Related papers: Exponential divided differences via Chebyshev poly…
Learning from data in the presence of outliers is a fundamental problem in statistics. Until recently, no computationally efficient algorithms were known to compute the mean of a high dimensional distribution under natural assumptions in…
It is shown in "SIAM J. Sci. Comput. 39 (2017):B424-B441" that free-form curves used in computer aided geometric design can usually be represented as the solutions of linear differential systems and points and derivatives on the curves can…
This paper proposes a Chebyshev polynomial expansion framework for the recovery of a continuous angular power spectrum (APS) from channel covariance. By exploiting the orthogonality of Chebyshev polynomials in a transformed domain, we…
A large class of machine learning techniques requires the solution of optimization problems involving spectral functions of parametric matrices, e.g. log-determinant and nuclear norm. Unfortunately, computing the gradient of a spectral…
Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
Gradient-based algorithms, popular strategies to optimization problems, are essential for many modern machine-learning techniques. Theoretically, extreme points of certain cost functions can be found iteratively along the directions of the…
We introduce $\textsf{gradOL}$, the first gradient-based optimization framework for solving Chebyshev center problems, a fundamental challenge in optimal function learning and geometric optimization. $\textsf{gradOL}$ hinges on…
It is known that the statistical properties of the spectrum provide an essential characterization of quantum chaos. The computation of a large group of interior eigenvalues at the middle spectrum is thus an important problem for quantum…
Recent theoretical results in quantum machine learning have demonstrated a general trade-off between the expressive power of quantum neural networks (QNNs) and their trainability; as a corollary of these results, practical exponential…
Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their…
For many hard computational problems, simple algorithms that run in time $2^n \cdot n^{O(1)}$ arise, say, from enumerating all subsets of a size-$n$ set. Finding (exponentially) faster algorithms is a natural goal that has driven much of…
We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which…
This paper introduces the exponential substitution calculus (ESC), a new presentation of cut elimination for IMELL, based on proof terms and building on the idea that exponentials can be seen as explicit substitutions. The idea in itself is…
Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to $\mathcal{O}(d^{k})$ scaling of the derivative tensor size and the…
This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…
Interior eigenvalue problems for large-scale sparse Hermitian matrices are fundamental in computational science. We propose an adaptive polynomial filtering strategy based on Chebyshev expansion of a step function, integrated into a…
Change-point problems have appeared in a great many applications for example cancer genetics, econometrics and climate change. Modern multiscale type segmentation methods are considered to be a statistically efficient approach for multiple…
Approximating adequate number of clusters in multidimensional data is an open area of research, given a level of compromise made on the quality of acceptable results. The manuscript addresses the issue by formulating a transductive…