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This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank…

We establish the convergence of a class of numerical algorithms, known as Dynamic Mode Decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data…

Volumetry is one of the principal downstream applications of 3D medical image segmentation, for example, to detect abnormal tissue growth or for surgery planning. Conformal Prediction is a promising framework for uncertainty quantification,…

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are…

Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite-dimensional, and…

Although EDM aims to unify the design space of diffusion models, its reliance on fixed Gaussian noise prevents it from explaining emerging flow-based methods that diffuse arbitrary noise. Moreover, our study reveals that EDM's forcible…

A key challenge to performing effective analyses of high-dimensional data is finding a signal-rich, low-dimensional representation. For linear subspaces, this is generally performed by decomposing a design matrix (via eigenvalue or singular…

Black-box discrete optimization (BB-DO) problems arise in many real-world applications, such as neural architecture search and mathematical model estimation. A key challenge in BB-DO is epistasis among parameters where multiple variables…

We propose a method for obtaining rigorous and accurate upper and lower bounds on the eigenvalues of ordinary and partial differential operators in bounded regions of Euclidean space. It uses a boundary condition homotopy method starting…

Rank minimization can be converted into tractable surrogate problems, such as Nuclear Norm Minimization (NNM) and Weighted NNM (WNNM). The problems related to NNM, or WNNM, can be solved iteratively by applying a closed-form proximal…

Dimension reduction is often needed in the area of data mining. The goal of these methods is to map the given high-dimensional data into a low-dimensional space preserving certain properties of the initial data. There are two kinds of…

Differential equations are pivotal in modeling and understanding the dynamics of various systems, offering insights into their future states through parameter estimation fitted to time series data. In fields such as economy, politics, and…

We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly…

Large-scale Vision-Language models have achieved remarkable results in various domains, such as image captioning and conditioned image generation. Nevertheless, these models still encounter difficulties in achieving human-like compositional…

Discrete-action reinforcement learning algorithms often falter in tasks with high-dimensional discrete action spaces due to the vast number of possible actions. A recent advancement leverages value-decomposition, a concept from multi-agent…

The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the…

In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of…

We rigorously derive novel error bounds for extended dynamic mode decomposition (EDMD) to approximate the Koopman operator for discrete- and continuous time (stochastic) systems; both for i.i.d. and ergodic sampling under non-restrictive…

The Dynamic Mode Decomposition (DMD) and the more general Extended DMD (EDMD) are powerful tools for computational analysis of dynamical systems in data-driven scenarios. They are built on the theoretical foundation of the Koopman…

Dynamical spectral estimation is a well-established numerical approach for estimating eigenvalues and eigenfunctions of the Markov transition operator from trajectory data. Although the approach has been widely applied in biomolecular…