Distance between configurations in MCMC...

Distance between configurationsin MCMC simulations and

the emergence of AdS geometryin the simulated tempering algorithm

N. Matsumoto (Kyoto Univ) and N. Umeda (PWC)

Masafumi Fukuma (Dept of Phys, Kyoto Univ)

＠ESI, July 9, 2018“Matrix Models for Noncommutative Geometry

and String Theory ”

[arXiv:1705.0609, JHEP 1712 (2017) 001 (FMN1), arXiv:1806.10915 (FMN2)]

based on work with

1. Introduction

Motivation (1/2)There has long been an expectationthat quantum mechanics has its origin in randomness.

Question: Can quantum gravity be treated in such a framework?

Main purpose of my talk is to show:

• One can introduce a geometry to any stochastic systemwhich is based on Markov-chain Monte Carlo (MCMC), s.t. it reflects the difficulty of transitions betweentwo configurations.

• Such geometry possesses a larger (or largest) symmetryif the algorithm is optimized s.t. distances are minimized.

• This distance gives another method to introducea geometry to matrix models

[Nelson, Parisi-Wu, ...]

( )2 2( ) ( 1) 12

To think more concretely, let us consider the action

S x xβ β= −

Separation of and that of are almost the same in space.

B B Cx

A −−

A B C

can be reached from easily cannot be reached from easily

BC BA “close” in MC

“far” in MC

We introduce a measure that enumerates this “distance”.

However, in MCMC simulations,

potential barrier

x

( )S xMotivation (2/2)

- This definition is universal for MCMC algorithmsthat generate local moves in configuration space

- The distance gives an AdS geometrywhen a simulated tempering is implemented for multimodal distributions with optimized parameters

x x

β

1−1−

1+1+

{ }original config space x {( , )}extended config space x β

geodesic in AdS space

Main resultsMain results

( )2 2( ) ( 1) 12

S x xβ β= −

- This gives another method to introduce a distancein matrix models

Plan

1. Introduction (done)2. Definition of distance

- preparation- definition of distance- universality of distance

3. Examples- unimodal case- multimodal case

4. Distance for simulated tempering- simulated tempering- emergence of AdS geometry- AdS geometry from matrix models

5. Conclusion and outlook

Plan

1. Introduction2. Definition of distance





{ }( )

: configuration space: action

xS x

=

( )( ) ( )

( ) :1( ) ( )

We want to estimate VEVs of operators S x S xdx

x

x e xZ

dxZ e− −≡ =∫ ∫

In MCMC simulations:

0

( )

1 1

1( )

( ) ( |( ) ( ) ( |) ) ( )( ) ( )

eq

eq s.t. converges uniquely to in the

- Regard as a PDF

- Introduce a Markov chain

li

mit

S x

nn n n

n

P x

p x eZ

p x p x dy p y dy P x y p yp x p n

yx

−

− −

≡

→ = =

→∞∫ ∫

Preparation 1: MCMC simulation (1/3)

( )0( | ) ( )eqi.e., n x y pP x n n≥

transition probability matrix

0 1 2

1

, ,...( | )

- Starting follow

from an initial valuing the transition m

e , generate atrix i i

x x xP x x −

1,...,{ }- Take a sample fter the system is well re a laxed m Mmy =

0 0 0 00 1 1 2n Mn n nx x x x x x+ + +→ → → → → →→ →

1y≡ 2y≡ My≡

relaxation ( )eqgenerated with p x

( )- Estimate VEVs of operators as a sample aver age:x

1

1( ) ( )M

mmx y

M =

≈ ∑

We first would like to establish a mathematical framework which enables the systematic understanding of relaxation


P P P P P P P P

We assume that

( )( ) ( )

( | )( | ) ( ) ( | ) ( ) ( | ) ( | )eq eq

satisfies the detailed balance condition:

are po

(1)

(2) all of the eigenvalues sitive of

S y S x

P x yP x y p y P y x ep x P P y x

Pex y − −⇔ ==

NB : (2) is not too restrictive

2

In fact, if as the elementary transition matrix,

for which - all the eigenvalues are positiv

has n

e -

egative eigenvalues, then we

the same detailed balance co

instea

ndition

d ca

is

n u

sa

se

i

tisf

PP

2 ( ) 2 ( )( | ) ( | ) :

d a

e s

S y S xxP

P e ey P y x− −=

NB : (1) can be written asˆ ˆ( ) ( )ˆ ˆS x S x TePe P− −=

ˆ( | ) | |ˆ | |P x y x P yx dx x x x

= ⟨ ⟩ ≡ ⟩⟨ ∫


ˆ ˆ( )/2 ( )/2ˆ ˆ S x S xT e Pe−≡

We introduce the “transfer matrix” :

properties:( )ˆ ˆ( ) ( )ˆ ˆ ˆ ˆ

ˆ(1)

(2) same eigenvalue se (thus t as a ll positive)

T S x S x TT T e eP P

P

− −== ⇔

0 1 21 0 We order the EVs as

λ λ λ= > ≥ ≥ >

spectral decomposition:

0 1| | 0ˆ | 0 | ||k k

k kT k kk kλ λ

≥ ≥

= ⟩⟨ = ⟩⟨ + ⟩⟨∑ ∑where

( )/21| 0 ( )eqS xx e x

Zp− =⟨ ⟩ =

Preparation 2: Transfer matrix (1/2)

( )( )/2 ( )/2(| ) |( )S x S yP x y eT x y e −⇔ =

[MF-Matsumoto-Umeda1]

Preparation 2: Transfer matrix (2/2)

relaxation to equilibrium

1

ˆ ˆ 0 | || 0 |n n nk

kP kT kλ

≥

= ⟩⟨ + ⟩⟨⇔ ∑Note that

ˆ | 0 0 || 1

in the limi relaxation of to decoupling of modes

t with

nT nk k

⇔⇔

⟩⟨ →∞⟩ ≥

NB:

decoupling occurs earlier for higher modes (i.e. for larger )kNB:

1/1 ~ can be estimated frelaxat rom ion time e ττ λ −

1 ~ 1slow relaxation λ⇔

( )0 1 21 0λ λ λ= > ≥ ≥ >

Preparation 3: Connectivity between configs (1/3)

(set of sequences of processes in )n n≡X

1 2

2 1

, )(set of sequences of processes in that start from and end at

n xn

x x

x

≡

X

1x2x

We define the connectivity between two configs as

1 21 2

( , )( , ) n

nn

xf

xx x ≡

XX

12

1 2 2

( ) ( )1 2 12 2 1

1 1( | ) ( | ) ( , )

(prob to obtain ) (prob to hav from e )S x S xn n

n

x x x

P x x e P x x e x xZ

fZ

− −

= ×

= = =

det balance



normalized connectivity (“half-time overlap”):

1 21 2

1 1 2 2

1 21 2 2 1

1 1 2 2 1 1 2 2

)

( , )( , )( , ) ( , )

( ,( | ) ( | )( | ) ( | ) ( , ) ( , )

nn

n n

n nn

n nn n

f x xx xf x x f x x

K x xP

F

x x P x xP x x P x x K x x K x x

≡

= =

( )1 2 1 2ˆ( , ) | |n

n x x x TK x≡ ⟨ ⟩

1 2( , ) is actually the overlap between two normalized "half-time" elapsed states:

n xF x

1 2 1 2( , ) , / 2 | , / 2n x x x n xF n≡ ⟨ ⟩

1x2x/ 2n

/ 2n

1

2

1 2( )

1 2

)2

(1

( , )1( | )1( | )

nS xn

S xn

f x x

P x x eZ

P x x eZ

−

−

=

=

( )/2 /2ˆ ˆ| , |/ 2 |||| |/n nx xx n T T⟩ ≡ ⟩ ⟩

1 1( , ) 1n xF x =


1 2( , )properties of nF x x

( )( )

1 2 2 1

1 2

1 2 1 2

1 2 1 2

( , ) ( , )0 ( , ) 1

( , ) 1lim ( , ) 1 ,

(1) (2) (3) when is finite(4)

n n

n

n

nn

F x x F x xF x x

F x x x x nx x x xF

→∞

= ≤ ≤ = ⇔ =

= ∀

1 2

1 2

1 2

1 2

( , ) 1

( , ) 1

(A) If can be easily reached from in

(B) If and are separated by high potential barr

steps, then

, then

ie s

rn

n

x x nF x x

x xF x x

1 2 1 2 1 1 22 1 2

:ˆ | 0 0 |

ˆ( , ) | | | 0 0 | ( , ) ( , ).

proof of (4)In the limit , , and thus,n

nn n n

n TK x x x T x x x K x x K x x

→ ∞ → ⟩⟨ = ⟨ ⟩ → ⟨ ⟩⟨ ⟩ =

1 21 2

1 1 2 2

1 2

( ,( , )( , ) ( , )

, / 2 / 2

)

| ,

nn

n n

K x xx xK x x K x x

x n x n

F =

= ⟨ ⟩

Definition of distance

1 2( , )properties of n x xθ

( )( )

1 2 2 1

1 2

1 2 1 2

1 2 1 2

1 2 2 3 1 3

( , ) ( , )( , ) 0( , ) 0

lim ( , ) 0 ,( , ) ( , ) ( , )

(1) (2) (3) when is finite(4) (5)

n n

n

n

nn

n n n

x x x xx xx x x x n

x x x xx x x x x x

θ θθθ

θ

θ θ θ→∞

=

≥ = ⇔ = = ∀ + ≥

1 2

1 2

1 2

1 2

( , )

( , )

(A) If can be easily reached from in

(B) If and are separated by high potential ba

steps, then : small

,

rrithen : large

ersn

n

x x nx x

x xx x

θ

θ

1 2 1 2( , ) arccos( ( , ))n nx x F x xθ ≡[MF-Matsumoto-Umeda1]

Alternative definition of distance

21 2 1 2( , ) 2 ln ( , )n nd x x F x x≡ −

1 2 1 2( , ) arccos( ( , ))Instead of ,one can also use the following as distance:

n nx x F x xθ =

21 2 1 2( , ) 2[1 ln ( , )]n nD x x F x x≡ −or

we will mainly use this

21 2( , ) 2

1 2 1

1

(12

2

/2)

( ,1( , ) ( , )cos2

)

1

0They agree when

n

n

n

d x xn nx x D

x x

x

F

xeθ

θ

−

= = − ≈

=

NB: analogy in quantum information

1 21,2 1,2 1,2

1 2

( , )| , / 2 , / 2 |

( , ) : Bures length for two pure states : Bures distance

n

n

x xx n n

x xDx

θρ = ⟩⟨


Universality of distance (1/4)The above distance is expected to be universalfor MCMC algorithms that generate local moves in config space.

"universal" in the sense that differences of distance between two such local MCMC algorithmscan always be absorbed into a rescaling of n

In fact, 2

1 2 1 2 1 2ˆ

ˆ( , ) , |ˆ

) |( universality of univ. o

f

univ. of

nn n

H

xd x x K x x T xT e−

⇔ =

⇔

⟨ ⟩

≡

ˆthen are expected to be local operatorsacting on fun

If algorithm

ctions over

s are sufficientl

in almost the

y l

sam

ocal,

e way.H

and,

| must be almosThe wave functi t the same for smalon l s x k k⟨ ⟩


Universality of distance (2/4)This expectation can be explicitly checked using a simple model.

algorithm 1: Langevin

1 ,( ) 2 with n n n n n m n mxx x S νν ν ν δ+ + − ′= ⟨ ⟩ =

( )

21 ( )ˆ 4 2

2

1ˆ| | | |4

( ) (1/ (1/ 2)4) ( ) ( ) with

x yx y VHx T y x e y e

V x S x S xπ

+ − − − − ⟨ ⟩

= −

= ⟨ ⟩

′ ′′

2(with Gaussian proposal of variance )σalgorithm 2: Metropolis

( )2

2

22

( )/2 ( )/2

1 ( )( ) ( ) ( )/2 ( )/222

1 1( ) ( ) ( )22

2

ˆ ˆ| | | |11,

21

2

min

S x S y

x yS x S y S x S y

x y S x S y

x T y x P y e

e e e

e

σ

σ

πσ

πσ

−

− −− +

− −−

−

−

=

⟨

=

⟩ = ⟨ ⟩ ×

×

Universality of distance (3/4)2 ~1ˆ ˆln 0,

| | | ( ) ( ) |

,

both Hamiltonians become local in the limit

and have the same tendency to enhance transitionswhen

With the identificati

and are small.

on

H T

x y S x S y

σ ≡ − →

− −

ˆThe l show e ouldnergy st be almoructu st thre e o .f sameH

The global structure of distan should be almost the sce ame.

In fact,the universality actually holds more than expectedeven for a single DOF

The argument for universality is more trustworthyas the DOF of the system become larger.

0 0 0 0 01 7.81 x 10^(-4) 1 7.62 x 10^(-4) 12 36.2 4.63 x 10^4 34.2 4.49 x 10^43 58.2 7.45 x 10^4 54.7 7.17 x 10^4

Universality of distance (4/4)

(Lang)kE (Met)kE1/ (Lang)k EE 1/ (Met)k EE

eigenvalues :

eigenfunctions :| 0x⟨ ⟩ |1x⟨ ⟩ | 2x⟨ ⟩ | 3x⟨ ⟩

: Langevin: Metropolis (almost indistinguishable)

( )2 2( ) ( 1) 202

S x xβ β= − =

Plan






Transfer matrix for Langevin

Langevin equation (continuum)0 0

'( )

2 ( ') witht

t tt t

tx xx S x

t tνν

ν ν δ= =′ −

=

⟨ ⟩ = −

ˆ( )/2 ( )/2( , ) ( | ) | |S x S y Ht tK x y e x y e x eP y− −= ⟨= ⟩

0 ,[ ])(t tx xx ν=

( ) ˆ0 0 0( | ) ,[ ]) | |( FPtH

t tP xx x x x x e xν

δ ν −⟨=≡ − ⟩

ˆ ( )][FPwith x xH S x′∂ ∂ += −

( )

ˆ ˆ( )/2 ( )/2 2

2

( )

(1/

ˆ ˆ ˆ

( ) (1/ 4) ( ) 2 () )FPwith S x S x

xH H x

V x S S

e

x

V

x

e−= = − +

−

∂ ′ ′′=

21 2

1 ( , )1 2 21 2

1 1 2 2

( , )( , )( , ) ( , )

td x xtt

t t

K x xx x eFK x x K x x

−= =

Example 1: Unimodal distribution (Gaussian)

ˆ

2 21 2 1 22

( , ) | |

exp [( ) cosh 22 (1 ) 4sinh

]

tHt

t

K x y x y

x x t x xe t

e

ω

ω ω ωπ ω

−

−

= ⟨ ⟩

= − +−

−

( )22

22

ˆ ˆ) ( ) (1/( (1/ 24) ( ) ( )

4

)

2

with

xH xV V x S x S x

xω ω′ ′′= − +∂

=

−

−

=

2 2 21 2 1 2 1 2( , ) | | ~ | |

2sinht

t x x x x e x xt

d ωωω

−= − −

2(2

)S x xω=

subtractszero-point energy

We see that:

2~ 1/ ~ ( )- geometry is flat and translationally invariant- is given by relaxation tim e V xτ τ ω ω ′′

Example 2: Unimodal dist. (non-Gaussian)

3 2 32 2 2

3 21 2

3

1

2 21 2

2 1 2 4

2

[12( 3 3 2 )

( 3 3 )( )3 ( 3 3 3 3 2 )( ) ]

( , ) | |2 8

( )

{

}

t s s c t ts ts c

s s tc x xs s tc t sc

x x x xs s

Ots x x

d ω λω

ω ω ω

ω ωω ω ω ω λ

− + + −

+ +

= −

+

− −

+ + − + +

−

− +

2 4

2( )

4xS xx ω λ

= +

We see that:

2~ 1/ ~ ( )- geometry is no longer flat or translationally invariant- is again give relaxa n by tion time V xτ τ ω ω ′′

perturbative expansion in :λ

( )cosh , sinht tc sω ω≡ ≡

Example 3: Multimodal dist. (double well) (1/2)

2 2 12

( ) ( 1) ( )S xx β β= −

2ˆ ˆ( ) x VH x+= −∂

2 6 2 4 2 2

2 2 2 2

2 ( )( ))( (

31)

x xV x xx x O

β β β β ββ β

− + ++

= −= −

with

( )S x ( )V x

2( )O β

)(O βx x

Example 3: Multimodal dist. (double well) (2/2)20For :β =

04

1

2

3

07.81 1036.258.2

EE

E

E

−

== ×==

( )V x

2( )O β

x

( )instanton Oe β−

( )| 2,3,|1

: decouple quickly : decouples very slowly

x k kx⟨ ⟩ = …⟨ ⟩

In fact,10 39.150 19.2100 16.9500 13.2

1,000 11.75,000 8.46

2ˆ ˆ( ) x VH x+= −∂

n 2 ( 1, 1)nd − +

decreases only very slowly

Plan






Simulated tempering (1/3)

x x

β

1−1−

1+1+

{ }original config space x {( , )}extended config space x β

0( ; )( ; )

.

is multimodal,it often happens that becomes lesEven when th

s multimodale original

if we t

acti

ake

on

smaller

S xS x

ββ

β

Basic idea of tempering :

We extend the configuration spaces.t. configurations in different modes can be reachedfrom each other by passing through small 's.β

( )2 200 0( ; ) ( 1) 1

2S x x

ββ β= −

[Marinari-Parisi]


{ }{ ( , )}( 0,1,..., ) to

- Extend the config spac ;

e

a

xX x x a Aβ

=× = = ∈ =

Realization

1( ; )

( ) ( )( ) ( ) ( , )eq eq

s.

- Introduce a stochastic process

t. a

n nS x

n a an

P X P XP X P X P x w e ββ

+−→∞

→→ = =

x

β

0β1β

Aβ

NB :( ; )( ( ), )eq

-th subsampl(appearance probability e)

o

f aS x

a a a a

adx P x w dxZ eZ ββ −= = =∫ ∫

1/

( 1/ ( 1))

is often set as ,which ensures that the desired 0-th configs appearwith nonvanishing probability

a a aw w Z

A

∝

= +

0- Estimate the VEV by only using the subsample with aβ =

considerationnot necessaryfor paralleltempering


( , ) ( , ) direction,

with some proper algorithm (such as Langevin or Metropo

(1) Generate a transition in th

i

e

l s)

a a

xX x X xβ β′ ′= → =

Algorithm

NB :

0 0{( , )} ( 1,..., )(3) Extract a subsample with ; a mx m Mβ β= =

1( , )

( , )

( , ) ( , )

1,

direction, (2) Generate a transition in

with the probability mi

th

n

e

a

a

a a aS x

aS x

a

X x X xw ew e

β

β

ββ β

′

′= ±−

′−

′= → =

01

1( ) ( )(4) Evaluate VEVs as M

mm

x xMβ

=∑

-dependence of -direction is easy.

This adjustment is usually done manually or adaptively

should be chosen s.t. the transition in the

aa ββ

.We will show that this can be done geometrically.

x

β

0β

aβ

xAβ

(1)(2)

0x

Distance for simulated tempering

The introduction of tempering should be seenas the reduction of distance.

In fact,

10 39.150 19.2100 16.9500 13.2

1,000 11.75,000 8.46

n 2 ( 1, 1)nd − +26.57.164.350.7080.106

2.78 x 10^(-8)

2 ( 1, 1)nd − +

w/o tempering w/ tempering

rapid decreasing


Coarse-grained configuration space (1/4)In MCMC simulations,the most expensive part is the transitionsbetween configs in different modes,and thus, configs in the same mode can beeffectively treated as a point.

This leads us to the idea of "coarse-grained config space"

We would like to show that

when the original config space is multimodal with high degeneracy, the extended coarse-grainined config space naturally has an AdS geometry

×

x

x1− 1+

1− 1+

[MF-Matsumoto-Umeda1,2]

Coarse-grained configuration space (2/4)[MF-Matsumoto-Umeda2]

21 2

1 2/2

1 2

1 2

( , )

1 2

1 2

( , )

( ,

( ,( , )

( o )s

)

, ) cthat are related with the half-time overlap as

We have two different distances, an

d

n

n

d x xn

n

n

n

F x xF x x e

x x

x

d x x

x

θ

θ −= =

1 2( , ) always satisfies triangle inequalitybut takes a complicated form even for Gaussian distribution.

n x xθ

1 2( , ) does not satisfy triangle ineq generically,but gives a flat geometry for Gaussian distribution. This does satisfy triangle ineq in the coarse-grained config space.

nd x x

[ ]( )( ,0)

1 cos2 w/o tempering

for the actiondistanc

e nd x

xS x β π= −

1 2( , )We will regard as more fundamentalwhen discussingcoarse-grained config space

nd x x

NB

Coarse-grained configuration space (3/4)

0 02( ; ) cos1action: xS x πβ β

− =

= original config space:

x

( )S x

coarse-grained config space: (1D lattice with spacin g )=

sim temp

[ ]{ ( , )}extended coarse-grained config space: (1D lattice with spacin ) g aX x xβ× = = ∈

x

⊕

Coarse-grained configuration space (4/4)

( )2 2( , ), ( , ) const. qn x x dx dxd β β β+ =

and set( )2 2( , ), ( , ) ( )nd x x d f dβ β β β β+ =

( )2 2

2 2

( , ), ( , ). )( const

Then we have n

q

ds d x x dx dd fx d

β β ββ β β

≡ +

= +

+

2( ) 1/If (#) is scale invariant (i.e., ),this gives an AdS metric:

f β β∝

---- (#)

( )2

2 2 2 22 2. .const const constq dds d dzdx x

zβββ

+= =+ ( )/2qz β −∝

We assume that

x

β

0β

(This is actually an asymptotic AdS with a horizon)

∞

horizon

boundary

xx dx+x11 when not tempered when tempered

qq=

<

aβ

a

AdS geometry as a result of optimization (1/4)

/

00

( 0,1,..., )

,

is chosen as

di

If

one can show that geometry in becomes scale invariant, so that we will obtain an Ad

reS geometry,

as we saw befor

ction

e.

aa A

Aa

a Aβ

ββ ββ

β

=

=

------------------- (##)

One can actually confirm that (##) is the best choicefor minimizing the distance in simulated tempering:

[ ]

( )

0 01

2

1 2 82

0 0

( ; ) cos(2 )

{ , ,..., }(0, ), ( 1 )

1

,

Consider the action :

tha

Search fort minimize

i

n

iS x

d

β β π

β β ββ β

=

−

=

+

∑x

( )50 10β =


“horizon”1

10

310210

410

510

0 2 4 6 8

AdS geometry as a result of optimization (2/4)That is,

( ) ( )

/

00

22 2 2 2 /2

2 2

( 0,1, 2,..., )

. .

for large

AdS metric :

optimize s.t. the distance is minimized

coconst const nst

a

a A

Aa a

q qi i d

a

dds dx dx zz

z

β

ββ β ββ

ββ ββ

−

= =

+= ∝+

=

This is the first example of the “emergence of AdS geometry” in nonequilibrium systems.


Algorithm to determine the metric2

2 2 22

1

(1) Set an ansatz: iq

i

N

d dxds β αββ =

=

+

∑

( )( , ), ( , ) for various and (2) Calculate distances n a a ad β β β0 x x

( , ) ( , ) :geodesic distance betw and een β β0 x

( )2

/2

| | 16 | |( 4, ; ), , ln

4

q

q

q qq

qI

α β αβ α

β

−

−=+ +x x

x

( , , ) that m(3) Find th inimize thee val costues of functionqα

( ) 2, , ( , ), ( , ) ( , ; , )( ) ,n a a a

aC q d I qα β β β α ≡ − ∑∑

x0 x x

x

β0β

β

x0geodesic


Result

2 22 2 2

21

q

ii

dds dxβ αββ =

=

+

∑

3

4

(5.3 0.3) 10(4.1 1.0) 10

0.020.39qα

− =

= ± ×= ± ×

±

0 2 4 6 8 10�y�

1

2

3

4

dn

a=0

0 2 4 6 8 10�y�

1

2

3

4

dn

a=1

0 2 4 6 8 10�y�

1

2

3

4

dn

a=2

| |x | |x | |x

nd nd nd0a = 1a = 2a =0( 0)10β = 1( 0)5β 2( 5)2β

( 2)N =

Plan






Conclusion and outlookWhat we have done:- We introduced the concept of “distance between configs”

in MCMC simulations- The distance satisfies desired properties as distance- This can be used for the optimization of parameters,

and AdS geometry appears as a result of optimization

Future work:- Establish a systematic method for optimization

- Investigate the relationship between the obtained resultand the Einstein equation

- Investigate the geometry of matrix models further

( )[ ( )]

would be found by simply solving the E-L eq of some functionale.g.,

a a

I aβ β

β=

Thank you.

Date post:	30-Oct-2019
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Distance between configurations in MCMC...

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