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The problem of finding a path between two points while avoiding obstacles is critical in robotic path planning. We focus on the feasibility problem: determining whether such a path exists. We model the robot as a query-specific rectangular…
Reeb spaces of (continuous) real-valued functions on (nice) topological spaces are the spaces whose underlying sets consist of all connected components (contours) of their level sets and seen naturally as quotient spaces of the spaces. They…
Fast computation of three-dimensional gravity and magnetic forward models is considered. Measurement data is assumed to be obtained on a uniform grid which is staggered with respect to the discretization of the parameter volume. Then, the…
In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating…
Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical…
In this article, we study the question of the statistical convergence of the 1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that the Mapper is an optimal estimator of the Reeb graph, which gives, as a byproduct, a…
Sparse neural networks are shown to give accurate predictions competitive to denser versions, while also minimizing the number of arithmetic operations performed. However current hardware like GPU's can only exploit structured sparsity…
Let $X$ be a simplicial complex with a piecewise linear function $f:X\to\mathbb{R}$. The Reeb graph $Reeb(f,X)$ is the quotient of $X$, where we collapse each connected component of $f^{-1}(t)$ to a single point. Let the nodes of…
Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation…
We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in $X$ and $Y$ are low with respect to their genera. The finite base fields $\FF_q$ are arbitrary, but their sizes…
Most signal processing problems involve the challenging task of multidimensional probability density function (PDF) estimation. In this work, we propose a solution to this problem by using a family of Rotation-based Iterative…
Bilevel optimization is an important formulation for many machine learning problems. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz. However, recent studies reveal that certain…
One of the prevailing ideas in geometric and topological data analysis is to provide descriptors that encode useful information about hidden objects from observed data. The Reeb graph is one such descriptor for a given scalar function. The…
We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, for checking node fathoming,…
The integer factorization problem (IFP) underpins the security of RSA, yet becomes efficiently solvable on a quantum computer through Shor's algorithm. Regev's recent high-dimensional variant reduces the circuit size through lattice-based…
Many of the classic graph problems cannot be solved in the Massively Parallel Computation setting (MPC) with strongly sublinear space per machine and $o(\log n)$ rounds, unless the 1-vs-2 cycles conjecture is false. This is true even on…
We consider jointly estimating the coefficient matrix and the error precision matrix in high-dimensional multivariate linear regression models. Bayesian methods in this context often face computational challenges, leading to previous…
Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric…
Large datasets represented by multidimensional data point clouds often possess non-trivial distributions with branching trajectories and excluded regions, with the recent single-cell transcriptomic studies of developing embryo being notable…
The paper is devoted to the development of a methodology for evaluating the scalability of compute-intensive iterative algorithms used in simulating complex physical processes on supercomputer systems. The proposed methodology is based on…