Related papers: Approximability Models and Optimal System Identifi…
In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms,…
In this paper, we employ fixed point theory and semidefinite programming to compute the performance bounds on convex block-sparsity recovery algorithms. As a prerequisite for optimal sensing matrix design, a computable performance bound…
This paper is concerned with the ubiquitous inverse problem of recovering an unknown function u from finitely many measurements possibly affected by noise. In recent years, inversion methods based on linear approximation spaces were…
Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence,…
The question of fast convergence in the classical problem of high dimensional linear regression has been extensively studied. Arguably, one of the fastest procedures in practice is Iterative Hard Thresholding (IHT). Still, IHT relies…
As with classic statistics, functional regression models are invaluable in the analysis of functional data. While there are now extensive tools with accompanying theory available for linear models, there is still a great deal of work to be…
This work explores multi-modal inference in a high-dimensional simplified model, analytically quantifying the performance gain of multi-modal inference over that of analyzing modalities in isolation. We present the Bayes-optimal performance…
We describe a probabilistic, {\it sublinear} runtime, measurement-optimal system for model-based sparse recovery problems through dimensionality reducing, {\em dense} random matrices. Specifically, we obtain a linear sketch $u\in \R^M$ of a…
The goal of optimal transport (OT) is to find optimal assignments or matchings between data sets which minimize the total cost for a given cost function. However, sometimes the cost function is unknown but we have access to (parts of) the…
We consider the joint problem of system identification and inverse optimal control for discrete-time stochastic Linear Quadratic Regulators. We analyze finite and infinite time horizons in a partially observed setting, where the state is…
Maximum entropy models provide the least constrained probability distributions that reproduce statistical properties of experimental datasets. In this work we characterize the learning dynamics that maximizes the log-likelihood in the case…
We study the recovery of functions in various norms, including $L_p$ with $1\le p\le\infty$, based on function evaluations. We obtain worst case error bounds for general classes of functions in terms of the best $L_2$-approximation from a…
Thresholding based iterative algorithms have the trade-off between effectiveness and optimality. Some are effective but involving sub-matrix inversions in every step of iterations. For systems of large sizes, such algorithms can be…
The indirect approach to continuous-time system identification consists in estimating continuous-time models by first determining an appropriate discrete-time model. For a zero-order hold sampling mechanism, this approach usually leads to a…
Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by…
Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This…
We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data is not sufficient for estimating time derivatives of state variables but enough to…
We consider the nonlinear inverse problem of learning a transition operator $\mathbf{A}$ from partial observations at different times, in particular from sparse observations of entries of its powers…
Optimal recovery is a mathematical framework for learning functions from observational data by adopting a worst-case perspective tied to model assumptions on the functions to be learned. Working in a finite-dimensional Hilbert space, we…
This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from full knowledge of the albedo operator at the boundary of a bounded domain of interest. We present optimal…