833 research outputs found
Continued fraction solution of Krein's inverse problem
The spectral data of a vibrating string are encoded in its so-called
characteristic function. We consider the problem of recovering the distribution
of mass along the string from its characteristic function. It is well-known
that Stieltjes' continued fraction provides a solution of this inverse problem
in the particular case where the distribution of mass is purely discrete. We
show how to adapt Stieltjes' method to solve the inverse problem for a related
class of strings. An application to the excursion theory of diffusion processes
is presented.Comment: 18 pages, 2 figure
Violation of the entropic area law for Fermions
We investigate the scaling of the entanglement entropy in an infinite
translational invariant Fermionic system of any spatial dimension. The states
under consideration are ground states and excitations of tight-binding
Hamiltonians with arbitrary interactions. We show that the entropy of a finite
region typically scales with the area of the surface times a logarithmic
correction. Thus, in contrast to analogous Bosonic systems, the entropic area
law is violated for Fermions. The relation between the entanglement entropy and
the structure of the Fermi surface is discussed, and it is proven, that the
presented scaling law holds whenever the Fermi surface is finite. This is in
particular true for all ground states of Hamiltonians with finite range
interactions.Comment: 5 pages, 1 figur
Maximal violation of Bell inequalities by position measurements
We show that it is possible to find maximal violations of the CHSH-Bell
inequality using only position measurements on a pair of entangled
non-relativistic free particles. The device settings required in the CHSH
inequality are done by choosing one of two times at which position is measured.
For different assignments of the "+" outcome to positions, namely to an
interval, to a half line, or to a periodic set, we determine violations of the
inequalities, and states where they are attained. These results have
consequences for the hidden variable theories of Bohm and Nelson, in which the
two-time correlations between distant particle trajectories have a joint
distribution, and hence cannot violate any Bell inequality.Comment: 13 pages, 4 figure
Processing and Transmission of Information
Contains reports on three research projects.Lincoln Laboratory, Purchase Order DDL B-00368U. S. ArmyU. S. NavyU. S. Air Force under Air Force Contract AF19(604)-7400National Institutes of Health (Grant MH-04737-03)National Science Foundation (Grant G-16526
Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem
Injective multiset functions have a key role in the theoretical study of
machine learning on multisets and graphs. Yet, there remains a gap between the
provably injective multiset functions considered in theory, which typically
rely on polynomial moments, and the multiset functions used in practice, which
rely on \unicode{x2014} whose injectivity on
multisets has not been studied to date.
In this paper, we bridge this gap by showing that moments of neural networks
do define injective multiset functions, provided that an analytic
non-polynomial activation is used. The number of moments required by our theory
is optimal essentially up to a multiplicative factor of two. To prove this
result, we state and prove a , which is of
independent interest.
As a corollary to our main theorem, we derive new approximation results for
functions on multisets and measures, and new separation results for graph
neural networks. We also provide two negative results: (1) moments of
piecewise-linear neural networks cannot be injective multiset functions; and
(2) even when moment-based multiset functions are injective, they can never be
bi-Lipschitz.Comment: NeurIPS 2023 camera-read
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