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Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm

Carl Henrik Ek, Oisin Kim, Challenger Mishra

TL;DR

A novel approach to obtaining Ricci-flat approximations to K\"ahler metrics, applying machine learning within a `principled' framework using gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric.

Abstract

Motivated by recent progress in the problem of numerical Kähler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired by his work, we present a novel approach to obtaining Ricci-flat approximations to Kähler metrics, applying machine learning within a `principled' framework. In particular, we use gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric. We combine this approach with both Donaldson's algorithm and learning on the $h$-matrix itself (the latter method being equivalent to gradient descent on the fibre bundle of Hermitian metrics on the tautological bundle over the Grassmannian). We implement our methods on the Dwork family of threefolds, commenting on the behaviour at different points in moduli space. In particular, we observe the emergence of nontrivial local minima as the moduli parameter is increased.

Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm

TL;DR

A novel approach to obtaining Ricci-flat approximations to K\"ahler metrics, applying machine learning within a `principled' framework using gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric.

Abstract

Motivated by recent progress in the problem of numerical Kähler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired by his work, we present a novel approach to obtaining Ricci-flat approximations to Kähler metrics, applying machine learning within a `principled' framework. In particular, we use gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric. We combine this approach with both Donaldson's algorithm and learning on the -matrix itself (the latter method being equivalent to gradient descent on the fibre bundle of Hermitian metrics on the tautological bundle over the Grassmannian). We implement our methods on the Dwork family of threefolds, commenting on the behaviour at different points in moduli space. In particular, we observe the emergence of nontrivial local minima as the moduli parameter is increased.

Paper Structure

This paper contains 32 sections, 10 theorems, 48 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Complex and Kähler geometry
  4. Algebraic metrics
  5. Existing methods
  6. Neural network approaches
  7. Obstructions to learning
  8. Problems with error composition
  9. Approximate Kähler metrics?
  10. Donaldson's algorithm, in brief
  11. Energy functionals
  12. No free lunch
  13. Donaldson's algorithm
  14. Balanced metrics
  15. Convergence results
  16. ...and 17 more sections

Key Result

Theorem 2.4

Given $(M,J, \omega)$, a compact Kähler manifold, and $\psi$, a real $(1,1)$-form representing $c_1(M)$, there exists a unique Kähler form $\tilde{\omega}$ such that $[\tilde{\omega}]=[\omega]$ and $\rho_{\tilde{\omega}}=2\pi \psi$.

Figures (3)

  • Figure 1: Final test $\sigma$-errors for Grassmann-Donaldson and bundle optimisation on the Fermat quintic and a range of line bundles.
  • Figure 2: $(a)$ Final test error for bundle optimisation and a fixed fraction $N_s=\frac{1}{2}N_k$, on the Fermat quintic. The $y$-axis has a linear scale. $(b)$ Final test errors using bundle optimisation for $\mathcal{O}(5)$ and a range of real moduli parameters on the Dwork. The initial positive matrix is chosen by a QR decomposition and the $y$-axis has a logarthmic scale.
  • Figure 3: All plots are for the Dwork equipped with the $\mathcal{O}(5)$ bundle, with a logarithmic scale on the $y$-axis. (a) Final test error for Grassmann-Donaldson optimisation for a range of real moduli parameters, initialising with QR distribution. $(b)$ Comparison of Grassmann-Donaldson, joint, and $T$-initialised joint optimisations for the $\phi=4$ case. $(c)$ Final test error for joint optimisation with a $T$-initialisation, for a range of real moduli parameters.

Theorems & Definitions (15)

  • Definition 2.1: Fundamental 2-form
  • Definition 2.2: Kähler manifold
  • Definition 2.3: Calabi-Yau manifold
  • Theorem 2.4: Calabi, 1957, Calabi
  • Theorem 2.5: Yau, 1977, Yau
  • Theorem 2.6
  • Proposition 2.7
  • Theorem 3.1
  • Definition 4.1
  • Theorem 4.2
  • ...and 5 more