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Probability metrics have become an indispensable part of modern statistics and machine learning, and they play a quintessential role in various applications, including statistical hypothesis testing and generative modeling. However, in a…
A theorem of McCann shows that for any two absolutely continuous probability measures on R^d there exists a monotone transformation sending one probability measure to the other. A consequence of this theorem, relevant to statistics, is that…
Image set-based visual classification methods have achieved remarkable performance, via characterising the image set in terms of a non-singular covariance matrix on a symmetric positive definite (SPD) manifold. To adapt to complicated…
Medical image segmentation is a fundamental task for medical image analysis and surgical planning. In recent years, UNet-based networks have prevailed in the field of medical image segmentation. However, convolution-neural networks (CNNs)…
Statistical shape modeling (SSM) enables population-based quantitative analysis of anatomical shapes, informing clinical diagnosis. Deep learning approaches predict correspondence-based SSM directly from unsegmented 3D images but require…
Statistical shape modeling (SSM) is an essential tool for analyzing variations in anatomical morphology. In a typical SSM pipeline, 3D anatomical images, gone through segmentation and rigid registration, are represented using…
Recent works have demonstrated promising performances of neural networks on hyperbolic spaces and symmetric positive definite (SPD) manifolds. These spaces belong to a family of Riemannian manifolds referred to as symmetric spaces of…
Scanning Probe Microscopy (SPM) is the primary tool for exploring nanoscale functionality, yet standard single-frequency operation is fundamentally limited, because the dynamic tip-sample interaction is mathematically underdetermined. While…
Matrix-valued optimization tasks, including those involving symmetric positive definite (SPD) matrices, arise in a wide range of applications in machine learning, data science and statistics. Classically, such problems are solved via…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
Merging the two cultures of deep and statistical learning provides insights into structured high-dimensional data. Traditional statistical modeling is still a dominant strategy for structured tabular data. Deep learning can be viewed…
Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing interest in various applied areas. For instance, DNNs on spherical and hyperbolic manifolds have been designed to solve a wide range of computer vision and nature…
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…
Statistical shape modeling is an important tool to characterize variation in anatomical morphology. Typical shapes of interest are measured using 3D imaging and a subsequent pipeline of registration, segmentation, and some extraction of…
Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the…
Stochastic gradient descent based algorithms are typically used as the general optimization tools for most deep learning models. A Restricted Boltzmann Machine (RBM) is a probabilistic generative model that can be stacked to construct deep…
Fine-grained high-resolution remote sensing mapping typically relies on localized visual features, which restricts cross-domain generalizability and often leads to fragmented predictions of large-scale land covers. While global geospatial…
Deep state-space models (SSMs) have gained increasing popularity in sequence modelling. While there are numerous theoretical investigations of shallow SSMs, how the depth of the SSM affects its expressiveness remains a crucial problem. In…
Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data. For…
Stochastic mirror descent (SMD) is a fairly new family of algorithms that has recently found a wide range of applications in optimization, machine learning, and control. It can be considered a generalization of the classical stochastic…