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We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data…

Accurate, fast, and reliable multiclass classification of electroencephalography (EEG) signals is a challenging task towards the development of motor imagery brain-computer interface (MI-BCI) systems. We propose enhancements to different…

Classification models used in brain-computer interface (BCI) are usually designed for a single BCI paradigm. This requires the redevelopment of the model when applying it to a new BCI paradigm, resulting in repeated costs and effort.…

Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian…

Brain-computer interface (BCI) aims to establish and improve human and computer interactions. There has been an increasing interest in designing new hardware devices to facilitate the collection of brain signals through various…

Imposing additional constraints on low-rank optimization has garnered growing interest. However, the geometry of coupled constraints hampers the well-developed low-rank structure and makes the problem intricate. To this end, we propose a…

Brain decoding is a field of computational neuroscience that uses measurable brain activity to infer mental states or internal representations of perceptual inputs. Therefore, we propose a novel approach to brain decoding that also relies…

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…

This paper presents a Bayesian regression model relating scalar outcomes to brain functional connectivity represented as symmetric positive definite (SPD) matrices. Unlike many proposals that simply vectorize the matrix-valued connectivity…

We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties…

Reformulating computer vision problems over Riemannian manifolds has demonstrated superior performance in various computer vision applications. This is because visual data often forms a special structure lying on a lower dimensional space…

A substantial amount of research has demonstrated the robustness and accuracy of the Riemannian minimum distance to mean (MDM) classifier for all kinds of EEG-based brain--computer interfaces (BCIs). This classifier is simple, fully…

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…

Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in…

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…

This paper advocates Riemannian multi-manifold modeling in the context of network-wide non-stationary time-series analysis. Time-series data, collected sequentially over time and across a network, yield features which are viewed as points…

We propose a new Riemannian gradient descent method for computing spherical area-preserving mappings of topological spheres using a Riemannian retraction-based framework with theoretically guaranteed convergence. The objective function is…

Brain-Computer Interface (BCI) is a rapidly developing technology that allows direct communications between the human brain and external devices, such as robotic arms and computers. Bayesian Networks is a powerful tool in machine learning…

Brain-Computer Interfaces (BCI) help patients with faltering communication abilities due to neurodegenerative diseases produce text or speech output by direct neural processing. However, practical implementation of such a system has proven…

Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks.…