyperspherical nature of feature embeddings, enhancing both generation quality and class consistency. The consistency of these trends across datasets confirms the generalizability angular reverse denoising in hyperspherical diffusion models.
Table 3. Comparative analysis of Euclidean and Angular based reverse denoising step for CIFAR-I0 10 dataset. Various score functions are also used for evaluation.
Euclidean with MSE
Angular with Cosine
Angular with Geodesic
FID
3.52
3,28
3.35
HCR
0.20
0.20
0.19
HDS
0.48
0.51
0.47
Table 4. Comparative analysis of Euclidean and Angular based reverse denoising step for MNIST dataset.
Euclidean with MSE
Angular with Cosine
Angular with Geodesic
FID
1.86
1.79
1.77
HCR
0.14
0.15
0.14
HDS
0.52
0.47
0.48
Table 5. Comparative analysis of Euclidean and Angular based reverse denoising step for D-LORD dataset.
Euclidean with MSE
Angular with Cosine
Angular with Geodesic
FID
9.27
9.01
8.97
HCR 0.21
0.19
0.20
HDS
0.62
0.61
0.62
Effect of Scheduling Scheduling controls the rate which class structure degrades, ensuring smooth transition to uniform noise. Without scheduling, any fixed x eventually results in a uniform distribution on the hypersphere, particularly as T increases. Gradually decaying preserves intra-class structure longer, aiding recovery during the reverse process. Formally, for di ~ (dl - 1, ke), the marginal distribution approaches uniformity as KT - 0 p(dr) R p--T Empirical results on the effect of scheduling are shown in Table 6.
Feature Representation Figure 9 illustrates the feature representations of conditional samples generated from the 10 classes of the CIFAR- 10 dataset. The figure highlights that the based reverse sampling effectively converges samples within class-specific hypercones, capturing the angular geometry of the data. In contrast, Gaussian based reverse sampling produces samples that converge within Euclidean space, failing to adhere to the hyperspherical structure.
Facial Data Generation The Figure 10 illustrates the generated images with diversity and occlusion facial challenges present, which is critical for robust face recognition under real-world conditions. The top section presents multiple views of the same subject generated without occlusion, showing typical intra-subject variation. The middle section focuses on cases with eye-region occlusion caused by sunglasses, while the bottom section includes examples of full occlusion from multiple accessories such as hats and scarves.
Hypercone specific generation Figure 12 and Figure 13 shows samples generated from inner and outer hypercone based on class-specife Og that determines the boundary of class. As demonstrated by the figure, the images sampled from inner hypercone are sharp and realistic while samples generated around the boundary are noisy. Also, various samples generated are shown in Figure 11
I. Applications
The applications of the proposed vMF-baaed angular diffusion are:
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Table Rffect of using kappa scheduler. Comparative analysis of using the kappa scheduler on datasets for various evaluation metrics
Metric
CIFAR-10
MNIST
With scheduling
Without scheduling
With scheduling
Without scheduling
Class-wise Accuracy
89.35
72.59
96.01
86.48
FID
3.52
6.28
1.86
2.11
HCR
0.20
0.37
0.14
0.21
HDS
0.48
0.63
0.52
0.75
1000
1.00
0.75
0.25
0.00
0.25
-025
0.50
0.50
0.75
0075
1.00
1.00
1.00
0.75
0.75
0.50
0.50
0.25
-1.00
-1.00
-0.50
-0225
0.75
0.50
-0.25
0.00
0.00
-0.50
-025
0.00
0.25
-0.55
0.25
0.50
-0.75
0.75
0.75
-1.00
1.00
(a) Gaussian Diffusion
(b) vMF based Diffusion
Figure 9. Feature representation of the CIFAR-10 10 datasct generated using Gaussian-baas based diffusion (left) and vMFF-bsed diffusion (right). The basod sampling aligns generated sample features within class-specific 3D hypercones, while Gaussian based sampling results scattered features outside the class-hypercones.
Few-shot learning: Our approach improves performance by generating more diverse and classsconsistent samples from limited data.
Fairness and bias mitigation: Manifold-aware generation allows controlled augmentation to rebalance datasets across demographics, reducing biases.
Face recognition robustness: Explicitly preserving directional structures helps models robustly handle variations (occlusion, illumination, pose).
Difficult sample generation: Controlled angular diffusion produces challenging samples near class boundaries, refining decision boundaries and improving model reliability.
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Same subject
occlusion
Occlusion:
Sunglasses
Occlusion:
All
Figure 10. Facial data synthesis demonstrates variations across occlusion, pose and resolution.
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(b) CIFAR-10
Figure 11. Generated samples from proposed -aa-ha diffusion model trained on (a) MNIST and (b) CIFAR-10 datasets. The samples effectively preserve class class-specific information while maintaining high visual quality.
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Inner Hypercone
Outer Hypercone
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Figure 12. Generated samples from the inner and outer hypercones using the proposed vMF-based diffusion model trained on the MNIST datasct. Samples from the outer hypercone exhibit noticeable noise and distortions.
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Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres
Inner Hypercone
Outer Hypercone
Figure 13, Generated samples from the inner and outer hypercones using the proposed vMF-based diffusion model trained on the CIFAR-10 10 dataset. Inner hypercone samples exhibit superior quality and effectively preserve class-specific information.
26


Reviews:
[{"review_id": "LG6R2UgsN2", "reviewer": "Reviewer_pxpN", "summary": "This paper introduces a diffusion model on hyperspherical space with hypersperical data and hypersperical noise distribution (von Mises-Fisher, vMF distribution). The forward process with vMF noises keeps the latent samples on the hypersphere. The reverse process is designed accordingly.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for constructive feedback and for recognizing the motivation of rethinking the noise distribution. Please find our detailed responses below. All Tables and Figures are available at: https://tinyurl.com/44sftcu8.\n\n**Advantage of Matching Noise and Data Manifold:** Unlike traditional Gaussian diffusion, which distorts angular relationships intrinsic to hyperspherical data (e.g., ArcFace embeddings), our proposed vMF-based diffusion explicitly preserves angular semantics and class structures, maintaining semantic consistency, class separability, and stability during generation, significantly improving downstream recognition performance.\n\n**Preserving Class Boundaries at t=(0,T):** Maintaining class boundaries throughout the diffusion process retains intra-class consistency and reduces ambiguity, preventing mode collapse by keeping samples within their semantic hypercones, facilitating controlled semantic interpolation.\n\n**Scheduling $\\kappa$** controls the rate at which class structure degrades, ensuring a smooth transition to uniform noise. Without scheduling, any fixed $\\kappa$ leads to a uniform distribution on the hypersphere as T increases. Gradually decaying $\\kappa_t$​ preserves intra-class structure longer, aiding recovery during the reverse process. Formally, for $\\mathbf{d}t \\sim \\mathrm{vMF}(\\mathbf{d}{t-1}, \\kappa_t)$, the marginal distribution approaches uniformity as $\\kappa_T \\to 0 \\quad p(\\mathbf{d}_{T}) \\approx \\frac{1}{|\\mathbb{S}^{d-1}|}$​. Empirical results on the effect of κ scheduling are shown in Tab. 1 (https://tinyurl.com/44sftcu8).\n\n**Angular addition in reverse denoising**\nThank you for the insightful question. Our choice of Euclidean addition in the reverse process was motivated by simplified optimization and ensuring faster convergence. However, angular operations may better preserve hyperspherical geometry. To investigate, we tested an alternative angular addition formulation:\n$$z_{t-1} \\sim \\text{vMF} \\left( \\Pi \\left( \\cos(\\theta_t) z_t + \\sin(\\theta_t) \\frac{\\nabla_{z_t} \\log f(z_t; \\mu_c)}{\\|\\nabla_{z_t} \\log f(z_t; \\mu_c)\\|} \\right), \\kappa_t \\right)$$\nHere, interpolation using $\\cos(\\theta_t)$ and $\\sin(\\theta_t)$ maintains angular relationships, while the normalized score function preserves directional consistency.\nTo further align optimization with hyperspherical geometry, we tested two alternative loss functions:\n\nCosine Loss: Encourages angular alignment between the score function and noise direction.\n$$\\mathcal{L}{\\text{c}} = 1 - \\mathbb{E} \\left[ \\nabla{z_t} \\log f(z_t; \\mu_c)^\\top \\epsilon_t \\right]$$\nGeodesic Loss: Penalizes angular deviations.\n$$\\mathcal{L}{\\text{g}} = \\mathbb{E} \\left[ \\arccos^2 \\left( \\frac{\\nabla{z_t} \\log f(z_t; \\mu_c)^\\top \\epsilon_t}{\\| \\nabla_{z_t} \\log f(z_t; \\mu_c) \\| \\|\\epsilon_t\\|} \\right) \\right]$$\n\nEuclidean addition was chosen for computational efficiency and stable convergence with standard diffusion loss formulations. However, our additional experiments (Tables 2–4 at  https://tinyurl.com/44sftcu8) comparing angular additions (including cosine/geodesic losses) confirm that while angular addition is more geometrically faithful, Euclidean addition maintains comparable performance at significantly reduced computational complexity.\n\n**Relation between embedding and image:** Our approach applies diffusion in the latent embedding space, not directly on images. Given an image $I$, a feature extractor $\\phi$ maps it to an embedding $x = \\phi(I) \\in \\mathbb{R}^d$ , decomposed into magnitude $\\|x\\|$ (capturing intensity variations) and direction $x / \\|x\\| \\in S^{d-1}$  (encoding class-relevant semantics). \n\n**Uncertainty** is degree of stochastic diffusion, controlled by $\\kappa$.\n\n**Ambiguity** Points near class boundaries with unclear class assignments.\n\n**Noise** refers to Embeddings displaced significantly due to diffusion.\n\n**HDS** quantifies sample difficulty based on angular deviation from the class mean—samples near the mean are “easy,” while distant ones are “hard,” reflecting variations like pose or occlusion. As shown in L438left, vMF diffusion achieves a balanced spread across difficulty levels, whereas Gaussian diffusion favors easy samples clustered in inner sub-cones. Validation (Fig. 1: https://tinyurl.com/44sftcu8) shows Gaussian diffusion yields tightly clustered samples (mean cosine similarity = 0.90, std = 0.05), while vMF diffusion produces a broader spread (mean = 0.72, std = 0.13), effectively capturing hard cases.\n\nFig. 1(a) in paper shows solid cones indicating natural class concentration and dashed cones denoting diffusion-induced uncertainty, increasing data spread. While vanilla hyperspherical models preserve tight class separation, diffusion introduces uncertainty at intermediate steps.\n\nIn Fig. 1(b) in paper, the points represent generated samples during the sampling process. For simplicity, we performed diffusion over three classes. We will revise the figure description to improve clarity.", "claims_and_evidence": "L055left, L098left Motivation of the need for rethinking the noise distribution is interesting. I agree that it is beautiful to match the underlying distribution of the data and the noise distribution. Still, a question remains. What is the advantage of matching them and preserving the class boundaries / class-wise structures at t=(0, T]? Respecting the geometric properties of hyperspherical data (L217right) is not enough.\n\nL059left Adding the definitions of uncertainty level, ambiguity, and noisy data points would help understanding the motivation. The hint at L139left is not enough for me to understand what the uncertainty is. Currently, I understand it as stochasticity from the context.\n\nL138right Arcface embedding is a proper example of hyperspherical space.", "methods_and_evaluation_criteria": "L204left The reason for scheduling $\\kappa$ should be described because forward diffusion with any non-infinite $\\kappa$ with large T would reach uniform distribution on the hypersphere. \n\nL247left I think increasing $\\kappa$ makes the vMF sampling become less stochastic, rather than more concentrated around class means because vMF in L229left is centered at $z_t+…$. The concentration could be true if $\\kappa$ defines the class cone, but in L229left, $\\kappa$ defines the score and stochastic reverse process. I understand that $z_0$ should be close to $\\mu_c$ but it is not guaranteed to reach $\\mu_c$, as written in L269right (arbitrarily close to $\\mu_c$).\n\nL265left Eq.(1) I wonder why we should add z_t and score linearly followed by projection rather than simple angular addition. Maybe because the proposed method adopts the simple denoising loss in Euclidean space? I think the loss should be also measured in angular space.\n\nI understand the dual diffusion processes for the magnitude and direction in Section 4.4 in the face embedding space. However, the connection between the embedding and the images are not explained. Adding an explanation would help readers understand the connection.\n\n\nThe performance is measured by\n* Hypercone Coverage Ratio (HCR) which measures preservation of the class structure. It makes sense.\n* Hypercone Difficulty Skew (HDS) which measures skewness toward easy samples. Explanation of the samples near the mean being easy and the samples far from the mean being hard would help understanding. I checked Appendix H.1 but it is not enough. In this regard, I do not understand L438left.", "theoretical_claims": "Theorem 3.1 (b) and (c) are correct to my knowledge. I did not check (a) but I suppose it is correct.\n\nI did not check Theorem 4.3.", "experimental_designs_or_analyses": "The target datasets include faces and simple images: MNIST, CIFAR-10, CUB-200, Cars-196, CelebA, and D-LORD. I wonder the connection between the embeddings on the hyperspheres and the images. It is reasonable for such theoretic content.", "relation_to_broader_scientific_literature": "n/a", "essential_references_not_discussed": "Related work is thoroughly discussed.", "other_strengths_and_weaknesses": "Clarity could be improved as written above.", "other_comments_or_suggestions": "None", "questions_for_authors": "What do the solid cones and the dashed cones mean in Figure 1a? I understand the points the data points but the cones are not described. If they are hypercones in Section 4.3, adding the vanilla hyperspherical scenario would help understanding.\n\nWhat are the points in Figure 1b? Maybe the generated samples from the learned models? I do not understand the term three-class \"diffusion\".\n\nI like the idea in this paper in general. This paper could be be accepted if the rebuttal explains the following:\n* The advantage of matching the data manifold (hypersphere) and the noise distribution (vMF)\n* The advantage of preserving the class boundaries at t=(0,T)\n* The connection between the hyperspherical embeddings and the images\n* The effect of increasing $\\kappa$\n* The reason for choosing Euclidean addition in the reverse process (Eq. 1) instead of angular addition,\nand the minor weaknesses and questions above are fixed or proved ok in the rebuttal.", "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": "KklKvSKhkW", "reviewer": "Reviewer_BHWr", "summary": "The paper introduced an idea to generate data defined on hyperspheres. When data is decomposed into magnitude and direction components, the generation results can be improved.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for acknowledging the thorough evaluation and the relevance of our proposed metrics for hyperspherical geometry. We also appreciate the recognition of our diverse dataset selection, which demonstrates the robustness of our approach. Please refer to the corresponding Tables and Figures at: https://tinyurl.com/44sftcu8\n\n**Results on Facial vs General datasets (Figure 7 https://tinyurl.com/44sftcu8):** Our proposed method is general and can indeed handle arbitrary datasets, as the reviewer correctly points out. However, facial datasets like CelebA and D-LORD are particularly illustrative because facial embeddings naturally form structured hyperspherical manifolds.\nTo clarify this visually, we now explicitly provide Figure 7 (see provided link), which illustrates facial variations, demonstrating how our method effectively handles diverse occlusions (glasses, hats, scarves) and maintains clear semantic consistency on a hyperspherical embedding manifold.\nAdditionally, our method is validated broadly on non-facial datasets such as MNIST, CIFAR-10, Cars-196, and CUB-200, confirming its general applicability beyond faces.\n\n**Relation to Progressive Distillation paper ([1], Eq 38):**\nWe appreciate the reviewer highlighting the structural similarity of equation in (L215) to Eq. (38) from \"Progressive Distillation for Fast Sampling of Diffusion Models\" (DDIM update rule). We clarify explicitly:\n- While the DDIM rule in [1] describes an angular parameterization in a Euclidean latent space, our method differs fundamentally in operating strictly on hyperspherical manifolds. This requires projecting each intermediate step back onto the hypersphere to preserve directional consistency.\n- Our angular update formulation specifically respects the geometry of spherical manifolds (unit-norm constraints), a key distinction absent in the Euclidean-based DDIM and EDM formulations.\n\n**Explicit Comparison to EDM (Variance-Preserving Property):** We thank the reviewer for raising this critical comparison. We explicitly clarify:\n- EDM enforces variance preservation explicitly through dataset-specific scaling factors (e.g., $\\sigma_data​$), adjusting Euclidean diffusion to maintain unit variance. However, EDM is agnostic to directional information and ignores data manifold geometry.\n- Our approach implicitly ensures variance preservation via the angular update parameter $\\theta_t​$, which naturally preserves unit norm constraints on a hyperspherical manifold. Thus, unlike EDM, we do not require dataset-specific scaling. Our method inherently respects both variance preservation and the underlying manifold structure, crucial for manifold-oriented data (like embeddings).\n\nTherefore, the reviewer’s suggestion that EDM could easily solve our addressed problem overlooks the critical aspect of directional geometry, which our method specifically addresses.\n\n**Visualization on 2D circles (Figure 2: https://tinyurl.com/44sftcu8):** As recommended, we explicitly illustrate training and sampling behaviors visually on simple 2D circles (Figure 2 provided at the link):\n\n **Figure 2a** demonstrates final embeddings: Gaussian diffusion ignores angular boundaries, while vMF-based diffusion clearly preserves angular structure.\n\n**Figure 2(b, left)** explicitly depicts embeddings during training, where classes progressively cluster into distinct angular regions.\n\n**Figure 2(b, right)** illustrates sampling from noise, showing vMF diffusion’s clear angular convergence, preserving directional semantics throughout sampling steps.\n\nThis visualization explicitly demonstrates the benefits of respecting manifold geometry in diffusion processes, as suggested by the reviewer.\n\n**Importance and Applications:** Regarding the concern of importance of the approach, we would like to highlight that diffusion method’s explicit manifold-awareness has significant practical implications:\n- Few-shot learning: Our approach improves performance by generating more diverse and class-consistent samples from limited data.\n- Fairness and bias mitigation: Manifold-aware generation allows controlled augmentation to rebalance datasets across demographics, reducing biases.\n- Face recognition robustness: Explicitly preserving directional structures helps models robustly handle variations (occlusion, illumination, pose).\n- Difficult sample generation: Controlled angular diffusion produces challenging samples near class boundaries, refining decision boundaries and improving model reliability.\n\nThus, the explicit consideration of hyperspherical geometry significantly enhances practical AI deployment, particularly in sensitive applications like facial recognition, fairness, and robustness.\n\nMinor clarifications: We will reference the equations and methods in the paper with the corresponding details in the supplementary material, We will also move the algorithm in the main paper as per the suggestion.", "claims_and_evidence": "In Sec 3, the authors mentioned facial datasets several times. However, the method seems to be working with general image datasets. This makes me confused. It would be good if the authors could show some figures to illustrate the problem (maybe in supplemental).", "methods_and_evaluation_criteria": "The evaluation makes sense. Several datasets are used to show the effectiveness of the method, including digits, birds, humans, cars, ...\n\nSeveral metrics related to hypersphere geometries are also proposed to verify the claim. They are also well-discussed.", "theoretical_claims": "I commend the authors for their thorough derivation of the method. However, some equations in the main paper appear tangential to the core contribution. The central ideas can be effectively communicated through simplified text explanations or illustrative figures. I would suggest relocating peripheral theorems/lemmas to the supplemental material. Conversely, Algorithm 1 and 2, currently in the supplemental, are critical to understanding the workflow. Their absence in the main text creates confusion, as the algorithm descriptions are essential for reproducibility.", "experimental_designs_or_analyses": "The experiments can prove the core idea of this paper.", "relation_to_broader_scientific_literature": "The idea is very interesting. I believe the decomposition is not explored in the image diffusion models. The diffusion models for data defined on hyperspheres are also interesting.", "essential_references_not_discussed": "The equation in L215 is also the sampling equation in [1, Eq 38], which satisfies the variance preserving property.\n\n[1] Progressive distillation for fast sampling of diffusion models.", "other_strengths_and_weaknesses": "I believe the exposition can be improved. The method starts from hyperspheres and all the descriptions are about hyperspheres. However, since a circle (von Mises distribution) is a special case of a hypersphere, I would suggest the authors show some illustrations using circles, for example, how the training and sampling process look like on 2d circles.", "other_comments_or_suggestions": "No.", "questions_for_authors": "Another concern is about the importance of the application. The authors showed some results to prove the effectiveness of the method. However, I am not quite convinced.\n\nThere is another property of DDPM, which is commonly called Variance Preserving. Specifically, when the data has unit variance, the noised sample will also retain unit variance. \n\nIn EDM [1], a scaling (sigma_{data} in [1,Tab 1]) is applied to ensure this property even if the data does not have unit variance. The equation in L215 also guarantees variance preserving during each adding noise step. Thus I would like the authors can discuss this thoroughly. I believe that the motivation of this paper is weak, considering the problem can be easily solved in [1].\n\n[1] Elucidating the Design Space of Diffusion-Based Generative Models", "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": "gtEQXzoPe1", "reviewer": "Reviewer_VDyd", "summary": "This paper explores the distributional assumptions made by denoising diffusion models and proposes exchanging the traditional Gaussian noise for a von Mises-Fisher distribution on a d-1-dimensional hypersphere. This choice somewhat improves the performance of the diffusion model in generative tasks, especially in those in which class relations and boundaries are particularly relevant.\n\n## update after rebuttal\n\n I am satisfied by the clarifications and will raise my score on the assumption that the promised changes are implemented in the paper.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's recognition of our well-reasoned and justified claims, as well as our intuition on magnitude and direction in hyperspherical space. We have added responses to the reviewer’s comments, and for respective Tables and Figures, kindly follow the link: https://tinyurl.com/44sftcu8.\n\n**Fundamental Theoretical Justification (spherical vs. Euclidean):** The reviewer raised an insightful question about the fundamental reasons for preferring spherical latent spaces over Euclidean ones. Indeed, spherical representations inherently emerge from neural embeddings through unit normalization (as in common embedding networks like SphereFace, ArcFace), explicitly disentangling direction (semantic information) from magnitude (intensity or scale). Such disentanglement directly matches perceptual similarity measures (cosine similarity), making spherical embeddings naturally robust to magnitude variations (e.g., lighting, contrast). Empirically, spherical latent spaces are widely validated in domains like face recognition precisely due to these properties.\n\n**Clarification of \"Classes\" in datasets (CelebA):**  We clarify explicitly: in facial datasets like CelebA, the \"classes\" refer specifically to **distinct individual identities**, derived using pre-trained identity-embedding models (e.g., ArcFace embeddings). Thus, \"in-class\" generation refers explicitly to maintaining identity consistency, ensuring generated samples remain within the identity's angular semantic region, and preserving identity integrity throughout diffusion.\n\nWe will explicitly clarify this identity-based definition in the revised manuscript for better readability.\n\n**Experimental Results Clarification:** The reviewer correctly notes the experimental improvements are modest. We emphasize explicitly:\n\n- The primary contribution of our method is theoretical—introducing manifold-aware diffusion explicitly designed for structured, hyperspherical embedding spaces, rather than solely maximizing benchmark numbers.\n- Achieving comparable performance demonstrates that our theoretically grounded approach is viable, interpretable, and practically useful without sacrificing performance relative to widely-used Euclidean methods. This validation is essential, showing our method as a robust alternative offering additional interpretability, control, and semantic consistency that Euclidean approaches inherently lack.\n\nWe will explicitly strengthen this point in the discussion, highlighting practical advantages beyond mere quantitative metrics.\n\n**Clarification on norm after forward noise addition:** Thank you for pointing this out. You are correct that the norm of  $z_t$  may deviate due to the cross-term $\\sin(\\theta_t) \\cos(\\theta_t) z_{t-1} \\cdot v$. Our implementation explicitly uses projection back onto the hypersphere after each step, maintaining the integrity of our approach. We will clearly and explicitly correct this mathematical statement and clarify the role of projection in the revised manuscript. As correctly pointed out, this correction does not impact any subsequent theoretical or practical results.\n\n\n**Additional Ablation Study (Magnitude vs. Direction separation): (Table 5 https://tinyurl.com/44sftcu8):**  We appreciate the reviewer’s suggestion to explicitly validate the benefit of separating magnitude (Gaussian) and direction (spherical). We provide an explicit ablation study (Table 5 at the provided link), clearly comparing our proposed decomposition to a fully spherical alternative. This experiment confirms explicitly that the magnitude-direction composition significantly improves generation quality, diversity, and semantic consistency, validating our proposed approach quantitatively and empirically.\n\n**Structure of the paper:** Thank you for your understanding. We aimed to balance completeness and clarity by keeping the main text focused while providing additional results in the supplemental. However, we will consider bringing key results into the main paper to improve accessibility without compromising readability.", "claims_and_evidence": "Most of the paper’s claims are well reasoned and justified. One major overarching claim that justifies the entire text is the fact that real data distributions are better explained as being contained in hyperspheres than in Euclidean space. As someone relatively unfamiliar with this area of research, I was initially surprised by this claim, which in the current manuscript is mostly justified by reference to prior work (see second paragraph in Sec. 1.2) as well as indirectly through performance improvements. A part of me wishes this had been explored further: is there a fundamental theoretical reason why one should believe that a spherical latent space will represent the space of “pictures of birds” better than a Euclidean one?", "methods_and_evaluation_criteria": "Evaluation criteria make sense for the problem at hand. I was initially confused by the relation between the datasets used and the insistence of in-class vs out-of-class generation. What are the “classes” in a dataset of portrait images like Celeb-A? In general, I did not find the experimental results to be conclusive proof of performance improvement, but just the fact that this new architecture works as well as others is an interesting experimental result on its own.", "theoretical_claims": "Overall, the method seems theoretically grounded, with one glaring exception. I could be wrong, but in Sec. 4.1 and Lemma 4.1., the vector \n\nz_t = cos(\\theta_t)z_{t-1} + sin(\\theta_t)v\n\ndoes not, in general, have norm 1, and thus does not belong to the sphere. Indeed (and unlike stated in the proof of the Lemma),\n\n||z_t||^2 = cos^2(\\theta_t)||z_{t-1}||^2 + sin^2(\\theta_t)||v||^2 + sin(\\theta_t)cos(\\theta_t) z_{t-1} \\cdot v = 1 + sin(\\theta_t)cos(\\theta_t) z_{t-1} \\cdot v\n\nWhich, unless I am missing something, can take any value between 0 and 1 (for example, take v=-z_{t-1} and \\theta = pi/4, then z_t=0). If the authors agree that this is a mistake, I do not believe that this majorly impacts the contribution of the paper; as far as I can tell, this fact is not used later in the work and is merely used as an intuition. Still, it should be corrected.", "experimental_designs_or_analyses": "See above. Additionally, I enjoyed the intuition about facial generation having a “magnitude” Gaussian component and a “directional” spherical component. I would have liked an ablation study showing whether this separation does indeed lead to a better output than assuming the entire distribution to be spherical.", "relation_to_broader_scientific_literature": "I am not very familiar with the literature on this topic, but I followed the discussion of related work and did not miss any related works that I know of. I defer to more expert reviewers to evaluate the novelty of this work and its placement in the literature.", "essential_references_not_discussed": "None that I know of.", "other_strengths_and_weaknesses": "In general, the main strength of this paper is that it introduces a hypersphere-based diffusion model instead of the traditional Euclidean one. This is a good contribution, and while it may be more useful for some tasks than for others, I can easily see how the community can benefit from having this in its toolbox. At the same time, the performance improvements are modest, the (theoretical and experimental) justification for the need for these hyper spherical distributions could be stronger, and the text should be reviewed for mathematical errors. A combination of the latter make me be less enthusiastic about accepting this work, although I will carefully read the other reviews and the authors’ rebuttal, fully acknowledging that I may have missed important aspects of the manuscript.", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "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": "IkHTsF4mxa", "reviewer": "Reviewer_9BTW", "summary": "Standard diffusion models have relied heavily on the simple isotropic Gaussian noise in the forward process to effectively transform an unknown complex data distribution to this simple Gaussian distribution and has proven to be effective for a large variety of tasks. However, despite this effectiveness, many real world problems involve non-Euclidean distributions, e.g., hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. If modeled in a Euclidean space, the angular geometry is not preserved and thus, the angular subtleties between classes are lost. To tackle this fundamental problem, the work proposes a new forward process involving angular noise injection which effectively transforms the unknown data distribution into a von Mises-Fisher (vMF) distribution over time.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for appreciating our visual inspections, especially  Figure (6) of the submitted paper. Further, we thank the reviewer for the insightful questions and feedback. We have addressed all the questions asked by the reviewer; kindly follow this link: https://tinyurl.com/44sftcu8 to refer to the corresponding Figures and Tables.\n\n**Baseline Description and Clarity in Table 1:** We appreciate the reviewer’s observation and apologize for the lack of clarity. We confirm that our Gaussian baseline is implemented using variance-preserving (VP) diffusion, consistent with the noise schedule used in DDPM. Our proposed method modifies this by replacing Gaussian noise with von Mises-Fisher (vMF) noise, enabling angular diffusion aligned with the hyperspherical manifold.\nWe did not compare with variance-exploding (VE) diffusion because its unbounded noise scaling is incompatible with hyperspherical geometry, where constrained angular relationships must be maintained. We will explicitly clarify this baseline setting in the main text, and will also update Table 1 (paper) to bold the best-performing results for better readability.\n\n**Spacing Mistakes:** We thank the reviewer for pointing out these writing mistakes. We have corrected them, which will also be reflected in the paper. We will carefully review the paper for any other errors.\n\n**Figure 6 (paper) for CIFAR-10:** We appreciate your positive feedback on Figure 6 of paper and agree that expanding this visualization to more datasets strengthens our message. We have added a **t-SNE visualization for CIFAR-10** (see Figure 3 in https://tinyurl.com/44sftcu8), which again demonstrates that vMF-based diffusion preserves class structure effectively in the feature space, similar to the results seen with MNIST. The clusters are well-separated, and generated samples align closely with their corresponding class clusters, confirming generalizability.\n\n**Improvement in Figure-4a wrt Resolution and Subjectivity:** We understand the concern regarding the subjective nature and low resolution of Figure 4(a) (paper) due to the use of surveillance-quality face data. We have improved the image quality and clarity in the updated Figure 4 (included at https://tinyurl.com/44sftcu8) by using higher-resolution samples and clearer labeling. While image-based qualitative comparisons can be subjective, this version more clearly illustrates the structural differences captured by vMF-based diffusion compared to Gaussian diffusion.\n\n\n**Memorization Concern and Nearest Neighbour Analysis:** \nThank you for raising the concern about potential memorization in generated samples. We have addressed this explicitly through a nearest-neighbor analysis in Figure 5 (https://tinyurl.com/44sftcu8). For CIFAR-10, we compare real training samples with their corresponding generated samples (after denoising). The analysis shows that:\n\n- While generated samples are semantically similar, they are not exact replicas of training data.\n- The variation in distance between real and generated samples indicates that the model captures structural features without memorizing input examples.\n- This supports the generative model's generalization rather than overfitting.\n\nThis visualization illustrates the trade-off between fidelity and diversity and confirms that our model produces meaningful variation rather than duplication.\n\n\n**t-SNE plot (Figure 6 https://tinyurl.com/44sftcu8):** We provide further explanation using the t-SNE visualization in Figure 6. It presents a comparison between real and generated samples across 10 distinct classes, with each color representing a different class.\n\n- Real samples are shown as circular markers, and the generated samples are shown as crosses.\n- The close alignment between generated and real sample clusters validates the model’s ability to **preserve class structure** in the latent space.\n- Mild dispersion across some classes shows that the model introduces **diversity** without sacrificing identity.\n\nWe believe this plot presents strong evidence that vMF-based diffusion maintains angular consistency and avoids collapse or overfitting.", "claims_and_evidence": "This section summarizes the claims and evidences provided in the paper. See the section below for my critiques.\n\nTwo fundamental claims are made by the authors: \n\n(1). vMF distribution has been employed in various generative modeling approaches, including VAEs and GANs, where it has shown to be effective at in face recognition, outlier detection, and even representation learning. ***However, the incorporation of vMF noise into diffusion models has been not done according to the authors.*** This claim is made in the related work section.\n\n(2). The second claim is obvious since it is the main narrative of the paper. By employing vMF noise or specifically directional noise aligned with hyperspherical structures, the authors claim that diffusion models are able to preserve class geometry and effectively capturing angular uncertainty.\n\nThe evidences for these claims are detailed in:\n\nTable (1) which illustrates a comparison between vMF-based and Gaussian-based noise injections using FID, ***Hypercone Difficulty Skew (HDS)***, and ***Hypercone Coverage Ratio (HDR)*** as comparison metrics. **Lower is better for all of the three metrics**. Note, the two metrics, HDR and HDS, are developed by this work. For this particular experiment, six datasets are used. They are MNIST, CIFAR10, CUB-200, Cars-196, CelebA, and D-LORD (a face recognition dataset). Overall, vMF-based diffusion models are shown to perform much better than its Gaussian-based counterparts for most of the datasets in Table (1). \n\nFigures (2), (3) and (4, a) show qualitative results of the generations done by vMF-based diffusion models, which are quite good but not surprising since the FID scores from Table (1) are pretty good. \n\nFigures (4, b) illustrates a regression experiment attempting to further contrast Gaussian-based and vMF-based diffusion models.\n\nLastly, perhaps the best evidence is Figure (6) in the Appendix, which illustrates a comparison between Gaussian-based and vMF-based processes in feature representation for MNIST classes. vMF-based process clusters points from the same class much better (visually).", "methods_and_evaluation_criteria": "Regarding the utilized metrics in Table (1), they are sound and fit nicely to the setting. However, the descriptions or the details about the baseline (Gaussian diffusion) are not clear in the main text.\n\nDid you compare to Gaussian-based VE/VP diffusion models? Or, is your baseline description located in Appendix (E), which is not referred to in the main text, by the way? Also, **please bold which is a better result in Table (1)**.", "theoretical_claims": "A fundamental claim made by the paper is that Gaussian noise does not preserve angular relationship. The full proof is shown in Appendix (A) and Appendix (B). I do not see any fundamental problems with the proof provided in the Appendix sections, I mentioned. However, I must note that there are some incorrect spacing. \n\n(1) For example, in the proof of Appendix (A) for the conclusion, \"Because isotropic Gaussian noise in $\\mathbb{R}^d$ shifts points off thehypersphere...\"\n\n(2) In the implications in proof on Appendix (B). Implications (1) and (2) are not spaced down properly.", "experimental_designs_or_analyses": "I really like Figure (6) from the Appendix as it illustrates the main point of the work very well. If this same figure can be produced for other datasets, this would be wonderful as it confirms the underlying message of the paper. \n\nHowever, I have the opposite message for Figure (4), I think (4)(a) is too subjective, and it's difficult to tell due to the low resolution of the images.  Thus, I don't think (4)(a) is convincing at all. Meanwhile, (4)(b) does illustrate a better message for vMF model, but I don't think this result is any stronger than Figure (6).\n\nLastly, regarding the visualizations of the generated samples in the Appendix, I understand or am aware of class conditioning model is prone to memorization. However, the generated samples from CIFAR10 are very similar/close to the training samples. They look very duplicated. I would like the authors to comment about this.", "relation_to_broader_scientific_literature": "As laid out in the paper, there have been attempts in formulating alternative noise injection approaches for diffusion models. As far as I am aware, the approaches, which work well, do not deviate much from Gaussian-based noise injection.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "See Above.", "other_comments_or_suggestions": "See Above.", "questions_for_authors": "Please provide more generated samples for visual inspection. Moreover, if Figure (6) can be produced for other datasets, that would be fantastic.", "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:

Meta-review: This paper proposes a method wherein the Gaussian noise commonly used in diffusion models is replaced with a von Mises-Fisher distribution. When the data lives in a hyperspherical manifold, this ensures that angular geometry is preserved throughout the forward process. Reviewers agree that this is a theoretically solid paper with potentially interesting applications. Although the empirical results are modest, the proposed method is novel enough that I believe it warrants acceptance due to potential future uses or extensions.
Decision: Accept (poster)