30,888 research outputs found
Geometric Distance Between Positive Definite Matrices of Different Dimensions
We show how the Riemannian distance on n++, the cone of n×n real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that n++ also parameterizes n-dimensional ellipsoids, and inner products on ℝn, n×n covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.ER
Positivity, entanglement entropy, and minimal surfaces
The path integral representation for the Renyi entanglement entropies of
integer index n implies these information measures define operator correlation
functions in QFT. We analyze whether the limit , corresponding
to the entanglement entropy, can also be represented in terms of a path
integral with insertions on the region's boundary, at first order in .
This conjecture has been used in the literature in several occasions, and
specially in an attempt to prove the Ryu-Takayanagi holographic entanglement
entropy formula. We show it leads to conditional positivity of the entropy
correlation matrices, which is equivalent to an infinite series of polynomial
inequalities for the entropies in QFT or the areas of minimal surfaces
representing the entanglement entropy in the AdS-CFT context. We check these
inequalities in several examples. No counterexample is found in the few known
exact results for the entanglement entropy in QFT. The inequalities are also
remarkable satisfied for several classes of minimal surfaces but we find
counterexamples corresponding to more complicated geometries. We develop some
analytic tools to test the inequalities, and as a byproduct, we show that
positivity for the correlation functions is a local property when supplemented
with analyticity. We also review general aspects of positivity for large N
theories and Wilson loops in AdS-CFT.Comment: 36 pages, 10 figures. Changes in presentation and discussion of
Wilson loops. Conclusions regarding entanglement entropy unchange
Including parameter dependence in the data and covariance for cosmological inference
The final step of most large-scale structure analyses involves the comparison
of power spectra or correlation functions to theoretical models. It is clear
that the theoretical models have parameter dependence, but frequently the
measurements and the covariance matrix depend upon some of the parameters as
well. We show that a very simple interpolation scheme from an unstructured mesh
allows for an efficient way to include this parameter dependence
self-consistently in the analysis at modest computational expense. We describe
two schemes for covariance matrices. The scheme which uses the geometric
structure of such matrices performs roughly twice as well as the simplest
scheme, though both perform very well.Comment: 17 pages, 4 figures, matches version published in JCA
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
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