Related papers: Data-driven optimal approximation on Hardy spaces …
In this paper we consider functions in the Hardy space $\mathbf{H}_2^{p\times q}$ defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably…
When the underlying conditional density is known, conditional expectations can be computed analytically or numerically. When, however, such knowledge is not available and instead we are given a collection of training data, the goal of this…
Large optimal transport problems can be approached via domain decomposition, i.e. by iteratively solving small partial problems independently and in parallel. Convergence to the global minimizers under suitable assumptions has been shown in…
For various Hilbert spaces of analytic functions on the unit disk, we characterize when a function $f$ has optimal polynomial approximants given by truncations of a single power series. We also introduce a generalized notion of optimal…
We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in…
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…
The problems of computational data processing involving regression, interpolation, reconstruction and imputation for multidimensional big datasets are becoming more important these days, because of the availability of data and their widely…
Effective optimization is essential for interactive systems to provide a satisfactory user experience. However, it is often challenging to find an objective to optimize for. Generally, such objectives are manually crafted and rarely capture…
We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in…
We develop a new computing paradigm, which we refer to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and conservation laws, such as compatibility…
Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…
We develop a new approximation theory for linear and quadratic interpolation models, suitable for use in convex-constrained derivative-free optimization (DFO). Most existing model-based DFO methods for constrained problems assume the…
This paper is devoted to the investigation of inertial dynamical systems with implicit Hessian-driven damping for strongly quasiconvex optimization which is a specific class of nonconvex optimization problems. We first establish exponential…
We consider least squares approximation of a function of one variable by a continuous, piecewise-linear approximand that has a small number of breakpoints. This problem was notably considered by Bellman who proposed an approximate algorithm…
This paper provides an $H_2$ optimal scheme for reducing diffusively coupled second-order systems evolving over undirected networks. The aim is to find a reduced-order model that not only approximates the input-output mapping of the…
We show an optimal data-dependent hashing scheme for the approximate near neighbor problem. For an $n$-point data set in a $d$-dimensional space our data structure achieves query time $O(d n^{\rho+o(1)})$ and space $O(n^{1+\rho+o(1)} +…
Optimization methods have been broadly applied to two classes of objects viz. (i) modeling and description of data and (ii) the determination of the stationary points of functions. Here, a theoretical basis is developed that optimizes an…
We introduce an interpolation framework for H-infinity model reduction founded on ideas originating in optimal-H2 interpolatory model reduction, realization theory, and complex Chebyshev approximation. By employing a Loewner "data-driven"…
The constrained minimization (respectively maximization) of directed distances and of related generalized entropies is a fundamental task in information theory as well as in the adjacent fields of statistics, machine learning, artificial…
In this paper an autonomous analytical system of ordinary differential equations is considered. For an asymptotically stable steady state x0 of the system a gradual approximation of the domain of attraction DA is presented in the case when…