Related papers: Geometry of the Minimum Volume Confidence Sets
Regularized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Usually the low-dimensional structure is encoded by the presence of the (unknown) parameters…
In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite, in the case $p=1$.…
Let $V$ be any vector space of multivariate degree-$d$ homogeneous polynomials with co-dimension at most $k$, and $S$ be the set of points where all polynomials in $V$ {\em nearly} vanish. We establish a qualitatively optimal upper bound on…
Medical segmentation models are evaluated empirically. As such an evaluation is based on a limited set of example images, it is unavoidably noisy. Beyond a mean performance measure, reporting confidence intervals is thus crucial. However,…
This study intends to introduce kernel mean embedding of probability measures over infinite-dimensional separable Hilbert spaces induced by functional response statistical models. The embedded function represents the concentration of…
Statistical depth is the act of gauging how representative a point is compared to a reference probability measure. The depth allows introducing rankings and orderings to data living in multivariate, or function spaces. Though widely applied…
When query evaluation produces too many tuples, a new approach in query answering is to retrieve a diverse subset of them. The standard approach for measuring the diversity of a set of tuples is to use a distance function between tuples,…
Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in a purely topological persistence diagram (also termed as barcode). In our earlier work, we showed that computing…
Multi-frame depth estimation improves over single-frame approaches by also leveraging geometric relationships between images via feature matching, in addition to learning appearance-based features. In this paper we revisit feature matching…
We prove that at differentiability points $r_0>0$ of the volume function of a compact set $A\subset {\mathbb R}^d$ (associating to $r$ the volume of the $r$-parallel set of $A$), the surface area measures of $r$-parallel sets of $A$…
The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the…
Subsampling and block-based bootstrap methods have been used in a wide range of inference problems for time series. To accommodate the dependence, these resampling methods involve a bandwidth parameter, such as subsampling window width and…
We study the Convex Set Disjointness (CSD) problem, where two players have input sets taken from an arbitrary fixed domain~$U\subseteq \mathbb{R}^d$ of size $\lvert U\rvert = n$. Their mutual goal is to decide using minimum communication…
In this paper, we propose and study construction of confidence bands for shape-constrained regression functions when the predictor is multivariate. In particular, we consider the continuous multidimensional white noise model given by $d…
This paper describes one objective function for learning semantically coherent feature embeddings in multi-output classification problems, i.e., when the response variables have dimension higher than one. In particular, we consider the…
During the past two decades there has been a lot of interest in developing statistical depth notions that generalize the univariate concept of ranking to multivariate data. The notion of depth has also been extended to regression models and…
Conformal Prediction is a widely studied technique to construct prediction sets of future observations. Most conformal prediction methods focus on achieving the necessary coverage guarantees, but do not provide formal guarantees on the size…
It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations.…
The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a…
The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we introduce…