Related papers: Determinant Dynamics under Low-Rank Perturbations:…
We propose a unified framework for estimating low-rank matrices through nonconvex optimization based on gradient descent algorithm. Our framework is quite general and can be applied to both noisy and noiseless observations. In the general…
The structured singular value $\mu$ was introduced independently by Doyle and Safanov as a tool for analyzing robustness of system stability and performance in the presence of structured uncertainty in the system parameters. While the…
We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions,…
Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing com plexity. Recently, fast and reliable eigensolvers dealing with low…
We study implications of unitarity for pseudo-orbit expansions of the spectral determinants of quantum maps and quantum graphs. In particular, we advocate to group pseudo-orbits into sub-determinants. We show explicitly that the…
We introduce a supervised dimensionality reduction methodology for categorical (and discretized mixed-type) data based on a density-matrix construction induced by class-conditional frequencies. Given a labeled dataset encoded in a one-hot…
We study a multiscale stochastic optimal control problem subject to state constraints on the slow variable. To address this class of problems, we develop a rigorous theoretical framework based on singular perturbation analysis, tailored to…
We develop a data-driven optimal shrinkage algorithm for matrix denoising in the presence of high-dimensional noise with a separable covariance structure; that is, the noise is colored and dependent across samples. The algorithm, coined…
In this article, we present an extension of the formulation recently developed by the authors (A Framework for Data-Driven Computational Mechanics Based on Nonlinear Optimization, arXiv:1910.12736 [math.NA]) to the structural dynamics…
The analytic structure of the S-matrix of singular quantum mechanics is examined within a multichannel framework, with primary focus on its dependence with respect to a parameter ($\Omega$) that determines the boundary conditions.…
Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical…
This paper studies the stability of low-rank implicit regularization in perturbed deep matrix factorization, where the target matrix is corrupted by a noise matrix. We first derive sufficient spectral conditions under which gradient descent…
The Error-in-Variables model of system identification/control involves nontrivial input and measurement corruption of observed data, resulting in generically nonconvex optimization problems. This paper performs full-state-feedback…
The fundamental lemma by Willems and coauthors facilitates a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays an important role in data-driven simulation and…
We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding…
We propose a method for learning dynamical systems from high-dimensional empirical data that combines variational autoencoders and (spatio-)temporal attention within a framework designed to enforce certain scientifically-motivated…
Controllability and observability Gramians, along with their inverses, are widely used to solve various problems in control theory. This paper proposes spectral decompositions of the controllability Gramian and its inverse based on system…
We introduce a Bayesian perspective for the structured matrix factorization problem. The proposed framework provides a probabilistic interpretation for existing geometric methods based on determinant minimization. We model input data…
Distributionally robust optimization is used to tackle decision making problems under uncertainty where the distribution of the uncertain data is ambiguous. Many ambiguity sets have been proposed for continuous uncertainty that build on…
This note presents a unified analysis of the identification of dynamical systems with low-rank constraints under high-dimensional scaling. This identification problem for dynamic systems are challenging due to the intrinsic dependency of…