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We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set $S$ of points, while…
Deep learning has demonstrated the power of detailed modeling of complex high-order (multivariate) interactions in data. For some learning tasks there is power in learning models that are not only Deep but also Broad. By Broad, we mean…
We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is…
We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of…
We show that the VC-dimension of a graph can be computed in time $n^{\log d+1} d^{O(d)}$, where $d$ is the degeneracy of the input graph. The core idea of our algorithm is a data structure to efficiently query the number of vertices that…
Given a plane curve $\gamma: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(\gamma)$ of inflections which curves $\mbox{diff}(\gamma)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms…
We introduce new rounding methods to improve the accuracy of finite precision quantum arithmetic. These quantum rounding methods are applicable when multiple samples are being taken from a quantum program. We show how to use multiple…
There are various automotive applications that rely on correctly interpreting point cloud data recorded with radar sensors. We present a deep learning approach for histogram-based processing of such point clouds. Compared to existing…
We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves $S^1 \to R^2$. They are a direct analog of similar techniques for knot invariants and are based on the calculus of {\em…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
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…
The extent to which the projected distribution of stars in a cluster is due to a large-scale radial gradient, and the extent to which it is due to fractal sub-structure, can be quantified -- statistically -- using the measure ${\cal Q} =…
In this work, we propose a generalization of the current most widely used quantum computing hardware metric known as the quantum volume. The quantum volume specifies a family of random test circuits defined such that the logical circuit…
A bounded curvature path is a continuously differentiable piece-wise $C^2$ path with bounded absolute curvature connecting two points in the tangent bundle of a surface. These paths have been widely considered in computer science and…
Sampling edges from a graph in sublinear time is a fundamental problem and a powerful subroutine for designing sublinear-time algorithms. Suppose we have access to the vertices of the graph and know a constant-factor approximation to the…
Dropout is conventionally used during the training phase as regularization method and for quantifying uncertainty in deep learning. We propose to use dropout during training as well as inference steps, and average multiple predictions to…
Let C be a non-hyperelliptic algebraic curve of genus at least 3. Enriques and Babbage proved that its canonical image is the intersection of the quadrics that contain it, except when C is trigonal (that is, it has a linear system of degree…
A large component of the building material concrete consists of aggregate with varying particle sizes between 0.125 and 32 mm. Its actual size distribution significantly affects the quality characteristics of the final concrete in both, the…
Modern depth sensors such as LiDAR operate by sweeping laser-beams across the scene, resulting in a point cloud with notable 1D curve-like structures. In this work, we introduce a new point cloud processing scheme and backbone, called…
We introduce algorithms for robustly computing intrinsic coordinates on point clouds. Our approach relies on generating many candidate coordinates by subsampling the data and varying hyperparameters of the embedding algorithm (e.g.,…