Effective dimension for weighted ANOVA and anchored spaces

24 Pages·2015·0.32 MB·English

Effective dimension for weighted ANOVA and
anchored spaces
Chenxi Fan12
[email protected]
1SchoolofMathematicsandStatistics
UniversityofNewSouthWales
2SchoolofMathematicalSciences
ZhejiangUniversity
Annual NSW/ACT ANZIAM Meeting 2015
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 1/24
Outline
1 Motivation
High Dimensional Integral: Effective dimension
Compare ANOVA and Anchored Decompositions and Spaces
2 Main Results
Relate Variances to Norms
Implications
3 Summary
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 2/24
Outline
1 Motivation
High Dimensional Integral: Effective dimension
Compare ANOVA and Anchored Decompositions and Spaces
2 Main Results
Relate Variances to Norms
Implications
3 Summary
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 3/24
High Dimensional Integral
(cid:90)
f(x)dx
[0,1]d
Applications: e.g., in finance d = 360 = 12×30.
Some integration problems are easier than others, e.g.,
d
(cid:88)
f(x) = f(x).
i i
i=1
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 4/24
Element I: ANOVA Decomposition
Multivariate Decomposition
(cid:88)
f(x) = f (x),
u
u⊆[1:d]
where u is a subset of {1,2,...,d} := [1 : d], and f (x) depends only
u
on x for j ∈ u.
j
ANOVA Decomposition
(cid:90) (cid:90)
(cid:88)
f = f(x)dx and f (x) = f(x)dx − f , (1)
A,∅ A,u uc A,v
[0,1]d [0,1]|uc| v⊂u
where x = (x) .
u j j∈u
Integrate out coordinates not in u (hard to evaluate)
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 5/24
Element I: ANOVA Decomposition
Example
d
(cid:88)(cid:0) 1(cid:1)
f(x) = x −
i
2
i=1
ANOVA decomposition
f = 0,
A,∅
1
f = x − ,
A,{i} i 2
f = 0 if |u| ≥ 2.
A,u
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 6/24
Element II: Variance of Functions
Definition
The variance of an integrable function
f : [0,1]d → R
is defined as
(cid:90) (cid:104)(cid:90) (cid:105)2
σ2(f) := [f(x)]2dx− f(x)dx .
[0,1]d [0,1]d
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 7/24
Definition of Effective Dimension
Caflischetal. (1997)
Truncation dimension
The smallest integer k such that
(cid:16) (cid:88) (cid:17)
σ2 f ≥ (1−ε)σ2(f), (2)
A,u
u⊆[1:k]
where ε is small, e.g., ε = 0.01.
Superposition dimension
The smallest integer k such that
(cid:16) (cid:88) (cid:17)
σ2 f ≥ (1−ε)σ2(f). (3)
A,u
|u|≤k
Previous example Why is it important? How to calculate it?
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 8/24
Outline
1 Motivation
High Dimensional Integral: Effective dimension
Compare ANOVA and Anchored Decompositions and Spaces
2 Main Results
Relate Variances to Norms
Implications
3 Summary
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 9/24
Anchored Decomposition
Definition
(cid:88)
f = f(0) and f (x) = f(x ,0 )− f , (4)
a,∅ a,u u uc a,v
v⊂u
(cid:12)
where f(xv,0vc) = f(x)(cid:12)x=0, j∈vc.
j
Fix some coordinates at 0 (the anchor).
Cf. the ANOVA case which integrates out other components.
Cf. ANOVA Decomposition
(cid:90) (cid:90)
(cid:88)
f = f(x)dx and f (x) = f(x)dx − f .
A,∅ A,u uc A,v
[0,1]d [0,1]|uc| v⊂u
ChenxiFan (UNSW) NSW/ACTANZIAM2015 November26,2015 10/24