We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions' 1. the minimal errors for the univariate case decay polynomially fast to zero, 2. the largest singular value for the univariate case is simple, 3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces. Joint work with Erich Novak

1. Henryk Wo´zniakowski QPT of LTP for Λstd
Quasi-polynomial tractability
of linear tensor products
using function values
Henryk Wo´zniakowski
Columbia University and University of Warsaw
joint work with Erich Novak
SAMSI 2017, 1

2. Henryk Wo´zniakowski QPT of LTP for Λstd
Multivariate Problems
S = {Sd}d∈N with Sd : Fd → Gd
Sd compact, linear, nonzero, Fd and Gd Hilbert
Sdf ≈ Ad,n(f) = φd,n(L1(f), L2(f), . . . , Ln(f))
Λall
: Lj(f) = f, η Fd
for any η ∈ Fd
Λstd
: Lj(f) = f(xj) for any xj, i.e., Fd RKHS
SAMSI 2017, 2

3. Henryk Wo´zniakowski QPT of LTP for Λstd
Minimal Worst Case Errors
e(n, Sd) = inf
Ad,n
sup
f Fd
≤1
Sdf − Ad,n(f) Gd
For Λall
, well known
e(n, Sd) = nth Gelfand/Kolmogorov width = λd,n+1
where
Wd = S∗
dSd : Fd → Fd, Wd ηd,n = λd,nηd,n
ηd,i, ηd,j Fd
= δi,j, λd,1 ≥ λd,2 ≥ · · · ≥ λd,n → 0
Best algorithm:
Ad,n(f) =
n
j=1
f, ηd,j Fd
Sdηd,j
For Λstd, e(n, Sd) =?
SAMSI 2017, 3

4. Henryk Wo´zniakowski QPT of LTP for Λstd
Information Complexity
n(ε, Sd) = min{n ∈ N : e(n, Sd) ≤ ε e(0, Sd)}
where
e(0, Sd) = Sd = the initial error
Tractability:
How does n(ε, Sd) depend on both ε−1
and d ?
SAMSI 2017, 4

5. Henryk Wo´zniakowski QPT of LTP for Λstd
QPT=Quasi-Polynomial Tractability
M. Gnewuch+H.W. [2011]:
S = {Sd}d∈N is QPT iff there are positive C and t such that
n(ε, Sd) ≤ C exp t(1 + ln d)(1 + ln ε−1
) ∀ ε ∈ (0, 1), d ∈ N
or equivalently
n(ε, Sd) ≤ C (e ε−1
)t(1+ln d)
∀ ε ∈ (0, 1), d ∈ N
Comment: weaker notion than PT=polynomial tractability
n(ε, Sd) ≤ C dp1
ε−p2
∀ ε ∈ (0, 1), d ∈ N.
SAMSI 2017, 5

6. Henryk Wo´zniakowski QPT of LTP for Λstd
LTP=Linear Tensor Products
d = 1, S1 : F1 → G1
S1 compact, linear, nonzero, F1 and G1 Hilbert
Sd = S ⊗d
1 , Fd = F ⊗d
1 , Gd = G⊗d
1
Now
W1 = S∗
1 S1 with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn → 0
Then
Wd = S∗
dSd has eigenvalues
{λd,n}n∈N = {λj1
λj2
· · · λjd
}(j1,j2,...,jd)∈Nd
SAMSI 2017, 6

7. Henryk Wo´zniakowski QPT of LTP for Λstd
QPT of LTP for Λall
For Λall, QPT of LTP depends only on λ = {λn}n∈N
decayλ := sup { r ≥ 0 : lim
n→∞
nr
λn = 0 }
Theorem (M. Gnewuch+H.W. [2011]:)
Let S = {S ⊗d
1 }d∈N be a LTP. Then S is QPT for Λall
iff
• λ1 > λ2
• decayλ > 0
Comments:
• For λ2 = 0, n(ε, S) = 1.
• For λ2 > 0, S is not PT.
• For λ1 = λ2 > 0, S is not QPT and suffers the curse of dimensionality.
SAMSI 2017, 7

8. Henryk Wo´zniakowski QPT of LTP for Λstd
QPT of LTP for Λstd
What do we have to assume for Λstd
?
• λ1 > λ2 since it is needed for Λall
• decayλ > 0 is needed for Λall. But we can now compute only f(xj ).
For d = 1, QPT=PT so we must now have for e = {e(n, S1)}
decaye := sup{ r ≥ 0 : lim
n→∞
nr
e(n, S1) = 0 } > 0
Comment: For many spaces
decayλ = decaye.
But there are spaces, see A. Hinrichs, E. Novak, J. Vybiral [2008], for which
decayλ = 2 and decaye = 0,
so we have QPT for Λall and no QPT for Λstd.
• Is it enough? No.
SAMSI 2017, 8

9. Henryk Wo´zniakowski QPT of LTP for Λstd
QPT of LTP for Λstd
Theorem (E. Novak+H.W. [2016]:)
Let
F1 = F1(K∗
1 ) small K∗
1 (x, t) = 1 + min(x, t) ∀ x, t ∈ [0, 1].
and let S = {S ⊗d
1 }d∈N be a LTP with λ2 > 0. Then
S suffers the curse of dimensionality for Λstd
iff
η1 = ± (K∗
1 (t, t))1/2
K∗
1 (·, t) = ±(1 + t)1/2
(1 + min(·, t)) ∀ t ∈ [0, 1]
It turns out the last condition is essential
SAMSI 2017, 9

10. Henryk Wo´zniakowski QPT of LTP for Λstd
Main Result
Theorem
Let S = {S ⊗d
1 } be a LTP for which
• λ1 > λ2
• decaye > 0
• η1 = ± K1(t, t)−1/2
K1(·, t) for some t
Then
S is QPT for Λstd
Comment:
L1(f) = f, η ⊗d
1 Fd
= (±)d
K1(t, t)−d/2
f(t, t, . . . , t)
can be computed for Λstd
SAMSI 2017, 10

11. Henryk Wo´zniakowski QPT of LTP for Λstd
Sketch of the Proof
Wlog let λ1 = 1. Decomposition
S1 = V1 + V2
V1f = ± K1(t, t)−1/2
f(t) S1η1
V2f =
j>2
f, ηj F1
S1ηj
with
V1 = 1 and V2 = λ2 < 1 = λ1
We have
Sd = (V1 + V2)⊗d
=
(j1,j2,...,jd)∈{1,2}d
Vj1 ⊗ Vj2 ⊗ · · · ⊗ Vjd
SAMSI 2017, 11

12. Henryk Wo´zniakowski QPT of LTP for Λstd
Sketch of the Proof
There is a linear Smolyak/sparse grid algorithm Ad,n,
see G. W. Wasilkowski+H.W. [1999], such that
e(Ad,n) = V ⊗d
2 − Ad,n ≤ α n−r
∀ d, n ∈ N
for some positive α and r independent of d and n.
Then
V
⊗(d−k)
1 ⊗ V ⊗k
2 ≈ V
⊗(d−k)
1 ⊗ Ak,n
SAMSI 2017, 12

13. Henryk Wo´zniakowski QPT of LTP for Λstd
Particular Space
Theorem
Let F1 = f1(K∗
1 ) with K1(x, t) = 1 + min(x, t), ∀ x, t ∈ [0, 1].
Let S = {S ⊗d
1 }d∈N be a LTP.
Then S is QPT
for Λall
iff for Λstd
iff
λ1 > λ2 λ1 > λ2
decayλ > 0 decaye > 0
η1 = ± (1 + t)−1/2
(1 + min(·, t))
for some t ∈ [0, 1]
SAMSI 2017, 13

14. Henryk Wo´zniakowski QPT of LTP for Λstd
Modified Problem
What to do if η1 = ± K1(t, t)−1/2
K1(·, t) ∀ t ?
Let
˜f = η1 −
K1(·, t∗
)
η1(t∗)
with η1(t∗
) = 0
Change F1 to
˜F1 = f ∈ F1 : f, ˜f
F1
= 0
Then
˜F1 = ˜F1( ˜K1) with ˜K1(x, t) = K1(x, t) −
˜f(x) ˜f(t)
˜f 2
F1
and
η1 = ˜K1(t∗
, t∗
)−1/2 ˜K1(·, t∗
)
SAMSI 2017, 14

15. Henryk Wo´zniakowski QPT of LTP for Λstd
Modified Problem
Let
˜S1 = S1 ˜F1
, ˜Sd = ˜S ⊗d
1
Theorem
˜S = { ˜Sd}d∈N is QPT
for Λall
iff for Λstd
iff
λ1 > λ2 λ1 > λ2
decayλ > 0 decaye > 0
SAMSI 2017, 15

16. Henryk Wo´zniakowski QPT of LTP for Λstd
Exponent of QPT
M. Gnewuch+H.W. [2011]:
S = {Sd}d∈N is QPT iff there are positive C and t such that
n(ε, Sd) ≤ C exp t(1 + ln d)(1 + ln ε−1
) ∀ ε ∈ (0, 1), d ∈ N
Exponent of QPT:
t∗
= inf{ t : t satisfies the bound above }
For Λall
t∗
= max
2
decayλ
,
2
ln λ1
λ2
For Λstd
t∗
= ?
SAMSI 2017, 16