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This paper presents a proof of an uncertainty principle of Donoho-Stark type involving $\varepsilon$-concentration of localization operators. More general operators associated with time-frequency representations in the Cohen class are then…
The constituents of a complex system exchange information to function properly. Their signalling dynamics often leads to the appearance of emergent phenomena, such as phase transitions and collective behaviors. While information exchange…
Recently we introduced mixed-state localization operators associated to a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a…
We propose a pair of completely data-driven algorithms for unsupervised classification and dimension reduction, and we empirically study their performance on a number of data sets, both simulated data in three-dimensions and images from the…
In complex networks, nodes that share similar structural characteristics often exhibit similar roles (e.g type of users in a social network or the hierarchical position of employees in a company). In order to leverage this relationship, a…
We introduce superdensity operators as a tool for analyzing quantum information in spacetime. Superdensity operators encode spacetime correlation functions in an operator framework, and support a natural generalization of Hilbert space…
Koopman analysis provides a general framework from which to analyze a nonlinear dynamical system in terms of a linear operator acting on an infinite-dimensional observable space. This theoretical framework provides a rigorous underpinning…
A framework for data assimilation combining aspects of operator-theoretic ergodic theory and quantum mechanics is developed. This framework adapts the Dirac--von Neumann formalism of quantum dynamics and measurement to perform sequential…
Over the last few years, several works have proposed deep learning architectures to learn dynamical systems from observation data with no or little knowledge of the underlying physics. A line of work relies on learning representations where…
We study a simple nonlocal-in-time dynamic system proposed for the effective modeling of complex diffusive regimes in heterogeneous media. We present its solutions and their commonly studied statistics such as the mean square distance. This…
Time averaging over the trajectory of a wavepacket up to time T defines a statistical operator (density matrix). The corresponding (Von Neumann) entropy is proven to asymptotically increase with time like D.log T, with D the Hausdorff…
We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the…
When complex systems with nonlinear dynamics achieve an output performance objective, only a fraction of the state dynamics significantly impacts that output. Those minimal state dynamics can be identified using the differential geometric…
Deep neural networks have achieved remarkable success, yet our understanding of how they learn remains limited. These models can learn high-dimensional tasks, which is generally statistically intractable due to the curse of dimensionality.…
For a quantum state undergoing unitary Schr\"odinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum…
Hierarchical tree structures are common in many real-world systems, from tree roots and branches to neuronal dendrites and biologically inspired artificial neural networks, as well as in technological networks for organizing and searching…
We develop a new generalization of Koopman operator theory that incorporates the effects of inputs and control. Koopman spectral analysis is a theoretical tool for the analysis of nonlinear dynamical systems. Moreover, Koopman is intimately…
Conditional von Neumann entropy is an intriguing concept in quantum information theory. In the present work, we examine the effect of global unitary operations on the conditional entropy of the system. We start with the set containing…
Dynamics of a periodically time dependent quantum system is reflected in the features of the eigenstates of the Floquet operator. Of the special importance are their localization properties quantitatively characterized by the eigenvector…
Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge. Koopman operator theory circumvents some of these issues by considering the dynamics in the space of…