The capacity of tuples of point clouds and generalized radial isotropy
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In this thesis, we provide both a qualitative and quantitative study of the capacity of tuples of point clouds through the notion of generalized radial isotropy. This is in part motivated by the celebrated Brascamp-Lieb ("BL") constants, and more generally, the Anantharam-Jog-Nair ("AJN") constants that are associated with a tuple of point clouds. Let m, n, and d1, . . . , dn be positive integers and let X = (X^1, . . . X^n) be an n-tuple of point clouds, with each X^i [subset] Rdi being a point cloud of m vectors in R^di. This tuple of point clouds can be encoded in a representation Vx of the complete bipartite quiver with n source vertices and m sink vertices. Let c = (c1, . . . , cm) [element] Qm [greater than] 0 be a vector of positive rational exponents that sum to the total of the dimensions di. The main object of study in this thesis is the capacity, denoted by Cap(X, c), of the quiver datum (Vx, c). When n = 1, the constants of the form Cap(X, c)^-12 are precisely the best constants that occur in the classical BL inequalities associated with single point cloud of m vectors. For arbitrary n, taking the natural logarithm of Cap(X, c)^-12 yields the best constants in the AJN inequalities in Information Theory (in the rank one case). Our first result gives necessary and sufficient conditions for a tuple of point clouds X to be put in generalized c-radial isotropic position. This result, combined with a general character formula for the capacity, allows us to obtain closed formulas for the capacity in several examples. We also describe the capacity Cap(X, c) in a way that enables us to use gradient descent to compute approximations of the capacity. The results in this thesis generalize the work of Barthe and Hardt-Moitra on radial isotropy from a single point cloud (the case of classical BL constants) to tuples of point clouds (the case of AJN constants).
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Ph. D.