Related papers: The Scaled Relative Graph of a Linear Operator
The scaled graph has been introduced recently as a nonlinear extension of the classical Nyquist plot for linear time-invariant systems. In this paper, we introduce a modified definition for the scaled graph, termed the signed scaled graph…
SVRG and its variants are among the state of art optimization algorithms for large scale machine learning problems. It is well known that SVRG converges linearly when the objective function is strongly convex. However this setup can be…
Many iterative methods in optimization are fixed-point iterations with averaged operators. As such methods converge at an $\mathcal{O}(1/k)$ rate with the constant determined by the averagedness coefficient, establishing small averagedness…
Scaled relative graphs have been originally introduced in the context of convex optimization and have recently gained attention in the control systems community for the graphical analysis of nonlinear systems. Of particular interest in…
A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, Bloch and…
It is known that convergence of l.s.b. closed symmetric sesquilinear forms implies norm resolvent convergence of the associated self-adjoint operators and this in turn convergence of discrete spectra. In this paper in both cases sharp…
Real linear operators between two complex Banach spaces unify naturally two important classes of linear operators and antilinear operators. We give a survey of basic geometric, spectral and duality properties of real linear operators. The…
The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random…
It is shown that the numerical range of a linear operator operator in a Hilbert space is a (complete) $(1{+}\sqrt2)$-spectral set. The proof relies, among other things, in the behavior of the Cauchy transform of the conjugates of…
Let G be a group acting on a tree with cyclic edge and vertex stabilizers. Then stable commutator length (scl) is rational in G. Furthermore, scl varies predictably and converges to rational limits in so-called "surgery" families. This is a…
The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this…
Let $\sigma(A)$, $\rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$\rho(AB)\le r(A)r(B)…
We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is…
In this paper, we treat some weighted line digraphs which are induced by a connected and undirected graph. For a given graph $G$, the adjacency matrix of the weighted line digraph $W$ is determined by a boundary operator from an arc-based…
Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational…
Joint spectra of tuples of operators are subsets in complex projective space. The corresponding tuple of operators can be viewed as an infinite dimensional analog of a determinantal representation of the joint spectrum. We investigate the…
Straight lines are common features in human made environments, which makes them a frequently explored feature for control applications. Many control schemes, like Visual Servoing, require the 3D parameters of the features to be estimated.…
For finite-dimensional operator systems $\mathcal{S}_{\mathsf T}$, ${\mathsf T} \in B({\mathcal H})^d$, we show that the local lifting property and $1$-exactness of $\mathcal{S}_{\mathsf T}$ may be characterized by measurements of the…
Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of Nonlinear (NL) systems. In this paper, we restrict the SRG to particular input spaces to compute frequency-dependent incremental gain bounds…
With the prevalence of social media, the connectedness between people has been greatly enhanced. Real-world relations between users on social media are often not limited to expressing positive ties such as friendship, trust, and agreement,…