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When solving the time-dependent radiative transport equation (RTE), implicit time discretization is often employed for its robustness and stability. This results in a sequence of steady-state RTEs with identical cross-sections but varying…
Recent years have seen a growing interest in understanding acceleration methods through the lens of ordinary differential equations (ODEs). Despite the theoretical advancements, translating the rapid convergence observed in continuous-time…
Classical neural ODEs trained with explicit methods are intrinsically limited by stability, crippling their efficiency and robustness for stiff learning problems that are common in graph learning and scientific machine learning. We present…
While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation…
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…
Recent focus on robustness to adversarial attacks for deep neural networks produced a large variety of algorithms for training robust models. Most of the effective algorithms involve solving the min-max optimization problem for training…
Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a…
Ordinary differential equations (ODEs) are a mathematical model used in many application areas such as climatology, bioinformatics, and chemical engineering with its intuitive appeal to modeling. Despite ODE's wide usage in modeling, the…
Optimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of…
This paper considers a natural fault-tolerant shortest paths problem: for some constant integer $f$, given a directed weighted graph with no negative cycles and two fixed vertices $s$ and $t$, compute (either explicitly or implicitly) for…
This paper proposes a redundancy resolution algorithm for a redundant manipulator based on dynamic programming. This algorithm can compute the desired joint angles at each point on a pre-planned discrete path in Cartesian space, while…
A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface and estimating distances on the polygon. We show that exact geodesic distances restricted to the polygon…
By restricting the iterate on a nonlinear manifold, the recently proposed Riemannian optimization methods prove to be both efficient and effective in low rank tensor completion problems. However, existing methods fail to exploit the easily…
This paper formulates the problem of Extremum Seeking for optimization of cost functions defined on Riemannian manifolds. We extend the conventional extremum seeking algorithms for optimization problems in Euclidean spaces to optimization…
We propose a stochastic recursive momentum method for Riemannian non-convex optimization that achieves a near-optimal complexity of $\tilde{\mathcal{O}}(\epsilon^{-3})$ to find $\epsilon$-approximate solution with one sample. That is, our…
This paper studies robust regression for data on Riemannian manifolds. Geodesic regression is the generalization of linear regression to a setting with a manifold-valued dependent variable and one or more real-valued independent variables.…
The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data…
Many applications require minimizing a family of optimization problems indexed by some hyperparameter $\lambda \in \Lambda$ to obtain an entire solution path. Traditional approaches proceed by discretizing $\Lambda$ and solving a series of…
We introduce two block coordinate descent algorithms for solving optimization problems with ordinary differential equations (ODEs) as dynamical constraints. The algorithms do not need to implement direct or adjoint sensitivity analysis…
Isomap algorithm is a representative manifold learning algorithm. The algorithm simplifies the data analysis process and is widely used in neuroimaging, spectral analysis and other fields. However, the classic Isomap algorithm becomes…