Multilinear Time Invariant System Theory

Abstract

In biological and engineering systems, structure, function and dynamics are highly coupled. Such interactions can be naturally and compactly captured via tensor based state space dynamic representations. However, such representations are not amenable to the standard system and controls framework which requires the state to be in the form of a vector. In order to address this limitation, recently a new class of multiway dynamical systems has been introduced in which the states, inputs and outputs are tensors. We propose a new form of multilinear time invariant (MLTI) systems based on the Einstein product and even-order paired tensors. We extend classical linear time invariant (LTI) system notions including stability, reachability and observability for the new MLTI system representation by leveraging recent advances in tensor algebra.

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

“…Data-driven approaches for the analysis of complex dynamical systems-be it methods to approximate transfer operators for computing metastable or coherent sets, methods to learn physical laws, or methods for optimization and control-have been steadily gaining popularity over the last years. Algorithms such as DMD [1,2], EDMD [3,4], SINDy [5], and their various kernel- [3,6,7], tensor- [8,9,10], or neural network-based [11,12,13] extensions and generalizations have been successfully applied to a plethora of different problems, including molecular and fluid dynamics, meteorology, finance, as well as mechanical and electrical engineering. An overview of different applications can be found, e.g., in [14].…”

Deep learning-based segmentation of high-resolution computed tomography image data outperforms commonly used automatic bone segmentation methods

“…Instead, several automated methods exist (e.g., grid search, random search, and Bayesian optimization) and could be applied to concisely and accurately determine the hyper-parameters that optimize model performance 41 . Also, other tensor-based approaches for segmentation provide promise, but the feasibility of CT scans has yet to be assessed 42,43 . Finally, our FCNNs were trained to segment entire cross sections, even though microstructural analyses are typically quantified on bone biopsies due to the field-of-view to resolution limitations with micro-CT 1 .…”

Cellular reprogramming: Mathematics meets medicine

“…Looking forward, a vast arena of mathematical principles with the potential to further sufficient understanding and facilitation of cellular reprogramming still remains. The high dimensional nature of cellular states, for example, invites an opportunity to examine cellular data in a tensor state space (C. Chen, Surana, Bloch, & Rajapakse, 2019). Tensors are multidimensional arrays generalized from vectors and matrices, and have wide applications in many domains such as social sciences, biology, applied mechanics, machine learning, and signal processing (Cichocki et al, 2016; Lu, Plataniotis, & Venetsanopoulos, 2008; W. Wang, Aggarwal, & Aeron, 2019; Williams et al, 2018).…”

“…Classical linear control systems, as used in Ronquist et al's work, often fail to fully capture the dynamics of cellular reprogramming because state vectors only represent gene expression, neglecting structural information. Chen et al generalized the classical systems notion of controllability into multilinear systems in which the states, inputs, and outputs are preserved as tensors (C. Chen et al, 2019). Multilinear control systems can significantly relieve the difficulty of describing genome‐wide structure and gene expression simultaneously, and will be beneficial in analyzing the dynamics of cellular reprogramming more comprehensively.…”