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.

In this paper, we investigate the universal approximation property of deep, narrow multilayer perceptrons (MLPs) for $C^1$ functions under the Sobolev norm, specifically the $W^{1, \infty}$ norm. Although the optimal width of deep, narrow MLPs for approximating continuous functions has been extensively studied, significantly less is known about the corresponding optimal width for $C^1$ functions. We demonstrate that \textit{the optimal width} can be determined in a wide range of cases within the $C^1$ setting. Our approach consists of two main steps. First, leveraging control theory, we show that any diffeomorphism can be approximated by deep, narrow MLPs. Second, using the Borsuk-Ulam theorem and various results from differential geometry, we prove that the optimal width for approximating arbitrary $C^1$ functions via diffeomorphisms is $\min(n + m, \max(2n + 1, m))$ in certain cases, including $(n,m) = (8,8)$ and $(16,8)$, where $n$ and $m$ denote the input and output dimensions, respectively. Our results apply to a broad class of activation functions.

Robust 6d object pose estimation by learning rgb-d features.

In this work we consider the problem of discretization of neural operators in a general setting. Using category theory, we give a no-go theorem that shows that diffeomorphisms between Hilbert spaces may not admit any continuous approximations by diffeomorphisms on finite spaces, even if the discretization is non-linear. This shows how infinite-dimensional Hilbert spaces and finite-dimensional vector spaces fundamentally differ. A key take-away is that to obtain discretization invariance, considerable effort is needed to ensure that finite-dimensional approximations of neural operator converge not only as sequences of functions, but that their representations converge in a suitable sense as well. With this perspective, we give several positive results. We first show that strongly monotone diffeomorphism operators always admit finite-dimensional strongly monotone diffeomorphisms. Next we show that bilipschitz neural operators may always be written via the repeated alternating composition of strongly monotone neural operators and invertible linear maps. We also show that such operators may be inverted locally via iteration provided that such inverse exists. Finally, we conclude by showing how our framework may be used `out of the box' to prove quantitative approximation results for discretization of neural operators.

We consider the problem of discretization of neural operators between Hilbert spaces in a general framework including skip connections. We focus on bijec-tive neural operators through the lens of diffeomorphisms in infinite dimensions.

Other concurrent works address, e.g., learning from soft interventions with polynomial mixing [

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty.

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