Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry
Authors: Antoine Collas
Ce Ju
Nicolas Salvy
Bertrand Thirion
Venue: NeurIPS 2025 poster ()
Keywords: brain connectivity
Riemannian geometry
covariance matrix
fmri
diffeomorphism
generative model
Primary area: neuroscience_and_cognitive_science

Abstract: Generating realistic brain connectivity matrices is key to analyzing population heterogeneity in brain organization, understanding disease, and augmenting data in challenging classification problems. Functional connectivity matrices lie in constrained spaces—such as the set of symmetric positive definite or correlation matrices—that can be modeled as Riemannian manifolds. However, using Riemannian tools typically requires redefining core operations (geodesics, norms, integration), making generative modeling computationally inefficient. In this work, we propose DiffeoCFM, an approach that enables conditional flow matching (CFM) on matrix manifolds by exploiting pullback metrics induced by global diffeomorphisms on Euclidean spaces. We show that Riemannian CFM with such metrics is equivalent to applying standard CFM after data transformation. This equivalence allows efficient vector field learning, and fast sampling with standard ODE solvers. We instantiate DiffeoCFM with two different settings: the matrix logarithm for covariance matrices and the normalized Cholesky decomposition for correlation matrices. We evaluate DiffeoCFM on three large-scale fMRI datasets with more than $4600$ scans from $2800$ subjects (ADNI, ABIDE, OASIS‑3) and two EEG motor imagery datasets with over $30000$ trials from $26$ subjects (BNCI2014‑002 and BNCI2015‑001). It enables fast training and achieves state-of-the-art performance, all while preserving manifold constraints.
Code: https://github.com/antoinecollas/DiffeoCFM

Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry
Antoine Collas", Ce Ju, Nicolas Salvy, Bertrand Thirion
Inria, CEA, Université Paris-Saclay Palaiseau, France antoine. wow.oseeeclllllaiiii.....
Abstract
Generating realistic brain connectivity matrices is key analyzing population heterogeneity in brain organization, understanding disease, and augmenting data in challenging classification problems. Functional connectivity matrices lie in constrained spaces such as the set of symmetric positive definite or correlation matrices that be modeled Riemannian manifolds. However, using Rie- mannian tools typically requires redefining core operations (geodesics, norms, integration). making generative modeling computationally inefficient. In this work, we propose DIFFROCFM, an approach that enables conditional flow matching (CFM) matrix manifolds by exploiting pullback metries induced by global diffeomorphisms on Euclidean spaces, We show that Riemannian CFM with such metrics is equivalent applying standard CFM after data transformation. This equivalence allows efficient vector field learning, and fast sampling with standard ODE solvers. We instantiate DIFFEOCFM with two different settings: the matrix logarithm for covariance matrices and the normalized Cholesky decomposition for correlation matrices. We evaluate DIFFEOCFM on three large-scale fMRI datasets with more than 4600 scans from 2800 subjects (ADNI, ABIDE, OASIS-3) and two EEG motor imagery datasets with over 30000 trials from subjects (BNCI2014 1-02200--002220 and BNCI2015-001), It enables fast training and achieves state- of-the-art performance, all while preserving manifold constraints. Code: https: //g gighhu..... com/ http///ninncolll////fffe.......
Introduction
Brain imaging connectivity and Riemannian geometry Modern neuroimaging analyses map brain functional signals toward connectivity matrices-covariince or correlation estimates between [37]. These structured representations are used in many applications, such as motor imagery classification [4, 51, 41, 44], brain age prediction [16, 22, 54, 55]. or disease diagnosis [15]. They are central to the analysis of signals from many neuroimaging modalities, such functional magnetic resonance imaging (IMRI), alectreencephalography (EEG), and magnetoencephalograpl (SPD) or lie in the open elliptope of full-rank correlation matrices, and thus belong to smooth matrix manifolds S and its submanifold Corr, defined respectively by
regions of interest or sensor channels
(MEG). Brain connectivity matrices are symmettic positive definite
St+ + Σ E pddd Σ Σ, Σ > 0} and Corr. = {> E S diag ( 11.
(I)
Several Riemannian metrics have been proposed to analyse these data such as the inffrrninvaiiant metric [65, 7]. the log-Euclidean metric [3], or the Euclidean-Cholee Cholesky metric [68]. They enable the definition of Riemannian operations such as geodesics, exponential maps, and parallel transport, which extend standard Euclidean operations the manifold. Building these manifolds, wide
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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range of machine learning algorithms have been designed for classification [4], regression [16, 55], or dimension reduction [27].
Deep generative models on manifolds Deep generative modeling has rapidly advanced with the success of generative adversarial networks (28),) variational autoencoders (39), autoregressive models such as PixeIRNN [70], normalizing flows [21, 40], large-scale autoregressive models [9], and more recently, diffusion models [31, 66] and flow matching [49, 47, 2. 48). These last two methods learn to synthesize data by estimating continuous-time stochastic (diffusion) or deterministic (flow) dynamics that interpolate between a source and a target distribution. Diffusion models do so by reversing noise injection process, while flow matching aligns Icarned time- dependent vector field to the velocity field of a straight-line path. Both paradigms offer good scalability and state-of-the-ar results on diverse domains, from natural images [23], to speech [43], and protein structure generation [32]. More recent efforts have extended generative modeling to Riemannian manifolds. Riemannian score-based generative modeling [17] and Riemannian flow matching [12] provide general formulations for sampling on manifolds. The latter learns time- dependent vector fields on the tangent bundle that match the velocity of geodesic paths between source and target distributions. These approaches offer principled tools for generating data with geometric constraints, by treating the data domain as a manifold. These novel frameworks have already been applied to several applications such as materials discovery [56], robotics [20], or climate science [12].
Contributions In this work, we address a novel and challenging issue in neuroimaging; generating realistic brain connectivity data from actual human neuroimaging data. This challenge stems from the unique structure of brain connectivity data, which is represented by SPDs or correlation matrices that lie on -ucluuli manifolds. Moreover, neuroimaging datasets typically have limited sample sizes, making realistic data generation particularly valuable. We propose a novel Riemannian flow matching method, DIFFEOCFM, based on pullback geometry, defined by a global diffeomorphism
M + E, where E is Euclidean space. This method is an efficient framework that guarantees manifold-constrrineed outputs by construction while avoiding computationally expensive operations specific to SPD or correlation manifolds. We instantiate DIFFEOCFM with two diffeomorphisms tailored to different neuroimaging data: the matrix logarithm for SPD matrices and the normalized Cholesky decomposition for correlation matrices. Finally, we evaluate DIFFEOCFM on three large- scale fMRI datasets (ADNL, ABIDE, and OASIS-3; 3; over 4600 scans from 2800 subjects) and two EEG motor imagery datasets (BNC12014-002 and BNCI201 100000 30000 trials from 26 subjects), demonstrating that DIFFFOCFM is capable of generating realistic, neurophysiologicaly meaningful samples, as validated by multiple statistical metrics.
Notations MM is a smooth manifold with tangent space TrM and Riemannian norm || I|a.. We write for geodesics, and +(t 2 dt (t) for curve's speed. Let 6 : M + E be a global diffcomorphism to Euclidean space E, with differential D á(z): T.M + E and inverse (D d(z))=1 The pushforward of a distribution p on M is Φ.p. defined via I d( = [(f o @) dp for any f continuous on E. We denote x y ~ p(- y) for conditional sampling with label y E y. Let S be the space of symmetric d x d matrices, f the SPD cone, and CORT. the set of correlation matrices. Let LT be the set of lower-triangular matrices with unit diagonal. We define vec,1 Sa Rd(d-1)/2 as the operator extracting strictly lower-triangular entries, and vecn, for the full lower-triangular part (diagonal included), with /2 scaling off-diagonal terms.
2 Background
Pullback Manifolds with Euclidean Spaces and ø M + E a global diffeomorphism (a smooth bijection with a smooth inverse). The diffeomorphism induces Riemannian metric on M by pulling back the Euclidean metric gE E
Let M be smooth manifold, F Fuclidean space,
(E,,)) _ gE (D D(()())) E
This metric induces a Riemannian norm on the tangent space T.M each point x E M: |1El gF) (E, The pair (M, " gE) is then called a pullback manifold, and Φ" gE is the pullback metric of g8 by d. The pullback manifold is geodesically complete and admits globally unique
2

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- Cory.
M= ST"
Correlation
41
)(=μ) ~ 64p
- (De
Wrapped Gauseian
Wrapped Geussien
my
Φ
o(a)
dept
Φ --
(
M
Figure 1: Overview of DIFFEOCFM. DIFFBOCFM is a principled framework for deep generative modeling on matrix manifolds. reformulates Riemannian conditional flow matching (CFM) on pullback manifold (M, ('g)) as conventional CFM in Euclidean space E, via global diffeomor- phism é: M E. The reformulation preserves geometry two ways: (i) the learned Euclidean vector field uy satisfies (7), ensuring that training o/ is equivalent to training ua"; (üi) the flow trajectories obey ((x(t)) = z(t), so that integrating in E' and pulling back via Φ yields the same samples as integrating directly on M. This allows both training and sampling be carried out efficiently in E, while remaining equivalent to operating on M. On the left, fMRI correlation and EEG spatial covariance matrices lie on M Corr, and M = $ respectively. These matrices are mapped to E through performed in E before mapping back via é yield connectivity manifold constrained matrices.
dimendependen vector field ug is trained E, and integration is
geodesics 168, Chap. 7]. Moreover, many Riemannian operations reduce simple computations in E. Given two points No, æ1 E M, the geodesic y [0, 1] + M connecting xo to æ and associated riemannian distance are given by
(3) ie., the pullbacks of the Euclidean straight line and distance in joining ((xo) and 6(x1). The Fréchet mean [29] of a set of points (,(") C CM with respect to the Riemannian distance is
(t) = 6- ((1 ())(z)) + to(x1)) and (<< æ1) ) = ((((x)) - <(æ1)]l
N Σ
Z arg min
dm dd(z, 0-1
p(a(m)))
ZEM n=1
N
1=1
Riemannian CFM CFM [49, 47, 2, 48] was recently extended to Riemannian manifolds [12], providing principled framework for learning time-dependent vector fields that transport samples between probability distributions defined on such spaces. Given a manifold M, a vector field ugM [0, 1] M y - TM trained to match the velocity of geodesics connecting samples from source and target distributions. The Riemannian CFM loss is defined as
((0) A Er, molys ailv u M (t, (t)) y) 100
(5)
where y y is a condition variable (such a disease status), and coll p(-ly), 2i q(-ly) and ~ 44[0 1]). Hence, this loss is computationally intensive, as it requires evaluating geodesics yt) between Zo and ø1, their derivatives (t), and Riemannian norms || Il Once trained, new samples on M are generated by solving the Riemannian ODE
i(l) = un (t), x00) zo p(- y)
(6)
and returning x(1) sample from the learned approximation of g(- y).
3 DIFFEOCFM: Conditional Flow Matching on Pullback Manifolds
Pullback manifolds provide natural setting for using Riemannian CFM [12] in practical generative modeling tasks. Indeed, when M is equipped with pullback metric o" gg induced by global
3

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diffeomorphism Φ M E, both training and sampling can be performed entirely in the Euclidean space E. We prove that this equivalence is exact: the learned vector field in E' corresponds one on M, and the ODE solutions in E map to those on M via Φ. This result motivates DIFFEOCFM, conditional flow matching framework that performs all computations E. avoiding costly geometric operations--uee as computing geodesics, Riemannian norms, or manifold integration guaranteeing munifold-constrineee outputs. An overview of the method shown in Figure
3.1 Training and sampling with diffeomorphism
Training Rather than learning a vector field uz directly on the manifold M, DIFFEOCFM trains its Euclidean counterpart up via the pullback:
ug (t, 2, v) & D (z) (1)
Indeed, in this case, the loss function (5) can be expressed in terms the Euclidean vector field up as shown in the following proposition.
Proposition (Riemannian CFM loss function on pullback manifolds). The Riemannian CFM loss (5) can be re-expressed in terms of the Euclidean vector field up (7) as
L( = E2, solu, 2s/y [ng (t, (1 z)=o () (=1 z0)
where zoly- Φ# (-y)) and 21yy 2# (--v))
should be noted that this new loss function is much simpler compute than the original Riemannian CFM one, as it does not require computing geodesics, their derivatives, or Riemannian norms.
Sampling DIFFEOCFM generates samples by solving the ODE
(t) u (t,=(t), v), z(0) 2g ~ Φ ())
(8)
Despite this simple form, the procedure is fully Riemannian: the generated trajectory corresponds exactly to a camifold-valued solution under Φ-1. Indeed, the following proposition states the equivalence of the solutions of the ODEs (6) and (8).
Proposition 2 (Equivalence ODE solutions). The solution x(t) to (6) satisfies
at)) = -- for all t E [0, 1],
where 2(t) is the solution of the ODE (8) with initial condition Zg = ((x)).
The previous result establishes that the Riemannian and ODEs define equivalent flows through the diffeomorphism é. In practice, these ODEs solved numerically using explicit Runge-Kutt integrators. The next proposition shows that the equivalence also holds at the discrete level: applying the same Runge-Kutta Kutta scheme in E or M yields iterates related by d.
Proposition (Equivalence of Runge Kutta iterates). Let {xt} } be the iterates produced on M by an explicit Riemannian Runge-Kulte -Kltte scheme applied the ODE (6). Then, the ilerates are
Φb (ze) for all E N, where { are the iterates obtained by applying the same scheme (same coefficients and step size) to the ODE (8) with initial condition zg ((zo).
Overall, the training and sampling algorithms for DIFFEOCFM are summarized in Algorithm 1 and 2, respectively.
Algorithm 1: DIFFEOCFM: Training
Algorithm DIFFEOCFM: Sampling
Input: step size h;; samplers wy, d.P. d# Output: Trained parameters 0*
Input:
label y; steps L; step size h;; trained 0* Output: Generated sample.x
Initialize
Sample 2n Φ g);
while not converged do
for = 0 to 1 do
Sample ~ Ty. É L4(0 1]):
Σo41 Runge- Kutte--Ktttttttuueee v.:);
Sample za @= 1)) Φ ww;
+ |n (f, 111-t)apg t21. v) - (21 20)
x (ZL);
0 < optimizer-sten((C;
end

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3.2 Diffeomorphic Embeddings for Generative Modeling of Brain Connectivity Matrices
To generate brain connectivity matrices, we map $ and Corra to Euclidean spaces via global diffeomorphisms: the matrix logarithm for SPD matrices and the normalized Cholesky map for correlation matrices. These maps define pullback metries on the tangent spaces TyS d Su and TyCorra (E E S4 diag(g) 0), of respective dimensions d(d + 1)/2 and d(d 1)/2. This allows DIFFFOCFM to perform efficient, geometry-aware generation for both matrix types. Note that we selected these two diffeomorphisms for their simplicity and ease of implementation. However, other choices are possible. For correlation matrices, alternative parameterizations are discussed in [68, Chapter 7]. For SPD matrices, one can use the Cholesky factor with logarithm applied to the diagonal, leading to the log-Cholesky metric [46], which also defines a global diffeomorphism.
Covariance matrices: Log-Euclidean metric On $ we define the global diffeomorphism $5zt : + + Rd(d+1//2 by composing the matrix logarithm with the vectorization map vecu.
+ (m)
vecie (log( Σ)) and
(1) exp (veci (m)))
where log and exp denote the matrix logarithm and exponential, respectively. This mapping induces the Log- metric on + by pulling back the standard Fuclidean inner product from Rd((d+1)/2 1)/2; 9x.5t+ (E. n) tr log D) (n]), as introduced in Arsigny et al. [3].
Correlation matrices: Euclidean-Cholesk metric On Corra, the matrix logarithm is no longer diffeomorphism. More generally, defining a log- based diffeomorphism nontrivial example, the Riemannian logarithm associated with the affine-invarian metric does not admit closed-fome expression in this setting (see Appendix F). Instead, we use the normalized Cholesky map, a global diffeomorphism onto LT d. the space of lower-triangular matrices with unit diagonal:
(chol(E) diag(chol(E))) chol(E),
(10)
where chol(E) is the unique Cholesky factor with positive diagonal. Its inverse is
nchol (L) D D-
LL D D-2, with D diag(LL").
(11)
Then, we define the diffeomorphism @coo,, Corra + Rd(d-1)/2 and its inverse as
Corra Σ) = veca and @ (n) = nchol (vec, (17)
(12)
This map induces the Euclidean- metric, Riemannian metric obtained by pulling back the Euclidean metric from the vector space R (d)-1)) [68]] 93,con, (E. n) D o [E] D
Label, source, and target distributions To train DIFFEOCFM on labeled brain connectivity data, we define class-conditional source and target distributions in the Euclidean space E' induced by the diffeomorphism Φ. Given a dataset (x((") ) of manifold-valuee matrices, we map each sample to E via ((m) = ()). For each class y. we fit Gaussian distribution to the embedded
samples (2(m), (()) = y} to define the class- cass-condiiional source distribution d+p(- y). The target distribution Φ y) defined as the empirical distribution over the same class-y samples.
3.3 Related work
Denoising Diffusion Probabilistic Models (DDPMs) [31] and CFM [47, 2] have emerged as robust, state-of-the-ard generative models in Euclidean spaces. Several extensions have been proposed to handle data that lie on Riemannian manifolds. These include Riemannian Score-Based Generative Modeling [17], SPD-DDPM [45], and Riemannian CFM [12]. These methods tailor the loss function and ODE/SDE solvers to the geometry of a specific manifold. However, this geometric fidelity comes cost: manifold-spa -peccffic operations such as Riemannian gradients, exponemiia//lgaarthm maps, parallel transport must be implemented to compute the loss function and the integration on the manifold. For instance, SPD-DDPM requires a specialized neural architecture, an SPDNet, which is computationally expensive and significantly slower to train than Fuclidean counterparts; see Figure in Appendix for a comparison. References [35] and [38] explore more general settings by learning bridge matches on arbitrary manifolds data-driven Riemannian metrics, whereas
5

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approach focuses on Riemannian geometries defined via known pullback diffeomorphisms. In [26], the authors leverage reparameterisation with normalising flows learn probability densities on Lie groups Mon-Fuclideen spaces), and thus can be seen as an early precursor to our approach.
In contrast, CONrGAN [52] offers a pragmatic approach generating correlation matrices by training and sampling entirely in Euclidean space using GAN. Geometric constraints, such as positive definiteness and unit diagonal, are enforced via post-processing. While the method is simple and fast, the post-processing step can change the generated data in unwanted ways and reduce their quality.
The proposed method combines the simplicity of Euclidean training with the rigor of Riemannian geometry by using a diffeomorphism 6 to embed structured matrices, apply standard CFM, and map samples back. To the best of our knowledge, it is the only method that enables generation of both SPD) and correlation matrices within a unified framework. detailed comparison of baseline meth- ods highlightt their assumptions, strengths, and limitations is provided in Appendix Section H.
Empirical benchmarks
These benchmarks were designed to evaluate whether the generated data (i.) match the test distribution and (ii.) enable classifiers trained on them to generalize to real data. This evaluation uses two human brain imaging modalities: three fMRI datasets and two EEG datasets.
4.1 Metrics
The metrics used are method -ageood-agnoti;;; that is, they are computed solely from the generated samples. They fall into two categories: (i) quality metrics, which assess how well the generated data approxi- mate the real data distribution; and (ii) classification accuracy score (CAS) metrics, which evaluate the usefulness of generated data for training classifiers.
We assess how closely the generated samples align with the real data distribution using the a-precision and 3-recall metrics introduced by [1]. We compute these metrics using the EvaGeM library'. In contrast to [1], which uses Deep neural network, EvaGeM employs One-Clast SVM, providing a more stable and hyperparameter-roobeee robust estimator across datasets. These metrics quantify the fidelity (how realistic the generated samples are) and the diversity (how well they span the true data distribution). We also report the harmonic mean of the two, denoted o, 9-F1.
Quality Metrics
Classification accuracy score metries We follow the CAS protocol [64], training a classifier on generated samples and evaluating it on real test data. Specifically, we assess classification utility using a logistic regression with liblinear solver, balances class weights and a 5-fold eross- pross-validation to select the inverse regularization strength C from the grid {10- 10 ..... 10*). High scores indicate that the generated data preserves task-relevant information. On fMRI datasets, the task is disease classification (control vs. patient), while for EEG, it is two-class motor imagery problem in a brain-computer interface setting. We report ROC-AUC and F1 scores.
4.2 Datasets
The experiments include both fMRI and EEG datasets; additional details are provided in Appendix G.
IMRI datasets We use three publicly available resting-state fMRI datasets. The ABIDE dataset [53] consists of 900 subjects (one scan each), including both neurotypical and autistic individuals with mean age of 17 years, collected across 19 international sites. The ADNI dataset [73] comprises 1,900 scans from 900 older adults (mean age 74), covering normal ageing, mild cognitive impairment, and Alzheimer's 's disease. The OASIS-3 3 dataset [42] includes 1,000 subjects and 1,800 longitudinal sessions collected over 10 years, targeting healthy ageing and neurodegenerative conditions, with a mean participant age of 71 years. We z-score the time series and then compute correlation matrices (Corra) using the OAS estimator [15]. We report mean and standard deviations computed across 10 random train-test splits with subject-level grouping, ensuring that scans from the same subject do not. appear in both training and test sets.

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EEG datasets We use two publicly available EEG motor imagery datasets from the BCI competition. The BNC12014-002 dataset [67] includes 13 subjects performing right- hand and feet imagery, recorded with 15 channels over 1 session, with 80 trials per class. The BNCI201 comprises 12 subjects, 13 channels over 2 or sessions, and 100 trials per class for the same motor imagery tasks. To process the raw data, we compute covariance matrices ($ using the OAS estimator [13]. following standard practices [31]. We report performance on a leave-one-session-e protocol on ENCI2014-002 and on cross-session experiments BNCI201 The reported standard deviations are computed over inner splits and averaged over sessions.
dataset [25]
4.3 Baselines
We present the baselines used for comparison with DIFFROCFM, including an oracle baseline (REAL DATA) that treats test data as if it were generated samples.
REAL DATA This oracle baseline treats the test set as if it were generated data when computing metrics. provides an upper bound on the achievable performance, both for quality and CAS metrics, showing the best any generative model could hope match.
DIFFEOGAUSS Given a diffeomorphic embedding & .M + E, we model each class-conditional distribution q(- y) as the push-forward of a Gaussian Ma Mny) fitted to the embedded training data 2(n) d(r(")). Samples are drawn in E and mapped back to M via o yielding wrapped Gaussian distribution on the manifold [62, 18].
TPPM and TRIANGCFM These baselines apply generative models to the lower- triangular part of SPD or correlation matrices, a common heuristic in manifold modeling [52]. For fMRI, they use the strictly lower-triangular entries (Ø Corra + Rd(d-1)/2); for EEG, all lower- triangular entries (ø $
pd(d+1//2). TRIANGDDPM uses a DDPM [2], while TRIANGCFM trains a vector field using the standard CFM loss. Since @ is not diffeomorphism, generated matrices that do not lie on the manifold are projected back onto it. particular, generated matrices are not necessarily positive definite, so we apply projection to ensure all eigenvalues are at least >> 0 with Epnj = (1 x)EE + aI; see Appendix M for more details. These methods trade geometric fidelity for simplicity, relying on post hoc projections to enforce constraints.
RIEMCFM This method applies Riemannian CFM [12] directly on the SPD manifold using the affine-invariant metric [65]. The target conditional vector field is computed analytically from the Riemannian logarithm and exponential maps. Unlike WWANNNDDPM/RRANGGCSS
RIEMCFM preserves the intrinsic geometry of 5 throughout training and sampling, yielding valid SPD matrices at every time step without post-hoc corrections. However, it requires computing geodesies, their derivatives and Riemannian norms under the intar-iivvrriatt metric, making it substantially more computationally expensive than all other presented methods. The reference implementation" focuses exclusively on the SPD manifold under the affine-invariant metric, and does not provide corresponding construction for correlation matrices.
DIFFEOCFM (proposed) We apply the log-Euclidean map Φ3 + (9) for EEG and the normalized Cholesky map ÓCony (12) for fMRI in Algorithm and 2. These diffeomorphisms allow Euclidean training with CFM while ensuring manifold-valid samples without post-processing.
Deep learning and training/sampling setups TRIANGDDPM, TRIANGCFM and DIFFEOCFM employ a two-layer MLP with 512 hidden units, trained using AdamW [50] with a learning rate of 10-3 and batch size 64. Training runs for 200 epochs on fMRI and 2000 epochs on EEG. RIEMCFM has a 6-layer MIP with 512 hidden units trained with Adam W (learning rate of 10-1), as recommended in [12]. These four methods the dopriš method from the torchdiffeq [11] library for time integration. All experiments were run within 10 hours on single Nvidia A40 GPU with a 32-cores cpu.
htttps://githuu.coo//// http:/hhhh...aeecccoooe//eeeeear
7

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5 Results
We report both quantitative results and neurophysiological plausibility study. Quantitative compar- isons are summarized in Table 1, with additional analysis of projection effects in Table 2. Neurophys- iological relevance is assessed in Figure 2, which shows cans-connitional fMRI connectomes via Fréchet means and topographic maps of EEG Common Spatial Patterns (CSP) filters. To complement Table I, we provide visual summary of our findings in Figure Appendix D). Together, these results show that DIFFEOCFM produces realistic, class-conditionss samples that preserve key features of brain connectivity. For a more detailed discussion of the baselines, please refer Appendix H.
Table 1: Performance of generative models 3 fMRI and 2 EEG datasets, evaluated with quality and Classification Accuracy Score (CAS) metrics. Quality metrics (a-precision, /3recall, and a, B-F1) assess alignment with the real distribution. CAS metrics (ROC-AUC and F1) evaluate downstream performance: a classifier is trained on generated data to predict disease status (IMRI) or motor imagery class (EEG), and tested on held-out real samples. Real Data rows use real samples to compare training and test distributions, serving as empirical upper bounds. The proposed method is denoted DIFFFOCFM mean std are reported. Bold values denote the best method and any methods that are not significantly worse than it (one-sided paired Wilcoxon signed-rank test, 0.05).
Quality Metrics
CAS Metrics
Time (s.)
a-PPecision t 3-Recall ox 3-FFI RROC-AUCT t FI f
Training 4 Sampling
Dataset
Method
Real Data
0,80 0.08
0.79 0.088 0.79 a 0.03
0.67 0.06
0.59 1.077
N/A
N/A
ABIDE
DIFFEOGAUSS
0.06
0.29
0.38 1 0.06
0.66
0.53 2,06
0.07 0.03
0.06
TRIANGDDPM
0.04 0.02
0.00
0.00 - 0.00
0.53 0006
0.47 +0.12
33.80 1.19
0.37 0.05
TRIANGCFM
0.04 a 0.02
0.00
0.00 a 0.00
0.52 1 0.05
0.40
48.78 127
0.79 1 0.78
DIFFEOCFM
0.77 0.09
0.48 x
0.59 0.08
0.64 + 0.06
0.58 e.17
32.78 +096
0.40 + 0.04
Real Data
0.91 0.03
0.85 0.066
0.88 0.03
0.62 1 0.05
0.62 1005
N/A
N/A 0.18 2 0.01
ADNI
DIFFEOGAUSS
0.02
0.51 1008
0.04 0.02
0.60
0.29 10.33
2.001
TRIANGDDPM
0.02
0.00
0.00
0.53 +
0.18
90.03
0.62 0.08
TRIANGCFM
0.00
0.00
0.01
0.01
0.56
0.34.0..1
87.37
22133
0.63 1 0.09
DIFFEOCFM
0.12
0.77 + 0.02
0.68 = 0.06
0.04
0.47 0.10
88.01
+2.90
0.69 0.11
Real Duta
0.88
0.87 0:08
0.88 0.02
0.73
0.63
N/A
N/A 0.13 x 0.00
OASIS-3
DIFFEOGAUSS
0.51 0.04
0.38 0.30
0.04
0.70 + 0.05
0.41 2007
0.10 0.01
TRIANGDDPM
0.03
0.00
0.00 = 0.00
0.54
0.41 #0.44
70.39 a 1.99
0.50 0.06
TRIANGCFM
0.06
0.00
0.00 0.00
0.52
0.41 +0.1
6
0.52 = 0.07
DIFFEOCFM
0.60 = 0.05
0.35 0.04
0.44 = 0.04
0.67 a 0.06
67.83 a 1.83
0.57 a 0.05
Real Data
2.00
0.60
0.64 0.03
0.83 0.01
0.75 2.02
N/A
N/A 0.08 + 0.01
2014-002
DIFFEOGAUSS
0.46 0.04
0.77 0.02 0.57 0.03
0.00
0.73 o.02
0.06
BNCI
TRIANGDDPM
4.03
0.16 0.10
0.03
0.52
0.20
257.88
1014
0.30 0.05
TRIANGCFM
0.48 0.05
0.22 000
0.30 0.03
0.55
0.24 +.11
251.78 +.85
0.35 0.09
RIEMCFM
0.67 0.07
0.62
0.63 1 0.03
0,81 0.02
0.72 0.02 1983.58
1 0.97
5,28 1 0.58
DIFFEOCFM
0.62 0.04
0.63 0.04
0.62 - 0.02
0.81 + 0.02
0.74
253.04 +0.33
0.59 0.08
Real Data
0.09
0.89 0.89
0.00
0.73
0.67 +001
N/A
N/A 0.16 0.01
2015-001
DIFFEOGAUSS
0.01
0.90
0,86 : 0.01
0.73
0.68 0.01
0.07 2.01
BNCI
TRIANGDDPM
0.03
0.55 0003
0.63 0.02
0.00
0.59
+.9.94
0.37 + 0.07
TRIANGCEM
2.79
0.73
0.76 2 0.02
0.61 0.00
0.59 1007
11312212.1
0,38 1 0.08
RIEMCFM
0.93 a 0.04
0.84
0.88 0.01
0.73 0.01
0.66
16661002773.999
11.02 1 0.59
DIFFEOCFM
0.92
0.86
0.89 0.01
0.65 0.03
+033
1.02 0.08
5.1 Quantitative Study
Quality Metries As shown in Table 1, DIFFEOCFM consistently matches or outperforms all baseline generative models across datasets in terms of a, 3-F1, establishing itself as the most robust method for aligning with the true data distribution. It also achieves the highest a-precision and B-recall across the three iMRI datasets. On EEG datasets, DIFFEOGAUSS achieves higher
8

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3-recall but at the cost of much lower a-precision, leading to weaker overall a, 3-F1 scores. In contrast, TRIANGDDPM and TRIANGCFM perform poorly on all quality metrics after projection. due to a substantial degradation in sample quality. As detailed in Table 2, projecting onto the manifold-Cone for fMRI and $ for EEG-introduces a significant performance drop. This is because TRIANGDDPM and TRIANGCFM generate matrices that visually resemble realistic connectivity patterns but contain negative eigenvalues, making them invalid. The projection step corrects these matrices but distorts their structure, leading to sharp decreases in a, 3-F1- up -0.74 on ADNI and 76 on OASiS-3-rendering TRIANGCFM impractical for use. This degradation is also visible in Figure 5, where post- pest-projection alter the structure of fMRI connectomes (see Appendix M). Compared to RIEMCFM, DIFFEOCFM delivers similar performance on the two EEG datasets, while training 8x faster and sampling 10x faster. Finally, EEG datasets, DIFFEOCFM nearly matches the performance of REAL DATA in terms of a, 8-F1, suggesting high sample realism.
CAS Metrics For the CAS metrics, which evaluate downstream predictive performance using ROC-AUC and F1 scores, DIFFEOCFM consistently achieves strong results, often approaching the performance of REAL DATA. It obtains the highest ROC-AUC and Fl scores across all fMRI and EEG datasets. While DIFFOG remains competitive in terms of ROC- AUC, DIFFEOCFM substantially outperforms it on Fl scores, with absolute gains of +0.05, +0.18, and +0.12 on ABIDE, ADNI, and OASIS-3, respec- tively. TRIANGDDPM and TRIANGCFM perform poorly across both CAS metrics on all datasets. This underperformance reflects the effect of the projection step, which alters generated matrices in ways detri- mental to downstream classification. It is important to note that the classification pipeline requires inputs to be valid SPD matrices (i.e., elements of
Table 2: Impact of projection onto the man- ifolds $ and Corræ: performance differ- ence ö TRIANGCFM TRIANGCFM without projections. Negative A indi- cate degraded sample quality after projection onto the manifold. It enforces geometric con- straints but severely reduces sample fidelity.
Dataset
a-precision B-recall a,
ABIDE ADNI
-0.34
-0.69
-0.50
-0.63
-0.74
-0.69
OASIS-3
-0.52
-0.76
-0.64
BNC12014-002
+0.13
-0.56
-0.19
BNC12015-001
+0.00
-0.19
-0.09
s ). As a result, CAS metries cannot be computed for TRIANGDDPM and TRIANGCFM without projection. For this reason, their performance without projection is not reported in Table 2.
Plotting of Generated Samples in Real Data Neighborhoods To qualitatively assess fidelity, we show Figure 10-15 (fMRI) and Figure 16-19 (EEG) in Appendix P the generated samples closest to real ones in Frobenius distance. fMRI results are grouped by control (CN) and patient (non-CN); EEG by motor imagery class. DIFFEOFFM reliably populates the neighborhood of real data, capturing dlass-conditional structure. We also show TRIANGCFM samples before projection, which appear realistic but not SPD.
5.2 Neurophysiological Plausibility Study
fMRI Connectome Plotting Figure 2a shows group-level functional connectomes from the ADNI dataset, computed as the Fréchet mean (4) of correlation matrices conditioned on disease status. For comparison, Figure in Appendix O presents the corresponding group-level connectomes derived from the ABIDE and OASIS3 datasets. The Fréchet mean is defined with respect to OCor, (12), the diffeomorphism used for the generation. For each class (CN and non-CN), we compare real connectomes (from held-out test subjects) those generated by DIFFEOCFM. In both cases, non- CN subjects exhibit reduced connectivity across hemispheres and between frontal and posterior regions patterns commonly associated with mild cognitive impairment and Alzheimer's disease [19].
EEG Topographic Map Figure 2b presents the group-level topographies of the first CSP filter across all subjects, derived from EEG generated by DIFFEOCFM, alongside those from real EEG recordings in the BNCI2015-001 dataset. Subject-level topographies of the first CSP filter from the same dataset are further detailed in Figure 7 (Subjects 1-6) and Figure 8 (Subjects 7-12) in Appendix N. providing a detailed comparison across individuals. We visualize CSP spatial filters in the a (8-12 Hz) and 3 (13-30 Hz) bands that distinguish imagined right-hand from feet movements. The filters trained on real and on generated data concentrate on the same contralateral sensorimotor regions, mirroring the close CAS scores between REAL DATA and DIFFEOCFM in Table 1. This confirms that the generative model preserves the physiologically relevant information.

---

Roal TMR (CN)
DiffecCFM (CN)
0,
6.7
40.6
Real 1MRI (non-CN)
DiffecCFM (non-CN)
01
(a) Class-conditional MMRI functional connectome plotting using the Fréchet mean (ADNI). Each panel displays class-conditional IMRI functional connectomes using the Fréchet mean (4) of correlation matrices computed with respect to the generation diffeomorphism dCor, (12). Left: real data from held-out test subjects; right: samples generated DIFFEOCFM. The comparison illustrates both the fidelity of generated connectomes and the disease- disease-specific connectivity structure preserved by the model.
Real EEGs (8-12 Ha: 0.5-2.5
DiffeCCMM (8-12 Hz: 0.5-25 st
Real EEGs
DiflecCMM (13-30 Hz, 0.5-2.5
(13 20 Ha: 0.5 1)
MB
(b) Group-level topographic map using the first CSP's spatial filter derived from real EEGs (BNC12015- 001) and generated data by DIFFEOCFM Each map shows the first CSP's spatial filter across 12 subjects the a 12 Hz) and 3 (13 30 Hz) frequeney bands during the first seconds following stimulus onset. Filters from DIFFEoCFM at the group-level closely resemble real EEGs, preserving discriminative patterns of motor imagery classification.
Figure Neurophysiological Plausibility Study of DIFFBOCFM.
6 Conclusions, Limitations, and Future Works
We introduced DUFFECCFM. an efficient framework for generating brain connectivity matrices. By reformulating Riemannian flow matching through global diffeomorphisms, DIFFEOCFM enabled fast training and sampling while ensuring fMRI and EEG data, it outperformed existing baselines with neurophysiologically plausible samples.
outputs by construction. Applied to
Nonetheless, several limitations remain. First, DIFFEOCFM relies on global diffeomorphism to Euclidean space, which exists for SPD and correlation matrices but not for all manifolds (e.g.. Stiefel). Second, higher parcellation granularity makes the problem intrinsically hard: manifold dimension grows quadratically with region count, and the sample complexity grows exponentially [57]. Third, how alternative connectivity definitions (e.g., partial correlation or graphical-L generation quality remains an open question, since the choice of estimator directly defines the ground truth. Fourth, common evaluation metries like a-precision and 3-recall are geometry-agnostee and may miss neurophysiological structure.
precision) affect
Despite these challenges, generative modeling remains promising direction for neuroimaging and BCI research. For instance, sharing trained generative models, rather than raw data, can facilitate multi-site collaboration with privacy guarantees [10].
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Acknowledgment
This work was supported by grant ANR-22-PESN-001 under the France 2030 program, managed by the Agence Nationale de la Recherche (ANR). CJ supported by DATAIA Convergence Institute as part of the "Programme d' I'neestissement d' Avenir", (ANR-17-CONV-ON) 17-CONV-00000 operated by Inria. This work was performed using HPC resources from GENCI-IDRIS (Grant 2025-AD01
Numerical computation was enabled by the scientific Python ecosystem: Matplotlib [33], Scikit-learn [61], Numpy [30], Scipy [72], PyTorch [59], IMRIprep [24], Nilearn [14], joblib [36], PyRiemann [5], torchcfm [69], torchdiffeq [11], pandas [58], and moabb [34].
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A Pullback Manifolds with Euclidean Spaces
Section briefty introduced pullback manifolds with Euclidean spaces. We here go more into details to then prove the different propositions of the paper. Let M be a smooth manifold, E' a Euclidean space, and é : .M- + E' a global diffeomorphism smooth bijection with a smooth inverse). The diffeomorphism & induces a Riemannian metric on M by pulling back the Euclidean metric gE on E
(6 gE) v)) _ gr (D (() D (v)) E T.M.
(13)
This metric induces a Riemannian norm on the tangent space T.MM at each point x E M: ||[|== = ((0"g))=(E,{) E). The pair (M, o gE) is then called a pullback manifold, and ø'gE is the pullback metric of gg by Φ. The pullback manifold geodesically complete and admits globally unique geodesics [68, Chap. 7]. Moreover, many Riemannian operations distances, geodesics, exponential and logarithmic maps, parallel transport, Fréchet means- reduuce to simple computations in E. Given two points z, y E .M, the geodesic [0, 1] + M connecting I to y and its associated Riemannian distance are given by
(14) i.c. the pullbacks of the Euclidean straight line and distance in E' joining o(x) and dly). This structure also defines expressions for the exponential and logarithmic maps any point I E M. The exponential map exp, T + M and the logarithmic map log, .M- + T.M are
(t) = ((1 t))(2) + to()) and dm( v) = 11((z) -
ex) = a (o(x) + D ((x) and log, (v) = (D (D)())) (6(y) - x(x))
(15)
When D (x)) is not available in closed form, it can be computed via automatic differentiation using libraries such as JAX [8] or PyTorch [60]. Its inverse, (D ()x)) -1 can be obtained from the differential of Φ using the identity (D x(x)) o D (o(x)) = Id. The parallel transport of a tangent vector & T along the geodesic 7 the pullback of the Euclidean parallel transport from ((x) to (b)). which is given by
PT (7) = (Do(v)) (D (E))
(16)
The Fréchet mean [29] of set of points (z(()) J C M with respect to the Riemannian distance is
z arg min
Σ d =0-
1 Σ
o(a())))
(17)
N
REM
n=1
n=1
B Riemannian Conditional Flow Matching
This section provides concise overview of Flow Matching, Conditional Flow Matching, and their extension to Riemannian manifolds, complementing Section 2. For clarity, we omit conditioning on variables y such as disease status, age, or sex; the derivations extend naturally to the conditional case.
B.1 Flow Matching (Intractable Objective)
Flow Matching (FM) [49, 47, 2, 48] aims to learn time-dependent vector field (t, x) that transports samples from a simple source (prior) distribution p(x) (e.g.. a standard Gaussian (0, at t = 0 to a target data distribution (=x) at This transformation governed by an ordinary differential
equation (ODE):
(t) autt, (t)), x=0) ~ (w)
(18)
Training Flow Matching trains a neural network us (t, x) to approximate true time-depende denendetnnd vector field u(t, .) that transports a source distribution p(z) target ((x) along an evolving density path (pr (Pr) The objective is:
(0) E (ine(((())..xxx(ee)) (/ (1, - w(1, x
(19)
This formulation generally intractable, as both Pe (x) and u(t, x)) are unknown. Indeed, the time evolution of (x) is governed by the continuity equation:
@p((x) + div (p)(x) u(t.x)) 0,
(20)
Øt
16

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which expresses conservation of mass along the flow.
Sampling To generate sample #1 from learned model us (t, draws initial sample To ~ (o()) and then solves the learned ODE from = to = using a numerical solver such as Fuler Kunge-Kutta scheme:
2 To + L/ uo(t, (t,))(())
=(() (t(()
(21)
B.2 Conditional Flow Matching (CFM)
Training Conditional Flow Matching (CFM) makes the training of flow models tractable by defining explicit conditional paths and vector fields. We still consider source distribution p and target data distribution q. A common choice for the path connecting To to # is a linear interpolation:
Zz (z 21) = (1 - t) + tæ
The corresponding target conditional vector field u(t, xz 2o x1) 21 Zg. The CFM training loss for neural network (o(t,z) is:
(0) (0.1) (ng()))))))))))))))))))))))))))) ue (t. (1 f) txi) (x) mo)
(22)
Here, p(xo) is typically a simple noise distribution (e.g., (0 I)) and ((x1) is the empirical data distribution. Minimizing this CFM loss has been shown to be equivalent to minimizing the original intractable FM loss under certain conditions [48]. Other conditional path definitions can also be employed.
Sampling Sampling is performed by drawing zo ~ p(zo) and integrating the learned vector field ut (t)) from t = 0 to t 1:
a(t) us (t, (t))
The solution z(1) is then a sample from the learned approximation of q(x1).
B.3 Riemannian Conditional Flow Matching (RCFM)
CFM was recently extended to Riemannian manifolds [12]. providing a principled framework for learning time-dependent vector fields that transport samples between probability distributions defined on such spaces.
Training Given manifold M,, vector field ugM [0, x MM -> TM (where T TM denotes the tangent bundle of M) is trained to match the velocity of geodesics (t) connecting samples from a source distribution p(zo) on M and target distribution ((x)) on M. The Riemannian CFM loss is defined as
(23) where ytt) is the geodesic connecting to 21 such that (0) zo and 711 and +t))) is its time derivative (velocity vector) which lies in (. should be noted that this loss extends CCrM-
(()) 2 (1-21((,1)) sorp))))))__)))).)..)))) mig(())))))))))))))))) 1/ M (1)) -
Indeed, for M d then we get L = LCFM-
Sampling Once trained, new samples on M are generated by solving the Riemannian ODE
i(() ug ug (I)) xx0) zo p(xo) and returning x11) as sample from the learned approximation of q(x1). Numerical solution of this ODE typically involves manifold operations like the exponential map.
(24)
C Proof of Proposition 1: Riemannian CFM loss function pullback manifolds
Proposition (Riemannian CFM loss function pullback manifolds). The Riemannian CFM loss (5) can be re-expressed in terms of the Euclidean vector field uly (7) as
L(0) = E,, soly, suly 1ug / (1, (1 1)20 + (21.8) - (z1 - =n)
where zoly d+p(-ly) and 21ly (
17

---

Proof. Fix a label draw zo ~ p(- ()) and 21 ~ 9(- y). and set 20 - o(zo), = 0(x1). For any E [0, let 2e _ (1 t)zo + to and (t) = (z)).
Geodesic velocity Since b a straight line in E, the chain rule gives
(tt) = の (2))211 20).
Pull-back of the vector field By definition of up in (7),
ug (1, 2,,8 v) = D (()(()) (ug (t, 7(),
Norm preservation Because the metric on M is the pull-back of ge: we have [| Il D ((r(t) (e)lle for corevery& & M. Applying ((>(t)) to the difference (t, ((). (())
yields
(e(((()) (ug (1,7(4)))) -(1) = D
(t,
=o))
Furthermore, since (ø (z) 2, we have D (2) Id qHence, we get De(r(t)) (1g (t, (t), (()) up (t, -(21 - -0)
Taking squared Euclidean norms on both sides gives
(t, || (1, 7(), - -0 = 1|ug (t, 2t, 1/ (2) -20) l
Expectation Finally, averaging over ~ L4 1). y ~ My, zo ~ p(: v). and #1 ~ g(: 1 v) gives the desired equality of loss functions, proving Proposition 1.
D Proof of Proposition 2: Equivalence of ODE Solutions
Proposition 2 (Equivalence of ODE solutions). The solution x(t) to (6) satisfies
(t)) = x- (x(()) for all t [0,1], where z(1) is the solution of the ODE with o (7) and initial condition 2o v(x0).
Proof. Let y 3. xo E M and =0 Φ (2o). Let the Euclidean trajectory 10, 1] + E be
solution of
=(1) = 4
E (1, (1,=(1),v)
=(0) 20-
Then, we define the candidate solution
xx1) _ の (=(1))
(0$151).
By the chain rule,
+(t) =
D (
(=(t)) (2(1)) = DØ (z(t)) (uz (t, 2t))
For every 2 = & (z), we have (7)
E (t. 2, y) = D o (z) (ug (t,z,v))
Applying this with x = x(1) and = = =(1),
i(t) D - (Ø (z(())) (D)()(t)) (us (t,z(t),
Using the identity, - 1 (d(x)) o D (xx) = IdT.M. we get
*(t) ug (t,)(t),))-
Finally, we check the initial condition.
(0) = - (20) = o 1 (o (2o)) = Zo-
Consequently z(t) -- (2(t)) is solution on M of (6).
18

---

E Proof of Proposition 3: Equivalence of Runge-Kutta iterates
Proposition (Equivalence of Runge Kutta iterates). Let {aze) be the iterates produced M by explicit Riemannian Kunge-Kutte scheme applied to the ODE (6). Then, the iterates are
Ft. Φb cb (ze) for all E E,,
where {zz} are the iterates obtained by applying the same scheme (same coefficients and step size) to the ODE with vector field up (7) and initial condition 2g ((zo).
Proof. We prove this for the fourth-order Runge-Kutta method (RK4). Similar proofs can be done for the Euler (RKI) and midpoint (RK2) schemes. The ODEs are: (tt) u (f,, (t), in Euclidean space E
a(t) = ug (t, (t))
on manifold M
The key relationships are:
Vector field relationship: u (1, 2,
(1,
Exponential map on M: (5) o (d(x)) ())()(E)) for E
Parallel transport on M from y E M to M: yT (Délt)) for E M.M.
We induction. Assume 2₁ = (2) for some é. We show xe+1 (ze-1))
RK4 scheme in E: Given žạ at time te, and step size h:
k = up (te, že, !) kE up ( + h 28
h
11 y)
h
h M,E y) 2
K u (tz +
2. 22
K = up (te + h 2 + hkg y)
h
zt + (K 2kg + 2kg + k
ze+1
RK4 scheme on M: Riemannian RK4 method involves evaluating ugM intermediate points. transporting the resulting tangent vectors to M, combining them, and then using the exponential map. By hypothesis, ((x)) 24.
1. First stage (kM) We have
kiM = uM I2, v) E M.
So,
MM (D ((xx)) - (u (te, ((Fz), = (D)(a)))
The intermediate point is
h F exPxt
( k
Substituting the expressions, we get
h (( ))
F = 0 -1
(((x)) De(Øz)
= -1
24 + D ((x))
(Do(s)))
24 + h
0 -1
k
2
Thus,
d(xx) 2z + kf
19

---

2. Second stage (k²²);
kn = upe + h
2 TA.D) E ETAMM
h
1/2 = (DD((xn) ( (1e + 2: ((x)).)) =
Transport 1:24 to M:
221pp PT - (1:2)) = (D (D - (D
- ((E
The intermediate point
TB = expr: 2 k2ppp))
6(xx) + D 2 @x(z))) n1 k211
BB -
Φ-1
24 ( (Do(
1 k
h
Φ-1
21
2
Thus,
h ke 2
((xB)
ze
3. Third stage (kgM):
kgM uu M1+ + h
.B v) E M
2
h
KM = (Do((())) (u. (1x + マ.4
= -D
Transport hig to M:
Spp = PT p2 + (kg = -
The intermediate point
FC = exPx, (hAgg1p)
xC = x (d(Iz) + D (((x))))(((
- (2 +
-1 (zz + hkg)
Thus,
o(xc) = 24 + hkg
4. Fourth stage (k-M);
R:M = ug (t, + h, v) Toa
hM = =DD((ac)) (ug (1: + h (1/(1
h, = -(
Transport kij to Mo
4.p = (kM) - (D = -
5. Final update on M: The combined tangent vector in Tz, M is:
h (K-M + 2k + 2k:a + k
A Anungent
6
2 x(z+)) (((zz)))))))) (1660))))-(((40+200000)))))1---) + 2(D
- (Do(zz)) (())
6
h
= =DD(Zz)))
(xF + 2k + 22 + k))
20

---

Then, 2641 exPx. A Trangeut)
2(+1 x (Ø(z)) + +Do(fe) (Annna)))
(((ze)+ +D((zz) (D (o + 22 +2k + kf) ))
o
h1 (kif 2h + 2kg k
6
- (((+1)
The base case zo = 6 (20) given by the problem setup (zo = o(xo)). Therefore, by induction, It o (zz) for all E N when the RK4 scheme is applied as described.
F The affine-invariant metric for correlation matrices
The set of correlation matrices Corra can be viewed as a quotient manifold of symmetric positive definite matrices + by the action of positive diagonal matrices DJ [16, 68]:
Corr. = 3 /
(25)
where two matrices E, E' E $ are equivalent if there exists D E 1 such that E' = DED.
This quotient structure naturally induces Riemannian metric Corra from the affine-invrriant metric on S defined by
(E. E) s++ (E---E--1
(26)
The canonical projection (submersion)
S:S - Corra is given by diagonal normalization:
Σ = diag Σ) 1/25
1/2
(27)
The resulting geometry Corr, requires solving optimization problem compute the Riemannian logarithm:
log
Cora (B) = 1 (A)
logg (DBD))
s
(28)
where
D arg min dgt+ (A, DBD) Denj
(29)
Since no closed-form solution for D is known, the logarithmic map on (Corra, (, Comms) cannot be expressed analytically (contrary to log $1
21

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G Datasets and Preprocessing
Resting-state IMRI datasets and preprocessing We used three publicly available resting fMRI -BIIDE [53], ADNI [73], and OASIS-3 segn-nnnn a wide age range and neuro- logical conditions. ABIDE includes 900 subjects (one scan each; mean age 17). both neurotypical and autistic, from 19 international sites. ADNI comprises 1900 scans from 900 older adults (mean age 74), covering normal ageing, mild cognitive impairment, and Alzheimer's disease. OASIS-3 includes 1000 participants and 1800 longitudinal sessions collected over 10 years (mean age 71), targeting healthy and pathological ageing-
Preprocessing was performed with fMRIPrep [24], which applies bias-field correction, skull strip- ping, tissue segmentation, nonlinear normalization to MNI space, motion correction, and confound estimation. Regional time series were extracted with Nilearn [14] using the MSDL atlas [71], followed by nuisance regression (including motion, CSF/WM signals, and global signal). Time series were z-scored and screened for minimum length, numerical anomalies (e.g., zeros or extreme values), and the conditioning of their covariance matrices.
EEG datasets Sessions with missing or non-standard protocol (e.g., 2C in BNCI2015-001) were discarded. We further filtered time series whose covariance matrices have high values (max entry 10*) or statistical outliers based on Mahalanobis distance from the group mean. This retained the top 90% most consistent trials for downstream modeling.
BNCI2014-002: The ENCI2014-002" dataset, provided through the BNCI Horizon 2020 initiative, includes recordings from 13 subjects engaged in motor imagery tasks. Participants were instructed to imagine movements of either their right hand or both feet for a sustained duration of 5 seconds, guided by visual cues. EEG signals were captured using an amplifier and active Ag/AgCI electrodes at sampling rate of 512 Hz, with a total of 15 electrode channel applied to each subject. The experimental protocol comprised eight sessions per participant, each including 80 trials for hand and foot imagery, summing to 160 trials. EEG epochs were extracted from 3.0 to 8.0 seconds relative to cue onset, aligning with the motor imagery window.
The BNC12015-001* dataset, released as part of the BNCI Horizon 2020 project, contains EEG recordings from 12 individuals who performed motor imagery tasks involving either the right hand or both feet. EEG data were acquired at a 512 Hz sampling rate, preprocessed with 0.5-100 Hz bandpass filter and 50 Hz notch filter. Each trial spanned 5 seconds, beginning 3.0 seconds after cue onset and ending at 8.0 seconds, corresponding to the motor imagery phase. The dataset provides EEG data from 13 electrode channels arranged in the following order: FC3, FCz, FC4, C5, C3, CI. Cz, C2 C4, C6, CP3, CPz, and CP4. For most participants (Subjects 1-8), recordings were conducted over two sessions on consecutive days. In contrast, Subjects 9-12 completed three sessions. Each session consisted of 100 trials per class, resulting in 200 trials per session per subject.
BNCI2015-001:
http:////wnnco...ooooooooo//o/os 001.html
22

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H Comparison of Baseline Methods: Assumptions, Strengths, and Limitations
In this section, we summarize the key assumptions, strengths, and limitations of the baseline methods discussed throughout the paper, based on our experimental findings.
This is a simple yet practical generative approach that operates directly on SPD or correlation manifolds. While it yields competitive results in terms of the CAS metries, it generally underperforms compared to deep generative models in terms of sample quality, see Table 1.
DIFFEOGAUSS:
TRIANGDDPM or TRIANGCFM: They are deep generative models that achieve strong performance across both CAS and quality-based metries. However, their outputs limited to triangular entities, requiring a post-processing projection step to obtain valid positive definite matrices. Unfortunately, this step alters the eigenvalue spectrum significantly, which degrades the quality and structure of the generated matrices (see Table 2 and Appendix M).
SPD-DDPM: This work extends DDPM to the space SPD manifolds under the affine- invariant Riemannian metric. However, in experiments, found that its implementation repeatedly relies on sampling from Gaussian distributions on SPD manifolds, as well as backpropagating through matrix operations in SPDNet. Both components are computation- ally intensive theory and practice, making the method prohibitively slow. fact, for a number of training samples above 64, were unable obtain results within reasonable time frame (see Figure 4). However, the fMRI experiments, the data have at least 1000 samples.
This is foundational contribution in the family of flow matching methods, providing a general theoretical framework for generative modeling on Riemannian manifolds. However, its construction is geometry-specife and requires dedicated imple- mentation for each manifold. Furthermore, on SPD matrices, we observe that it roughly 8x slower train and 10x slower to sample from than DIFFEOCFM (see Table 1), due to the repeated computation of affine-invariant geodesics, their derivatives and Riemannian norms.
RIEMANNIAN CFM:
DIFFEOCFM: The proposed approach introduces a novel Riemannian CFM framework using the pullback geometry, enabling direct data generation on SPD and correlation manifolds, particularly helpful in neuroscience and neuroengineering tasks. It trains and sampels efficiently (see Figure 4 and Table 1), achieves strong performance on both quality and CAS metrics (see Table 1), and consistently demonstrates neurophysiological plausibility across two distinct evaluation protocols in both fMRI and EEG (see Section 5.2).
23

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I Summary Figure
While Table provides comprehensive numerical evaluation of all methods, visual representation can offer clearer insights into the practical trade-offs between model performance and computational efficiency. To this end, Figure 3 plots a unified performance metric,the Average Fl Score, against both training and sampling times. This allows for direct comparison the performance-cost profile of each method for both fMRI and EEG data.
DiffeoGauss
TriangCFM RiemCFM
DiffeoCFM
TriangDDPM
Real Data
Correlation Matrices (fMRI)
0.7
Average F1 Score
0.6
0.5
0.4
0.3
0.2
0.1
Covariance Matrices (EEG)
0.7
Average F1 Score
0.6
0.5
0.4
0.3
0.2
0.1
10-1
10a
10
10a
10a
10-1
10" Sampling Time (s)
10
Training Time (s)
Figure 3: Trade-off between generative performance and computational cost for IMRI (top) and EEG (bottom) data. The figure plots the Average Fl Score against Training Time (left) and Sampling Time (right). The Average Fl Score is the mean of the quality metric (a, 3-F1) and the CAS Fi-score from Table 1. Each point marks the mean performance across all splits and datasets for given modality, with error bars and shaded regions indicating the standard deviation. The dashed gray line represents the Real Data baseline, which serves as an empirical upper bound. Time shown logarithmic scale.
24

---

J
Scalability and Computational cost of SPD-DDPM [45] vs DIFFEOCFM
DiffeoCFM
SPD-DDPM
Fixed dimension (d 16)
Fixed sample size 128)
10
Training time (s)
10a
10a
10
10
512
8
16
32
64
Sample size n
Matrix dimension d
Figure 4: Training efficiency on $ (simulated data). Log-scale training time over 1000 epochs for generating SPD matrices. We vary either the number of samples the matrix dimension d, keeping the other fixed. DIFFEOCFM leverages the diffeomorphism Øs++ (9) and is at least x000 faster than SPD-DDPM [45]. Importantly, DIFFROCFM can also generate natively correlation matrices (Corra), and performs with same training time as on St²
K Ablation Study
We conduct ablation study analyze the sensitivity of proposed DIFFEOCFM model to its neural network architecture. The experiments are performed on the ABIDE dataset using the Cholesky diffeomorphism. We investigate hidden layers) and the depth (number of hidden layers).
key architectural hyperparameters: the width (size of
First, we fix the network depth to single hidden layer and vary its width from 8 to 1024, as detailed Table Second, we fix the layer width to 512 neurons and vary the depth from one to four hidden layers, with results presented in Table 4. For both experiments, performance is evaluated using the or 3-FF1 score, which
harmonic mean a-precision and 9-rrcall. We also report the average training and sampling times over 10 splits.
The results from Table 3 indicate that performance generally improves with network width, peaking at 256 and 512 hidden units. A wider network of 1024 units shows drop in performance. Regarding network depth, Table 4 clearly shows that single hidden layer achieves the best results. Deeper models exhibit consistent decline in performance. Based on these findings, a single hidden layer with width of 512 was chosen for our main experiments, it offers the best performance without unnecessary complexity.
Table 3: Ablation study on the width of the hidden layer for the DIFFEOCFM model on the ABIDE dataset. The network depth fixed to one hidden layer. We report the mean and standard deviation of the a, 3-F1 score, training time, and sampling time across 10 splits. Best values are in bold.
Hidden dim.
a, 3-F1
Training time (s.)
Sampling time (s.)
8
0.47 + 0.06
29.07 + 1.07
0.35 = 0.01
16
0.47 0.09
27.46
1.24
0.29 0.03
32
0.05
28.36 1.26
0.26 + 0.02
64
0.07
29.69 1.31
0.30 0.03
128
0.05
31.01 + 1.32
0.35 + 0.04
256
0.63
=1.39
0.32 + 0.04
512
0.63 0.06
31.36 1.14
0.31 # 0.02
1024
0.49 0.12
32.34 1.17
0.36 0.05
25

---

Table 4: Ablation study on the depth of the neural network for the DIFFEOCFM model on the ABIDE dataset. The width of all hidden layers is fixed to 512. We report the mean and standard deviation of the a, 3-FI score, training time, and sampling time across 10 splits. Best values are in bold.
Depth
Ot, 3-F1
Training time (s.)
Sampling time (s.)
1
0.62 = 0.05
33.06 + 1.05
0.38 + 0.03
2
0.52 # 0.09
32.93 1.00
0.36 + 0.02
3
0.46 # 0.05
34.36 + 1.00
0.46 + 0.04
4
0.50 = 0.08
35.68 1.01
0.48 + 0.08
Constraint Satisfaction ( and Corr,)
Table reports the fraction of generated samples that satisfy structural matrix constraints across datasets. Both DIFFEOGAUS: and DIFFEOCFM systematically generate matrices that are symmetric, positive definite, and (for correlation matrices) have unit diagonal, achieving perfect 1.00 score all datasets. In contrast, TRIANGCFM without projection produces matrices that are symmetric and have unit diagonal by construction but fail ensure positive definiteness. This issue is particularly evident fMRI datasets, where none of the generated samples satisfy the SPD constraint, and persists lesser extent on EEG datasets (c.g., 29% validity on BNCI2015-001), These results highlight the importance of geometry- awarr methods like DIFFEOCFM, which inherently respect the manifold structure of connectivity matrices without relying on post hoc projections.
Table 5:
Fraction of generated samples satisfying matrix constraints. We report the fraction of samples satisfying symmetry, positive definiteness, and unit diagonal constraints across datasets.
Sym.
Pos. def.
Unit diag.
Dataset
Method
ABIDE
DIFFEOG AUSS
1.00 + 0.00
1.00 = 0.00
1.00 + 0.00
TRIANGCFM (no proj.)
1.00 = 0.00
0.00 0.00
1.00 0.00
DiffeoCFM
1.00 = 0.00
1.00 # 0.00
1.00 + 0.00
ADNI
DITFEOGAUSS
1.00 = 0.00
1.00 # 0.00
1.00 0.00
TRIANGCFM (no proj.)
1.00 = 0.00
0.00 = 0.00
1.00 + 0.00
DiffeoCFM
1.00 = 0.00
1.00 = 0.00
1.00 = 0.00
2014-002 OASIS-3
DIFFEOGAUSS
1.00 = 0.00
1.00 0.00
1.00 0.00
TRIANGCFM (no proj.)
1.00 = 0.00
0.00 # 0.00
1.00 + 0.00
DiffeoCFM
1.00 # 0.00
1.00 0.00
1.00 0.00
DIFFROGAUSS
1.00 + 0.00
1.00 0.00
BNCI
TRIANGCFM (no proj.)
1.00 = 0.00
0.00 = 0.00
DiffeoCFM
1.00 = 0.00
1.00 = 0.00
DIFEEO AUSS
1.00 = 0.00
1.00 + 0.00
BNCI
2015-001
TRIANGCFM (no proj.)
1.00 = 0.00
0.29 0.02
DiffeoCFM
1.00 # 0.00
1.00 # 0.00
M Projection
Since TRIANGDDPM and TRIANGCFM operate in Euclidean spaces, the generated matrices not guaranteed to be positive definite. To address this, we apply projection to ensure all eigenvalues
least > O. Let (m) denote the smallest eigenvalue of Σ. If Amin Σ) < C, the matrix is projected using an affine transformation:
Σ Epoj (1 - a)Σ al
(30)
where I is the identity matrix. The parameter a chosen such that the smallest eigenvalue of Eproj becomes exactly e. Given that the eigenvalues of Epoi A (eppo)))) (1 aa) + a, setting
26

---

the new minimum eigenvalue to e yields é (1 Amin ( + a. Solving for a, we get:
mii
a
(31)
Amin [
This formulation guarantees that if Amin Σ) < C, the new minimum eigenvalue c. Typically, is small positive value (e.g., 10-5), ensuring S 1. In this regime, and given Amin < c. it follows that O < a S 1, making the projection an interpolation. An important property of this projection is that if the original matrix Σ has ones on its diagonal (diag(>) Σ) 1), this property is preserved in Eproj as diag ppmj = (1 a)1 + a1 1.
We randomly select 5 sample images from each generated dataset produced by the TRIANGCFM method across the 5 datasets. These images are not positive definite matrices, as they contain negative cigenvalues. We apply a projection algorithm to convert these matrices into valid correlation matrices, and both the original and projected versions are shown Figure 5 and Figure we annotate the maximum and minimum eigenvalues of the corresponding matrix. As can be seen, after the projection algorithm, all negative eigenvalues are eliminated, and the matrices become valid positive definite matrices. Unfortunately. although this step is necessary for TRIANGDDPM and TRIANGCFM, it alters the spectrum of the matrices, which usually crucial in the downstream classification tasks. The advantage of the proposed DIFFEOCFM lies in the fact that it does not require this step, and thus naturally preserves the matrix spectrum.
each image,
N DIFFEOCFM on EEGs
N.1 Filter's Topographic Map
Common Spatial Pattern
The Common Spatial Pattern (CSP) filter is supervised spatial filtering technique widely used in based signal analysis, particularly for motor imagery classification. It enhances class- discriminative information by projecting multi-channel EEG data onto low-dimensional space that maximizes variance differences between two classes [6]. Formally, for a signal sxt) E Re the CSP algorithm seeks a projection matrix W ReNcxN such that the transformed signal (t)r(t) = WTv(t yields maximally discriminative variance patterns across classes, where Ne is the number of electrode channels. Each column vector wj E g. (j=1, I, Nc) of matrix W is called the filter of CSP.
Generally, CSP is spatial filtering method that effectively enhances the discrimination of men- tal states characterized by event-related synnchrnizattio/syneeommzzz (ERD/ERS), which commonly occurs in the alpha (8-12 Hz) and beta (13-30 Hz) bands during motor imagery tasks [63].
Experimental Settings
In this experiment, we aim to demonstrate whether the proposed model, DIFFEOCFM, can learn similar spatial filter patterns {wy} No in the alpha and beta frequency bands during motor imagery, reflected in their corresponding topographies.
Specifically, we conduct this analysis only on the ENCI1015-001 dataset, as provides the predefined order of 13 EEG channels, allowing to correctly map each channel to its corresponding position in the topographic montage. In contrast, the BNCI2014-002 dataset does not offer such channel ordering information.
All filters were computed using data from session of each subject individually, in order remain consistent with the cross-session experimental setting. The filters for real data were computed directly using the CSP algorithm on session data from each subject, focusing on the 8- 6-12 Hz (alpha) and 13-30 Hz (beta) frequency bands during the first 2 seconds following the cue. For the generated data, filters were obtained by applying the same CSP procedure on generated datasets produced by DIFFEOCFM, with the same number of samples as in session 1. In both cases, only the spatial filter corresponding to the largest eigenvalue from CSP's generalized eigenvalue decomposition was retained for analysis.
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TriangCFM
TriangCFM
TriangCFM 1304
TriangCFM
TriangCFM
largest Fig. 8. lowest Eig. -1.22E+000 00
largest Fig. 6.98E+000 lowest Eig. -1009++00
largest Fig. 7.26E lowest Eig. -9.27E-01
largest Fig. 4.445+00
largest Fig. + lowest Eig. -9.835-01
lowest Eig. -8.60E-01
Projected ThaangCFM 131 largest Eig. 4.32E+00
Projected THannCFMM 664 largest Eig. 3.87E+00 lowest Eig. 1.00E 08
Projected THannCFM 1304 largest Eig. 4.255-000 -00 lowest Eig. 1.00E 008
Projected TMangCFM M 1348 largest Eig. 2.85E+00 lowest Eig. 1.00E 08
Projected THaangFFM 543 largest Eig. 3.18E+00 lowest Eig. 1.00E 08
lowest Eig. 1.00E 08
(a) ADNI
TriangCFM largest Eig. 5.38E+00
TriangCFM largest Eig. 4.30E+00
TriangFFM largest Eig. 5,86E+00 lowest Eig. -5.89--01
TriangCFM
TriangCFM
largest Eig. 5.77E+00 lowest Eig. -3.17E-01
largest Eig. 4,55E+00 lowest Eig. 4.16E-01
lewest Fig. -5.785-01
lowest Big -.8826-01
Projected TManggFFM 845 largest Eig. 3. lowest Eig. 1.00E-08
Projected TMaanCFMM 196 largest Fig. 3.23E lowest 1.00E-08
Projected THangCFM 755 largest Fig. 4.06E DG lowest Eig. 1.00E-08
Projected TriangFFM 838 largest Fig. 4.62E on lowest Eig. 1.00E-08
Projected TriangCFM 591 largast Fig. 3.51E+00 lowest Eig. 1.005-08
(b) ABIDE
TriangCFM
TriangCFM largest Eig. 7,94E+00 lowest Eig. 00.36E+00
TriangCFM largest Eig. 6,36E-00 lowest Eig. 4.94E 01
TriangCFM largest Eig. 9.04E+00 lowest Eig. -7.65-01
TriangCF
largest 60 lowest Eig. 8.34E
largest Eig. 9.88E+00
lowest Eig. -1.1E++0
Projected TrlangCFM 1605
Projected TriangCFM 1342
Projected TrlangCFM 1775
Projected TriangCFM 774 largest Eig. 5,48E+00 lowest Fig. 1. 1.00F-06
Projected TriangCFM 1756
largest Eig. 4.10E-00 lowest Fig. 1.00F-08
largest Eig. 3.91E+00 00 lowest Fig. 1. 1.00F-08
largest Eig. 4.60E-00 lowest Fig. 1.00F-08
largest Eig. 5,09E+00
lowest Fig. 1. 1.00E008
(c) OASIS3
Figure 5: Comparison of fMRI matrices before and after projection
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TriangCFM 5.86E+02 lowest Eig. -3.216+00
TrianGGFM largest Eig.
TrangCFM
TrianCFMM
TrangGFM largest Eig.
largest Eig. 3.16E+02
largest Eiq. 1.32E+02
Eig. -1003E+00
Elg. 11200E+00
Eig. -1.47E+00
Big. -1.73E+00
Projected ThiangCFM 360
Projected ThangCFMM 824 largest Eig. 7.68E++0 lowest Eig. 1.00E-08
Projected THaanCFFM 1257
Projected ThangCFM 593
Projected ThangCFM 26 largest Eig. 7.34E+001
largest Eig. 1.40E++02
largest Eig. 1,44E+02
largest Eig. 5.42E+00 lowest Eig. 1.00E-08
lowest 1.00E-08
lowest Eig. 1.00E-08
lowest Eig. 1.00E-08
(a) BNCI2014-002
TriangCFM
TrangCFM largest Elg. 1.90E-02
TrlanGCMM
largest 1.08E+02 Eig.-8.
largest Eig.
largest Eig. 1.86E-02
largest Eig. 1.91E+02
Eig. -9.30E-01
Eig.
lowest Eig. 9981E-01
Eig. -2.95E-01
Projected FriangCFM 208 largest Eig. 9.96E+01 lowest Eig. 1.00E-08
Projected FriangCFM 503 largest Eig. 6.8BE lowest Eig. 1.00E-08
Projected TriangCFM 2952 largest Eig. 1.63E-02 lowest Eig. 1.00E-06
Projected TriangCFM 2521
Projected TriangGFM 2354
largest Eig. 9.66E+01 lawest Eig. 1.80E-08
largest Eiq. 1.48E+02
lowest Eig. 1.00E
(b) BNCI2015-001
Figure 6: Comparison of EEG matrices before and after projection
N.2 Results and Analysis
Figure shows the topographic map using the first CSP filter for each subject in the BNC12015-001 dataset (Subjects 1-6), while Figure 8 displays the topographic map using the first CSP's filter for each subject in the same dataset (Subjects 7-12). We can observe that, except for Subjects 5, 6, 10, and 12, the topographies for the remaining subjects appear highly similar, with noticeable increase in signal amplitude in the C3 region. Moreover, the average topographic map of these subjects, shown in Figure 2b, also reveals a very similar filter pattern.
This results show that the model, DIFFEOCFM, used for generating the generated data has successfully captured the key characteristics and patterns of the real EEGs. This similarity in CSP's filters indicates that the generated data closely resembles the real data in terms of spatial patterns of brain activity, which is crucial for validating the effectiveness of the model in simulating realistic neural processes, particularly for motor imagery tasks. Additionally, it may imply that the model has learned to preserve the underlying structure and discriminative features that are typically seen in real EEG data during motor imagery.
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DiffecCFM
DiffeoCFM
0.5-2.5
[8-12 0.5-2.5
(13-30
Sub (8-12 Hz, 0.5-2.5 s)
DiffeCCFM (8-12 Hz; 0.5-2.5 s)
Sub 2 (13-30 Hz. 0.5-2.5 s)
DiffeoCFM (13-30 Hz; 0.5-2.5 s)
DiffecCFM (8-12 0.5-2.5
DiffeoCFM [13-30 0.5-2.5 s)
(8-12 Hz, 0.5 5)
(13-30 H2: 0.5-2.5 s)
Sub (8-12 Hz, 0.5- 2.5 s)
DiffecCEM (8-12 Hz; 0.5-2.5 s)
Sub (13-30 Hz. 0.5-2.5 s)
DiffeoCFM (13-30 Hz, 0.5-2.5 s)
DiffecCFM (8-12 0.5-2.5
DiffeoCFM (13-30 0.5-2.5 5)
(8-12 Hz, 0.5-2.5
113-30 H2: 0.5-2.5 5)
Sub (8-12 Hz: 0.5-2.5
DiffecCFM (8-12 0.5-2.5
Sub (13-30 H2: 0.5-2.5 5)
DiffeoCFM (13-30 Hz: 0.5-2.5
0.4
0,0
-0.4
Figure 7: Subject-level topographic map using the first CSP's filter derived from real EEGs (BNC12015-001, Sub 1-6) and generated data by DIFFEOCFM
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DiffecCFM
DiffeoCFM (13-30 0.5-2.5
0.5-2.5 s)
0.5-2.5
Sub 8 (8-12 Hz, 0.5-2.5 s)
DiffeoCFM (8-12 Hz; 0.5-2.5 s)
Sub (13-30 Hz, 0.5-2.5 s)
DiffeoCFM (13-30 Hz: 0.5-2.5 s)
DiffecCFM [8-12 0.5-2.5
DiffeoCFM [13-30 0.5-2.5 s)
(8-12 Hz, 2.5
(13-30 H2: 0.5-2.5 s)
Sub 10 (8-12 Hz, 0.5- s)
DiffecCEM (8-12 Hz, 0.5-2.5 s)
Sub 10 (13-30 Hz, 0.5-2.5 2.5 s)
DiffeoCFM (13-30 Hz, 0.5-2.5 s)
DiffecCFM (8-12 0.5-2.5
DiffeoCFM [13-30 0.5-2.5 5)
(8-12 Hz. 0.5 s)
113-30 H2: 0.5-2.5 s)
G
go.
Sub 12 (8-12 Hz; 0.5-2.5
DiffecCFM (8-12 0.5-2.5
Sub 12 (13-30 H2: 0.5-2.5 5)
DiffeoCFM (13-30 Hz; 0.5-2.5
0.4
0,0
Figure 8: Subject-level topographic map using the first CSP's filter derived from real EEGs (BNC12015-001,, Sub 7-12) and generated data by DIFFFOCFM
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O DIFFEOCFM on fMRI
0.1 IMRI Connectome Plotting
Experimental Settings
We Nilearn [14] for neuroimaging analysis and visualization based the MSDL brain atlas. The atlas is retrieved using the fetch atlas msdl function, which provides probabilistic maps of functional brain networks in the form of 4D NIITI image. The maps attribute extracts this image, while find toou__-bbbblisticcc __tla___t_______ out_coords computes representative 3D coordinates for each network, typically corresponding to the spatial center of the activation, to support anatomical localization. In addition, predefined region labels are extracted to annotate the corresponding brain areas. These components together enable the visualization of brain networks, including plotting the regions with spatial coordinates and labels.
The connectome plots grouped into categories: "CN" and "non-CN".. The adjacency_mand _mtrrix used in nilearn. peapttinggplo_cooneeeee of connectivity matrices in all the 5-fold experiments, and all edge thresholds in the plots are to 90%.
is computed as the Fréchet mean of the corresponding group
Results and Analysis
Figure 9a and Figure 9b visualize group-averaged functional connectomes derived from two distinct neuroimaging datasets using the Fréchet mean. By thresholding edges to retain only the strongest 10% of connections (90% edge threshold), these plots emphasize dominant, statistically reliable interactions between brain regions while filtering out noise. The use of the Fréchet mean ensures that the group-level adjacency matrices respect the intrinsic geometric structure of covariance data, avoiding distortions caused by arithmetic averaging. This comparison highlights both the fidelity of data generation methods by aligning generated and real data connectomes using the Fréchet mean and potential differences in functional network organization between the two cohorts. The workflow underscores the utility of geometric statistics and stringent thresholds for interpreting brain connectivity in heterogeneous populations.
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Real IMRI (CN)
DiffeCCMM M (CN)
Real IMRI (non-CN)
DiffecCFM (non-CN)
(a) Class-conditional fMRI functional connectomes using the Fréchet mean (ABIDE).
Real IMRI (CN)
DiffeCCFM (CN)
Real IMRI (non CN)
DiffecCFM CN)
02
02
(b) Class-conditional fMRI functional connectomes using the Fréchet mean (OASIS3).
from two separate neuroimaging datasets, are visualized after applying a 90% edge threshold to retain only the top 10% of strongest connections. This thresholding highlights dominant and statistically robust interactions between brain regions, effectively reducing noise. Leveraging the Fréchet mean preserves the underlying geometric structure of the covariance matrices, avoiding potential distortions introduced by conventional Euclidean averaging.
Figure 9: Group-level functional connectomes, computed Fréchet
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P Plotting of Generated Samples in Real Data Neighborhoods
Figure 10-15 present the selected generated IMRI samples that nearest to real data points in terms of Frobenius distance, categorized by "CN" and "non-CN" from the ADNI, ABIDE, and OASIS3 datasets.. Figure 16- show the selected generated EEG samples that are nearest to real samples in Frobenius distance, separately for hand and feet motor imagery tasks from the BNCI2014-002 and ANC12015-001 datasets. Each real sample annotated with its index in the original dataset. The first, second, and third columns correspond to samples generated by DIFFEOGAUSS, TRIANGCFM, and DIFFEOCFM, respectively. The numeric values on the generated samples indicate the Frobenius distance the corresponding real sample. Each generated sample shown is the nearest one (in Frobenius distance) to the real sample within its class. Due to its consistently highest recall across multiple datasets, DIFFEOCFM is more likely to produce generated samples that are closer to real samples, i.c. with the smallest Frobenius distance, compared to those generated by other methods. The matrices generated by TRIANGCFM shown in the figures are presented without applying the projection step. Before applying the projection, the outputs of TRIANGCFM already exhibit strong recall scores.
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Difse Dist. 10.24
TriangCFM Dist. 9.03
DiffeoCFM Dist. 8.38
Real 1MRI 286
DiffeoGauss Dist. 9.15
TriangCFM Dist. 8.01
DiffeoCFM Dist. 7.49
Real FMRI 744
DiffeoGauss Dist. 9.19
TriangCFM Dist. 8.43
DiffeoCFM Dist. 7.74
Real FMRI 410
DiffeoGauss Dist. 8.47
TriangCFM Dist. 7.83
DiffeoCFM Dist. 7.04
Real FMRI 101
DiffeoGauss Dist. 9.65
TriangCFM Dist. 9.03
DiffeoCFM Dist. 8.49
Real 1MRI 444
Figure 10: Nearest Generated Samples in Real Data Neighborhoods: ADNI-CN Cohort. The matrices generated by TRIANGCFM shown in the figures presented without applying the projection step.
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DiffeoGauss Dist. 10.29
TriangCFM Dist. 8.39
DiffeoCFM Dist. 7.53
Real 1MRI 778
DiffeoGauss Dist. 9.64
TriangCFM Dist. 8.16
DiffeoCFM Dist. 7.38
Real 1MRI 428
DiffeoGauss Dist. 10.15
TriangCFM Dist. 8.42
DiffeoCFM Dist. 7.81
Real (MRI 548
DiffeoGauss Dist. 10.41
TriangCFM Dist. 8.14
DiffeoCFM Dist. 7.17
Real (MRI 995
DiffeoGauss Dist. 10.29
TriangCFM Dist. 8.39
DiffeoCFM Dist. 7.53
Real IMRI 778
Figure 11: Nearest Generated Samples in Real Data Neighborhoods: ADNI-nonCN Cohort. The ma- trices generated by TRIANGCFM shown in the figures presented without applying the projection step.
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DiffeoGauss Dist. 5.76
TriangCFM Dist. 5.93
DiffeoCFM Dist. 5.43
Real 1MRI 267
DiffeoGauss Dist. 6.66
TriangCFM Dist. 6.70
DiffeoCFM Dist. 6.01
Real 1MRI 332
DiffeoGauss Dist. 6.17
TriangCFM Dist. 6.31
DiffeoCFM Dist. 5.87
Real (MRI 242
DiffeoGauss Dist. 6.57
TriangCFM Dist. 6.70
DiffeoCFM Dist. 6,20
Real FMRI 64
DiffeoGauss Dist. 6.29
TriangCFM Dist. 6.36
DiffeoCFM Dist. 5.63
Real 1MRI 237
Figure 12: Nearest Generated Samples in Real Data Neighborhoods: ABIDE-CN Cohort. The matri- generated by TRIANGCFM shown in the figures presented without applying the projection step.
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DiffeoGauss Dist. 6.05
TriangCFM Dist. 6.10
DiffeoCFM Dist. 5.26
Real 1MRI 136
DiffeoGauss Dist. 6.53
TriangCFM Dist. 6.43
DiffeoCFM Dist. 5.82
Real 1MRI 131
DiffeoGauss Dist. 7.12
TriangCFM Dist. 7.41
DiffeoCFM Dist. 6.55
Real (MRI 175
DiffeoGauss Dist. 6.05
TriangCFM Dist. 6.10
DiffeoCFM Dist. 5,26
Real (MRI 136
DiffeoGauss Dist. 7.07
TriangCFM Dist. 7.10
DiffeoCFM Dist. 6.41
Real IMRI 93
Figure 13: Nearest Generated Samples in Real Data Neighborhoods: ABIDE-nomCN Cohort. The matrices generated by TRIANGCFM shown the figures are presented without applying the projec- tion step.
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DiffeoGauss Dist. 9.45
TriangCFM Dist. 9.16
DiffeoCFM Dist. 6.63
Real IMRI 1004
DiffeoGauss Dist. 7.54
TriangCFM Dist. 7.72
DiffeoCFM Dist. 6.98
Real 1MRI 101
DiffeoGauss Dist. 8.61
TriangCFM Dist. 8.43
DiffeoCFM Dist. 7.77
Real fMRI 612
DiffeoGauss Dist. 7.16
TriangCFM Dist. 7.17
DiffeoCFM Dist. 6.48
Real (MRI 953
DiffeoGauss Dist. 6.24
TriangCFM Dist. 8.46
DiffeoCFM Dist. 7.71
Real 1MRI 445
Figure 14: Nearest Generated Samples in Real Data Neighborhoods: OASIS3-CN Cohort. The matri- generated by TRIANGCFM shown in the figures presented without applying the projection step.
39

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DiffeoGauss Dist. 8.12
TriangCFM Dist. 8.56
DiffeoCFM Dist. 7.24
Real 1MRI 136
DiffeoGauss Dist. 7.84
TriangCFM Dist. 8.04
DiffeoCFM Dist. 7.01
Real 1MRI 733
DiffeoGauss Dist. 7.44
TriangCFM Dist. 7.49
DiffeoCFM Dist. 6,64
Real AMRI 148
DiffeoGauss Dist. 7.68
TriangCFM Dist. 7.73
DiffeoCFM Dist. 6,85
Real (MRI 354
DiffeoGauss Dist. 7.65
TriangCFM Dist. 7.70
DiffeoCFM Dist. 6.72
Real (MRI 286
Figure 15: Nearest Generated Samples in Real Data Neighborhoods: OASIS3-nonCN Cohort. The matrices generated by TRIANGCFM shown in the figures are presented without applying the projection step.
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DiffeoGauss
TriangCFM 31.55
DiffeoCFM
DiffeoGauss
TriangCFM 28.56
DiffeoCFM
DiffeoGauss
TriangCFM
DiffeoCFM
DiffeoGauss Dist. 13.05
TriangCFM Dist. 13.36
DiffeoCFM Dist. 6.61
Real EEG 682
DiffeoGauss Dist. 22.67
TriangCFM Dist, 24.80
DiffeoCFM Dist. 14.62
Real EEG 496
Figure 16: Nearest Generated Samples in Real Data Neighborhoods: BNNC1201-0002 Hand.
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DiffeoGauss
TriangCFM 22.85
DiffeoCFM
DiffeoGauss
TriangCFM
DiffeoCFM
DiffeoGauss
TriangCFM
DiffeoCFM
169
DiffeoGauss Dist. 21.66
TriangCFM Dist, 26.11
DiffeoCFM Dist. 12.51
Real EEG 3
DiffeoGauss Dist. 41.90
TriangCFM Dist, 21.89
DiffeoCFM Dist. 13.46
Real EEG 468
Figure 17: Nearest Generated Samples in Real Data Neighborhoods: SEVC120140002-F5
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DiffeoGauss
TriangCFM
DiffeoCFM
DiffeoGauss
TriangCFM
DiffeoCFM
DiffeoGauss 31.77
TriangCFM
DiffeoCFM
DiffeoGauss Dist. 52.21
TriangCFM Dist, 34.47
DiffeoCFM Dist. 20.00
Real EEG 456
DiffeoGauss Dist. 36.99
TriangCFM Dist, 36.52
DiffeoCFM Dist. 26.37
Real EEG 1270
Figure 18: Nearest Generated Samples in Real Data Neighborhoods: SSCC12015-001RR0 Hand.
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DiffeoGauss
TriangCFM 25.49
DiffeoCFM
1411
DiffeoGauss
TriangCFM 25.
DiffeoCFM
DiffeoGauss
TriangGFM
DiffeoCFM
DiffeoGauss Dist. 16.41
TriangCFM Dist, 16.62
DiffeoCFM Dist. 9.45
Real EEG 556
DiffeoGauss Dist. 16.07
TriangCFM Dist, 15.58
DiffeoCFM Dist. 9.19
Real EEG 507
Figure 19: Nearest Generated Samples in Real Data Neighborhoods: |BNC12015-001-
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Justification: The abstract and Section clearly state the core contributions- maanttllonnnnaaml.......a conditional flow matching on pullback manifolds, its instantiation for SPD and correlation matrices, and a large-scale neuroimaging benchmark. These claims are substantiated theoretically in Section and empirically in Section and Section 5.
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Answer: [Yes]
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reproducible, they 5. Open access to data and code
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Justification: All datasets used in this study (ADNI, ABIDE, OASIS-3, BNC12014-002, BNCI2015-001) are publicly available but fMRI ones (ADNI, ABIDE, OASIS-3) -33) require registration data agreements. We do not redistribute these datasets, but provide detailed instructions and references in the supplementary material to guide users on how to access and preprocess them using standard libraries (e.g., fMRIPrep, Nilearn). Anonymized code, including scripts for training, evaluation, and reproducing experimental results, is included in the supplementary material under a permissive open-source license.
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Reviews:
[{"review_id": "5A89ThHmm3", "reviewer": "Reviewer_DK8R", "summary": "The paper introduces DIFFEOCFM, a method designed to address the challenge of generating realistic brain connectivity matrices, such as symmetric positive definite or correlation matrices, reside on Riemannian manifolds. Traditional manifold-aware generative models incur heavy computational overhead. DIFFEOCFM uses a global diffeomorphism to map these manifolds to Euclidean space, enabling efficient application of standard generative techniques while ensuring geometric constraints are maintained. Evaluated on 3 fMRI datasetsand 2  EEG datasets, DIFFEOCFM demonstrates superior performance in quality and classification metrics than baseline methods.", "questions": "1) Discussion on scalability to high-dimensional manifolds would be helpful.\n2) The choice of a two-layer 512-unit MLP is not fully justified. Could the authors try transformer-style vector fields or neural ODEs, and also explore alternative pullbacks?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper is well motivated and well written. The integration of pullback geometry with conditional flow matching is a novel approach that simplifies generative modeling on Riemannian manifolds while preserving geometric constraints.\n- DIFFEOCFM achieves state-of-the-art results on multiple fMRI and EEG datasets, with significant improvements in quality and classification metrics over baselines.\n- The biological plausibility analysis adds credibility, and generated samples are neurophysiologically meaningful, supporting applications in brain disease research and classification tasks.\n\nWeaknesses:\n- The impact of diffeomorphism choice (log-Euclidean vs. affine-invariant, other Cholesky normalizations) or vector-field capacity is not discussed.\n- Comparisons with other true manifold baselines such as SPD-DDPM in terms of quality and runtime would be helpful.\n- Additional literature could be discussed, such as Jo et al. and Kapusniak et al., which also generate data on manifolds through learned or mixture-based interpolants.\n\nJo et al. Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes. ICML 2024\n\nKapusniak et al. Metric Flow Matching for Smooth Interpolations on the Data Manifold. NeurIPS 2024", "was_revised": true, "final_justification": "I have carefully reviewed the author response and the other reviewers' comments. The responses have addressed several of my concerns, and I believe the additional comparison to SPD-CFM and the provided clarifications meaningfully strengthen the paper. I am therefore maintaining my Accept rating.", "rebuttal": "We thank the reviewer for their positive and encouraging feedback. We are glad that the **motivation, writing, and integration of pullback geometry** with conditional flow matching in our proposed method, DiffeoCFM, were appreciated. We are also pleased that the reviewer found our **state-of-the-art** results and the **biological plausibility analysis** convincing, as these are central to the impact we aim to achieve in neuroscience applications.\n\nWe now address the reviewer’s comments and questions, recalling each point in their own words before providing our response.\n\n## Impact of the choice of diffeomorphism and vector field\n> The impact of diffeomorphism choice (log-Euclidean vs. affine-invariant, other Cholesky normalizations) or vector-field capacity is not discussed.\n\n> The choice of a two-layer 512-unit MLP is not fully justified. Could the authors try transformer-style vector fields or neural ODEs, and also explore alternative pullbacks?\n \nFirst of all, the number of available diffeomorphisms for SPD and correlation manifolds is limited. We selected the normalized Cholesky map for correlation matrices and the logarithm map for SPD matrices because of their simplicity and ease of implementation. Then, identifying alternative mappings and systematically comparing their properties remains an open problem and we believe is beyond the scope of this work. We have added the following note about the choice of the two diffeomorphisms in Section 3.2:\n\n>> **Added in Section 3.2**: We selected these two diffeomorphisms for their simplicity and ease of implementation. However, other choices are possible. For correlation matrices, alternative parameterizations are discussed in [1]. For SPD matrices, one can use the Cholesky factor with a logarithm applied to the diagonal, leading to the log-Cholesky metric [2], which also defines a global diffeomorphism.\n\nRegarding the vector field, it is implemented as a two-layer MLP with 512 hidden units. This architectural choice is consistent with prior work [3, 4, 5], and in our experience, the results remained stable as long as the network was sufficiently large. In our setting, the pullback yields an unstructured vector representation, lacking spatial or sequential structure. As such, transformer architectures do not offer additional meaningful inductive bias while increasing model complexity. Similarly, replacing the MLP with a neural ODE backbone would duplicate the existing continuous-time formulation without a clear benefit.\n\n[1] Thanwerdas, *Riemannian and stratified geometries of covariance and correlation matrices*, Thesis 2022.\n\n[2] Lin et al., *Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition*, SIAM Journal on Matrix Analysis and Applications 2019.\n\n[3] Kotelnikov et al., *TabDDPM: Modelling Tabular Data with Diffusion Models*, ICML 2023. \n\n[4] Shi et al., *Diffusion Schrödinger Bridge Matching*, NeurIPS 2023. \n\n[5] Albergo et al., *Building Normalizing Flows with Stochastic Interpolants*, ICLR 2023.\n\n\n## Comparisons with other true manifold baselines\n\n> Comparisons with other true manifold baselines such as SPD-DDPM in terms of quality and runtime would be helpful.\n\n**SPD-DDPM** A comparison of training time of DiffeoCFM with SPD-DDPM is provided in Appendices G–H. First, we note that SPD-DDPM is designed specifically for SPD matrices (not correlation matrices) and operates entirely on the SPD manifold. As a result, it relies on expensive Riemannian training and sampling (Riemannian optimization of SPDNet, exp/log map, Riemannian sampling solver, ...), making it impractical even for medium-sized datasets (e.g., 100 samples of dimension 30). Indeed, as shown in Figure 3 (Appendix H), it is over 1000× slower than DiffeoCFM, which shares the same computational footprint as CFM, a method explicitly designed for scalability and widely adopted in computer vision. Due to this computational burden, we were unable to assess the sample quality of SPD-DDPM in our experimental setting.\n\n**SPD-CFM** We compare DiffeoCFM against SPD-CFM [6], which will be added to Table 1. We adapted the official implementation of SPD-CFM to use the same conditional prior as DiffeoCFM and evaluated on BNCI 2014‑002 by holding out each of the first two sessions in turn for testing while training on all remaining sessions. The results are shown in the next table; BNCI 2015‑001 yielded similar results.\n\nQuality metrics = $\\alpha$-Precision, $\\beta$-Recall, $\\alpha$,$\\beta$-F1\nClassification Accuracy Score (CAS) metrics: ROC-AUC, F1\n\n| Method   | $\\alpha$-Precision   | $\\beta$-Recall   | $\\alpha$,$\\beta$-F1   | ROC-AUC  | F1  | Training time (s.)\n|:-----------------|:---------------------------------------------------------|:-----------------------------------------------------|:-----------------------------------------------------------|:-----------------------------------------------|:------------------------------------------|:--------------------------------------------------------|\n| SPD-CFM          | 0.59 ± 0.08                                              | 0.59 ± 0.07                                          | 0.59 ± 0.10                                                | 0.82 ± 0.02                                    | 0.73 ± 0.03                               | 1656.78 ± 4.86                                          |\n| DiffeoCFM  (proposed)      | **0.62 ± 0.05**                                     | **0.69 ± 0.04**                                          | **0.65 ± 0.07**                                       | **0.84 ± 0.02**                           | **0.76 ± 0.03**                      | **265.94 ± 4.17**\n\n**Results interpretation** SPD-CFM [6] performs slightly worse than DiffeoCFM. Moreover, SPD-CFM is approximately 6x slower, which is expected since it relies on geodesic computation and its derivative, the Riemannian norm, and a Riemannian ODE solver. In contrast, DiffeoCFM first maps data to a Euclidean space via the diffeomorphism $\\phi$, trains a standard Euclidean CFM (which is very fast), and then maps generated data back to the manifold via $\\phi^{-1}$. We will complete Table 1 of the manuscript with this additional baseline.\n\n[6] Chen et al., *Flow Matching on General Geometries*, ICLR 2024\n\n## Additional literature\n\n> Additional literature could be discussed, such as Jo et al. and Kapusniak et al., which also generate data on manifolds through learned or mixture-based interpolants.\n\nWe have added a brief discussion of [7] and [8] in the Section 3.3 Related-work:\n\n>> **Added in Section 3.3**: References [7] and [8] explore more general settings by learning bridge matches on arbitrary manifolds or data-driven Riemannian metrics, whereas our approach focuses on Riemannian geometries defined via known pullback diffeomorphisms.\n\nKey distinctions are summarized below:\n* Reference [7] addresses a more general setting, learning bridge matching on manifolds where Riemannian logarithms may be undefined or costly to compute. They rely on the spectral decomposition of the Laplace-Beltrami operator to define distances and then reduce the simulation budget during training. It should be noted that they still require the Riemannian norm in their loss function during training and the geodesic (Riemannian exponential mapping) during sampling. In contrast, our work focuses on pullback manifolds such as SPD and correlation spaces, where known diffeomorphisms enable fast and scalable training and sampling without complex Riemannian operators. \n* Reference [8] replaces straight interpolants with curved ones derived from a learned Riemannian metric. While complementary, this line of work is orthogonal to our focus on leveraging existing pullback maps (and their associated Riemannian metrics) for efficient training and generation. \n\n[7] Jo et al., *Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes*, ICML 2024.\n\n[8] Kapusniak et al., *Metric Flow Matching for Smooth Interpolations on the Data Manifold*, NeurIPS 2024\n\n\n## Scalability to high-dimensional manifolds\n> Discussion on scalability to high-dimensional manifolds would be helpful.\n\nFirst, learning generative models when the ambient dimension $d$ approaches the sample size $n$ is statistically hard. For example, Theorem 3.1 of [9] shows that the estimation error of neural networks trained with diffusion decays no faster than $n^{-1/d}$, which is the curse of dimensionality. In our fMRI experiments we adopted the MSDL atlas with 39 regions, giving a manifold dimension of $39 \\times 38 / 2 = 741$, which remains below the number of available samples (e.g. 1900 ADNI scans). Using finer atlases would push us into the $n<d$ regime where theory predicts over-fitting and degraded generative quality. Designing priors or low-rank parameterisations to tame this regime is an important avenue for future work.\n\nWe have inserted the following clarification into the Section 6:\n\n>> **Added in Section 6 Conclusions, Limitations, and Future Works:**  \nLearning brain connectivities at higher parcellation granularity is challenging: the manifold dimension grows quadratically with the number of regions, while the number of required samples increases exponentially with dimension [9]. Addressing this high-dimensional regime, e.g. through latent representations, remains an important direction for future work.\n\nSecond, from a computational standpoint, DiffeoCFM inherits the scalability of CFM, which was originally developed for high-dimensional data and large sample sizes in vision tasks. In contrast, manifold-based methods like SPD-DDPM scale poorly in practice due to their reliance on costly geometric operations.\n\n[9] Oko, et al. *Diffusion models are minimax optimal distribution estimators*. ICML 2023"}, {"review_id": "mOvTvQ6bOp", "reviewer": "Reviewer_mBJ3", "summary": "This paper introduces DIFFEOCFM, a novel conditional flow matching framework designed for efficient generative modelling on Riemannian manifolds, specifically targeting brain connectivity matrices (SPD and correlation matrices). DIFFEOCFM utilises global diffeomorphisms to map manifold data to Euclidean spaces, thus enabling simpler computations and avoiding expensive manifold-specific operations. Experiments were conducted on three large-scale fMRI datasets (ADNI, ABIDE, OASIS-3) and two EEG motor imagery datasets (BNCI2014-002, BNCI2015-001), demonstrating superior performance in both realism and downstream classification tasks.", "questions": "1. The vector field transformation in Eq. (7) relies on computing the differential of the diffeomorphism and its inverse. What are the computational costs and numerical stability issues, particularly for high-dimensional Corr matrices? Have the authors observed any instability in Dϕ(Σ) when Σ is near singular?\n2. How would the method extend to manifolds without known diffeomorphic embeddings (e.g., the Stiefel manifold)? Could approximate embeddings or learned bijections (e.g., neural ODEs) be considered?\n3. During back-propagation through Dϕ, did you encounter exploding/vanishing gradients?\n4. The CSP maps in Figure 2(b) are presented with colour bars and topographic projections. It would be helpful to also report quantitative similarity metrics (e.g., structural similarity index) between real and synthetic filters for more rigorous comparison.\n5. α-precision and β-recall are geometry-agnostic. Could authors provide a discussion of a manifold-aware version (e.g., geodesic-based recall) in future work?", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nQuality: The manuscript is technically sound, and the mathematical framework is well-constructed. The pullback approach significantly reduces computational overhead without compromising geometric fidelity, which is convincingly supported by theoretical propositions (Propositions 1-3).\nClarity: The manuscript is clearly structured, thoroughly described, and easy to follow, with well-defined notation and coherent transitions between sections.\nSignificance: DIFFEOCFM significantly advances generative modelling on Riemannian manifolds by simplifying complex geometric computations. This methodological innovation is impactful, particularly in fields like neuroimaging, where preserving manifold constraints is crucial.\nOriginality: The paper's primary contribution—leveraging pullback geometry via global diffeomorphisms—is novel and well-motivated, clearly delineating itself from prior works such as SPD-DDPM and traditional Riemannian CFM.\n\nWeaknesses:\n1. Key implementation and experimental details—e.g., hyperparameters and projection analyses—are buried in the appendices (J, L); a concise summary in the main text would greatly improve self-containment.\n2. Algorithms 1 and 2 omit pseudocode for computing crucial terms such as Dϕ(x) and Dϕ⁻¹(z), which are necessary for understanding Eq. (7) and ensuring reproducibility.\n3. No ablation is reported on ODE step size h or network depth, so the robustness–efficiency trade-off of DIFFEOCFM remains unclear.", "was_revised": true, "final_justification": "The rebuttal response addresses previous concerns and clarifies the ambiguous part. I decided to keep my score as Borderline accept.", "rebuttal": "We thank the reviewer for their thoughtful and positive feedback. We are glad that the paper’s **technical soundness, clarity, and originality** were appreciated, that the pullback approach was recognized as **mathematically grounded and computationally efficient**, and that its relevance to neuroimaging was noted. We also appreciate the acknowledgment of our **clear exposition** and the significance of DiffeoCFM for generative modeling on manifolds.\n\nWe now address the reviewer’s comments and questions, recalling each point in their own words before providing our response.\n\n## Experimental details in Appendices \n>Key implementation and experimental details—e.g., hyperparameters and projection analyses—are buried in the appendices (J, L); a concise summary in the main text would greatly improve self-containment.\n\nWe moved from the appendices key implementation details in the main text, complementing Appendices J and L, to improve readability without sacrificing completeness.\n\nFirst, we added the following information for the metrics:\n>> **Added in Section 4.1 Metrics:**  \nWe follow the CAS protocol, training a classifier on generated samples and evaluating it on real test data. Specifically, we assess classification utility using a logistic regression [...].\n\nThen, we detailed TriangDDPM and TriangCFM and the deep learning setup:\n>> **Added in Section 4.3 Baselines:**  \nSince $\\phi$ is not a diffeomorphism, generated matrices that do not lie on the manifold are projected back onto it. In particular, generated matrices are not necessarily positive definite, so we apply a projection [...].\n\n>> TriangDDPM, TriangCFM and DiffeoCFM employ a two-layer MLP with $512$ hidden units, trained using AdamW with a learning rate of $10^{-3}$ and batch size $64$. Training runs for $200$ epochs on fMRI and $2000$ epochs on EEG. Time integration follows the midpoint method with a step size of $10^{-3}$.\n\n\n## Differentials $\\mathrm{D} \\phi(x)$, $(\\mathrm{D} \\phi)^{-1}(x)$: usage, cost, stability, backpropagation\n>Algorithms 1 and 2 omit pseudocode for computing crucial terms such as Dϕ(x) and Dϕ⁻¹(z), which are necessary for understanding Eq. (7) and ensuring reproducibility.\n\n>The vector field transformation in Eq. (7) relies on computing the differential of the diffeomorphism and its inverse. What are the computational costs and numerical stability issues, particularly for high-dimensional Corr matrices? Have the authors observed any instability in Dϕ(Σ) when Σ is near singular?\n\n>During back-propagation through Dϕ, did you encounter exploding/vanishing gradients?\n\nImplementing Riemannian CFM [1] with a pullback metric requires $\\mathrm{D}\\phi(x)$, $(\\mathrm{D}\\phi)^{-1}(x)$, and backpropagation through them—leading to numerical instability, gradient explosion/vanishing, and high computational cost, which demand careful implementation and tuning.\n\nThis motivated DiffeoCFM, which, as shown in Propositions 1–3, is mathematically equivalent to Riemannian CFM with the pullback metric induced by a diffeomorphism $\\phi$. In practice, DiffeoCFM proceeds by:\n1. apply $\\phi$ to data on $\\mathcal{M}$;  \n2. train and sample a classical CFM in the Euclidean space $E$;  \n3. map generated data back to $\\mathcal{M}$ via $\\phi^{-1}$.\n\nThese steps, detailed in Algorithms 1–2, avoid computing $\\mathrm{D}\\phi(x)$ or any Riemannian operations—only $\\phi$ and $\\phi^{-1}$ are needed, both easy to implement. It should be noted that Eq (7) is used solely to prove Propositions 1, 2, and 3.\n\n[1] Chen et al., *Flow Matching on General Geometries*, ICLR 2024\n\n[2] Lipman et al., *Flow Matching for Generative Modeling*, ICLR 2023\n\n\n## Step size and network depth\n>No ablation is reported on ODE step size h or network depth, so the robustness–efficiency trade-off of DIFFEOCFM remains unclear.\n\nWe studied the robustness–efficiency trade‑off by varying network depth and ODE step size on the fMRI ADNI dataset, reporting $\\alpha,\\beta$‑F1 (mean ± std), training, and sampling times over 5‑fold cross‑validation.\n\n\n|   Depth | $\\alpha,\\beta$-F1      | Training time (s.)   | Sampling time (s.)   |\n|-------------:|:------------|:---------------------|:---------------------|\n|            1 | **0.65 ± 0.06** | **35.07 ± 1.36**         | **3.75 ± 0.81**          |\n|            2 | 0.56 ± 0.04 | 37.23 ± 1.18         | 5.35 ± 0.54          |\n|            3 | 0.52 ± 0.05 | 44.73 ± 2.00         | 5.31 ± 0.93          |\n|            4 | 0.48 ± 0.09 | 49.60 ± 1.64         | 7.74 ± 1.28          |\n\n|   ODE step size | $\\alpha,\\beta$-F1  | Training time (s.)   | Sampling time (s.)   |\n|------------:|:------------|:---------------------|:---------------------|\n|      0.0001 | **0.64 ± 0.05** | 33.86 ± 2.25         | 52.12 ± 0.24         |\n|      0.001  | 0.63 ± 0.05 | 33.50 ± 1.97        | 2.17 ± 0.68          |\n|      0.01   | 0.61 ± 0.06 | 33.22 ± 1.02         | 0.20 ± 0.03          |\n|      0.1    | 0.55 ± 0.09 | 33.63 ± 0.43       | **0.06 ± 0.01**          |\n\n\nResults show that performance saturates with 1 hidden layer, while deeper networks increase computation without improving quality. For the ODE step size, $10^{-3}$ offers a good trade‑off between sample quality and sampling time. This confirms that **DiffeoCFM maintains high quality across a broad range of settings**, with a favourable robustness–efficiency balance.\n\n\n## Extension to manifold without global diffeomorphisms\n>How would the method extend to manifolds without known diffeomorphic embeddings (e.g., the Stiefel manifold)? Could approximate embeddings or learned bijections (e.g., neural ODEs) be considered?\n\nThis work targets brain‑connectivity data—SPD and correlation matrices—where known global diffeomorphisms $\\phi$ enable fast, stable training via Euclidean CFM [2] on embedded data.\n\nAs noted in the conclusion, DiffeoCFM requires such a diffeomorphism and does not apply to manifolds like the Stiefel. Even general algorithms [1] fail on manifolds without closed‑form geodesics. Two possible extensions are:\n1. **Approximate geodesic operators** using retractions/inverses, as in $R$/$RL$-barycentres on Stiefel and Grassmann manifolds [3, 4].  \n2. **Ambient‑space methods** that progressively map back to the manifold without explicit retractions, e.g., landing algorithms for the orthogonal group [5].\n\nExploring these directions is promising but beyond our present scope.\n\n[3] Kaneko, et al., *Empirical arithmetic averaging over the compact Stiefel manifold* IEEE Transactions on Signal Processing, 2012\n\n[4] Bouchard, et al., *Beyond $R$-barycenters: an effective averaging method on Stiefel and Grassmann manifolds* IEEE Signal Processing Letters, 2025\n\n[5] Ablin, et al., *Fast and accurate optimization on the orthogonal manifold without retraction*, ICML 2022\n\n\n## Quantitative evaluation of spatial filters\n>The CSP maps in Figure 2(b) are presented with colour bars and topographic projections. It would be helpful to also report quantitative similarity metrics (e.g., structural similarity index) between real and synthetic filters for more rigorous comparison.\n\nOur study includes such a task‑level test: Table 1 reports CAS [6] when training on real or synthetic data and testing on held‑out real sets. For EEG, real vs synthetic scores match within the standard deviation, showing CSP filters from both have the same discriminative power.\n\nFigure 2 visually confirms this result: in both $\\alpha$- (8–12 Hz) and $\\beta$- (13–30 Hz) bands, filters from synthetic data reproduce the contralateral C3/C4 patterns of real data during right-hand vs. feet imagery [7,8], as quantified in Table 1.\n\nTo clarify this point, we have added the following paragraph in the paper:\n>> **Added in Section 5.2 Neurophysiological Plausibility Study:**\nWe visualize CSP spatial filters in the $\\alpha$ (8–12 Hz) and $\\beta$ (13–30 Hz) bands that distinguish imagined right-hand from feet movements. The filters trained on real and on generated data concentrate on the same contralateral sensorimotor regions, mirroring the close CAS scores between Real data and DiffeoCFM in Table 1. This confirms that the generative model preserves the physiologically relevant decision boundaries.\n\nApplying SSIM to EEG topographies can be misleading: (1) topographies are computed at subject/group level, making SSIM across trials impossible to compute; (2) single‑trial maps contain high non‑neural noise, so pixel‑wise metrics risk capturing artifacts rather than $\\alpha$/$\\beta$ ERD/ERS patterns. This motivated using CAS [6] and CSP filter inspection.\n\n[6] Ravuri et al., *Classification accuracy score for conditional generative models*, NeurIPS 2019.\n\n[7] Pfurtscheller et al., *Event-related EEG/MEG synchronization and desynchronization: basic principles*, Clinical Neurophysiology, 1999.\n\n[8] Pfurtscheller et al., *Mu rhythm (de)synchronization and EEG single-trial classification of different motor imagery tasks*, NeuroImage, 2006. \n\n## Manifold-aware α-precision and β-recall\n>α-precision and β-recall are geometry-agnostic. Could authors provide a discussion of a manifold-aware version (e.g., geodesic-based recall) in future work?\n\nThis remark echoes our conclusion: these metrics [9], standard in generative modeling, assess generated–real pairwise correlation match via the Frobenius norm $||\\Sigma_{\\text{real}} - \\Sigma_{\\text{gen}}||_F^2$, a long‑standing choice in covariance analysis.\n\nA Riemannian distance could reveal other aspects—for instance, the affine‑invariant metric compares eigenvalues in log space, making multiplicative variance changes (e.g., fixed dB shifts in EEG power) linear in distance. While promising, this would change the meaning of $\\alpha$‑precision and $\\beta$‑recall and is beyond our scope.\n\n[9] Alaa et al., *How faithful is your synthetic data? sample-level metrics for evaluating and auditing generative models*, ICML 2022"}, {"review_id": "IQ7dW3VGQP", "reviewer": "Reviewer_P6e5", "summary": "The paper studies the problem of generating brain connectivity with Riemannian flow matching. Authors provide comprehensive descriptions on the methodology. Experiments on fMRI and EEG data demonstrate the effectiveness of the proposed work.", "questions": "Please see above", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The problem being resolved is interesting and of great value. \n- The idea of optimizing over objective in the Euclidean space from perspective of using pullback metric is interesting. It simplified the problem and better utilized the power of neural networks.\n- The paper is well-written with comprehensive details.\n- Experiments show the proposed work can generate decent results on real data.\n\nWeakness: \nSince the generation is on connectivity where the connectivity matrices are obtained from brain signal data (such as fMRI), the way of defining connectivity could possibly influence the \"ground-truth\" thus introduce bias. It would be great if authors can discuss more on it.", "was_revised": true, "final_justification": "Thanks authors for addressing the question regarding connectivity matrices. After reading comments, I decide to keep original rating.", "rebuttal": "We thank the reviewer for their thoughtful and encouraging feedback. We are pleased that they found the **addressed problem both interesting and valuable**, and that the proposed **use of the pullback metric to enable Euclidean optimization** was recognized as a **key strength of our approach**. We also appreciate their remarks on the **clarity and completeness** of the manuscript, as well as their **positive assessment of the experimental results on real data**.\n\nWe now address the reviewer’s comment, recalling the point in their own words before providing our response.\n\n\n## Estimation of Connectivity Matrices\n> Since the generation is on connectivity where the connectivity matrices are obtained from brain signal data (such as fMRI), the way of defining connectivity could possibly influence the \"ground-truth\" thus introduce bias. It would be great if authors can discuss more on it.\n\nOracle Approximating Shrinkage (OAS) [1] and Ledoit–Wolf [2] are the state-of-the-art shrinkage estimators in recent fMRI and EEG studies [3–5], where they consistently deliver the best classification performance. Accordingly, we estimate covariance matrices with OAS and obtain correlation matrices by z-scoring the time series before applying OAS. Indeed, we chose OAS to maximise the Classification Accuracy Score (CAS) of the “Real data” upper bound in Table 1. CAS is the accuracy of a classifier trained on generated data (conditioned on disease) and tested on real data; a high upper bound gives the strongest baseline for evaluating our generative model.\n\nWe agree that the estimator itself may influence the ground-truth and, hence, our results. Because our contribution lies in generative modelling rather than connectivity estimation, we fixed a community-standard pipeline to keep this factor constant.\nTo clarify the limitation, we added the following to the manuscript:\n>> **Added in Section 6 Conclusions, Limitations, and Future Work:**  \nA full study of how alternative connectivity definitions—e.g., partial correlation or graphical-Lasso precision—affect generation quality remains an open problem that needs to be addressed, as the choice of definition directly determines the ground truth used in the experiments.\n\n[1] Chen et al., *Shrinkage algorithms for MMSE covariance estimation*, IEEE Transactions on Signal Processing 2010\n\n[2] Ledoit et al., *A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices*, Journal of Multivariate Analysis 2004\n\n[3] Kamalaker et al., *Benchmarking functional connectome-based predictive models for resting-state fMRI*, NeuroImage 2019\n\n[4] Pervaiz, et al., *Optimising network modelling methods for fMRI*, NeuroImage 2020\n\n[5] Sabbagh et al., *Predictive regression modeling with MEG/EEG: from source power to signals and cognitive states*, NeuroImage, 2022"}, {"review_id": "NTnBy0ereV", "reviewer": "Reviewer_hktp", "summary": "This paper proposes a new method for generating realistic brain connectivity matrices, i.e. symmetric positive definite (SPD) and correlation matrices. Generative modeling with such data requires one to take into account their Riemannian geometries, which is a computationally expensive task. To reduce the computational complexity of the task, the authors focus on Pullback metrics which turns Riemannian computations into Euclidean computations. Generative modeling can then be easily and efficiently performed on Euclidean space. Experimental results on fMRI and EEG motor imagery datasets show the effectiveness of the proposed method.", "questions": "- State-of-the-art works for generative modeling on Riemannian manifolds such as [9,14] use NLL (Negative log-likelihood) scores \nand NFE (number of function evaluations) as evaluation metrics. I think those metrics are relevant in the context of the present work\nand are worth investigating (e.g. by using synthetic data as in [9,14]). This would further demonstrate the advantage of the proposed \nmethod w.r.t. [9,14]. Currently, those metrics are not considered in the experiments. Could the authors elaborate on this choice of \nevaluation metrics ?\n\n- The authors do not use the methods in [9,14] for comparison with the proposed method. I carefully checked those references [9,14] and noticed that both works are applicable (and can be easily adapted) to SPD manifolds, making them strong baselines. The authors of [9] even included SPD manifolds as part of their code. However, the authors  state in Appendix G that \"its current implementations are limited to a few specific manifolds such as spheres and tori, which are not directly applicable to our task.\". Could the authors elaborate on their choice of baselines ?\n\n- The work in [A] is closely related to the present work but is missing in the references. The proposed generative method \nfor SPD manifolds can be directly derived from [A], since SPD manifolds with Log-Euclidean metrics have Lie group structure. \nI think it should be cited in the related works and eventually be used in comparison with the proposed method.\n\n**References**\n\nLuca Falorsi, Pim de Haan, Tim R. Davidson, Patrick Forré: Reparameterizing Distributions on Lie Groups. AISTATS 2019: 3244-3253.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The proposed method relies on Pullback metrics which are well-established and have proven efficient in extending traditional machine learning models to the Riemannian settting. \n- Theoretical justifications are provided in the supplemental material. \n- The considered applications and datasets are relevant. Experimental results are promising.\n\nWeaknesses:\n- Important baselines are missing, making it hard to evaluate real benefits of the proposed method w.r.t. existing works.", "was_revised": true, "final_justification": "In my initial review, I raised a concern regarding the evaluation metrics and a lack of comparison of the proposed methods against some state-of-the-art methods. During the rebuttal, the authors clearly explained their choice of evaluation metrics. They also provided experimental results which demonstrate the advantage of their methods w.r.t state-of-the-art methods. For this reason, I have decided to update my initial score from 2 to 4.", "rebuttal": "We thank the reviewer for their detailed feedback and for highlighting the strengths of our work. We are pleased that the **use of pullback metrics**, the **theoretical justification** provided in the supplementary material, and the **relevance of the considered applications and datasets** were appreciated.\n\nWe now address the reviewer’s comments and questions, recalling each point in their own words before providing our response.\n\n\n## Choice of baselines\n> Important baselines are missing, making it hard to evaluate real benefits of the proposed method w.r.t. existing works.\n> The authors do not use the methods in [1,2] for comparison with the proposed method. [...] Could the authors elaborate on their choice of baselines ?\n\n**Adding baseline SPD-CFM [1].** DiffeoCFM (proposed) is, to our knowledge, the first generative model that handles both SPD and correlation matrices while respecting manifold structure and being computationaly efficient. We acknowledge that [1] includes an SPD-manifold implementation, which we initially overlooked, and thank the reviewer for pointing it out. We adapted the official implementation of SPD-CFM to use the same conditional prior as DiffeoCFM and evaluated on BNCI 2014‑002 by holding out each of the first two sessions in turn for testing while training on all remaining sessions. The results are shown in the next table; BNCI 2015‑001 yielded similar results.\n\nQuality metric: $\\alpha$,$\\beta$-F1\n\nClassification Accuracy Score (CAS) metrics: ROC-AUC, F1\n\n| Method   | $\\alpha$,$\\beta$-F1   | ROC-AUC  | F1  | Training time (s.)\n|:-----------------|:-----------------------------------------------------------|:-----------------------------------------------|:------------------------------------------|:--------------------------------------------------------|\n| Real Data         | 0.67 ± 0.07                                                | 0.86 ± 0.01                                    | 0.77 ± 0.02                               | N/A                                        |\n| DiffeoGauss                        | 0.56 ± 0.06                                                | 0.83 ± 0.01                                    | 0.75 ± 0.02                               | **0.09 ± 0.01**                                    |\n| TriangDDPM                            | 0.15 ± 0.04                                                | 0.51 ± 0.03                                    | 0.22 ± 0.12                               | 265.24 ± 1.96                                           |\n| TriangCFM                                   | 0.26 ± 0.04                                                | 0.54 ± 0.03                                    | 0.24 ± 0.14                               | 261.15 ± 1.15                                           |\n| SPD-CFM                                     | 0.59 ± 0.10                                                | 0.82 ± 0.02                                    | 0.73 ± 0.03                               | 1656.78 ± 4.86                                          |\n| DiffeoCFM   (proposed)                         | **0.65 ± 0.07**                                       | **0.84 ± 0.02**                           | **0.76 ± 0.03**                      | 265.94 ± 4.17\n\n**Results interpretation.** SPD-CFM [1] performs slightly worse than DiffeoCFM. Moreover, **SPD-CFM is 6x slower**, which is expected since it relies on geodesic computation and its derivative, the Riemannian norm, and a Riemannian ODE solver. In contrast, DiffeoCFM first maps data to a Euclidean space via the diffeomorphism $\\phi$, trains a standard Euclidean CFM [3] (which is very fast), and then maps generated data back to the manifold via $\\phi^{-1}$. **We will complete Table 1 of the manuscript with this additional baseline.**\n\n**SPD-DDPM [4].** The Riemannian diffusion framework of [2] is instantiated on the SPD manifold in SPD-DDPM [4]. We have already benchmarked this model in Appendices G–H. Figure 3 (Appendix H) shows that **SPD-DDPM is over 1000 ×** slower than DiffeoCFM due to the heavy computational cost of Riemannian manifold operations, particularly manifold sampling.\n\n**Choice of baselines.** The other baselines were chosen for their complementary properties:\n- DiffeoGauss fits a multivariate Gaussian distribution per class, which—despite its simplicity—performs surprisingly well, suggesting that simple yet effective priors can be valuable for neuroscience applications.\n- TriangCFM and TriangDDPM are straightforward parametrizations that learn the lower-triangular parts with a Euclidean CFM; they are as fast as DiffeoCFM but, lacking diffeomorphic parametrizations, show lower precision/recall.\n- The Real Data baseline compares real-vs-real samples with different splits, providing an upper bound on attainable performance. On EEG, DiffeoCFM reaches this bound, whereas on fMRI, there remains a gap to close.\n\nThis setup exposes key challenges in neuroscience and motivates the use of DiffeoCFM as a simple yet high-performing method.\n\n[1] Chen et al., *Flow Matching on General Geometries*, ICLR 2024\n\n[2] De Bortoli et al., *Riemannian score-based generative modelling*, NeurIPS 2022\n\n[3] Lipman et al., *Flow Matching for Generative Modeling*, ICLR 2023\n\n[4] Li et al., *Spd-ddpm: Denoising diffusion probabilistic models in the symmetric positive definite space*, AAAI 2024\n\n## Choice of evaluation metrics\n> State-of-the-art works for generative modeling on Riemannian manifolds such as [1,2] use NLL (Negative log-likelihood) scores and NFE (number of function evaluations) as evaluation metrics. [...] Currently, those metrics are not considered in the experiments. Could the authors elaborate on this choice of evaluation metrics ?\n\n**NFE.** First, all models in our experiments solve the ODE with the same fixed-step midpoint solver ($dt = 0.001$), yielding an **identical number of function evaluations (NFE)**. We therefore did not report NFE.  \n\n**NLL.** Then, regarding **negative log-likelihood (NLL)** versus sample-level metrics $\\alpha$‑precision, $\\beta$‑recall, and $\\alpha,\\beta$‑F1 [5]:  \n- **NLL feasibility:** Its computation **requires a diffeomorphism** between the prior and the learned distribution so that the change-of-variables log-det Jacobian is well defined. Two important baselines—TriangCFM and TriangDDPM—generate samples in ambient space and project them to the SPD (or correlation) manifold. The projection breaks injectivity, so the log-det Jacobian (and hence NLL) is undefined.\n- **Interpretability:** The sample-level metrics compare generated to real data on a 0–1 scale, with an empirical upper bound from real-vs-real comparisons. They are therefore directly interpretable: in Table 1, DiffeoCFM reaches this bound on EEG and approaches it on fMRI. Hence they highlight generative-model challenges in Neuroscience, whereas absolute NLL values are harder to interpret.\n- **Conditional generation:** Unlike [1, 2], our work targets **conditional generation**—essential in neuroscience, where data must be generated given patient-level variables (here, diagnosis). To assess conditional fidelity, we use the **Classification Accuracy Score (CAS)** [6], which trains a state-of-the-art classifier on generated data and evaluates it on real data, directly testing preservation of the conditional structure.  \n\n[5] Alaa et al., *How faithful is your synthetic data? sample-level metrics for evaluating and auditing generative models*, ICML 2022\n\n[6] Ravuri et al., *Classification accuracy score for conditional generative models*, NeurIPS 2019.\n\n\n## Additional reference: generation on Lie groups\n> The work in [7] is closely related to the present work but is missing in the references. The proposed generative method for SPD manifolds can be directly derived from [7], since SPD manifolds with Log-Euclidean metrics have Lie group structure. I think it should be cited in the related works and eventually be used in comparison with the proposed method.\n\n**Contributions of [7].** The authors extend the reparameterisation trick to Lie groups and demonstrate how normalising flows can leverage this approach to learn probability densities on such spaces.\n\n**Manuscript update.** Since both [7] and DiffeoCFM rely on reparameterisation tricks to estimate probability densities, [7] can be viewed as an early precursor to our approach. We have therefore added the following to the manuscript:\n>> **Added in Section 3.3 – Related Work**  \nIn [7], the authors leverage reparameterisation with normalising flows to learn probability densities on Lie groups (non-Euclidean spaces), and thus can be seen as an early precursor to our approach.\n\n**How we differ.** Our contribution differs in both method and scale. We adopt the newer flow-matching framework and show that Riemannian Flow Matching [7] with pull-back metrics reduces to our proposed DiffeoCFM. The latter performs all operations in Euclidean space, yielding greater numerical stability and improved training and sampling efficiency. Moreover, while [7] presents low-dimensional examples, our experiments reach 741-dimensional fMRI connectivity (39 × 39 matrices), a particularly challenging regime.\n\n[7] Luca Falorsi et al., *Reparameterizing Distributions on Lie Groups*, AISTATS 2019\n\nWe hope these results clarify our choices and demonstrate DiffeoCFM’s advantages. Thank you again for the constructive feedback."}]

Author rebuttal:

Meta-review: This paper studies flow matching on matrix manifolds using pullback metrics induced by global diffeomorphisms from Euclidean spaces. Motivated by generating brain functional connectivity matrices, which are symmetric positive definite, to study population heterogeneity and disease, the proposed method, DiffeoCFM, enables conditional flow matching on the matrix manifold. The approach is validated on large-scale fMRI and EEG datasets.

Reviewers found that this paper is well-motivated and well-written, the idea is interesting, the applications are important from the scientific point of view, and the practical performance is strong. Concerns include the lack of literature review, baseline methods, and technical details in experiments, many of which were resolved during the rebuttal and discussion phase. 

Finally, reviewers reached an agreement that this is a good paper, and I agree and recommend acceptance. However, the authors should revise the paper for the camera-ready version carefully based on reviewers' comments and promised changes during the discussion phase.
Decision: Accept (poster)