Caroline Moosmüller

Caroline Moosmüller, Nira Dyn

In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\ell}$ limits to another such scheme of the same dimension, generating $C^{\ell+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\ell}$ limit curves can be transformed to a new scheme producing $C^{\ell+1}$ limit curves by multiplying the scheme's symbol with the smoothing factor $\tfrac{z+1}{2}$. We extend this approach to vector and Hermite subdivision schemes, by manipulating symbols. The algorithms presented in this paper allow to construct vector (Hermite) subdivision schemes of arbitrarily high regularity from a convergent vector scheme (from a Hermite scheme whose Taylor scheme is convergent with limit functions of vanishing first component).

Mariantonia Cotronei, Caroline Moosmüller, Tomas Sauer et al.

We study many properties of level-dependent Hermite subdivision, focusing on schemes preserving polynomial and exponential data. We specifically consider interpolatory schemes, which give rise to level-dependent multiresolution analyses through a prediction-correction approach. A result on the decay of the associated multiwavelet coefficients, corresponding to a uniformly continuous and differentiable function, is derived. It makes use of the approximation of any such function with a generalized Taylor formula expressed in terms of polynomials and exponentials.

Alexander Cloninger, Keaton Hamm, Varun Khurana et al.

We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

Keaton Hamm, Caroline Moosmüller, Bernhard Schmitzer et al.

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ with $Ω$ a compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2 distance $\mathbb{W}$. We begin by introducing a construction of submanifolds $Λ$ in $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ equipped with metric $\mathbb{W}_Λ$, the geodesic restriction of $\mathbb{W}$ to $Λ$. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of $\mathbb{R}^d$. We then show how the latent manifold structure of $(Λ,\mathbb{W}_Λ)$ can be learned from samples $\{λ_i\}_{i=1}^N$ of $Λ$ and pairwise extrinsic Wasserstein distances $\mathbb{W}$ on $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ only. In particular, we show that the metric space $(Λ,\mathbb{W}_Λ)$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\{λ_i\}_{i=1}^N$ and edge weights $W(λ_i,λ_j)$. In addition, we demonstrate how the tangent space at a sample $λ$ can be asymptotically recovered via spectral analysis of a suitable ``covariance operator'' using optimal transport maps from $λ$ to sufficiently close and diverse samples $\{λ_i\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $Λ$ and numerical examples on the recovery of tangent spaces through spectral analysis.

Caroline Moosmüller

In this paper we study the connection between the spectral condition of an Hermite subdivision operator and polynomial reproduction properties of the associated subdivision scheme. While it is known that in general the spectral condition does not imply the reproduction of polynomials, we here prove that a special spectral condition (defined by shifted monomials) is actually equivalent to the reproduction of polynomials. We further put into evidence that the sum rule of order $\ell>d$ associated with an Hermite subdivision operator of order $d$ does not imply that the spectral condition of order $\ell$ is satisfied, while it is known that these two concepts are equivalent in the case $\ell=d$.

Amartya Banerjee, Xingyu Xu, Caroline Moosmüller et al.

In an inverse problem, the goal is to recover an unknown parameter (e.g., an image) that has typically undergone some lossy or noisy transformation during measurement. Recently, deep generative models, particularly diffusion models, have emerged as powerful priors for protein structure generation. However, integrating noisy experimental data from multiple sources to guide these models remains a significant challenge. Existing methods often require precise knowledge of experimental noise levels and manually tuned weights for each data modality. In this work, we introduce Adam-PnP, a Plug-and-Play framework that guides a pre-trained protein diffusion model using gradients from multiple, heterogeneous experimental sources. Our framework features an adaptive noise estimation scheme and a dynamic modality weighting mechanism integrated into the diffusion process, which reduce the need for manual hyperparameter tuning. Experiments on complex reconstruction tasks demonstrate significantly improved accuracy using Adam-PnP.

Scott Mahan, Caroline Moosmüller, Alexander Cloninger

We introduce Deep Set Linearized Optimal Transport, an algorithm designed for the efficient simultaneous embedding of point clouds into an $L^2-$space. This embedding preserves specific low-dimensional structures within the Wasserstein space while constructing a classifier to distinguish between various classes of point clouds. Our approach is motivated by the observation that $L^2-$distances between optimal transport maps for distinct point clouds, originating from a shared fixed reference distribution, provide an approximation of the Wasserstein-2 distance between these point clouds, under certain assumptions. To learn approximations of these transport maps, we employ input convex neural networks (ICNNs) and establish that, under specific conditions, Euclidean distances between samples from these ICNNs closely mirror Wasserstein-2 distances between the true distributions. Additionally, we train a discriminator network that attaches weights these samples and creates a permutation invariant classifier to differentiate between different classes of point clouds. We showcase the advantages of our algorithm over the standard deep set approach through experiments on a flow cytometry dataset with a limited number of labeled point clouds.

Varun Khurana, Harish Kannan, Alexander Cloninger et al.

In this paper we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into $L^2$ spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a pre-specified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.

Caroline Moosmüller, Alexander Cloninger

Discriminating between distributions is an important problem in a number of scientific fields. This motivated the introduction of Linear Optimal Transportation (LOT), which embeds the space of distributions into an $L^2$-space. The transform is defined by computing the optimal transport of each distribution to a fixed reference distribution, and has a number of benefits when it comes to speed of computation and to determining classification boundaries. In this paper, we characterize a number of settings in which LOT embeds families of distributions into a space in which they are linearly separable. This is true in arbitrary dimension, and for families of distributions generated through perturbations of shifts and scalings of a fixed distribution.We also prove conditions under which the $L^2$ distance of the LOT embedding between two distributions in arbitrary dimension is nearly isometric to Wasserstein-2 distance between those distributions. This is of significant computational benefit, as one must only compute $N$ optimal transport maps to define the $N^2$ pairwise distances between $N$ distributions. We demonstrate the benefits of LOT on a number of distribution classification problems.