1. No category

Gaussian kernels have also Gaussian
minimizers
Paweł Wolff
University of Warsaw & Polish Academy of Sciences
Probability and Analysis, Bedlewo,
˛
May 4, 2015
joint work with Franck Barthe (University of Toulouse)
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: Gaussian kernels have Gaussian maximizers
For p, q ≥ 1 consider T : Lp (Rn1 ) → Lq (Rn2 )
Z
Tf (x) =
K (x, y )f (y ) dy
Rn1
Question: kT k =?
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: Gaussian kernels have Gaussian maximizers
For p, q ≥ 1 consider T : Lp (Rn1 ) → Lq (Rn2 )
Z
Tf (x) =
K (x, y )f (y ) dy
Rn1
Question: kT k =?
Gaussian function: f (x) = exp(−Q(x) + `(x) + c)
Q is a psd quadratic form; if Q is pd then f is a
non-degenerate Gaussian function
` is a linear form; if ` ≡ 0 then f is centered
CG — non-degenerate and centered Gaussian functions
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: Gaussian kernels have Gaussian maximizers
For p, q ≥ 1 consider T : Lp (Rn1 ) → Lq (Rn2 )
Z
Tf (x) =
K (x, y )f (y ) dy
Rn1
Question: kT k =?
Gaussian function: f (x) = exp(−Q(x) + `(x) + c)
Q is a psd quadratic form; if Q is pd then f is a
non-degenerate Gaussian function
` is a linear form; if ` ≡ 0 then f is centered
CG — non-degenerate and centered Gaussian functions
Theorem (Lieb ’90)
If K (x, y ) = exp − Q1 (x) − Q2 (y ) − B(x, y ) then
kTf kLq
kT k = sup
: f ∈ CG
kf kLp
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: more general setting
H, H1 , . . . , Hm Euclidean spaces; c1 , . . . , cm > 0 exponents
R
fi : Hi → [0, ∞) measurable with 0 < Hi fi < ∞
Bi : H → Hi surjective linear maps
Q : H → H self-adjoint, psd (Q ≥ 0)
Example: H = Rn ,
Hi = R,
Paweł Wolff
Bi = h·, ui i,
ui ∈ H
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: more general setting
H, H1 , . . . , Hm Euclidean spaces; c1 , . . . , cm > 0 exponents
R
fi : Hi → [0, ∞) measurable with 0 < Hi fi < ∞
Bi : H → Hi surjective linear maps
Q : H → H self-adjoint, psd (Q ≥ 0)
Example: H = Rn ,
Hi = R,
Bi = h·, ui i,
ui ∈ H
The functional J
R
J(f1 , . . . , fm ) =
H
Q
ci
e−hQx,xi m
i=1 fi (Bi x) dx
ci
R
Qm
f
i=1
Hi i
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: more general setting
H, H1 , . . . , Hm Euclidean spaces; c1 , . . . , cm > 0 exponents
R
fi : Hi → [0, ∞) measurable with 0 < Hi fi < ∞
Bi : H → Hi surjective linear maps
Q : H → H self-adjoint, psd (Q ≥ 0)
Example: H = Rn ,
Hi = R,
Bi = h·, ui i,
ui ∈ H
The functional J
R
J(f1 , . . . , fm ) =
H
Q
ci
e−hQx,xi m
i=1 fi (Bi x) dx
ci
R
Qm
f
i=1
Hi i
Theorem (Brascamp-Lieb ’76 (Q = 0, dimHi = 1), Lieb ’90)
sup J(f1 , . . . , fm ) =
f1 ,...,fm
Paweł Wolff
sup
g1 ,...,gm ∈CG
J(g1 , . . . , gm )
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: remarks
hi ∈ Lpi (Hi ), pi = 1/ci , fi = |hi |pi
R −hQx,xi Q
hi (Bi x) dx
He
Q
≤ J(|h1 |p1 , . . . , |hm |pm )
khi kLpi
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: remarks
hi ∈ Lpi (Hi ), pi = 1/ci , fi = |hi |pi
R −hQx,xi Q
hi (Bi x) dx
He
Q
≤ J(|h1 |p1 , . . . , |hm |pm )
khi kLpi
J(g1 , . . . , gm ) for gi ∈ CG:
let A be positive definite and gA (x) = e−πhAx,xi
Z
gA (x) dx = (det A)−1/2
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: remarks
hi ∈ Lpi (Hi ), pi = 1/ci , fi = |hi |pi
R −hQx,xi Q
hi (Bi x) dx
He
Q
≤ J(|h1 |p1 , . . . , |hm |pm )
khi kLpi
J(g1 , . . . , gm ) for gi ∈ CG:
let A be positive definite and gA (x) = e−πhAx,xi
Z
gA (x) dx = (det A)−1/2
J(gA1 , . . . , gAm )
R
=
P
Q
1
det(Q + ci Bi∗ Ai Bi ) − 2
e−hQx,xi e−ci hAi (Bi x),(Bi x)i dx
Q
Q
=
(det Ai )ci
(det Ai )−ci /2
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: examples
Nelson’s hypercontractivity (’73): (Pt )t≥0 acting on f : R → R
Z
p
(Pt f )(x) = f (e−t x + 1 − e−2t y ) γ(dy )
[Pt : Lp (γ) → Lp (γ) has norm 1 for all t ≥ 0 and 1 ≤ p ≤ ∞]
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: examples
Nelson’s hypercontractivity (’73): (Pt )t≥0 acting on f : R → R
Z
p
(Pt f )(x) = f (e−t x + 1 − e−2t y ) γ(dy )
[Pt : Lp (γ) → Lp (γ) has norm 1 for all t ≥ 0 and 1 ≤ p ≤ ∞]
Theorem (Nelson ’73)
If 1 < p < q < ∞ and t > 0 satisfy e−2t ≤
Pt : Lp (γ) → Lq (γ)
Paweł Wolff
p−1
q−1
then
has norm 1
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: examples
Nelson’s hypercontractivity (’73): (Pt )t≥0 acting on f : R → R
Z
p
(Pt f )(x) = f (e−t x + 1 − e−2t y ) γ(dy )
[Pt : Lp (γ) → Lp (γ) has norm 1 for all t ≥ 0 and 1 ≤ p ≤ ∞]
Theorem (Nelson ’73)
If 1 < p < q < ∞ and t > 0 satisfy e−2t ≤
Pt : Lp (γ) → Lq (γ)
p−1
q−1
then
has norm 1
Equivalently:
R
1/p R
1/q 0
R
p dγ
q 0 dγ
(P
f
)(x)g(x)
γ(dx)
≤
|f
|
|g|
, or
t
R
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: examples
Nelson’s hypercontractivity (’73): (Pt )t≥0 acting on f : R → R
Z
p
(Pt f )(x) = f (e−t x + 1 − e−2t y ) γ(dy )
[Pt : Lp (γ) → Lp (γ) has norm 1 for all t ≥ 0 and 1 ≤ p ≤ ∞]
Theorem (Nelson ’73)
If 1 < p < q < ∞ and t > 0 satisfy e−2t ≤
Pt : Lp (γ) → Lq (γ)
p−1
q−1
then
has norm 1
Equivalently:
R
1/p R
1/q 0
R
p dγ
q 0 dγ
(P
f
)(x)g(x)
γ(dx)
≤
|f
|
|g|
, or
t
R
Z
Z
1/p Z
1/q 0
1/q 0
−Qt (x,y ) 1/p
e
f1 (x)f2 (y ) dx dy ≤
f1 dx
f2 dx
R2
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: examples
Young’s convolution inequality with sharp constant
(Beckner ’75, Brascamp-Lieb ’76)
Let p, q, r ∈ [1, ∞] such that
1
p
+
1
q
=1+
1
r
For all f ∈ Lp , g ∈ Lq ,
kf ∗ gkLr ≤ Cp,q kf kLp kgkLq
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: examples
Young’s convolution inequality with sharp constant
(Beckner ’75, Brascamp-Lieb ’76)
Let p, q, r ∈ [1, ∞] such that
1
p
+
1
q
=1+
1
r
For all f ∈ Lp , g ∈ Lq ,
kf ∗ gkLr ≤ Cp,q kf kLp kgkLq
Equivalently
Z
f (x)g(x − y )h(x) dx dy ≤ Cp,q kf kLp kgkLq khkLr 0
R2
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Lieb’s principle: examples
Young’s convolution inequality with sharp constant
(Beckner ’75, Brascamp-Lieb ’76)
Let p, q, r ∈ [1, ∞] such that
1
p
+
1
q
=1+
1
r
For all f ∈ Lp , g ∈ Lq ,
kf ∗ gkLr ≤ Cp,q kf kLp kgkLq
Equivalently
Z
f (x)g(x − y )h(x) dx dy ≤ Cp,q kf kLp kgkLq khkLr 0
R2
0 2
0 2
Sharp for f (x) = e−p x , g(x) = e−q x , h(x) = e−r
Paweł Wolff
0x 2
Gaussian kernels have also Gaussian minimizers
Converse inequalities: Borell’s reversed hypercontractivity
f ≥ 0 =⇒ Pt f > 0 for t > 0 (positivity improving)
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Converse inequalities: Borell’s reversed hypercontractivity
f ≥ 0 =⇒ Pt f > 0 for t > 0 (positivity improving)
Digression: for p ∈ R ∪ {±∞} and f ≥ 0 let
R
1/p
kf kLp (γ) =
f p dγ
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Converse inequalities: Borell’s reversed hypercontractivity
f ≥ 0 =⇒ Pt f > 0 for t > 0 (positivity improving)
Digression: for p ∈ R ∪ {±∞} and f ≥ 0 let
R
1/p
kf kLp (γ) =
f p dγ
for p < 0 if γ(f = 0) > 0 then kf kLp (γ) = 0
kf kL−∞ (γ) = ess inf f
p 7→ kf kLp (γ) is non-decreasing
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Converse inequalities: Borell’s reversed hypercontractivity
f ≥ 0 =⇒ Pt f > 0 for t > 0 (positivity improving)
Digression: for p ∈ R ∪ {±∞} and f ≥ 0 let
R
1/p
kf kLp (γ) =
f p dγ
for p < 0 if γ(f = 0) > 0 then kf kLp (γ) = 0
kf kL−∞ (γ) = ess inf f
p 7→ kf kLp (γ) is non-decreasing
p < 1, any measure, f ≥ 0
Z
nZ
o
0
kf kLp = inf
fg : g ≥ 0, g p = 1 ,
Paweł Wolff
p0 =
p
<1
p−1
Gaussian kernels have also Gaussian minimizers
Converse inequalities: Borell’s reversed hypercontractivity
f ≥ 0 =⇒ Pt f > 0 for t > 0 (positivity improving)
Digression: for p ∈ R ∪ {±∞} and f ≥ 0 let
R
1/p
kf kLp (γ) =
f p dγ
for p < 0 if γ(f = 0) > 0 then kf kLp (γ) = 0
kf kL−∞ (γ) = ess inf f
p 7→ kf kLp (γ) is non-decreasing
p < 1, any measure, f ≥ 0
Z
nZ
o
0
kf kLp = inf
fg : g ≥ 0, g p = 1 ,
p0 =
p
<1
p−1
Theorem (Borell ’82)
−∞ < q < p < 1. If e−2t ≤
1−p
1−q
then
kPt f kLq (γ) ≥ kf kLp (γ)
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Converse inequalities: more examples
Converse Young’s convolution ineq. (Brascamp-Lieb ’76)
Let p, q, r ∈ (0, 1] such that
f,g ≥ 0
=⇒
1
p
+
1
q
=1+
1
r
kf ∗ gkLr ≥ Cp,q kf kLp kgkLq
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Converse inequalities: more examples
Converse Young’s convolution ineq. (Brascamp-Lieb ’76)
Let p, q, r ∈ (0, 1] such that
f,g ≥ 0
=⇒
1
p
+
1
q
=1+
1
r
kf ∗ gkLr ≥ Cp,q kf kLp kgkLq
t
For λ ∈ (0, 1), taking p = λt , q = 1−λ
and letting t → 0+
1−λ
Z λ Z 1−λ
Z
y
x −y λ
g
dx ≥
g
f
ess sup f
λ
1−λ
y
(Prékopa-Leindler); more general inequalities by Barthe (’98)
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Converse inequalities: more examples
Converse Young’s convolution ineq. (Brascamp-Lieb ’76)
Let p, q, r ∈ (0, 1] such that
f,g ≥ 0
=⇒
1
p
+
1
q
=1+
1
r
kf ∗ gkLr ≥ Cp,q kf kLp kgkLq
t
For λ ∈ (0, 1), taking p = λt , q = 1−λ
and letting t → 0+
1−λ
Z λ Z 1−λ
Z
y
x −y λ
g
dx ≥
g
f
ess sup f
λ
1−λ
y
(Prékopa-Leindler); more general inequalities by Barthe (’98)
Converse Brascamp-Lieb ineq. (Chen-Dafnis-Paouris ’13)
Under certain assumptions on ci and Bi (some of ci ≥ 1, the
remaining ci < 0):
Z Y
m
H i=1
fici (Bi x) dx
≥
m Z
Y
i=1
Paweł Wolff
fi
ci
Hi
Gaussian kernels have also Gaussian minimizers
Main result
ci ∈ R \ {0},
c1 , . . . , cm+ > 0,
cm+ +1 , . . . , cm < 0
Q : H → H self-adjoint
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Main result
ci ∈ R \ {0},
c1 , . . . , cm+ > 0,
cm+ +1 , . . . , cm < 0
Q : H → H self-adjoint
R
J(f1 , . . . , fm ) =
H
Q
ci
e−hQx,xi m
i=1 fi (Bi x) dx
ci
R
Qm
f
i=1
Hi i
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Main result
ci ∈ R \ {0},
c1 , . . . , cm+ > 0,
cm+ +1 , . . . , cm < 0
Q : H → H self-adjoint
R
J(f1 , . . . , fm ) =
H
Q
ci
e−hQx,xi m
i=1 fi (Bi x) dx
ci
R
Qm
f
i=1
Hi i
Theorem (Barthe-W. ’14)
If
(i) x 7→ (B1 x, . . . , Bm+ x) from H to H1 × · · · Hm+ is onto
(ii) s+ (Q) + dim H1 + . . . + dim Hm+ ≤ dim H
then
inf J(f1 , . . . , fm ) =
f1 ,...,fm
inf
g1 ,...,gm ∈CG
J(g1 , . . . , gm )
Moreover, if (i) or (ii) fails then either J ≡ ∞ or inf J = 0.
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: the setting
Setting: Q = 0, H = Rn , Hi = R, Bi = h·, ui i, assume
Paweł Wolff
R
fi = 1
Gaussian kernels have also Gaussian minimizers
Proof: the setting
Setting: Q = 0, H = Rn , Hi = R, Bi = h·, ui i, assume
R
fi = 1
Rn
(thus m+ = n)
(i) and (ii) ⇐⇒ (u1 , . . . , um+ ) is a basis in
Z Y
J(f1 , . . . , fm ) =
fici (hx, ui i) dx
Rn
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: the setting
Setting: Q = 0, H = Rn , Hi = R, Bi = h·, ui i, assume
R
fi = 1
Rn
(thus m+ = n)
(i) and (ii) ⇐⇒ (u1 , . . . , um+ ) is a basis in
Z Y
J(f1 , . . . , fm ) =
fici (hx, ui i) dx
Rn
Example: converse Young’s convolution inequality
1
1
1
H = R2 , m = 3; c1 = , c2 = ≥ 1, c3 = 0 < 0
p
q
r
hx, u1 i = x1 , hx, u2 i = x1 − x2 , hx, u3 i = x2
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: the setting
Setting: Q = 0, H = Rn , Hi = R, Bi = h·, ui i, assume
R
fi = 1
Rn
(thus m+ = n)
(i) and (ii) ⇐⇒ (u1 , . . . , um+ ) is a basis in
Z Y
J(f1 , . . . , fm ) =
fici (hx, ui i) dx
Rn
Example: converse Young’s convolution inequality
1
1
1
H = R2 , m = 3; c1 = , c2 = ≥ 1, c3 = 0 < 0
p
q
r
hx, u1 i = x1 , hx, u2 i = x1 − x2 , hx, u3 i = x2
√
Gaussian functions: gai (x) = ai exp(−πai x 2 )
D X
E
Y c
Y c /2
gaii (hx, ui i) =
ai i exp − π
ci ai ui ⊗ ui x, x
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: the setting
Setting: Q = 0, H = Rn , Hi = R, Bi = h·, ui i, assume
R
fi = 1
Rn
(thus m+ = n)
(i) and (ii) ⇐⇒ (u1 , . . . , um+ ) is a basis in
Z Y
J(f1 , . . . , fm ) =
fici (hx, ui i) dx
Rn
Example: converse Young’s convolution inequality
1
1
1
H = R2 , m = 3; c1 = , c2 = ≥ 1, c3 = 0 < 0
p
q
r
hx, u1 i = x1 , hx, u2 i = x1 − x2 , hx, u3 i = x2
√
Gaussian functions: gai (x) = ai exp(−πai x 2 )
D X
E
Y c
Y c /2
gaii (hx, ui i) =
ai i exp − π
ci ai ui ⊗ ui x, x
P
If
ci ai ui ⊗ ui > 0 (is p.d.)
then
1/2
Q ci
a
P i

J(ga1 , . . . , gam ) = 
det
ci ai ui ⊗ ui
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: Gaussian computation
J(ga1 , . . . , gam ) =





Q
det
P
1/2
c
ai i
ci ai ui ⊗ui




+∞

if
P
ci ai ui ⊗ ui > 0
otherwise
Define
D 1/2 :=
inf
a1 ,...,am >0
J(ga1 , . . . , gam )
i.e. D ≥ 0 is the best (greatest) constant s.t. for all ai > 0
X
X
Y c
ci ai ui ⊗ ui ≥ 0 =⇒
ai i ≥ D · det
ci ai ui ⊗ ui
(1)
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: Gaussian computation
J(ga1 , . . . , gam ) =





Q
det
P
1/2
c
ai i
ci ai ui ⊗ui




+∞

if
P
ci ai ui ⊗ ui > 0
otherwise
Define
D 1/2 :=
inf
a1 ,...,am >0
J(ga1 , . . . , gam )
i.e. D ≥ 0 is the best (greatest) constant s.t. for all ai > 0
X
X
Y c
ci ai ui ⊗ ui ≥ 0 =⇒
ai i ≥ D · det
ci ai ui ⊗ ui
(1)
Goal: J(f1 , . . . , fm ) ≥ D 1/2
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: monotone mass transport
Fix fi : R → [0, ∞) regular
Ii := {fi > 0} bd open int. for i ≤ m+ and Ii = R for i > m+
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: monotone mass transport
Fix fi : R → [0, ∞) regular
Ii := {fi > 0} bd open int. for i ≤ m+ and Ii = R for i > m+
P −1
Fix ai > 0 s.t.
ci ai ui ⊗ ui > 0
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: monotone mass transport
Fix fi : R → [0, ∞) regular
Ii := {fi > 0} bd open int. for i ≤ m+ and Ii = R for i > m+
P −1
Fix ai > 0 s.t.
ci ai ui ⊗ ui > 0
Monotone transport fi (x)dx onto gai (y )dy by θi : Ii → R
Z x
Z θi (x)
fi (t) dt =
gai (y ) dy , i.e. fi (x) = gai (θi (x))θi0 (x)
−∞
−∞
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: monotone mass transport
Fix fi : R → [0, ∞) regular
Ii := {fi > 0} bd open int. for i ≤ m+ and Ii = R for i > m+
P −1
Fix ai > 0 s.t.
ci ai ui ⊗ ui > 0
Monotone transport fi (x)dx onto gai (y )dy by θi : Ii → R
Z x
Z θi (x)
fi (t) dt =
gai (y ) dy , i.e. fi (x) = gai (θi (x))θi0 (x)
−∞
Let Ω = {x ∈
−∞
Rn :
∀i≤m+ hx, ui i ∈ Ii } (bd, open, convex)
P
and define θ : Ω → Rn as θ(x) = ci ui θi (hx, ui i).
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: monotone mass transport
Fix fi : R → [0, ∞) regular
Ii := {fi > 0} bd open int. for i ≤ m+ and Ii = R for i > m+
P −1
Fix ai > 0 s.t.
ci ai ui ⊗ ui > 0
Monotone transport fi (x)dx onto gai (y )dy by θi : Ii → R
Z x
Z θi (x)
fi (t) dt =
gai (y ) dy , i.e. fi (x) = gai (θi (x))θi0 (x)
−∞
Let Ω = {x ∈
−∞
Rn :
∀i≤m+ hx, ui i ∈ Ii } (bd, open, convex)
P
and define θ : Ω → Rn as θ(x) = ci ui θi (hx, ui i).
Z Y
J(f1 , . . . , fm ) =
fici (hx, ui i) dx
n
ZR Y
Y 0
=
gacii θi (hx, ui i)
θi (hx, ui i)ci dx
Ω
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: monotone mass transport
Fix fi : R → [0, ∞) regular
Ii := {fi > 0} bd open int. for i ≤ m+ and Ii = R for i > m+
P −1
Fix ai > 0 s.t.
ci ai ui ⊗ ui > 0
Monotone transport fi (x)dx onto gai (y )dy by θi : Ii → R
Z x
Z θi (x)
fi (t) dt =
gai (y ) dy , i.e. fi (x) = gai (θi (x))θi0 (x)
−∞
Let Ω = {x ∈
−∞
Rn :
∀i≤m+ hx, ui i ∈ Ii } (bd, open, convex)
P
and define θ : Ω → Rn as θ(x) = ci ui θi (hx, ui i).
Z Y
J(f1 , . . . , fm ) =
fici (hx, ui i) dx
n
ZR Y
Y 0
=
gacii θi (hx, ui i)
θi (hx, ui i)ci dx
Ω
On Ω+ := {x ∈ Ω : Dθ(x) =
P
Paweł Wolff
ci θi0 (hx, ui i)ui ⊗ ui ≥ 0} use (1)
Gaussian kernels have also Gaussian minimizers
Proof: duality of quadratic forms
P
ci ui θi (hx, ui i)
Z Y
(1)
J(f1 , . . . , fm ) ≥ D ·
gacii θi (hx, ui i) · det Dθ(x) dx
Recall: θ(x) =
Ω+
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: duality of quadratic forms
P
ci ui θi (hx, ui i)
Z Y
(1)
J(f1 , . . . , fm ) ≥ D ·
gacii θi (hx, ui i) · det Dθ(x) dx
Recall: θ(x) =
Ω+
Z
≥D·
Ω+
P
inf
ci ui yi =θ(x)
Paweł Wolff
Y
gacii (yi ) · det Dθ(x) dx
Gaussian kernels have also Gaussian minimizers
Proof: duality of quadratic forms
P
ci ui θi (hx, ui i)
Z Y
(1)
J(f1 , . . . , fm ) ≥ D ·
gacii θi (hx, ui i) · det Dθ(x) dx
Recall: θ(x) =
Ω+
Z
≥D·
Ω+
≥D·
Y
c /2
ai i
Z
·
Ω+
P
inf
ci ui yi =θ(x)
Y
gacii (yi ) · det Dθ(x) dx
exp − π P sup
X
ci ui yi =θ(x)
{z
|
Gaussian f. with Cov=
Paweł Wolff
P
ci ai yi2 · det Dθ(x) dx
ci ai−1 ui ⊗ui
}
Gaussian kernels have also Gaussian minimizers
Proof: duality of quadratic forms
P
ci ui θi (hx, ui i)
Z Y
(1)
J(f1 , . . . , fm ) ≥ D ·
gacii θi (hx, ui i) · det Dθ(x) dx
Recall: θ(x) =
Ω+
Z
≥D·
Ω+
≥D·
Y
c /2
ai i
Z
·
Ω+
P
Y
inf
ci ui yi =θ(x)
gacii (yi ) · det Dθ(x) dx
exp − π P sup
ci ui yi =θ(x)
{z
|
Gaussian f. with Cov=
=D·
Y
c /2
ai i
X
Z
·
g
Ω+
P
ci ai−1 ui ⊗ui
Paweł Wolff
P
ci ai yi2 · det Dθ(x) dx
ci ai−1 ui ⊗ui
}
−1 θ(x) · det Dθ(x) dx
Gaussian kernels have also Gaussian minimizers
Proof: duality of quadratic forms
P
ci ui θi (hx, ui i)
Z Y
(1)
J(f1 , . . . , fm ) ≥ D ·
gacii θi (hx, ui i) · det Dθ(x) dx
Recall: θ(x) =
Ω+
Z
≥D·
Ω+
≥D·
Y
c /2
ai i
Z
·
Ω+
P
Y
inf
ci ui yi =θ(x)
gacii (yi ) · det Dθ(x) dx
exp − π P sup
ci ui yi =θ(x)
{z
|
Gaussian f. with Cov=
=D·
Y
c /2
ai i
θ : Ω+
Z
·
onto n
→R
≥
X
g
Ω+
P

ci ai−1 ui ⊗ui
det
P
ci ai yi2 · det Dθ(x) dx
ci ai−1 ui ⊗ui
}
−1 θ(x) · det Dθ(x) dx
1/2
ci ai−1 ui ⊗ ui

Q −1 c
(ai ) i
P
D·
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: optimizing over Gaussian functions
For any ai > 0 s.t.
P
ci ai−1 ui ⊗ ui > 0 we have proved

onto
J(f1 , . . . , fm )
θ : Ω+ → Rn
≥
D·
Paweł Wolff
det
1/2
ci ai−1 ui ⊗ ui

Q −1 c
(ai ) i
P
Gaussian kernels have also Gaussian minimizers
Proof: optimizing over Gaussian functions
For any ai > 0 s.t.
P
ci ai−1 ui ⊗ ui > 0 we have proved

onto
J(f1 , . . . , fm )
θ : Ω+ → Rn
≥
D·
det
1/2
ci ai−1 ui ⊗ ui

Q −1 c
(ai ) i
P
Take sup over ai > 0 to get J(f1 , . . . , fm ) ≥ D · D −1/2 = D 1/2 .
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: Surjectivity of θ(x) =
P
ci ui θi (hx, ui i) : Ω+ → Rn ...?
Let φi : Ii → R s.t. φ0i = θi ; then φi convex.
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: Surjectivity of θ(x) =
P
ci ui θi (hx, ui i) : Ω+ → Rn ...?
Let φi : Ii → R s.t. φ0i = θi ; then φi convex.
Note that θ(x) = ∇φ(x), where
X
X
|ci |φi (hx, ui i)
ci φi (hx, ui i) −
φ(x) = φ+ (x) − φ− (x) =
i≤m+
Paweł Wolff
i>m+
Gaussian kernels have also Gaussian minimizers
Proof: Surjectivity of θ(x) =
P
ci ui θi (hx, ui i) : Ω+ → Rn ...?
Let φi : Ii → R s.t. φ0i = θi ; then φi convex.
Note that θ(x) = ∇φ(x), where
X
X
|ci |φi (hx, ui i)
ci φi (hx, ui i) −
φ(x) = φ+ (x) − φ− (x) =
i>m+
i≤m+
φ+ : Ω → Rn is convex; extend to convex on Rn
φ+ (x0 ) = lim inf φ+ (x)
x→x0
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: Surjectivity of θ(x) =
P
ci ui θi (hx, ui i) : Ω+ → Rn ...?
Let φi : Ii → R s.t. φ0i = θi ; then φi convex.
Note that θ(x) = ∇φ(x), where
X
X
|ci |φi (hx, ui i)
ci φi (hx, ui i) −
φ(x) = φ+ (x) − φ− (x) =
i>m+
i≤m+
φ+ : Ω → Rn is convex; extend to convex on Rn
φ+ (x0 ) = lim inf φ+ (x)
x→x0
Goal: for any y0 ∈ Rn , find x0 ∈ Ω+ s.t. θ(x0 ) = ∇φ(x0 ) = y0
Paweł Wolff
Gaussian kernels have also Gaussian minimizers
Proof: Surjectivity of θ(x) =
P
ci ui θi (hx, ui i) : Ω+ → Rn ...?
Let φi : Ii → R s.t. φ0i = θi ; then φi convex.
Note that θ(x) = ∇φ(x), where
X
X
|ci |φi (hx, ui i)
ci φi (hx, ui i) −
φ(x) = φ+ (x) − φ− (x) =
i>m+
i≤m+
φ+ : Ω → Rn is convex; extend to convex on Rn
φ+ (x0 ) = lim inf φ+ (x)
x→x0
Goal: for any y0 ∈ Rn , find x0 ∈ Ω+ s.t. θ(x0 ) = ∇φ(x0 ) = y0
x0 = arg min φ+ (x) − φ− (x) − hx, y0 i
Note that x0 ∈ Ω. Therefore
∇(φ+ − φ− − h·, y0 i)(x0 ) = 0, i.e. ∇φ(x0 ) = y0
D 2 (φ+ − φ− − h·, y0 i)(x0 ) = Dθ(x0 ) ≥ 0, i.e. x0 ∈ Ω+
Paweł Wolff
Gaussian kernels have also Gaussian minimizers