 # Ido Cohen et al.

The space-discrete Total Variation (TV) flow is analyzed using several mode decomposition techniques. In the one-dimensional case, we provide analytic formulations to Dynamic Mode Decomposition (DMD) and to Koopman Mode Decomposition (KMD) of the TV-flow and compare the obtained modes to TV spectral decomposition. We propose a computationally efficient algorithm to evolve the one-dimensional TV-flow. A significant speedup by three orders of magnitude is obtained, compared to iterative minimizations. A common theme, for both mode analysis and fast algorithm, is the significance of phase transitions during the flow, in which the subgradient changes. We explain why applying DMD directly on TV-flow measurements cannot model the flow or extract modes well. We formulate a more general method for mode decomposition that coincides with the modes of KMD. This method is based on the linear decay profile, typical to TV-flow. These concepts are demonstrated through experiments, where additional extensions to the two-dimensional case are given. # Ido Cohen et al.

The space-discrete Total Variation (TV) flow is analyzed using several mode decomposition techniques. In the one-dimensional case, we provide analytic formulations to Dynamic Mode Decomposition (DMD) and to Koopman Mode Decomposition (KMD) of the TV-flow and compare the obtained modes to TV spectral decomposition. We propose a computationally efficient algorithm to evolve the one-dimensional TV-flow. A significant speedup by three orders of magnitude is obtained, compared to iterative minimizations. A common theme, for both mode analysis and fast algorithm, is the significance of phase transitions during the flow, in which the subgradient changes. We explain why applying DMD directly on TV-flow measurements cannot model the flow or extract modes well. We formulate a more general method for mode decomposition that coincides with the modes of KMD. This method is based on the linear decay profile, typical to TV-flow. These concepts are demonstrated through experiments, where additional extensions to the two-dimensional case are given. # Ido Cohen and Guy Gilboa

This work binds the existence of Koopman Eigenfunction (KEF), the geometric of the dynamics, and the validity of DMD to one coherent theory. Viewing the dynamic as a curve in the state-space allows us to formulate an existence condition of KEFs and their multiplicities. These conditions lay the foundations for system reconstruction, global controllability, and observability for nonlinear dynamics.  DMD can be interpreted as a finite dimension approximation of KMD. However, this method is limited to the case when KEFs are linear combinations of the observations. We examine the limitations of DMD through the analysis of Koopman theory. We propose a new mode decomposition technique based on the typical time profile of the dynamics. An overcomplete dictionary of decay profiles is used to sparsely represent the dynamic. This analysis is also valid in the full continuous setting of Koopman theory, which is based on variational calculus.
We demonstrate applications of this analysis, such as finding KEFs and their multiplicities, calculating KMD, dynamics reconstruction, global linearization, and controllability. # Ido Cohen and Guy Gilboa

This work binds the existence of Koopman Eigenfunction (KEF), the geometric of the dynamics, and the validity of DMD to one coherent theory. Viewing the dynamic as a curve in the state-space allows us to formulate an existence condition of KEFs and their multiplicities. These conditions lay the foundations for system reconstruction, global controllability, and observability for nonlinear dynamics.  DMD can be interpreted as a finite dimension approximation of KMD. However, this method is limited to the case when KEFs are linear combinations of the observations. We examine the limitations of DMD through the analysis of Koopman theory. We propose a new mode decomposition technique based on the typical time profile of the dynamics. An overcomplete dictionary of decay profiles is used to sparsely represent the dynamic. This analysis is also valid in the full continuous setting of Koopman theory, which is based on variational calculus.
We demonstrate applications of this analysis, such as finding KEFs and their multiplicities, calculating KMD, dynamics reconstruction, global linearization, and controllability. # Ido Cohen et al. code

In this work we analyze theTotal Variation(TV) flow ap-plied to one dimensional signals. We formulate a relation between Dynamic Mode Decomposition(DMD), a dimensionality reduction method basedon the Koopman operator, and the spectral TV decomposition. DMD isadapted by time rescaling to fit linearly decaying processes, such as theTV flow. For the flow with finite subgradient transitions, a closed formsolution of the rescaled DMD is formulated. In addition, a solution to theTV-flow is presented, which relies only on the initial condition and itscorresponding subgradient. A very fast numerical algorithm is obtainedwhich solves the entire flow by elementary subgradient updates. # Ido Cohen et al. SSVM

In this work, we analyze the Total Variation(TV) flow applied to one-dimensional signals. We formulate a relation between Dynamic Mode Decomposition(DMD), a dimensionality reduction method based on the Koopman operator and the spectral TV decomposition. DMD is adapted by time rescaling to fit linearly decaying processes, such as the TV flow. For the flow with finite subgradient transitions, a closed-form solution of the rescaled DMD is formulated. In addition, a solution to the TV-flow is presented, which relies only on the initial condition and its corresponding subgradient. A very fast numerical algorithm is obtained which solves the entire flow by elementary subgradient updates. # Ido Cohen et al. SIAM

Finding latent structures in data is drawing increasing attention in diverse fields such as image and signal processing, fluid dynamics, and machine learning. In this work we examine the problem of finding the main modes of gradient flows. Gradient descent is a fundamental process in optimization where its stochastic version is prominent in training of neural networks. Here our aim is to establish a consistent theory for gradient flows ψt=P(ψ), where P is a nonlinear homogeneous operator. Our proposed framework stems from analytic solutions of homogeneous flows, previously formalized by Cohen-Gilboa, where the initial condition ψ0 admits the nonlinear eigenvalue problem P(ψ0)=λψ0. We first present an analytic solution for \ac{DMD} in such cases. We show an inherent flaw of \ac{DMD}, which is unable to recover the essential dynamics of the flow. It is evident that \ac{DMD} is best suited for homogeneous flows of degree one. We propose an adaptive time sampling scheme and show its dynamics are analogue to homogeneous flows of degree one with a fixed step size. Moreover, we adapt \ac{DMD} to yield a real spectrum, using symmetric matrices. Our analytic solution of the proposed scheme recovers the dynamics perfectly and yields zero error. We then proceed to show that in the general case the orthogonal modes {ϕi} are approximately nonlinear eigenfunctions P(ϕ)λϕ. We formulate Orthogonal Nonlinear Spectral decomposition (\emph{OrthoNS}), which recovers the essential latent structures of the gradient descent process. Definitions for spectrum and filtering are given, and a Parseval-type identity is shown. # Ido Cohen and Guy Gilboa Signal Processing

In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p ∈ (1, 2). With this decomposition we can process signals of different degrees of smoothness.

We first analyze solutions of scale spaces, generated by γ-homogeneous operators, γ∈R. An analytic solution is formulated when the scale space is initialized with a nonlinear eigenfunction of the respective operator. We show that the flow is extinct in finite time for γ ∈ [0, 1).

A main innovation in this study is concerned with operators of fractional homogeneity, which require the mathematical framework of fractional calculus. The proposed transform rigorously defines the notions of decomposition, reconstruction, filtering and spectrum. The theory is applied to the p-Laplacian operator, where the tools developed in this framework are demonstrated. # Ido Cohen, Adi Falik and Guy Gilboa SSVM 2019

Implementation of nonlinear flows by explicit schemes can be very convenient, due to their simplicity and low-computational cost per time step. A well known drawback is the small time step bound, referred to as the CFL condition, which ensures a stable flow. For p-Laplacian flows, with 1<p<21<p<2, explicit schemes without gradient regularization require, in principle, a time step approaching zero. However, numerical implementations show explicit flows with small time-steps are well behaved. We can now explain and quantify this phenomenon.

In this paper we examine explicit p-Laplacian flows by analyzing the evolution of nonlinear eigenfunctions, with respect to the p-Laplacian operator. For these cases analytic solutions can be formulated, allowing for a comprehensive analysis. A generalized CFL condition is presented, relating the time step to the inverse of the nonlinear eigenvalue. Moreover, we show that the flow converges and formulate a bound on the error of the discrete scheme. Finally, we examine general initial conditions and propose a dynamic time-step bound, which is based on a nonlinear Rayleigh quotient. # Ido Cohen and Guy Gilboa HAL

We examine nonlinear scale-spaces in the general form ut = P (u(t)), where P is a bounded nonlinear operator. We seek solutions with separation of variables in space and time u(x, t) = a(t)f (x), where f is the initial condition. We term these as shape-preserving flows and provide necessary and sufficient conditions for their existence. We show that homogeneous operators admit the above conditions. It turns out that the initial condition must admit a nonlinear eigenvalue problem, with respect to the operator P , P (f) = λf , where λ is the eigenvalue. In this case we can formulate a closed form solution for any P which is homogeneous of positive degree. Consequently, we can determine if a finite extinction time exists. We show that in all cases the extinction time is inversely proportional to the eigenvalue λ. Following the above analysis, we generalize the total-variation and one-homogeneous transforms to a homogeneous spectral representation. The notions of spectrum, generalized Parseval’s theorem and filtering are defined. We apply these formulations to the p−Laplace operator for 1 < p < 2.