Related papers: ManifoldNorm: Extending normalizations on Riemanni…

Recent methods in geometric deep learning have introduced various neural networks to operate over data that lie on Riemannian manifolds. Such networks are often necessary to learn well over graphs with a hierarchical structure or to learn…

In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…

Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower…

Recent works have highlighted scale invariance or symmetry present in the weight space of a typical deep network and the adverse effect it has on the Euclidean gradient based stochastic gradient descent optimization. In this work, we show…

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

In the domain of image-set based classification, a considerable advance has been made by representing original image sets as covariance matrices which typical lie in a Riemannian manifold. Specifically, it is a Symmetric Positive Definite…

Deep neural networks (DNNs) have been successfully applied to a wide variety of acoustic modeling tasks in recent years. These include the applications of DNNs either in a discriminative feature extraction or in a hybrid acoustic modeling…

In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man…

Existing EEG foundation models mainly treat neural signals as generic time series in Euclidean space, ignoring the intrinsic geometric structure of neural dynamics that constrains brain activity to low-dimensional manifolds. This…

In recent years, with the large-scale expansion of graph data, there has been an increased focus on Riemannian manifold data spaces other than Euclidean space. In particular, the development of hyperbolic spaces has been remarkable, and…

Magnetic Resonance Imaging (MRI) is considered the gold standard of medical imaging because of the excellent soft-tissue contrast exhibited in the images reconstructed by the MRI pipeline, which in-turn enables the human radiologist to…

An important task in image processing and neuroimaging is to extract quantitative information from the acquired images in order to make observations about the presence of disease or markers of development in populations. Having a…

Manifold learning-based encoders have been playing important roles in nonlinear dimensionality reduction (NLDR) for data exploration. However, existing methods can often fail to preserve geometric, topological and/or distributional…

We present Manifold Diffusion Fields (MDF), an approach that unlocks learning of diffusion models of data in general non-Euclidean geometries. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on…

In recent studies, several asymptotic upper bounds on generalization errors on deep neural networks (DNNs) are theoretically derived. These bounds are functions of several norms of weights of the DNNs, such as the Frobenius and spectral…

The remarkable performance of deep neural networks (DNNs) currently makes them the method of choice for solving linear inverse problems. They have been applied to super-resolve and restore images, as well as to reconstruct MR and CT images.…

Manifold learning is a central task in modern statistics and data science. Many datasets (cells, documents, images, molecules) can be represented as point clouds embedded in a high dimensional ambient space, however the degrees of freedom…

The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of…

Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into…