22 research outputs found
Visible and hidden observables in super-linearization
We call a system super-linearizable if it admits finite-dimensional embedding
as a linear system -- known as a finite-dimensional Koopman embedding; said
otherwise, if its dynamics can be linearized by adding a finite set of
observables. We introduce the notions of visible and hidden observables for
such embeddings which, roughly speaking, are the observables that explicitly
appear in the original system and the ones that do not, but yet are necessary
for its embedding. Distinct embeddings can have different numbers of hidden and
visible observables. In this paper, we derive a tight lower bound for the
number of visible observables of a system among all its super-linearizations
On sparse representations of linear operators and the approximation of matrix products
Thus far, sparse representations have been exploited largely in the context
of robustly estimating functions in a noisy environment from a few
measurements. In this context, the existence of a basis in which the signal
class under consideration is sparse is used to decrease the number of necessary
measurements while controlling the approximation error. In this paper, we
instead focus on applications in numerical analysis, by way of sparse
representations of linear operators with the objective of minimizing the number
of operations needed to perform basic operations (here, multiplication) on
these operators. We represent a linear operator by a sum of rank-one operators,
and show how a sparse representation that guarantees a low approximation error
for the product can be obtained from analyzing an induced quadratic form. This
construction in turn yields new algorithms for computing approximate matrix
products.Comment: 6 pages, 3 figures; presented at the 42nd Annual Conference on
Information Sciences and Systems (CISS 2008
Graphons and the -property
A graphon satisfies the -property if graphs sampled from it contain a
Hamiltonian decomposition almost surely, which in turn implies that the
corresponding network topologies are, e.g., structurally stable and
structurally ensemble controllable. In recent papers, we have exhibited a set
of conditions that is essentially necessary and sufficient for the -property
to hold for the finite-dimensional class of step-graphons. The extension to the
infinite-dimensional case of general graphons was hindered by the fact that
said conditions relied on objects that do not admit immediate extensions to the
infinite-dimensional case. We outline here our approach to bypass this
difficulty and state conditions that guarantee that the -property holds for
general graphons.Comment: Extended abstrac