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Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing…
Clustering algorithms are one of the main analytical methods to detect patterns in unlabeled data. Existing clustering methods typically treat samples in a dataset as points in a metric space and compute distances to group together similar…
We propose a visualization method to understand the effect of multidimensional projection on local subspaces, using implicit function differentiation. Here, we understand the local subspace as the multidimensional local neighborhood of data…
Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may…
In this paper, we consider the following question: how many degree $d$ curves are there in $\mathbb{P}^3$ (passing through the right number of generic lines and points), whose image lies inside a $\mathbb{P}^2$, having $\delta$ nodes and…
In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…
We have developed a general method for finding apparent horizons in 3D numerical relativity. Instead of solving for the partial differential equation describing the location of the apparent horizons, we expand the closed 2D surfaces in…
The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider…
In this paper, we introduce discrete approximate circle bundles, a class of objects designed to serve as the data science analog of circle bundles from algebraic topology. We show that, under appropriate conditions, one can meaningfully and…
Given a smooth complex variety $X$, an algebraically skew embedding of $X$ is an embedding of $X$ into a complex projective space $\mathbb{P}^N$ such that for any two points $x,y\in X$, their embedded tangent spaces in $\mathbb{P}^N$ do not…
The `linear orbit' of a plane curve of degree d is its orbit in P^{d(d+3)/2} under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of…
We investigate geometric aspects of co-equational parametric resurgence, by studying physical problems whose formal asymptotic solutions give rise to Borel transforms lying on an algebraic curve. This perspective allows us to elucidate…
It is a key to construct a similarity graph in graph-oriented subspace learning and clustering. In a similarity graph, each vertex denotes a data point and the edge weight represents the similarity between two points. There are two popular…
A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a…
In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty, here proposed, is that curved elements are naturally handled without any degradation of the solution accuracy. In…
Different types of two- and three-dimensional representations of a finite metric space are studied that focus on the accurate representation of the linear order among the distances rather than their actual values. Lower and upper bounds for…
We propose a differentiable sphere tracing algorithm to bridge the gap between inverse graphics methods and the recently proposed deep learning based implicit signed distance function. Due to the nature of the implicit function, the…
The challenge of navigation in environments with dynamic objects continues to be a central issue in the study of autonomous agents. While predictive methods hold promise, their reliance on precise state information makes them less practical…
We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments.…
We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…