Related papers: Adaptive Metrics for Adaptive Samples
We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to…
In this article, we propose a class of test statistics for a change point in the mean of high-dimensional independent data. Our test integrates the U-statistic based approach in a recent work by \cite{hdcp} and the $L_q$-norm based…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
This paper derives identification, estimation, and inference results using spatial differencing in sample selection models with unobserved heterogeneity. We show that under the assumption of smooth changes across space of the unobserved…
Discrete forms of the mean and directed curvature are constructed on piecewise flat manifolds, providing local curvature approximations for smooth manifolds embedded in both Euclidean and non-Euclidean spaces. The resulting expressions take…
The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The…
We propose an approach for capturing the signal variability in hyperspectral imagery using the framework of the Grassmann manifold. Labeled points from each class are sampled and used to form abstract points on the Grassmannian. The…
The normalizing constant plays an important role in Bayesian computation, and there is a large literature on methods for computing or approximating normalizing constants that cannot be evaluated in closed form. When the normalizing constant…
We show that the Gauduchon metric $g_0$ of a compact locally conformally product manifold $(M,c,D)$ of dimension greater than $2$ is adapted, in the sense that the Lee form of $D$ with respect to $g_0$ vanishes on the $D$-flat distribution…
Domain adaptation techniques address the problem of reducing the sensitivity of machine learning methods to the so-called domain shift, namely the difference between source (training) and target (test) data distributions. In particular,…
How can we detect anomalies: that is, samples that significantly differ from a given set of high-dimensional data, such as images or sensor data? This is a practical problem with numerous applications and is also relevant to the goal of…
Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance…
Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing…
We propose novel randomized optimization methods for high-dimensional convex problems based on restrictions of variables to random subspaces. We consider oblivious and data-adaptive subspaces and study their approximation properties via…
We propose a new method to obtain landmark-matching transformations between n-dimensional Euclidean spaces with large deformations. Given a set of feature correspondences, our algorithm searches for an optimal folding-free mapping that…
We develop a skew-adaptive extension of split conformal prediction for regression. The method starts from an asymmetric interval family centered at a point prediction and uses the gauge approach to deduce the conformity score induced by…
We consider adaptive estimation and statistical inference for high-dimensional graph-based linear models. In our model, the coordinates of regression coefficients correspond to an underlying undirected graph. Furthermore, the given graph…
Recovering homological features of spaces from samples has become one of the central themes of topological data analysis, leading to many successful applications. Many of the results in this area focus on global homological features of a…
Distance metric learning (DML) approaches learn a transformation to a representation space where distance is in correspondence with a predefined notion of similarity. While such models offer a number of compelling benefits, it has been…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…