A complete representation of 3D objects requires characterizing the space deformations in an interpretable manner, from articulations a single instance to changes shape across categories. In this work, we improve on prior generative model geometric disentanglement for shapes, wherein object geometry is factorized into rigid orientation, non-rigid pose, and intrinsic shape. The resulting can be ...

Manin’s conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety X in terms of global geometric invariants of X. The strongest form of the conjecture implies certain inequalities among geometric invariants of X and of its subvarieties. We provide a general geometric framework explaining these phenomena, via the notion of balanc...

This paper presents a problem reduction scheme that converts geometric constraints in work space to a system of equations in parameter space. We demonstrate that this scheme can solve many interesting geometric problems that have been considered quite di cult to deal with using conventional techniques. An important advantage of our approach is that equations represented in the parameter space h...

We introduce, and investigate the properties of, the family of quantum Brègman distances, based on embeddings into suitable vector spaces (with the reflexive noncommutative Orlicz spaces over semi-finite W-algebras and noncommutative Lp spaces over any W-algebras providing two important examples). This allows us to define geometric categories for nonlinear quantum inference theory, with morphis...

We have already met covering spaces in Chapter 2, where our discussion was geometric and basically limited to definitions and examples. In this chapter, we return to this topic from a more algebraic point of view, which will allow us to produce numerous examples coming from group actions and to classify all covering spaces with given base (provided the latter is “nice” enough). The main tools w...

The notion of the shell of a Hilbert space operator, which is a useful generalization (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators ...

We present a new scale U p (s < −t < 0 and 1 ≤ p < ∞) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator Lgφ = (g · φ) ◦ T−1 associated to a hyperbolic dynamical system T has good spectral properties. When p = 1 and t is an integer, the spaces are analogous to the “geometric” spaces Bt,|s+t| considered by Gouëzel and Liverani [26]. When p > 1 and −1 + 1/p...

We consider smoothed versions of geometric range spaces, so an element of the ground set (e.g. a point) can be contained in a range with a non-binary value in [0, 1]. Similar notions have been considered for kernels; we extend them to more general types of ranges. We then consider approximations of these range spaces through ε-nets and εsamples (aka ε-approximations). We characterize when size ...

The notion of the shell of a Hilbert space operator, which is a useful generalization (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators ...

Using the Gauss-Manin connection (Picard-Fuchs differential equation) and a result of Malgrange, a special class of algebraic solutions to isomonodromic deformation equations, the geometric isomonodromic deformations, is defined from “families of families” of algebraic varieties. Geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surf...