Related papers: Study of Manifold Geometry using Multiscale Non-Ne…

Centered kernel alignment (CKA) is a popular metric for comparing representations, determining equivalence of networks, and neuroscience research. However, CKA does not account for the underlying manifold and relies on numerous heuristics…

Probabilistic models of data sets often exhibit salient geometric structure. Such a phenomenon is summed up in the manifold distribution hypothesis, and can be exploited in probabilistic learning. Here we present normal-bundle bootstrap…

High-dimensional datasets are well-approximated by low-dimensional structures. Over the past decade, this empirical observation motivated the investigation of detection, measurement, and modeling techniques to exploit these low-dimensional…

Machine learning problems have an intrinsic geometric structure as central objects including a neural network's weight space and the loss function associated with a particular task can be viewed as encoding the intrinsic geometry of a given…

Learning with kernels is an important concept in machine learning. Standard approaches for kernel methods often use predefined kernels that require careful selection of hyperparameters. To mitigate this burden, we propose in this paper a…

In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper we…

Point cloud data are widely used in manufacturing applications for process inspection, modeling, monitoring and optimization. The state-of-art tensor regression techniques have effectively been used for analysis of structured point cloud…

We introduce a nonparametric way to estimate the global probability density function for a random persistence diagram. Precisely, a kernel density function centered at a given persistence diagram and a given bandwidth is constructed. Our…

In recent years, manifold methods have moved into focus as tools for dimension reduction. Assuming that the high-dimensional data actually lie on or close to a low-dimensional nonlinear manifold, these methods have shown convincing results…

Polygon representation learning is essential for diverse applications, encompassing tasks such as shape coding, building pattern classification, and geographic question answering. While recent years have seen considerable advancements in…

Many successful learning algorithms have been recently developed to represent graph-structured data. For example, Graph Neural Networks (GNNs) have achieved considerable successes in various tasks such as node classification, graph…

Kernel matrices are crucial in many learning tasks such as support vector machines or kernel ridge regression. The kernel matrix is typically dense and large-scale. Depending on the dimension of the feature space even the computation of all…

Kernel-based non-linear dimensionality reduction methods, such as Local Linear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwise distances or similarity scores, with which one can construct and study a weighted graph…

Ensuring proper generalization is a critical challenge in applying data-driven methods for solving inverse problems in imaging, as neural networks reconstructing an image must perform well across varied datasets and acquisition geometries.…

Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…

Understanding the reasons for the success of deep neural networks trained using stochastic gradient-based methods is a key open problem for the nascent theory of deep learning. The types of data where these networks are most successful,…

Manifold learning (ML), known also as non-linear dimension reduction, is a set of methods to find the low dimensional structure of data. Dimension reduction for large, high dimensional data is not merely a way to reduce the data; the new…

Experimental sciences have come to depend heavily on our ability to organize and interpret high-dimensional datasets. Natural laws, conservation principles, and inter-dependencies among observed variables yield geometric structure, with…

We propose a new neural network framework, termed Neural Network Machine Regression (NNMR), which integrates trainable input gating and adaptive depth regularization to jointly perform feature selection and function estimation in an…

This article proposes an active learning method for high dimensional data, based on intrinsic data geometries learned through diffusion processes on graphs. Diffusion distances are used to parametrize low-dimensional structures on the…