168 research outputs found
A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone
This is an account of some aspects of the geometry of K\"ahler affine metrics
based on considering them as smooth metric measure spaces and applying the
comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a
version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By
a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric
on a proper convex domain, and the Schwarz lemma gives a direct proof of its
uniqueness up to homothety. The potential for this metric is a function
canonically associated to the cone, characterized by the property that its
level sets are hyperbolic affine spheres foliating the cone. It is shown that
for an -dimensional cone a rescaling of the canonical potential is an
-normal barrier function in the sense of interior point methods for conic
programming. It is explained also how to construct from the canonical potential
Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean
curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde
Interior-point methods on manifolds: theory and applications
Interior-point methods offer a highly versatile framework for convex
optimization that is effective in theory and practice. A key notion in their
theory is that of a self-concordant barrier. We give a suitable generalization
of self-concordance to Riemannian manifolds and show that it gives the same
structural results and guarantees as in the Euclidean setting, in particular
local quadratic convergence of Newton's method. We analyze a path-following
method for optimizing compatible objectives over a convex domain for which one
has a self-concordant barrier, and obtain the standard complexity guarantees as
in the Euclidean setting. We provide general constructions of barriers, and
show that on the space of positive-definite matrices and other symmetric
spaces, the squared distance to a point is self-concordant. To demonstrate the
versatility of our framework, we give algorithms with state-of-the-art
complexity guarantees for the general class of scaling and non-commutative
optimization problems, which have been of much recent interest, and we provide
the first algorithms for efficiently finding high-precision solutions for
computing minimal enclosing balls and geometric medians in nonpositive
curvature.Comment: 85 pages. v2: Merged with independent work arXiv:2212.10981 by
Hiroshi Hira
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
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