Related papers: Riemannian kernel based Nystr\"om method for appro…
We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time…
The ability to precisely quantify similarity between various entities has been a fundamental complication in various problem spaces specifically in the classification of cellular images. Contemporary similarity measures applied in the…
Nystr\"om approximation is a fast randomized method that rapidly solves kernel ridge regression (KRR) problems through sub-sampling the n-by-n empirical kernel matrix appearing in the objective function. However, the performance of such a…
Deep learning with Convolutional Neural Networks has shown great promise in various areas of image-based classification and enhancement but is often unsuitable for predictive modeling involving non-image based features or features without…
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…
Although Convolutional Neural Networks (CNNs) have achieved promising results in image classification, they still are vulnerable to affine transformations including rotation, translation, flip and shuffle. The drawback motivates us to…
We introduce Neural Representation of Distribution (NeRD) technique, a module for convolutional neural networks (CNNs) that can estimate the feature distribution by optimizing an underlying function mapping image coordinates to the feature…
These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…
Based on the theory of reproducing kernel Hilbert space (RKHS) and semiparametric method, we propose a new approach to nonlinear dimension reduction. The method extends the semiparametric method into a more generalized domain where both the…
Convolutional Neural Networks (CNN) have emerged as powerful tools for learning discriminative image features. In this paper, we propose a framework of 3-D fully CNN models for Glioblastoma segmentation from multi-modality MRI data. By…
We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which…
We present a method for metric optimization in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, by treating the induced Riemannian metric on the space of diffeomorphisms as a kernel in a machine learning context. For…
Motivated by the surge of interest in Koopman operator theory, we propose a machine-learning alternative based on a functional Bayesian perspective for operator-theoretic modeling of unknown, data-driven, nonlinear dynamical systems. This…
Deep learning models for complex-valued Synthetic Aperture Radar (CV-SAR) image recognition are fundamentally constrained by a representation trilemma under data-limited and domain-shift scenarios: the concurrent, yet conflicting,…
Euclidean representations distort data with intrinsic non-Euclidean structure. While Riemannian representation learning offers a solution by embedding data onto matching manifolds, it typically relies on an encoder to estimate densities on…
Quantum computation offers potential exponential speedups for simulating certain physical systems, but its application to nonlinear dynamics is inherently constrained by the requirement of unitary evolution. We propose the quantum Koopman…
When applying a convolutional kernel to an image, if the output is to remain the same size as the input then some form of padding is required around the image boundary, meaning that for each layer of convolution in a convolutional neural…
With the impressive capability to capture visual content, deep convolutional neural networks (CNN) have demon- strated promising performance in various vision-based ap- plications, such as classification, recognition, and objec- t…
Nonsmooth Riemannian optimization has attracted increasing attention, especially in problems with sparse structures. While existing formulations typically involve convex nonsmooth terms, incorporating nonsmooth difference-of-convex (DC)…
We propose a method for the approximation of high- or even infinite-dimensional feature vectors, which play an important role in supervised learning. The goal is to reduce the size of the training data, resulting in lower storage…