Related papers: Degree-optimal moving frames for rational curves
Gradient-related first-order methods have become the workhorse of large-scale numerical optimization problems. Many of these problems involve nonconvex objective functions with multiple saddle points, which necessitates an understanding of…
A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(\xi)$ defines an orthonormal basis for $\mathbb{R}^3$ in which ${\bf f}_1={\bf r}'/|{\bf r}'|$ is the curve tangent, and the normal-plane vectors ${\bf…
The simulation of the physical movement of multi-body systems at an atomistic level, with forces calculated from a quantum mechanical description of the electrons, motivates a graph partitioning problem studied in this article. Several…
The accuracy and complexity of machine learning algorithms based on kernel optimization are determined by the set of kernels over which they are able to optimize. An ideal set of kernels should: admit a linear parameterization (for…
In this paper, we consider optimizing a smooth, convex, lower semicontinuous function in Riemannian space with constraints. To solve the problem, we first convert it to a dual problem and then propose a general primal-dual algorithm to…
In this paper we obtain a formula for the number of rational degree d curves in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem can…
We define arithmetical and dynamical degrees for dynamical systems with several rational maps on projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial…
Robot design optimization, imitation learning and system identification share a common problem which requires optimization over robot or task parameters at the same time as optimizing the robot motion. To solve these problems, we can use…
Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in…
A new closed-form solver is proposed minimizing the algebraic error optimally, in the least-squares sense, to estimate the relative planar motion of two calibrated cameras. The main objective is to solve the over-determined case, i.e., when…
We solve the problem of characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column)…
Curves of maximal slope are a reference gradient-evolution notion in metric spaces and arise as variational formulation of a vast class of nonlinear diffusion equations. Existence theories for curves of maximal slope are often based on…
We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and…
Motion planning in the presence of multiple dynamic obstacles is an important research problem from the perspective of autonomous vehicles as well as space-constrained multi-robot work environment. In this paper, we address the motion…
In addition to the theoretical value of challenging optimal control problmes, recent progress in autonomous vehicles mandates further research in optimal motion planning for wheeled vehicles. Since current numerical optimal control…
The 4IM+1CM problem is determining all pairs (f,g) of meromorphic functions in the complex plane that are not Moebius transformations of each other and share five pairs of complex values, one of them counting multiplicities. It is shown…
Most existing motion planning algorithms assume that a map (of some quality) is fully determined prior to generating a motion plan. In many emerging applications of robotics, e.g., fast-moving agile aerial robots with constrained embedded…
We describe an approximate dynamic programming approach to compute lower bounds on the optimal value function for a discrete time, continuous space, infinite horizon setting. The approach iteratively constructs a family of lower bounding…
The continuous dynamical system approach to deep learning is explored in order to devise alternative frameworks for training algorithms. Training is recast as a control problem and this allows us to formulate necessary optimality conditions…
In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…