Sampling constrained probability distributions using ... · Sampling constrained probability distributions using Spherical Augmentation Shiwei Lan1 Babak Shahbaba2 1Department of

Sampling constrained probability distributions

using Spherical Augmentation

Shiwei Lan1 Babak Shahbaba2

1Department of Statistics, University of Warwick, Coventry, UK

2Department of Statistics, University of California, Irvine, US

WCPM, June 4, 2015

Lan, S. and Shahbaba, B. (UW and UCI) Spherical Augmentation 06/04/2015 1 / 51

Motivationq = Inf q = 4 q = 2

q = 1 q = 0.5 q = 0.1


Background

Sampling from probability distributions with constraints is common:

Lasso, Bridge, probit, copula, and Latent Dirichlet Allocation, etc.

Direct truncation is easily doable but computationally wasteful.

Neal (2010) discusses a modified HMC algorithm for which the

sampler bounces off the boundary once hitting it (Wall HMC).

Brubaker et al (2012, constrained HMC on implicit manifolds),

Pakman and Paninski (2012, exact HMC for truncated Gaussian),

Byrne and Girolami (2013, Geodesic Monte Carlo on embedded

manifolds), etc.


1 Review: from HMC to RHMC

2 Spherical Augmentation

Simple examples: ball and box

General q-norm constraints

Some functional constraints

3 Spherical Monte Carlo

Spherical HMC in the Cartesian coordinate

Spherical HMC in the spherical coordinate

Spherical LMC on the probability simplex

4 Experiments

5 Conclusion and future work


Review: from HMC to RHMC

Hamiltonian Monte Carlo

U()

=H

p

p = H

Position RD = variable of interest

Momentum p RD = fictitious, usually N (0,M)

Potential energy U() = minus log of target density f ()

Kinetic energy K (p) = minus log of momentum density

Hamiltonian H(,p) = U() + K (p) = constant.



Hamiltonian Monte Carlo

Definition 1 (Hamiltonian dynamics)

= pH(,p) = M1p

p = H(,p) = U()

Leapfrog: numerical integrator

p(t + /2) = p(t) (/2)U((t))

(t + ) = (t) + M1p(t + /2)

p(t + ) = p(t + /2) (/2)U((t + ))

Run for L steps and accept the joint proposal of z := (,p) with

= min{1, exp(H(z) + H(z))}



Riemannian Hamiltonian Monte Carlo

On the manifold {f (;)} with metric G () = Ex|[2 log f (x;)]:

H(,p) = U() + K (p,)

= log () + 12

log det G() +1

2pTG()1p

() + 12

pTG()1p

where p| N (0,G()). Girolami and Calderhead (2011) propose:

Definition 2 (Riemannian Hamiltonian dynamics)

=

pH(,p) = G()1p

p =

H(,p) =() +1

2pTG()1G()G()1p



Lagrangian Monte Carlo

To resolve the implicitness of RHMC, Lan et al. (2012) propose

Definition 3 (Lagrangian Dynamics)

= G()1p

p = () +1

2pTG()1G()G()1p

p vwww Lagrangian Dynamics

= v

v = vT()v G()1()

Not Hamiltonian dynamics of (, v)!

An explicit integrator can be found more efficient.



Geometry helps!RWM

1

-2 -1 0 1 2

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

HMC

1

-2 -1 0 1 2

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

RHMC

1

-2 -1 0 1 2

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

LMC

1

-2 -1 0 1 2

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


Spherical Augmentation










4 Experiments



Spherical Augmentation Simple examples: ball and box

Change of the domain: from unit ball BD0 (1) to sphere SD

A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B


Spherical Augmentation Simple examples: ball and box

Change of the domain: from rectangle RD0 to sphere SD

2

A

B

x1 = cos1x2 = sin1cos2xD = sin1sinD1cosDxD+1 = sin1sinD1sinD

x

x1

A

B


Spherical Augmentation General q-norm constraints

Mapping q-norm constrained domain to unit ball

= sgn()||(q 2)

||||q 1

||||2 1

0 < q <

= ||||||||2

|||| 1

||||2 1

q =


Spherical Augmentation Some functional constraints


linear M linear constraints l A u, with A an M D matrix, aD-vector and l,u both M-vectors.

Assume M = D and ADD invertible. A1l A1u not true.

Sample := X with l u and transform back to = A1.

quadratic Quadratic constraints l TA + bT u with l , u > 0 scalars.

Assume A = QQT > 0. Use 7 =QT( + 12A

1b):

} : l 22 = ()T u, l = l + 1

4bTA1b, u = u +

1

4bTA1b

It can further be mapped to unit ball:

T}B : BD0 (u)\BD0 (

l) BD0 (1), 7 =

22

l

u l


Spherical Augmentation Some functional constraints

An example of linear constraints0 0.51 + 2 2 and 0 1 + 2 2

1

2

2 1 0 1 2

0.0

0.5

1.0

1.5

2.0

0.0

8

0.0

8

0.1

0.1

0.1

2

0.1

2

0.14

0.14

0.16

0.1

8

Truth

1

22 1 0 1 2

0.0

0.5

1.0

1.5

2.0

Estimate by exact HMC

1

2

2 1 0 1 2

0.0

0.5

1.0

1.5

2.0

Estimate by cSphHMC

1

2

2 1 0 1 2

0.0

0.5

1.0

1.5

2.0

Estimate by sSphHMC

1

2

2 1 0 1 2

0.0

0.5

1.0

1.5

2.0

0.1

0.2

0.2

0.3

0.3

0.4

0.5

0.6

0.7

0.7

Truth

1

2

2 1 0 1 2

0.0

0.5

1.0

1.5

2.0


1

22 1 0 1 2

0.0

0.5

1.0

1.5

2.0

Estimate by cSphHMC

1

2

2 1 0 1 2

0.0

0.5

1.0

1.5

2.0

Estimate by sSphHMC

upper row: N (,) with =

[0

1

], =

[1 0.5

0.5 1

]lower row: f () sin

2 Q()Q() , Q() =

12 ( )

T1( )


Spherical Monte Carlo










4 Experiments



Spherical Monte Carlo

Change of variablesDenote the original parameter as and the constrained domain as D.We use to denote the coordinate of sphere SD . Change variables

Change of variablesDf ()dD =

Sf ()

dDdS dS (3.1)

The energy functions will be changed to

() = log f () logdDdS

= U(()) log dDdS

H(, v) = () +1

2v, vGSc ()

The Jacobian determinantdDdS can be used as weight afterwards.

We then consider partial Hamiltonian H(, v) = U() + 12v, vGSc ()Lan, S. and Shahbaba, B. (UW and UCI) Spherical Augmentation 06/04/2015 17 / 51

A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B

Spherical Monte Carlo Spherical HMC in the Cartesian coordinate

Spherical HMCin the Cartesian coordinate


A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B


Spherical HMC for ball type constraints

BD0 (1) := { RD :

2 =D

i=1 2i 1}

7=(,D+1)D+1=

122

SD := { RD+1 :2 = 1}

Change of variablesBD0 (1)

f ()dB =

SD+

f ()

dBdScdSc =

SD+f ()|D+1|dSc

where f () f ().

What We Want:

f ()dB

drop D+1weigh it by |D+1|

What We Sample:

f ()dSc


A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B


Canonical spherical metric

Here, the proper metric on SD is called canonical spherical metric:

Definition 4 (canonical spherical metric)

GSc () = ID +T

2D+1= ID +

T

1 22(3.2)

For any vector v = (v, vD+1) TSD := {v RD+1 : Tv = 0},

GSc () can be viewed as a way to express the length of v in v:

vTGSc ()v = v22 +vTTv

2D+1= v22 +

(D+1vD+1)2

2D+1

= v22 + v2D+1 = v22


A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B


Hamiltonian (Lagrangian) dynamics on sphere

On BD0 (1)

H(, v) = U() + K (v)

= log f () + 12

vTIv

7

On SD

H(, v) = U() + K (v)

= log f () + 12

vTGSc ()v

v N (0, ID)v 7v v (ID+1

T)N (0, ID+1)

= v

v = U()

2 1

= v

v = vTSc ()v GSc ()1U()

D+1 =

1 22, vD+1 = Tv/D+1


A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B


Split Lagrangian dynamics on sphere

= v

v = vTSc ()v GSc ()1U()(3.3)

= 0

v = 12

GSc ()1U()

= v

v = vTSc ()v

(t) = (0)

v(t) = v(0)

t2

[[ID

0T

] (0)(0)T

]U((0))

(t) =(0) cos(v(0)2t)

+v(0)

v(0)2sin(v(0)2t)

v(t) = (0)v(0)2 sin(v(0)2t)

+ v(0) cos(v(0)2t)


A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B


Error analysisDenote z := (, v), z(tn) as the true solution to (3.3) at time tn and z(n)

the numerical solution at n-th step. We have the following bound of the

error en = z(tn) z(n):

Proposition 1

Assume f(, v) := vTS()v + GS()1U() is smooth. Then

en+1 (1 + M1+ M22)en +O(3)

where Mk = ck supt[0,T ] k f((t), v(t)), k = 1, 2 for some constantsck > 0. = tn+1 tn is the discretization step size. Further accumulatingthe local errors for L = T/ steps yields the global error

eL+1 (eM1T + T )2


A B

~=(,D+1)D+1=(1||||

2)0.5

D+1

A

B


Algorithm 1 Spherical HMC in the Cartesian coordinate (c SphHMC )

Initialize (1)

at current after transformation TDSSample a new velocity value v(1) N (0, ID+1)

Set v(1) v(1) (1)((1))T

v(1)

Calculate H((1), v(1)) = U((1)) + K(v(1))

for ` = 1 to L do

v(`+12

) = v(`) 2

([ID0T

] (`)((`))T

)U((`))

(`+1)

= (`)

cos(v(`+12

)) + v(`+ 1

2)

v(`+12

)sin(v(`+

12

))

v(`+12

) (`)v(`+12

) sin(v(`+12

)) + v(`+12

) cos(v(`+12

))

v(`+1) = v(`+12

) 2

([ID0T

] (`+1)((`+1))T

)U((`+1))

end for

Calculate H((L+1)

, v(L+1)) = U((L+1)) + K(v(L+1))

Calculate the acceptance probability = min{1, exp[H((L+1), v(L+1)) + H((1), v(1))]}Accept or reject the proposal according to for the next state

Calculate TSD() and the corresponding weight |dTSD|


http://www.ics.uci.edu/~slan/animation/TMG_sphHMC.html

2

A

B


x

x1

A

B

Spherical Monte Carlo Spherical HMC in the spherical coordinate

Spherical HMCin the spherical coordinate


2

A

B


x

x1

A

B


Spherical HMC for box type constraints

RD0 :=[0, ]D1 [0, 2)

7xxd =cos d

d1i=1 sin i

SD := {x RD+1 :x2 = 1}

Change of measureRD0

f ()dR0 =

SD

f ()

dR0dSrdSr =

SDf ()

D1d=1

sindD ddSr

where f () = f ((x)) on SD .

What We Want:

f ()dR0

weigh sample by D1d=1 sindD d

What We Sample:

f ()dSr


2

A

B


x

x1

A

B


Round spherical metric

Here, the natural metric on SD is called round spherical metric:

Definition 5 (round spherical metric)

GSr () = diag

[1, sin2 1, ,

D1d=1

sin2 d

](3.4)

For any vector v TRD0 , we have

vTGSr ()v v22 v22 = vTGSc ()v


2

A

B


x

x1

A

B


Hamiltonian (Lagrangian) dynamics on sphere

On RD0

H(, v) = U() + K (v)

= log f () + 12

vTIv

7x

On SD

H(x, x) = U(x) + K (x)

= log f () + 12

vTGSr ()v

v N (0, ID)v 7x v(x, x) GSr ()

12N (0, ID)

= v

v = U()

l u

= v

v = vTSr ()v GSr ()1U()

= (x), v = v(x, x)


2

A

B


x

x1

A

B


Split Lagrangian dynamics on sphere

= v

v = vTSr ()v GSr ()1U()

= 0

v = 12

GSr ()1U()

= v

v = vTSr ()v

(t) = (0)

v(t) = v(0) t2

diag

[1, ,

D1d=1

sin2 d

]U((0))

((0), v(0)) (x(0), x(0))

(x(t), x(t)) = gr (x(0), x(0))

((0), v(0)) (x(0), x(0))


2

A

B


x

x1

A

B


Algorithm 2 Spherical HMC in the spherical coordinate (s-SphHMC)

Initialize (1) at current after transformation TDSSample a new velocity value v(1) N (0, ID )Set v

(1)d v

(1)d

d1i=1 sin

1((1)i ), d = 1, ,D

Calculate H((1), v(1)) = U((1)) + K(v(1))

for ` = 1 to L do

v(`+ 1

2)

d = v(`)d

d

2d

U((`))d1

i=1 sin2(

(`)i ), d = 1, ,D

((`+1), v(`+12

)) TSR0 g TR0S((`), v(`+

12

))

v(`+1)d = v

(`+ 12

)

d d

2d

U((`+1))d1

i=1 sin2(

(`+1)i ), d = 1, ,D

end for

Calculate H((L+1), v(L+1)) = U((L+1)) + K(v(L+1))

Calculate the acceptance probability = min{1, exp[H((L+1), v(L+1)) + H((1), v(1))]}Accept or reject the proposal according to for the next state

Calculate TSD() and the corresponding weight |dTSD|


Spherical Monte Carlo Spherical LMC on the probability simplex

Spherical LMCon the probability simplex




A class of models having probability distributions defined on simplex

K := { RD | k 0,K

k=1

d = 1}

Latent Dirichlet Allocation (LDA) (Blei et al., 2003) is a hierarchical

Bayesian model frequently used to model document topics.

1-norm constraint: identify the first (all positive) orthant with others.

T

: 7 = maps the simplex to the sphere

K:= { SK1|k 0, k = 1, ,K} SK1




Prototype example: Dirichlet-Multinomial distribution

p(xi = k |) = k , k = 1, ,K

p() K

k=1

k1k

p(|x) K

k=1

nk +k1k , nk =N

i=1

I (xi = k), n =K

k=1

nk

Fisher metric on

coincides GSc () on SK1 up to a constant.

G(K ) = n[diag(1/K ) + 11T/K ]

G() =dTKdK

G(K )dK

dTK= 4nGSc ()




Use G() instead of GSc () in c-SphHMC.

Include the volume adjustment term,dDdS in the Hamiltonian

H(, v) = () +1

2v, vG(), () = U() log

dDdS

No afterward re-weight: online learning

c-SphHMCabove modifications Spherical Lagrangian Monte Carlo.

SphLMC: stems from the Fisher metric on the simplex.




Category0 1 2 3 4 5 6 7 8 9 10

Estim

ate

d P

ro

ba

bility a

dju

ste

d b

y T

ru

th

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

RWMtWallHMCRLDSphLMCTruth

Lag0 10 20

Au

to

co

rre

latio

n o

f p

1

-0.2

0

0.2

0.4

0.6

0.8

RWMt

Lag0 10 20

Au

to

co

rre

latio

n o

f p

2

-0.2

0

0.2

0.4

0.6

0.8

Lag0 10 20

Au

to

co

rre

latio

n o

f p

9

-0.2

0

0.2

0.4

0.6

0.8

Lag0 10 20

Au

to

co

rre

latio

n o

f p

1

-0.2

0

0.2

0.4

0.6

0.8

WallHMC

Lag0 10 20

Au

to

co

rre

latio

n o

f p

2

-0.2

0

0.2

0.4

0.6

0.8

Lag0 10 20

Au

to

co

rre

latio

n o

f p

9

-0.2

0

0.2

0.4

0.6

0.8

Lag0 10 20

Au

to

co

rre

latio

n o

f p

1

-0.2

0

0.2

0.4

0.6

0.8

RLD

Lag0 10 20

Au

to

co

rre

latio

n o

f p

2

-0.2

0

0.2

0.4

0.6

0.8

Lag0 10 20

Au

to

co

rre

latio

n o

f p

9

-0.2

0

0.2

0.4

0.6

0.8

Lag0 10 20

Au

to

co

rre

latio

n o

f p

1

-0.2

0

0.2

0.4

0.6

0.8

SphLMC

Lag0 10 20

Au

to

co

rre

latio

n o

f p

2

-0.2

0

0.2

0.4

0.6

0.8

Lag0 10 20

Au

to

co

rre

latio

n o

f p

9

-0.2

0

0.2

0.4

0.6

0.8


Experiments










4 Experiments



Experiments

Experiments

Definition 6 (Effective Sample Size)

For N samples, effective sample size is calculated as follows:

ESS = N[1 + 2Kk=1(k)]1

where (k) is the autocorrelation function with lag k , and K 1.

Performance measured by time-normalized ESS.

Interpreted as number of nearly independent samples.

Use the minimum ESS normalized by CPU time: min(ESS)/s.

Compare RWMt, Wall HMC, exact HMC, c-SphHMC, s-SphHMC,

RLD and SphLMC.


Experiments

Truncated Multivariate Gaussian(12

) N

(0,

[1 .5

.5 1

]), 0 1 5, 0 2 1

1

2

2 1 0 1 2

21

01

2

0.0

2

0.0

6

0.1

0.12

0.16

Truth

1

2

2 1 0 1 2

21

01

2

Estimate by RWMt

1

2

2 1 0 1 2

21

01

2

Estimate by Wall HMC

1

2

2 1 0 1 2

21

01

2


1

2

2 1 0 1 2

21

01

2

Estimate by cSphHMC

1 2

2 1 0 1 22

10

12

Estimate by sSphHMC


Experiments

Truncated Multivariate GaussianTo evaluate efficiency, we increase the dimensionality for D = 10, 100

N (0,), ij = 1/(1+|ij |); 0 1 5, 0 i 0.5, i 6= 1.

RWM: > 95% of times proposals rejected due to constraint violation.

Wall HMC: average wall hits 3.81 (L=2, D=10), 6.19 (L=5, D=100).

Dim Method AP s/iter ESS(min,med,max) Min(ESS)/s spdup

RWMt 0.62 5.72E-05 (48,691,736) 7.58 1.00

Wall HMC 0.83 1.19E-04 (31904,86275,87311) 2441.72 322.33

D=10 exact HMC 1.00 7.60E-05 (1e+05,1e+05,1e+05) 11960.29 1578.87

c-SphHMC 0.82 2.53E-04 (62658,85570,86295) 2253.32 297.46

s-SphHMC 0.79 2.02E-04 (76088,1e+05,1e+05) 3429.56 452.73

RWMt 0.81 5.45E-04 (1,4,54) 0.01 1.00

Wall HMC 0.74 2.23E-03 (17777,52909,55713) 72.45 5130.21

D=100 exact HMC 1.00 4.65E-02 (97963,1e+05,1e+05) 19.16 1356.64

c-SphHMC 0.73 3.45E-03 (55667,68585,72850) 146.75 10390.94

s-SphHMC 0.87 2.30E-03 (74476,99670,1e+05) 294.31 20839.43


Experiments

Bayesian Lasso: regularized regression

30

20

10

010

2030

Shrinkage Factor

Coeff

icien

ts

0 0.2 0.4 0.6 0.8 1

21

108

46

Bayesian Lasso Gibbs Sampler

30

20

10

010

2030

Shrinkage Factor

Coeff

icien

ts

0 0.2 0.4 0.6 0.8 1

52

71

104

3

Bayesian Lasso Wall HMC

30

20

10

010

2030

Shrinkage Factor

Coeff

icien

ts

0 0.2 0.4 0.6 0.8 1

52

71

106

43

9

Bayesian Lasso Spherical HMC

Obtain the coefficients by minimizing the residual sum of squares

(RSS) subject to a constraint on the magnitude of

min1t

RSS(), RSS() :=

i

(yi 0 xTi )2

Park and Casella (2008) use a Laplace prior: P() exp(||)


Experiments

Bayesian Lasso

05

00

10

00

15

00

Shrinking Factor

Min

(ES

S)/

s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gibbs SamplerWall HMCSpherical HMC


Experiments

Bayesian Bridge: regularized regression

30

20

10

010

2030

Shrinkage Factor

Coeff

icien

ts

0 0.2 0.4 0.6 0.8 1

52

71

106

43

9

Beysian Bridge Lasso (q=1)

30

20

10

010

2030

Shrinkage Factor

Coeff

icien

ts

0 0.2 0.4 0.6 0.8 1

52

110

86

39

Beysian Bridge q=1.2

30

20

10

010

2030

Shrinkage Factor

Coeff

icien

ts

0 0.2 0.4 0.6 0.8 1

75

14

9

Beysian Bridge q=0.8

Obtain the coefficients by minimizing the residual sum of squares

(RSS) subject to a constraint on the magnitude of

minqt

RSS(), RSS() :=

i

(yi 0 xTi )2

Polson et al (2013) have Bayesian Bridge with complicated priors


Experiments

Reconstruction of quantized stationary Gaussian process

0 10 20 30 40

1.5

0.5

0.5

1.5

0 10 20 30 40

21

01

2 TruthRWMtWall HMC

exact HMCcSphHMCsSphHMC

Given N values of a function {f (xi )}Ni=1, taken values in a set {qk}Kk=1Assume this is a quantized projection of y(xi ) from a stationary GP

f (xi ) = qk , if zk y(xi ) < zk+1The objective is to sample from the posterior distribution

p(y|f) T N (0, ), ij = 2 exp{|xi xj |2

22

}, 2 = 0.6, 2 = 0.2


Experiments

Reconstruction of quantized stationary Gaussian process

Method AP s/iter ESS(min,med,max) Min(ESS)/s spdup

RWMt 0.70 7.11E-05 (2,9,35) 0.22 1.00

Wall HMC 0.69 9.94E-04 (12564,24317,43876) 114.92 534.48

exact HMC 1.00 1.00E-02 (72074,1e+05,1e+05) 65.31 303.76

c-SphHMC 0.72 1.73E-03 (13029,26021,56445) 68.44 318.32

s-SphHMC 0.80 1.09E-03 (14422,31182,81948) 120.59 560.86

Table: Comparing efficiency of RWMt, Wall HMC, exact HMC, c-SphHMC and

s-SphHMC in reconstructing a quantized stationary Gaussian process. AP is

acceptance probability, s/iter is seconds per iteration, ESS(min,med,max) is the

(minimal,median,maximal) effective sample size, and Min(ESS)/s is the minimal

ESS per second.


Experiments

LDA on Wikipedia corpus

LDA(Blei et al. 2003) is a popular Bayesian model for topic modeling.

The model consists of K topicks k , which are distributions over the

words in the collection, drawn from a Dirichlet prior Dir().

A document d is modeled by a mixture of topics, with mixing

proportion d Dir().

Documents are produced by drawing a topic assignment zdi i.i.d from

d for each word wdi in document d , and then drawing the word wdi

from the assigned topic zdi .


Experiments


Conditioned on , the documents are i.i.d, and the joint distribution

can be factorized (Patterson and Teh, 2013)

p(w , z , |, ) = p(|)D

d=1

p(wd , zd |, )

p(wd , zd |, ) =K

k=1

( + ndk)

()

Ww=1

ndkwkw

To compare with sg-RLD (Patterson and Teh, 2013), apply SphLMC

to update = with stochastic gradient for L = 1 decreasing

gkw = [(nkw + 1/2)/kw + kw (nk + W ( 1/2))]/(2 nk)

where nkw =|D||Dt |

dDt Ezd |wd ,,[ndkw ], and |Dt | = 50.


Experiments


Online learn 50000 documents randomly downloaded from Wikipedia.

Vocabulary consists of approx. 8000 words from Project Gutenburg.

Evaluate the performance in perplexity on 1000 held-out documents.

10000 20000 30000 40000 50000

7.0

7.5

8.0

8.5

number of documents

log

perp

lexi

ty (c

ircle

)

sgRLDsgwallLMCsgSphHMCsgSphLMC

Tim

e (s

quar

e)

0

2000

4000

6000

8000

10000


Conclusion and future work










4 Experiments




Conclusion

Spherical Augmentation (SA) is a natural and efficient framework to

handle norm related constraints in statistical inference.

Spherical HMC and Spherical LMC demonstrate substantial

advantage over existing methods. SA can have more extensions.

Based on change of variables, SA defines the dynamics on sphere in 1

higher dimension by slack variable or embedding map. The resulting

sampler moves on sphere freely while implicitly handling constraints.

To account for the change of geometry, volume adjustment is needed

to re-weight samples (SphHMC) or added to Hamiltonian (SphLMC).



Future work

Instead of Euclidean metric I on BD0 (1), we can start from Fishermetric GF(), and consider metric like GF() +

T/2D+1 for

augmented space to facilitate exploring complicated structures.

Derive an acceptance rule that does not drop quickly as dimension

increases (Beskos et al., 2011).

Develop tune-free algorithms for spherical HMC (Hoffman and

Gelman, 2011).


Thank you !

Web: http://www.ics.uci.edu/~slan/SphHMC/Intro.html


http://www.ics.uci.edu/~slan/SphHMC/Intro.html~

Review: from HMC to RHMCSpherical AugmentationSimple examples: ball and boxGeneral q-norm constraintsSome functional constraints

Spherical Monte CarloSpherical HMC in the Cartesian coordinateSpherical HMC in the spherical coordinateSpherical LMC on the probability simplex

ExperimentsConclusion and future workAppendix