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Register a new userOn non-parametric density estimation on linear and non-linear manifolds using generalized Radon transforms
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Here we present a new non-parametric approach to density estimation and classification derived from theory in Radon transforms and image reconstruction. We start by constructing a forward problem in which the unknown density is mapped to a set of one dimensional empirical distribution functions computed from the raw input data. Interpreting this mapping in terms of Radon-type projections provides an analytical connection between the data and the density with many very useful properties including stable invertibility, fast computation, and significant theoretical grounding. Using results from the literature in geometric inverse problems we give uniqueness results and stability estimates for our methods. We subsequently extend the ideas to address problems in manifold learning and density estimation on manifolds. We introduce two new algorithms which can be readily applied to implement density estimation using Radon transforms in low dimensions or on low dimensional manifolds embedded in $mathbb{R}^d$. We test our algorithms performance on a range of synthetic 2-D density estimation problems, designed with a mixture of sharp edges and smooth features. We show that our algorithm can offer a consistently competitive performance when compared to the state-of-the-art density estimation methods from the literature.
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Let $G_{n,r}(bbK)$ be the Grassmannian manifold of $k$-dimensional $bbK$-subspaces in $bbK^n$ where $bbK=mathbb R, mathbb C, mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, $mathcal R_{r^prime, r}$, $mathcal C_{r^prime, r}$ and $mathcal S_{r^prime, r}$, from the $L^2$ space $L^2(G_{n,r}(bbK))$ to the space $L^2(G_{n,r^prime}(bbK))$, for $r, r^prime le n-1$. The $L^2$ spaces are decomposed into irreducible representations of $G$ with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus $mathbb R^r/{2pi Q^vee}$ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space $mathbb R^r$. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Steins complementary series in the compact picture.
In this report we consider the parameterization of low-dimensional manifolds that are specified (approximately) by a set of points very close to the manifold in the original high-dimensional space. Our objective is to obtain a parameterization that is (1-1) and non singular (in the sense that the Jacobian of the map between the manifold and the parameter space is bounded and non singular).
In this paper we set a framework in which experiments whose goal is to test QED predictions can be used in a more general way to test non-linear electrodynamics (NLED) which contains low-energy QED as a special case. We review some of these experiments and we establish limits on the different free parameters by generalizing QED predictions in the framework of NLED. We finally discuss the implications of these limits on bound systems and isolated charged particles for which QED has been widely and successfully tested.
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
We study the different horospherical Radon transforms that arise by regarding a homogeneous tree T as a simplicial complex whose simplices are vertices V, edges E or flags F (flags are oriented edges). The ends (infinite geodesic rays starting at a reference vertex) provide a boundary $Omega$ for the tree. Then the horospheres form a trivial principal fiber bundle with base $Omega$ and fiber $mathZ$. There are three such fiber bundles, consisting of horospheres of vertices, edges or flags, but they are isomorphic: however, no isomorphism between these fiber bundles maps special sections to special sections (a special section consists of the set of horospheres through a given vertex, edge or flag). The groups of automorphisms of the fiber bundles contain a subgroup $A$ of parallel shifts, analogous to the Cartan subgroup of a semisimple group. The normalized eigenfunctions of the Laplace operator on T are boundary integrals of complex powers of the Poisson kernel, that is characters of $A$, and are matrix coefficients of representations induced from $A$ in the sense of Mackey, the so-called spherical representations. The vertex-horospherical Radon transform consists of summation over V in each vertex-horosphere, and similarly for edges or flags. We prove inversion formulas for all these Radon transforms, and give applications to harmonic analysis and the Plancherel measure on T. We show via integral geometry that the spherical representations for vertices and edges are equivalent. Also, we define the Radon back-projections and find the inversion operator of each Radon transform by composing it with its back-projection. This gives rise to a convolution operator on T, whose symbol is obtained via the spherical Fourier transform, and its reciprocal is the symbol of the Radon inversion formula.
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