Related papers: Convergence of a Normalized Gradient Algorithm for…
Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite…
We propose a new gradient descent algorithm with added stochastic terms for finding the global optimizers of nonconvex optimization problems. A key component in the algorithm is the adaptive tuning of the randomness based on the value of…
In this work, we develop an $\mathcal{O}(N)$ implicit real space method in 1D and 2D for the Cahn Hilliard (CH) and vector Cahn Hilliard (VCH) equations, based on the Method Of Lines Transpose (MOL$^\text{T}$) formulation. This formulation…
We consider the ground states of the nonlinear Schr{\"o}dinger equation, which stand for radially symmetric and exponentially decaying solutions on the full space. We investigate their behaviors at both endpoint powers of the nonlinearity,…
Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a…
Many important machine learning applications involve regularized nonconvex bi-level optimization. However, the existing gradient-based bi-level optimization algorithms cannot handle nonconvex or nonsmooth regularizers, and they suffer from…
We consider the problem of learning observation models for robot state estimation with incremental non-differentiable optimizers in the loop. Convergence to the correct belief over the robot state is heavily dependent on a proper tuning of…
Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. We give the first randomized polynomial-time algorithm for finding ground states of gapped one-dimensional Hamiltonians: it outputs an…
We present a semiclassical phase-space method to calculate thermal and ground states of large interacting spin systems. To this end, we extend the recently developed truncated Wigner approximation for spins (TWA) to the imaginary time,…
In this paper, we propose an implicit gradient descent algorithm for the classic $k$-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch…
Policy gradient algorithms have been widely applied to Markov decision processes and reinforcement learning problems in recent years. Regularization with various entropy functions is often used to encourage exploration and improve…
We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to…
It is not possible, using standard lattice techniques in Euclidean space, to calculate the complete fermionic spectrum of a quantum field theory. Algorithms running on quantum computers have the potential to access the theory with real-time…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
We consider stochastic gradient methods under the interpolation regime where a perfect fit can be obtained (minimum loss at each observation). While previous work highlighted the implicit regularization of such algorithms, we consider an…
It is well-known that accelerated gradient first order methods possess optimal complexity estimates for the class of convex smooth minimization problems. In many practical situations, it makes sense to work with inexact gradients. However,…
The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant…
We consider the proximal-gradient method for minimizing an objective function that is the sum of a smooth function and a non-smooth convex function. A feature that distinguishes our work from most in the literature is that we assume that…
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schr\"odinger equation provides full knowledge of all dynamical quantities of the physical system. However…
This article derives lower bounds on the convergence rate of continuous-time gradient-based optimization algorithms. The algorithms are subjected to a time-normalization constraint that avoids a reparametrization of time in order to make…