Related papers: A homogeneous Rayleigh quotient with applications …
An appealing property of the natural gradient is that it is invariant to arbitrary differentiable reparameterizations of the model. However, this invariance property requires infinitesimal steps and is lost in practical implementations with…
We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the…
Bilevel optimization has been developed for many machine learning tasks with large-scale and high-dimensional data. This paper considers a constrained bilevel optimization problem, where the lower-level optimization problem is convex with…
This paper provides a comprehensive and detailed analysis of the local convergence behavior of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for computing the extreme eigenvalue of a…
The paper introduces a new efficient nonlinear one-class classifier formulated as the Rayleigh quotient criterion optimisation. The method, operating in a reproducing kernel Hilbert space, minimises the scatter of target distribution along…
For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has established new local general convergence results, independent of iterative solvers for inner linear systems. The theory shows that the method locally converges…
Understanding treatment effect heterogeneity is vital to many scientific fields because the same treatment may affect different individuals differently. Quantile regression provides a natural framework for modeling such heterogeneity. We…
Although it is relatively easy to apply, the gradient method often displays a disappointingly slow rate of convergence. Its convergence is specially based on the structure of the matrix of the algebraic linear system, and on the choice of…
The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for applications such as graph classification, clustering, or eigenmode analysis. Recently, the Hodge…
We compare the well known Rayleigh-Ritz variational method (RRVM) with a recently proposed approach based on supersymmetric quantum mechanics and the Gram-Schmidt orthogonalization method (SSQMGS). We apply both procedures to a particular…
The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients…
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the…
A universal energy eigenvalue equation is proposed in this paper. It is proven that the unique set of eigenfunctions or preferred basis exists for any non-isolated sub-system. Applying the new eigenvalue equation to the relative motion of a…
Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of…
We analyze the behavior of a quantum system described by a one-dimensional asymmetric potential consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integral representation method, which allows us to…
For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit…
Conventional incoherent imaging based on measuring the spatial intensity distribution in the image plane faces the resolution hurdle described by the Rayleigh diffraction criterion. Here, we demonstrate theoretically using the concept of…
We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the…
A remarkable extension of Rayleigh-Schroedinger perturbation method is found. Its (N+q) x (N+1) - dimensional Hamiltonians (as emerging, e.g., during quasi-exact constructions of bound states) are non-square matrices at q > 1. The role of…
This article is about the quantitative homogenization theory of linear elliptic equations in divergence form with random coefficients. We derive gradient estimates on the homogenization error, i.e. on the difference between the actual…