GRAPH LAPLACIANS FOR ROTATION EQUIVARIANT NEURAL NETWORKS Master - - PowerPoint PPT Presentation

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graph laplacians for rotation equivariant neural networks
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GRAPH LAPLACIANS FOR ROTATION EQUIVARIANT NEURAL NETWORKS Master - - PowerPoint PPT Presentation

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GRAPH LAPLACIANS FOR ROTATION EQUIVARIANT NEURAL NETWORKS Master thesis in Computational Science and Engineering School of Basic Sciences Department of Mathematics cole Polytechnique Fdrale de Lausanne Supervised by: Michal

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GRAPH LAPLACIANS FOR ROTATION EQUIVARIANT NEURAL NETWORKS

Supervised by: Michaël Deferrard (EPFL) Nathanaël Perraudin (SDSC – ETH) Piercesare Secchi (Politecnico di Milano) Pierre Vandergheynst (EPFL)

16.07.19 Martino Milani 1

Master thesis in Computational Science and Engineering School of Basic Sciences Department of Mathematics École Polytechnique Fédérale de Lausanne

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Rotation Equivariant Filtering

Martino Milani 2 16.07.19

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Martino Milani 3 [1]

[1] Starry: analytic occultation light curves, Luger R. et al., https://rodluger.github.io/starry/v0.3.0/tutorials/basics1.html

Fourier analysis on the sphere and the spherical harmonics

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  • Convolutions can be done in the spectral domain as usual:
  • Complexity of FFT algorithms is

.

  • Convolutions are ro

rotation equivariant operations

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Convolution on the sphere

Definition: .

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[2] Computing Fourier Transforms and Convolutions on the 2-Sphere, Driscoll J.R. and Healy D.M., 1994.

[2]

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Our tool to perform fast spherical convolutions: graphs.

Adjacency matrix

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  • The graph Laplacian is symmetric:

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Graph convolutions

  • Given a weighted adjacency matrix , define the graph Laplacian to be
  • Convolving on the graph a signal with a kernel

is defined as: Notice the similarity with

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KEY CONCEPT: if if th then the graph Fourier transform will be rotation equivariant.

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How to build a rotation equivariant graph

Heat Kernel Graph (HKG): Belkin et al. proved that with a random sampling scheme, in probability [3]

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[3] Convergence of Laplacian Eigenmaps, Belkin M. and Nyiogi P,, in Advances in Neural Information Processing Systems 19, 2007.

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  • The HKG Laplacian eigenvectors well approximate the spherical

harmonics:

  • The dot product of the HKG Laplacian eigenvectors and any sampled

signal well approximates the corresponding Fourier coefficient:

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HEALPix: equiarea sampling scheme

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  • DeepSphere[4] is a Spherical Graph Convolutional Neural Network
  • Each layer implements a po

polynomial filter of a sp sparse se HKG Laplacian of a spherical signal sampled with HEALPix.

  • Filtering is

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DeepSphere 1.0

0.10

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[4] DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications, Perraudin N., Defferrard M., Kacprzak T., Sgier R., 2018

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  • DeepSphere 2.0 is an improved version of DeepSphere
  • Each layer implements a po

polynomial filter of a sparse (but no not to too sp sparse se) HKG Laplacian of a spherical signal sampled with HEALPix.

  • Filtering is

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DeepSphere 2.0

0.10

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Non-equiarea sampling schemes

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Figure: left, FEM diffusion of a point source signal. Right, diffusion obtained with the HKG Laplacian

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The Ga Galer erkin pr probl blem is a discretized version of the weak eigenvalue problem, where the ambient space is finite dimensional:

Towards the Finite Element Method (FEM)

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The Finite Element Method (FEM) is a method to solve the Galerkin problem where the space 𝑊

" is the space of continuous piecewise linear

functions defined on a triangulation of the sphere.

The Finite Element Method (FEM)

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Writing the Galerkin problem times, each time with we obtain the following al algebrai aic generalized eigenvalue problem:

The Finite Element Method (FEM)

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FEM convergence

Remark: it’s the on

  • nly case (together with the random HKG Laplacian) where

we have a convergence theorem.

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[5] A priori estimates for the FEM approximations to eigenvalues and eigenfunctions of the Laplace-Beltrami operator, Bonito et al., 2017.

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  • Equivalent to finding the decomposition
  • The eigenvectors are such that:
  • FEM Fourier transform:

FEM Fourier transform

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Filtering a signal with a kernel is defined as the following matrix multiplication, where is the FEM Fourier matrix.

FEM filtering

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  • is not symmetric.
  • is full.
  • .

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FEM polynomial filtering

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  • is not symmetric.
  • is sparse.

Lumped FEM Laplacian

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Lumped FEM Laplacian as a graph Laplacian

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Equivariance error and computational time

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Conclusions

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1) We gained a better understanding of the HKG. 2) We put that knowledge in practice, improving the Equivariance Error of DeepSphere. 3) We investigated different Discrete Laplacians, from Differential Geometry to Numerical Mathematics 4) We used this knowledge to better understand the advantages and limitations of Graph Laplacians when it comes to non uniform sampling of the sphere.