Gaussian kernel regression as an improved estimator for - - PowerPoint PPT Presentation

Gaussian kernel regression as an improved estimator for magnetization curve and spin gap Tota Nakamura (Shibaura Institute of Technology) (arXiv:1902.02941) “How much information can we extract from (poor) numerical data?”

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 H(0) H(0) 0.501 M/Ms H difference 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 H(0) 0.501 M/Ms H kernel

It also replaces conventional

• numerical differentiation

beyond the difference

• data extrapolation

beyond the least-square method →

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Outline

• 1. The Gaussian kernel regression and the Bayesian inference
• 2. Magnetization curve

(a) S = 1/2 bond-alternation XY spin chain (b) kagome antiferromagnet

• 3. Spin gap and the size extrapolation

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§1. The Gaussian kernel regression and the Bayesian inference (Problem) Obtain a model function y = F(x) for data xi, yi, and ∆yi

0.21 0.22 0.23 0.24 0.02 0.04 0.06 0.08 y x data model function

(Answer) F(x) = d

• ij K(xi, x)C−1

ij yj

with K(xi, x): Gauss kernel function Cij : covariance matrix of data

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§1. The Gaussian kernel regression and the Bayesian inference Recipe

• 1. Define a covariance matrix Cij (i, j = 1, · · · d)

Cij = (∆yi)2δij + K(xi, xj) with a Gaussian kernel function K(xi, xj) = θ2

1 exp

      −(xi − xj)2

2θ2

2

       + θ2

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(xi = xj)

• 2. Calculate determinant and inverse of C
• 3. Find (θ1, θ2, θ3) that maximize the log-likelihood function

log L(θ1, θ2, θ3) = −1 2 log |C| −

• ij

1 2yiC−1

ij yj

• 4. Using (θ1, θ2, θ3), C−1

ij , and data (xi, yi),

F(x) = d

• ij K(xi, x)C−1

ij yj

This is analytically differentiable!

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§1. The Gaussian kernel regression and the Bayesian inference Bayesian inference in the scaling analysis of critical phenomena

• K. Harada, PRE 84, 056704 (2011)
• Include unknown parameters, Tc and ν, in xi and yi

xi = (1/T − 1/Tc)(L/256)1/ν yi = U(T, L) (Binder ratio)

• Find (Tc, ν) and (θ1, θ2, θ3) that maximize log L

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§1. The Gaussian kernel regression and the Bayesian inference

• 1. obtain analytically-differentiable model function

↓ improves numerical differentiation → magnetization curve by H(M) = ∂E(M) ∂M

• 2. estimate unknown parameters if included in (xi, yi)

↓ improves data extrapolation → plateau magnetization → spin gap → extrapolation of N → ∞

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Outline

• 1. The Gaussian kernel regression and the Bayesian inference
• 2. Magnetization curve
• 3. Spin gap and the size extrapolation
• 4. Other applications

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§2 Magnetization curve :Recipe for a standard numerical evaluation

• 1. Calculate the ground state energy E(M)
• 2. Estimate H such that E − HM is minimum

↔ H(M) = ∂E(M) ∂M by the difference

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§2 Magnetization curve :Recipe for the kernel method

• 1. We have E(M) of ∆-chain with 30 spins by using Titpack ver2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.2 0.4 0.6 0.8 1 E/N M/Ms

• riginal data

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§2 Magnetization curve :Recipe for the kernel method

• 2. Assume a plateau point (Mp, E(Mp)) and divide data at Mp
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.2 0.4 0.6 0.8 1 plateau point (trial) E/N M/Ms

• riginal data

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§2 Magnetization curve :Recipe for the kernel method

• 3. Define mirror data with respect to the plateau point
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.2 0.4 0.6 0.8 1 plateau point (trial) E/N M/Ms

• riginal data

mirror data

Now, xi and yi include Mp and E(Mp).

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§2 Magnetization curve :Recipe for the kernel method

• 4. Obtain the model function by the kernel method
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.2 0.4 0.6 0.8 1 plateau point (trial) E/N M/Ms

• riginal data

mirror data model functions

bad guess gives a non-smooth function: low value of log L

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§2 Magnetization curve :Recipe for the kernel method

• 5. Find a plateau point that gives the largest log L

6 8 10 12 14 16 18 0.4 0.45 0.5 0.55 0.6 log L Mp all(800) select(80)

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§2 Magnetization curve :Recipe for the kernel method

• 6. Final result
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.2 0.4 0.6 0.8 1 plateau point (final) (0.5015,-0.2546) E/N M/Ms

• riginal data

mirror data model functions

differentiable function of E(M) is now obtained

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§2 Magnetization curve :Recipe for the kernel method

• 7. The magnetization curve is obtained by

H(M) = ∂E(M) ∂M

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 (b) 0.501 M/Ms H kernel(0.4<Mp<0.6) kernel(Mp=0) kernel(Mp=1) difference

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§2 Magnetization curve :Compare with exact results S = 1/2 Bond-alternation XY chain with 30 spins

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 λ=0.1 λ=0.5 λ=0.3 M/Ms H kernel exact difference

we set Mp = 0

1+λ 1+λ 1+λ 1−λ 1−λ 1−λ

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§2 Magnetization curve :kagome antiferromagnet Using mixed data of 30 spins and 27 spins

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 (b) M/Ms H

kernel difference DMRG

we set Mp = 0, 1/9, 1/3, 5/9,7/9 DMRG(Nishimoto et al. 2013)

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Outline

• 0. Motivation
• 1. The Gaussian kernel regression and the Bayesian inference
• 2. Magnetization curve
• 3. Spin gap and the size extrapolation
• 4. Other applications

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§3 Spin gap Critical field H(0) is another definition for the spin gap 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 H(0) 0.501 M/Ms H kernel

• 0.38
• 0.36
• 0.34
• 0.32
• 0.3
• 0.28
• 0.26

0.1 0.2 0.3 0.4 0.5 E/N M/Ms N=30 26 model function

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§3 Spin gap :S = 1/2 bond-alternation XY model, λ = 0.1 case

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.02 0.04 0.06 0.08 λ=0.1 (b) Spin Gap 1/N H(0) E(1)-E(0) Model Least-sq

exact gap is 0.1 Spin gap(N = ∞)=0.0995(1)

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§3 Spin gap :kagome antiferromagnet 0.05 0.1 0.15 0.2 0.25 0.3 0.01 0.02 0.03 0.04 0.05 (a) Spin Gap 1/N

E(1)-E(0)[PBC] E(1)-E(0)[TBC] H(0)[PBC] H(0)[TBC] Model

PBC: periodic boundary condition TBC: twisted boundary condition Spin gap(N = ∞)=0.0276(2)

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§3 Spin gap :Tips ”Cross Validation by random noise works fine” ∆-chain case

0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.02 0.04 0.06 0.08 Spin Gap 1/Neff H(0) E(1)-E(0) with CV without CV noiseful

0.215 0.22 0.225 0.23 10-6 10-5 10-4 10-3 10-2 Spin gap(N=∞) ∆E(M)

• ver fitting
• ptimal

noiseful Search parameters by data with a noise Validate the results(calculate log L) by data with another noise

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Summary

• We can extract more physical information by using the kernel method.
• It replaces numerical differentiation and data extrapolation.
• Finite but small spin gap in kagome antiferromagnet
• We cannot use machine learning as a black box

– No-Free-Lunch theorem: we need some extra tips – There is no “grand truth”: it works well when it works

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