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Quantum Chemistry Regression x = { z k (charge) , r k (position) } k - - PowerPoint PPT Presentation

Dec 07, 2022 147 likes •502 views

Quantum Chemistry Energy Regression with Scattering Transforms Matthew Hirn, Stphane Mallat cole Normale Suprieure Nicolas Poilvert Penn-State 1 Quantum Chemistry Regression x = { z k (charge) , r k (position) } k d R 4 d

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Quantum Chemistry Energy Regression

with Scattering Transforms

Matthew Hirn, Stéphane Mallat École Normale Supérieure Nicolas Poilvert Penn-State

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Quantum Chemistry Regression

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x = {zk(charge) , rk(position)}k≤d ∈ R4d without quantum chemistry: very fast.

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Linear Regression

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Energy Properties

  • Invariant to permutations of the index k.

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Overview

  • Coulomb kernel representations
  • Density functional approach to representation
  • From Fourier to wavelet energy regressions
  • Wavelet scattering dictionaries: deep networks without learning
  • Numerical energy regression results
  • Relations with image classification and deep networks

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Coulomb Kernel

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with Organic molecules Hydrogne, Carbon Nitrogen, Oxygen Sulfur, Chlorine H4C6OS H9C7NO H3C6NO2 H9C8N

Density Functional Theory

  • Computes the energy of a molecule x from

its electronic probability density ρx(u) for u ∈ R3

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Kohn-Sham model: E(ρ) = T(ρ) + Z ρ(u) V (u) + 1 2 Z ρ(u)ρ(v) |u − v| dudv + Exc(ρ)

Molecular energy

At equilibrium:

Density Functional Theory

Kinetic energy electron-electron Coulomb repulsion electron-nuclei attraction Exchange

  • correlat. energy

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  • Coulomb potential energy:

Coulomb Interactions in Fourier

Diagonalized in Fourier:

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Coulomb in Fourier Dictionary

ω1

ω2

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Large Scale Instabilities

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(Rocklin, Greengard)

Coulomb Multiscale Factorizations

  • Multiscale regroupment of interactions:

For an error ✏, interactions can be reduced to O(log ✏) groups Fast multipoles

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Scale separation with Wavelets

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ω2

rotated and dilated:

real parts imaginary parts

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Wavelet Interference for Densities

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Sparse Wavelet Regression

For any ✏ > 0 there exists wavelets with Theorem:

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ω2

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Dictionaries for Quantum Energies

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Atomization Density

Electronic density ρx(u) Approximate density ˜ ρx(u)

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Sparse Linear Regressions

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x

Regression:

Fourier and Wavelets Regressions

log2 M

1 2 3 4 5 6 7 8 9 10

Model Complexity log (M)

1 2 3 4 5 6 7 8

  • |

− |

  • Fourier

Wavelet Scattering Coulomb

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Energy Regression Results

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Wavelet Dictionary

|ρ ∗ ψj1,θ1(u)| Rotations θ1 Scales j1 ρ(u)

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Wavelet Dictionary

Rotations θ1 Scales j1 |ρ ∗ ψj1,θ1(u)|

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Scattering Dictionary

Recover translation variability: Recover rotation variability: |ρ ∗ ψj1,·(u)| ~ ψl2(θ1) |ρ ∗ ψj1,θ1| ∗ ψj2,θ2(u)

Rotations θ1 Scales j1 |ρ ∗ ψj1,θ1(u)|

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Scattering Dictionary

Recover translation variability: Recover rotation variability: |ρ ∗ ψj1,·(u)| ~ ψl2(θ1) |ρ ∗ ψj1,θ1| ∗ ψj2,θ2(u) Combine to recover roto-translation variabiltiy:

Rotations θ1 Scales j1 |ρ ∗ ψj1,θ1(u)| ||ρ ∗ ψj1,·| ∗ ψj2,θ2(u) ~ ψl2(θ1)|

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Scattering Second Order

|ρ ∗ ψj1,θ1(u)|, j1 fixed Rotations θ2 S c a l e s j2

j1, l2 fixed ||ρ ∗ ψj1,·| ∗ ψj2,θ2(u) ~ ψl2(θ1)|

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Scattering Dictionary

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1 2 3 4 5 6 7 8 9 10

Model Complexity log (M)

1 2 3 4 5 6 7 8

  • |

− |

  • Fourier

Wavelet Scattering Coulomb

log2 M

x Scattering Regression

Regression:

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Quantum Chemistry Energy Regression Results

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From 2D to 3D Scattering

ω1

ω2

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M .Eickenberg, M .Hirn

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Original images of N 2 pixels:

Order m = 2

Reconstruction from Scattering

Joan Bruna

Reconstruction from {kxk1 , kx ? λ1k1 , k|x ? λ1| ? λ2k1} : O(log2

2 N) coeff.

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Ergodic Texture Reconstructions

Joan Bruna Original Textures

2D Turbulence E(|x ? λ1|) , E(||x ? λ1| ? λ2|)

Second order Gaussian Scattering: O(log N 2) moments

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LeCun et. al.

Classification Errors Joan Bruna

Digit Classification: MNIST

Linear Classifier SJx y = f(x)

x

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Classification Accuracy SJx Data Basis Deep-Net Scat.-2 CalTech-101 85% 80% CIFAR-10 90% 80%

Rigid Mvt.

computes invariants

Complex Image Classification

Bateau Nénuphare Metronome Castore Arbre de Joshua Ancre

CalTech 101 data-basis:

Linear Classif. y x

Edouard Oyallon

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Conclusion

  • Quantum energy regression involves generic invariants to rigid

movements, stability to deformations, multiscale interactions

  • These properties require scale separations, hence wavelets.
  • Multilayer wavelet scattering create large number of invariants
  • Equivalent to deep networks with predefined wavelet filters
  • Knowing physics provides the invariants: can avoid learning

representations

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