Quantum Chemistry Energy Regression

Transcription

1 Quantum Chemistry Energy Regression with Scattering Transforms Matthew Hirn, Stéphane Mallat École Normale Supérieure Nicolas Poilvert Penn-State 1

2 Quantum Chemistry Regression 2

3 Linear Regression 3

4 Energy Properties Invariant to permutations of the index k. 4

5 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 5

6 Coulomb Kernel 6

7 Density Functional Theory Computes the energy of a molecule x from its electronic probability density x (u) for u 2 R 3 Organic molecules with Hydrogne, Carbon Nitrogen, Oxygen Sulfur, Chlorine H 4 C 6 OS H 9 C 7 NO H 9 C 8 N 7 H 3 C 6 NO 2

8 Density Functional Theory Kohn-Sham model: E( )=T ( )+ Z (u) V (u)+ 1 2 Z (u) (v) u v dudv + E xc( ) Molecular energy Kinetic energy electron-nuclei attraction electron-electron Coulomb repulsion Exchange correlat. energy At equilibrium: 8

9 Coulomb Interactions in Fourier Coulomb potential energy: Diagonalized in Fourier: 9

10 Coulomb in Fourier Dictionary

11 Large Scale Instabilities 11

12 Coulomb Multiscale Factorizations Multiscale regroupment of interactions: For an error, interactions can be reduced to O(log ) groups Fast multipoles (Rocklin, Greengard) 12

13 Scale separation with Wavelets rotated and dilated: real parts imaginary parts

14 Wavelet Interference for Densities 14

15 Sparse Wavelet Regression 2 1 Theorem: For any > 0 there exists wavelets with 15

16 Dictionaries for Quantum Energies 16

17 Atomization Density Electronic density x (u) Approximate density x (u) 17

18 Sparse Linear Regressions 18

19 xfourier and Wavelets Regressions Regression: Fourier Wavelet Scattering Coulomb Model Complexity log 19 (M) log 2 M

20 Energy Regression Results 20

21 Wavelet Dictionary Rotations 1 (u) j1, 1 (u) Scales j1 21

22 Wavelet Dictionary Rotations 1 Scales j1 j1, 1 (u) 22

23 Scattering Dictionary Rotations 1 Recover translation variability: j1, 1 j2, 2 (u) Recover rotation variability: Scales j1 j1, (u) ~ l2 ( 1 ) j1, 1 (u) 23

24 Scattering Dictionary Rotations 1 Recover translation variability: j1, 1 j2, 2 (u) Recover rotation variability: Scales j1 j1, (u) ~ l2 ( 1 ) Combine to recover roto-translation variabiltiy: j1, j2, 2 (u) ~ l2 ( 1 ) j1, 1 (u) 24

25 Scattering Second Order j1, 1 (u), j 1 fixed j1, j2, 2 (u) ~ l2 ( 1 ) j 1,l 2 fixed Scales j2 25 Rotations 2

26 Scattering Dictionary 26

27 x Scattering Regression Regression: Fourier Wavelet Scattering Coulomb Model Complexity log (M) log 2 M

28 Quantum Energy Regression ChemistryResults 28

29 From 2D to 3D Scattering

30 Reconstruction from Scattering Joan Bruna Original images of N 2 pixels: Order m =2 Reconstruction from {kxk 1, kx? 1 k 1, k x? 1? 2 k 1 } : O(log 2 2 N) coe.

31 Ergodic Texture Reconstructions Original Textures Joan Bruna m = 2, 2 J = N: reconstruction from O(log 2 2 N) scattering coe.

32 Digit Classification: MNIST Joan Bruna x SJ x Linear Classifier Classification Errors LeCun et. al. y = f (x)

33 Complex Image Classification Arbre de Joshua CalTech 101 data-basis: Ancre Metronome Castore Edouard Oyallon Nénuphare Bateau x S J x Rigid Mvt. Classification Accuracy Linear Classif. computes invariants Data Basis Deep-Net Scat.-2 CalTech % 80% CIFAR-10 90% 80% y

34 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 34