The Universal Approximation Property

The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models' potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation functions allow the feed-forward architecture to maintain its universal approximation capabilities when multiple constraints are imposed on its final layers and its remaining layers are only sparsely connected. These include a rescaled and shifted Leaky ReLU activation function but not the ReLU activation function. Our construction and representation result is used to exhibit a simple modification of the feed-forward architecture, which can approximate any continuous function with non-pathological growth, uniformly on the entire Euclidean input space. This improves the known capabilities of the feed-forward architecture

Fuzzy Euclidean wormholes in anti-de Sitter space

This paper is devoted to an investigation of Euclidean wormholes made by fuzzy instantons. We investigate the Euclidean path integral in anti-de Sitter space. In Einstein gravity, we introduce a scalar field with a potential. Because of the analyticity, there is a contribution of complex-valued instantons, so-called fuzzy instantons. If we have a massless scalar field, then we obtain Euclidean wormholes, where the probabilities become smaller and smaller as the size of the throat becomes larger and larger. If we introduce a non-trivial potential, then in order to obtain a non-zero tunneling rate, we need to tune the shape of the potential. With the $O(4)$ symmetry, after the analytic continuation to the Lorentzian time, the wormhole throat should expand to infinity. However, by adding mass, one may obtain an instant wormhole that should eventually collapse to the event horizon. The existence of Euclidean wormholes is related to the stability or unitarity issues of anti-de Sitter space. We are not conclusive yet, but we carefully comment on these physical problems.Comment: 20 pages, 9 figure

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds

This paper proposes to generalize the histories included in Euclidean functional integrals from manifolds to a more general set of compact topological spaces. This new set of spaces, called conifolds, includes nonmanifold stationary points that arise naturally in a semiclasssical evaluation of such integrals; additionally, it can be proven that sequences of approximately Einstein manifolds and sequences of approximately Einstein conifolds both converge to Einstein conifolds. Consequently, generalized Euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and conifold histories that approach a stationary point of the Einstein action. Therefore sums over conifold histories provide a useful and self-consistent starting point for further study of topological effects in quantum gravity. Postscript figures available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.