Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Genralized means in manifolds
Existence, uniqueness, robustness and algorithms.
Marc Arnaudon
Université de Bordeaux, France
Wien, 19 February 2015
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
joint works with
Frédéric Barbaresco (Thalès Air Systems)
Clément Dombry (Université de Franche-Comté, Besançon)
Laurent Miclo (Institut de Mathématiques de Toulouse)
Frank Nielsen (Ecole Polytechnique et Sony (Tokyo))
Anthony Phan (Université de Poitiers)
Le Yang (Université de Poitiers)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
M Riemannian manifold with Riemannian distance ρ
µ a probability measure on M
κ : M × M → R continuous
U:M→R
Z
x 7→
κ(x, y ) µ(dy )
M
Pb: Find a global minimizer of U
Z
for p ∈ [1, ∞),
Hp,µ (x) :=
ρp (x, y ) µ(dy );
M
H∞,µ (x) := kρ(x, ·)kL∞ (µ) .
Qp,µ : set of minimizers of Hp,µ
if Qp,µ has a unique element, we denote it ep,µ and call it the p-mean of µ.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
M Riemannian manifold with Riemannian distance ρ
µ a probability measure on M
κ : M × M → R continuous
U:M→R
Z
x 7→
κ(x, y ) µ(dy )
M
Pb: Find a global minimizer of U
Z
for p ∈ [1, ∞),
Hp,µ (x) :=
ρp (x, y ) µ(dy );
M
H∞,µ (x) := kρ(x, ·)kL∞ (µ) .
Qp,µ : set of minimizers of Hp,µ
if Qp,µ has a unique element, we denote it ep,µ and call it the p-mean of µ.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
For p = 2, e2 is the barycenter or the Fréchet mean of µ
In Rd
Z
e2 =
In convex domains
Z
y µ(dy ).
Rd
−→
e2 y µ(dy ) = 0.
M
For p = 1, e1 is the median of µ. In convex domains it is characterized by
Z
−→
e1 y
−→ µ(dy ) ≤ µ({e1 }).
M e1 y For p = ∞, e∞ is the center of the smallest enclosing ball of supp(µ).
In statistics we are mainly interested in
µN (ω) = µ(X1 (ω), . . . , XN (ω))
with
µ(x1 , . . . , xN ) =
N
1 X
δxk .
N
k =1
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
For p = 2, e2 is the barycenter or the Fréchet mean of µ
In Rd
Z
e2 =
In convex domains
Z
y µ(dy ).
Rd
−→
e2 y µ(dy ) = 0.
M
For p = 1, e1 is the median of µ. In convex domains it is characterized by
Z
−→
e1 y
−→ µ(dy ) ≤ µ({e1 }).
M e1 y For p = ∞, e∞ is the center of the smallest enclosing ball of supp(µ).
In statistics we are mainly interested in
µN (ω) = µ(X1 (ω), . . . , XN (ω))
with
µ(x1 , . . . , xN ) =
N
1 X
δxk .
N
k =1
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
For p = 2, e2 is the barycenter or the Fréchet mean of µ
In Rd
Z
e2 =
In convex domains
Z
y µ(dy ).
Rd
−→
e2 y µ(dy ) = 0.
M
For p = 1, e1 is the median of µ. In convex domains it is characterized by
Z
−→
e1 y
−→ µ(dy ) ≤ µ({e1 }).
M e1 y For p = ∞, e∞ is the center of the smallest enclosing ball of supp(µ).
In statistics we are mainly interested in
µN (ω) = µ(X1 (ω), . . . , XN (ω))
with
µ(x1 , . . . , xN ) =
N
1 X
δxk .
N
k =1
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
For p = 2, e2 is the barycenter or the Fréchet mean of µ
In Rd
Z
e2 =
In convex domains
Z
y µ(dy ).
Rd
−→
e2 y µ(dy ) = 0.
M
For p = 1, e1 is the median of µ. In convex domains it is characterized by
Z
−→
e1 y
−→ µ(dy ) ≤ µ({e1 }).
M e1 y For p = ∞, e∞ is the center of the smallest enclosing ball of supp(µ).
In statistics we are mainly interested in
µN (ω) = µ(X1 (ω), . . . , XN (ω))
with
µ(x1 , . . . , xN ) =
N
1 X
δxk .
N
k =1
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
M = Sym+ (d) positive definite real symmetric matrices
v
u d
uX
ρ(P, Q) = t
ln2 λi , λi eigenvalues of P −1/2 QP −1/2
i=1
Riemannian metric
hV , W iP = tr(VP −1 WP −1 )
is invariant by the action of Gl(d): A ? P = APAT ,
and P 7→ P −1 is an isometry.
M is a Cartan-Hadamard manifold endowed with the Fisher information metric for the
centered Gaussian model.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
Auto-regressive centered Gaussian stationary process: Z = (Z1 , ..., Zn )T
Covariance matrix

Rn =
E[Zi Zj ]
1≤i, j≤n


=

r0
r1
..
.
rn−1
r1
r0
..
.
...
...
...
..
.
r1

r n−1
r n−2 

.. 
. 
r0
Rn ∈ M = THPDn is a Toeplitz Hermitian Positive Definite matrix
An estimate of Rn is our observation (radar or echo-doppler)
Kähler potential function: Φ(Rn ) = − ln(det Rn ) − n ln(πe) yields a Riemannian
metric close to Fisher information metric
In suitable coordinates (r0 , µ1 , . . . , µn−1 ) = ϕ(Rn ),
ds2 = n
dr02
r02
+
n−1
X
(n − k )
k =1
|dµk |2
,
(1 − |µk |2 )2
THPDn is a Cartan-Hadamard manifold satisfying −4 ≤ K ≤ 0.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
Auto-regressive centered Gaussian stationary process: Z = (Z1 , ..., Zn )T
Covariance matrix

Rn =
E[Zi Zj ]
1≤i, j≤n


=

r0
r1
..
.
rn−1
r1
r0
..
.
...
...
...
..
.
r1

r n−1
r n−2 

.. 
. 
r0
Rn ∈ M = THPDn is a Toeplitz Hermitian Positive Definite matrix
An estimate of Rn is our observation (radar or echo-doppler)
Kähler potential function: Φ(Rn ) = − ln(det Rn ) − n ln(πe) yields a Riemannian
metric close to Fisher information metric
In suitable coordinates (r0 , µ1 , . . . , µn−1 ) = ϕ(Rn ),
ds2 = n
dr02
r02
+
n−1
X
(n − k )
k =1
|dµk |2
,
(1 − |µk |2 )2
THPDn is a Cartan-Hadamard manifold satisfying −4 ≤ K ≤ 0.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
Auto-regressive centered Gaussian stationary process: Z = (Z1 , ..., Zn )T
Covariance matrix

Rn =
E[Zi Zj ]
1≤i, j≤n


=

r0
r1
..
.
rn−1
r1
r0
..
.
...
...
...
..
.
r1

r n−1
r n−2 

.. 
. 
r0
Rn ∈ M = THPDn is a Toeplitz Hermitian Positive Definite matrix
An estimate of Rn is our observation (radar or echo-doppler)
Kähler potential function: Φ(Rn ) = − ln(det Rn ) − n ln(πe) yields a Riemannian
metric close to Fisher information metric
In suitable coordinates (r0 , µ1 , . . . , µn−1 ) = ϕ(Rn ),
ds2 = n
dr02
r02
+
n−1
X
(n − k )
k =1
|dµk |2
,
(1 − |µk |2 )2
THPDn is a Cartan-Hadamard manifold satisfying −4 ≤ K ≤ 0.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
Auto-regressive centered Gaussian stationary process: Z = (Z1 , ..., Zn )T
Covariance matrix

Rn =
E[Zi Zj ]
1≤i, j≤n


=

r0
r1
..
.
rn−1
r1
r0
..
.
...
...
...
..
.
r1

r n−1
r n−2 

.. 
. 
r0
Rn ∈ M = THPDn is a Toeplitz Hermitian Positive Definite matrix
An estimate of Rn is our observation (radar or echo-doppler)
Kähler potential function: Φ(Rn ) = − ln(det Rn ) − n ln(πe) yields a Riemannian
metric close to Fisher information metric
In suitable coordinates (r0 , µ1 , . . . , µn−1 ) = ϕ(Rn ),
ds2 = n
dr02
r02
+
n−1
X
(n − k )
k =1
|dµk |2
,
(1 − |µk |2 )2
THPDn is a Cartan-Hadamard manifold satisfying −4 ≤ K ≤ 0.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
Auto-regressive centered Gaussian stationary process: Z = (Z1 , ..., Zn )T
Covariance matrix

Rn =
E[Zi Zj ]
1≤i, j≤n


=

r0
r1
..
.
rn−1
r1
r0
..
.
...
...
...
..
.
r1

r n−1
r n−2 

.. 
. 
r0
Rn ∈ M = THPDn is a Toeplitz Hermitian Positive Definite matrix
An estimate of Rn is our observation (radar or echo-doppler)
Kähler potential function: Φ(Rn ) = − ln(det Rn ) − n ln(πe) yields a Riemannian
metric close to Fisher information metric
In suitable coordinates (r0 , µ1 , . . . , µn−1 ) = ϕ(Rn ),
ds2 = n
dr02
r02
+
n−1
X
(n − k )
k =1
|dµk |2
,
(1 − |µk |2 )2
THPDn is a Cartan-Hadamard manifold satisfying −4 ≤ K ≤ 0.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
Auto-regressive centered Gaussian stationary process: Z = (Z1 , ..., Zn )T
Covariance matrix

Rn =
E[Zi Zj ]
1≤i, j≤n


=

r0
r1
..
.
rn−1
r1
r0
..
.
...
...
...
..
.
r1

r n−1
r n−2 

.. 
. 
r0
Rn ∈ M = THPDn is a Toeplitz Hermitian Positive Definite matrix
An estimate of Rn is our observation (radar or echo-doppler)
Kähler potential function: Φ(Rn ) = − ln(det Rn ) − n ln(πe) yields a Riemannian
metric close to Fisher information metric
In suitable coordinates (r0 , µ1 , . . . , µn−1 ) = ϕ(Rn ),
ds2 = n
dr02
r02
+
n−1
X
(n − k )
k =1
|dµk |2
,
(1 − |µk |2 )2
THPDn is a Cartan-Hadamard manifold satisfying −4 ≤ K ≤ 0.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
General setting
Particular cases
Example 1
Example 2
Auto-regressive centered Gaussian stationary process: Z = (Z1 , ..., Zn )T
Covariance matrix

Rn =
E[Zi Zj ]
1≤i, j≤n


=

r0
r1
..
.
rn−1
r1
r0
..
.
...
...
...
..
.
r1

r n−1
r n−2 

.. 
. 
r0
Rn ∈ M = THPDn is a Toeplitz Hermitian Positive Definite matrix
An estimate of Rn is our observation (radar or echo-doppler)
Kähler potential function: Φ(Rn ) = − ln(det Rn ) − n ln(πe) yields a Riemannian
metric close to Fisher information metric
In suitable coordinates (r0 , µ1 , . . . , µn−1 ) = ϕ(Rn ),
ds2 = n
dr02
r02
+
n−1
X
(n − k )
k =1
|dµk |2
,
(1 − |µk |2 )2
THPDn is a Cartan-Hadamard manifold satisfying −4 ≤ K ≤ 0.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Assume the sectional curvatures on M satisfy −β 2 ≤ Kσ ≤ α2 .
Fix a geodesic ball B(a, r ) ⊂ M.
Assume suppµ ⊂ B(a, r ).
Assumption ∗
The support of µ is not a singleton. p > 1 or the support of µ is not contained in a
geodesic, the radius r satisfies
(
π
if p ∈ [1, 2)
rα,p = 21 min inj(M), 2α
r < rα,p with
1
π
if p ∈ [2, ∞)
rα,p
= 2 min inj(M), α
Theorem (B. Afsari, 2010)
Under Assumption ∗, the function Hp has a unique minimizer ep in M, the p-mean of µ.
Moreover ep ∈ B(a, r ).
Case p = 2: W. Kendall (91)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Assume the sectional curvatures on M satisfy −β 2 ≤ Kσ ≤ α2 .
Fix a geodesic ball B(a, r ) ⊂ M.
Assume suppµ ⊂ B(a, r ).
Assumption ∗
The support of µ is not a singleton. p > 1 or the support of µ is not contained in a
geodesic, the radius r satisfies
(
π
if p ∈ [1, 2)
rα,p = 21 min inj(M), 2α
r < rα,p with
1
π
if p ∈ [2, ∞)
rα,p
= 2 min inj(M), α
Theorem (B. Afsari, 2010)
Under Assumption ∗, the function Hp has a unique minimizer ep in M, the p-mean of µ.
Moreover ep ∈ B(a, r ).
Case p = 2: W. Kendall (91)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Assume the sectional curvatures on M satisfy −β 2 ≤ Kσ ≤ α2 .
Fix a geodesic ball B(a, r ) ⊂ M.
Assume suppµ ⊂ B(a, r ).
Assumption ∗
The support of µ is not a singleton. p > 1 or the support of µ is not contained in a
geodesic, the radius r satisfies
(
π
if p ∈ [1, 2)
rα,p = 21 min inj(M), 2α
r < rα,p with
1
π
if p ∈ [2, ∞)
rα,p
= 2 min inj(M), α
Theorem (B. Afsari, 2010)
Under Assumption ∗, the function Hp has a unique minimizer ep in M, the p-mean of µ.
Moreover ep ∈ B(a, r ).
Case p = 2: W. Kendall (91)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Assume the sectional curvatures on M satisfy −β 2 ≤ Kσ ≤ α2 .
Fix a geodesic ball B(a, r ) ⊂ M.
Assume suppµ ⊂ B(a, r ).
Assumption ∗
The support of µ is not a singleton. p > 1 or the support of µ is not contained in a
geodesic, the radius r satisfies
(
π
if p ∈ [1, 2)
rα,p = 21 min inj(M), 2α
r < rα,p with
1
π
if p ∈ [2, ∞)
rα,p
= 2 min inj(M), α
Theorem (B. Afsari, 2010)
Under Assumption ∗, the function Hp has a unique minimizer ep in M, the p-mean of µ.
Moreover ep ∈ B(a, r ).
Case p = 2: W. Kendall (91)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Assume the sectional curvatures on M satisfy −β 2 ≤ Kσ ≤ α2 .
Fix a geodesic ball B(a, r ) ⊂ M.
Assume suppµ ⊂ B(a, r ).
Assumption ∗
The support of µ is not a singleton. p > 1 or the support of µ is not contained in a
geodesic, the radius r satisfies
(
π
if p ∈ [1, 2)
rα,p = 21 min inj(M), 2α
r < rα,p with
1
π
if p ∈ [2, ∞)
rα,p
= 2 min inj(M), α
Theorem (B. Afsari, 2010)
Under Assumption ∗, the function Hp has a unique minimizer ep in M, the p-mean of µ.
Moreover ep ∈ B(a, r ).
Case p = 2: W. Kendall (91)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Assume the sectional curvatures on M satisfy −β 2 ≤ Kσ ≤ α2 .
Fix a geodesic ball B(a, r ) ⊂ M.
Assume suppµ ⊂ B(a, r ).
Assumption ∗
The support of µ is not a singleton. p > 1 or the support of µ is not contained in a
geodesic, the radius r satisfies
(
π
if p ∈ [1, 2)
rα,p = 21 min inj(M), 2α
r < rα,p with
1
π
if p ∈ [2, ∞)
rα,p
= 2 min inj(M), α
Theorem (B. Afsari, 2010)
Under Assumption ∗, the function Hp has a unique minimizer ep in M, the p-mean of µ.
Moreover ep ∈ B(a, r ).
Case p = 2: W. Kendall (91)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 1
Take p ∈ [1, ∞). Let (tk )k ≥1 ⊂ (0, Cp,µ,r ] a sequence of positive numbers satisfying
P∞
P∞ 2
k =1 tk = +∞ and
k =1 tk < ∞. Let x0 ∈ B(a, r ); define the sequence (xk )k ≥0
by
xk +1 = expxk −tk +1 gradxk Hp (·) , k ≥ 0;
. Then xk −→ ep .
For 1 ≤ p < 2, the proof relies on
ρ2 (xk +1 , ep ) ≤ ρ2 (xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
For p ≥ 2, similar inequality with Hp (xk +1 ) − Hp (ep ).
D. Groisser (2004 and 2005) and Huiling Le (2005): p = 2, tk constant, slightly smaller
domains.
B. Afsari, R. Tron, R.Vidal (2012): p ≥ 2, optimal result in the sphere: tk constant as
large as possible, domain as large as possible.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 1
Take p ∈ [1, ∞). Let (tk )k ≥1 ⊂ (0, Cp,µ,r ] a sequence of positive numbers satisfying
P∞
P∞ 2
k =1 tk = +∞ and
k =1 tk < ∞. Let x0 ∈ B(a, r ); define the sequence (xk )k ≥0
by
xk +1 = expxk −tk +1 gradxk Hp (·) , k ≥ 0;
. Then xk −→ ep .
For 1 ≤ p < 2, the proof relies on
ρ2 (xk +1 , ep ) ≤ ρ2 (xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
For p ≥ 2, similar inequality with Hp (xk +1 ) − Hp (ep ).
D. Groisser (2004 and 2005) and Huiling Le (2005): p = 2, tk constant, slightly smaller
domains.
B. Afsari, R. Tron, R.Vidal (2012): p ≥ 2, optimal result in the sphere: tk constant as
large as possible, domain as large as possible.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 1
Take p ∈ [1, ∞). Let (tk )k ≥1 ⊂ (0, Cp,µ,r ] a sequence of positive numbers satisfying
P∞
P∞ 2
k =1 tk = +∞ and
k =1 tk < ∞. Let x0 ∈ B(a, r ); define the sequence (xk )k ≥0
by
xk +1 = expxk −tk +1 gradxk Hp (·) , k ≥ 0;
. Then xk −→ ep .
For 1 ≤ p < 2, the proof relies on
ρ2 (xk +1 , ep ) ≤ ρ2 (xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
For p ≥ 2, similar inequality with Hp (xk +1 ) − Hp (ep ).
D. Groisser (2004 and 2005) and Huiling Le (2005): p = 2, tk constant, slightly smaller
domains.
B. Afsari, R. Tron, R.Vidal (2012): p ≥ 2, optimal result in the sphere: tk constant as
large as possible, domain as large as possible.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 1
Take p ∈ [1, ∞). Let (tk )k ≥1 ⊂ (0, Cp,µ,r ] a sequence of positive numbers satisfying
P∞
P∞ 2
k =1 tk = +∞ and
k =1 tk < ∞. Let x0 ∈ B(a, r ); define the sequence (xk )k ≥0
by
xk +1 = expxk −tk +1 gradxk Hp (·) , k ≥ 0;
. Then xk −→ ep .
For 1 ≤ p < 2, the proof relies on
ρ2 (xk +1 , ep ) ≤ ρ2 (xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
For p ≥ 2, similar inequality with Hp (xk +1 ) − Hp (ep ).
D. Groisser (2004 and 2005) and Huiling Le (2005): p = 2, tk constant, slightly smaller
domains.
B. Afsari, R. Tron, R.Vidal (2012): p ≥ 2, optimal result in the sphere: tk constant as
large as possible, domain as large as possible.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 2
Take p = ∞. Let x0 ∈ B(a, r ); define the sequence (xk )k ≥0 by
−−−−→
xk +1 = expxk −tk +1 xk yk +1 , k ≥ 0,
where yk +1 is one point of supp(µ) which realizes the maximum of the distance to xk .
Then xk −→ e∞ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 3
Let (Pk )k ≥1 an independent sequence of random variables taking their values in
B(a, r ), and with law µ.
let x0 ∈ B(a, r ); define a random walk (Xk )k ≥0 with X0 = x0
−−−−→
Xk +1 = expXk −tk +1 gradXk ρp (·, Pk +1 ) = expXk tk +1 pρp−1 Xk Pk +1 , k ≥ 0;
where by convention gradx ρ1 (·, x) = 0.
Then Xk −→ ep in L2 and a.s.
The main advantage of this method is that it does not require the computation of
grad Hp .
For 1 ≤ p < 2 the proof relies on inequality
h
i
E ρ2 (Xk +1 , ep )|Fk ≤ ρ2 (Xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
and then convergence of bounded supermartingale.
For p ≥ 2, similar inequality with Hp (Xk +1 ) − Hp (ep ).
Related results :
p = 2, Cartan-Hadamard manifold : K.T. Sturm 2004
p = 1, M a Hilbert space : H. Cardot, P. Cenac, P.A. Zitt 2012
convex manifold, C 3 cost function : S. Bonnabel 2011
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 3
Let (Pk )k ≥1 an independent sequence of random variables taking their values in
B(a, r ), and with law µ.
let x0 ∈ B(a, r ); define a random walk (Xk )k ≥0 with X0 = x0
−−−−→
Xk +1 = expXk −tk +1 gradXk ρp (·, Pk +1 ) = expXk tk +1 pρp−1 Xk Pk +1 , k ≥ 0;
where by convention gradx ρ1 (·, x) = 0.
Then Xk −→ ep in L2 and a.s.
The main advantage of this method is that it does not require the computation of
grad Hp .
For 1 ≤ p < 2 the proof relies on inequality
h
i
E ρ2 (Xk +1 , ep )|Fk ≤ ρ2 (Xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
and then convergence of bounded supermartingale.
For p ≥ 2, similar inequality with Hp (Xk +1 ) − Hp (ep ).
Related results :
p = 2, Cartan-Hadamard manifold : K.T. Sturm 2004
p = 1, M a Hilbert space : H. Cardot, P. Cenac, P.A. Zitt 2012
convex manifold, C 3 cost function : S. Bonnabel 2011
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 3
Let (Pk )k ≥1 an independent sequence of random variables taking their values in
B(a, r ), and with law µ.
let x0 ∈ B(a, r ); define a random walk (Xk )k ≥0 with X0 = x0
−−−−→
Xk +1 = expXk −tk +1 gradXk ρp (·, Pk +1 ) = expXk tk +1 pρp−1 Xk Pk +1 , k ≥ 0;
where by convention gradx ρ1 (·, x) = 0.
Then Xk −→ ep in L2 and a.s.
The main advantage of this method is that it does not require the computation of
grad Hp .
For 1 ≤ p < 2 the proof relies on inequality
h
i
E ρ2 (Xk +1 , ep )|Fk ≤ ρ2 (Xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
and then convergence of bounded supermartingale.
For p ≥ 2, similar inequality with Hp (Xk +1 ) − Hp (ep ).
Related results :
p = 2, Cartan-Hadamard manifold : K.T. Sturm 2004
p = 1, M a Hilbert space : H. Cardot, P. Cenac, P.A. Zitt 2012
convex manifold, C 3 cost function : S. Bonnabel 2011
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 3
Let (Pk )k ≥1 an independent sequence of random variables taking their values in
B(a, r ), and with law µ.
let x0 ∈ B(a, r ); define a random walk (Xk )k ≥0 with X0 = x0
−−−−→
Xk +1 = expXk −tk +1 gradXk ρp (·, Pk +1 ) = expXk tk +1 pρp−1 Xk Pk +1 , k ≥ 0;
where by convention gradx ρ1 (·, x) = 0.
Then Xk −→ ep in L2 and a.s.
The main advantage of this method is that it does not require the computation of
grad Hp .
For 1 ≤ p < 2 the proof relies on inequality
h
i
E ρ2 (Xk +1 , ep )|Fk ≤ ρ2 (Xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
and then convergence of bounded supermartingale.
For p ≥ 2, similar inequality with Hp (Xk +1 ) − Hp (ep ).
Related results :
p = 2, Cartan-Hadamard manifold : K.T. Sturm 2004
p = 1, M a Hilbert space : H. Cardot, P. Cenac, P.A. Zitt 2012
convex manifold, C 3 cost function : S. Bonnabel 2011
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 3
Let (Pk )k ≥1 an independent sequence of random variables taking their values in
B(a, r ), and with law µ.
let x0 ∈ B(a, r ); define a random walk (Xk )k ≥0 with X0 = x0
−−−−→
Xk +1 = expXk −tk +1 gradXk ρp (·, Pk +1 ) = expXk tk +1 pρp−1 Xk Pk +1 , k ≥ 0;
where by convention gradx ρ1 (·, x) = 0.
Then Xk −→ ep in L2 and a.s.
The main advantage of this method is that it does not require the computation of
grad Hp .
For 1 ≤ p < 2 the proof relies on inequality
h
i
E ρ2 (Xk +1 , ep )|Fk ≤ ρ2 (Xk , ep )(1 − Cp,µ,r tk +1 ) + C(β, r , p)tk2+1 .
and then convergence of bounded supermartingale.
For p ≥ 2, similar inequality with Hp (Xk +1 ) − Hp (ep ).
Related results :
p = 2, Cartan-Hadamard manifold : K.T. Sturm 2004
p = 1, M a Hilbert space : H. Cardot, P. Cenac, P.A. Zitt 2012
convex manifold, C 3 cost function : S. Bonnabel 2011
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
How does the algorithm work?
A1 , A2 , A3 and A4 are data points, M is the p-mean.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Exemple of simulation: the map p 7−→ ep .
e∞
mean
4
median
3
2
1
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Existence and uniqueness
Deterministic algorithms
Stochastic algorithms
Stochastic algorithms, speed of convergence
Theorem 4
Let (Xk )k ≥0 the time inhomogeneous Markov process defined in Theorem 3 with
tk = min δk , Cp,µ,r . Assume that Hp is C 2 in a neighbourhood of ep and that
δ > Cp,µ,r .
Then the sequence of processes
[nt] −−−−→
√ ep X[nt]
n
t≥0
converges weakly in D((0, ∞), Tep M) to a diffusion process yδ with generator
δ2 h
i
1
y − δ∇dHp (y , ·)] +
E gradep ρp (·, P1 ) ⊗ gradep ρp (·, P1 ) .
t
2
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Assume now that M is a complete Riemannian manifold of dimension ≥ 2.
Let ∆ be an upper bound for sectional curvatures
1
Let r > 0 such that δ := µ B̄(a, r ) > .
2
Theorem 5
If ∆ > 0 and
2δr
≤ rδ,1 then
2δ − 1
Q1,µ ⊂ B̄
1
a, √ arcsin
∆
If ∆ = 0 then
Q1,µ ⊂ B̄
a, √
√ !!
δ sin( ∆r
√
.
2δ − 1
δr
2δ − 1
.
If ∆ < 0 then
Q1,µ ⊂ B̄
a, √
1
argsinh
−∆
Marc Arnaudon
!!
√
δ sinh( −∆r
√
.
2δ − 1
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Assume now that M is a complete Riemannian manifold of dimension ≥ 2.
Let ∆ be an upper bound for sectional curvatures
1
Let r > 0 such that δ := µ B̄(a, r ) > .
2
Theorem 5
If ∆ > 0 and
2δr
≤ rδ,1 then
2δ − 1
Q1,µ ⊂ B̄
1
a, √ arcsin
∆
If ∆ = 0 then
Q1,µ ⊂ B̄
a, √
√ !!
δ sin( ∆r
√
.
2δ − 1
δr
2δ − 1
.
If ∆ < 0 then
Q1,µ ⊂ B̄
a, √
1
argsinh
−∆
Marc Arnaudon
!!
√
δ sinh( −∆r
√
.
2δ − 1
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Assume now that M is a complete Riemannian manifold of dimension ≥ 2.
Let ∆ be an upper bound for sectional curvatures
1
Let r > 0 such that δ := µ B̄(a, r ) > .
2
Theorem 5
If ∆ > 0 and
2δr
≤ rδ,1 then
2δ − 1
Q1,µ ⊂ B̄
1
a, √ arcsin
∆
If ∆ = 0 then
Q1,µ ⊂ B̄
a, √
√ !!
δ sin( ∆r
√
.
2δ − 1
δr
2δ − 1
.
If ∆ < 0 then
Q1,µ ⊂ B̄
a, √
1
argsinh
−∆
Marc Arnaudon
!!
√
δ sinh( −∆r
√
.
2δ − 1
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Assume now that M is a complete Riemannian manifold of dimension ≥ 2.
Let ∆ be an upper bound for sectional curvatures
1
Let r > 0 such that δ := µ B̄(a, r ) > .
2
Theorem 5
If ∆ > 0 and
2δr
≤ rδ,1 then
2δ − 1
Q1,µ ⊂ B̄
1
a, √ arcsin
∆
If ∆ = 0 then
Q1,µ ⊂ B̄
a, √
√ !!
δ sin( ∆r
√
.
2δ − 1
δr
2δ − 1
.
If ∆ < 0 then
Q1,µ ⊂ B̄
a, √
1
argsinh
−∆
Marc Arnaudon
!!
√
δ sinh( −∆r
√
.
2δ − 1
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Assume that M is complete.
N
1 X
δxk
Recall µ(x1 , . . . , xN ) =
N
k =1
Theorem 6
Fix p ∈ [1, ∞). Then for λ⊗N almost all (x1 , . . . , xN ) ∈ M N the p-mean ep of
µ(x1 , . . . , xN ) is unique.
Sketch of proof: if y1 (x1 , . . . , xN ) and y2 (x1 , . . . , xN ) are two p-means then the gradient
of the map
(x1 , . . . xN ) 7→ Hp,µ(x1 ,...,xN ) (y1 (x1 , . . . , xN ))
is not equal to the gradient of the map
(x1 , . . . xN ) 7→ Hp,µ(x1 ,...,xN ) (y2 (x1 , . . . , xN )).
The difficulty comes from the fact that there are many sets on which these gradients
are not well-defined.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Assume that M is complete.
N
1 X
Recall µ(x1 , . . . , xN ) =
δxk
N
k =1
Theorem 6
Fix p ∈ [1, ∞). Then for λ⊗N almost all (x1 , . . . , xN ) ∈ M N the p-mean ep of
µ(x1 , . . . , xN ) is unique.
Sketch of proof: if y1 (x1 , . . . , xN ) and y2 (x1 , . . . , xN ) are two p-means then the gradient
of the map
(x1 , . . . xN ) 7→ Hp,µ(x1 ,...,xN ) (y1 (x1 , . . . , xN ))
is not equal to the gradient of the map
(x1 , . . . xN ) 7→ Hp,µ(x1 ,...,xN ) (y2 (x1 , . . . , xN )).
The difficulty comes from the fact that there are many sets on which these gradients
are not well-defined.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Assume that M is complete.
N
1 X
Recall µ(x1 , . . . , xN ) =
δxk
N
k =1
Theorem 6
Fix p ∈ [1, ∞). Then for λ⊗N almost all (x1 , . . . , xN ) ∈ M N the p-mean ep of
µ(x1 , . . . , xN ) is unique.
Sketch of proof: if y1 (x1 , . . . , xN ) and y2 (x1 , . . . , xN ) are two p-means then the gradient
of the map
(x1 , . . . xN ) 7→ Hp,µ(x1 ,...,xN ) (y1 (x1 , . . . , xN ))
is not equal to the gradient of the map
(x1 , . . . xN ) 7→ Hp,µ(x1 ,...,xN ) (y2 (x1 , . . . , xN )).
The difficulty comes from the fact that there are many sets on which these gradients
are not well-defined.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Corollary 1
Fix p ∈ [1, ∞). If (Xn )n≥1 is a sequence of i.i.d. M-valued random variables with
absolutely continuous laws, then the process of empirical p-means ep,µ(X1 (ω),...,Xn (ω))
is a.s. well defined.
For M = T, p = 2, continuous law: Bhattacharya and Pantagreanu (2003)
For M = T, limit theorems for e2,µ(X1 (ω),...,Xn (ω)) : Hotz and Huckemann (2011)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Corollary 1
Fix p ∈ [1, ∞). If (Xn )n≥1 is a sequence of i.i.d. M-valued random variables with
absolutely continuous laws, then the process of empirical p-means ep,µ(X1 (ω),...,Xn (ω))
is a.s. well defined.
For M = T, p = 2, continuous law: Bhattacharya and Pantagreanu (2003)
For M = T, limit theorems for e2,µ(X1 (ω),...,Xn (ω)) : Hotz and Huckemann (2011)
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Back to general situation. Assume M is a compact manifold.
Z
U(x) =
κ(x, y )µ(dy ).
M
M set of minimizers of U.
PB: find Xt → M
Z
Need for δ > 0 small, κδ (x, y ) =
Z
Uδ (x) =
κδ (x, y )µ(dy ).
M
p(δ, x, z)κ(z, y )λ(dz)
M
1
exp (−βUδ (x)) λ(dx)
Zβ,δ
1
1
= ∆ − β∇Uδ
2
2
Gibbs measure νβ,δ =
Generator Lβ,δ
Facts
For large β, νβ,δ concentrates around minimal values of Uδ .
νβ,δ is the invariant measure for the diffusion process associated to Lβ,δ
β
dXt = ”dBt ” − ∇Uδ (Xt ) dt.
2
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Back to general situation. Assume M is a compact manifold.
Z
U(x) =
κ(x, y )µ(dy ).
M
M set of minimizers of U.
PB: find Xt → M
Z
Need for δ > 0 small, κδ (x, y ) =
Z
Uδ (x) =
κδ (x, y )µ(dy ).
M
p(δ, x, z)κ(z, y )λ(dz)
M
1
exp (−βUδ (x)) λ(dx)
Zβ,δ
1
1
= ∆ − β∇Uδ
2
2
Gibbs measure νβ,δ =
Generator Lβ,δ
Facts
For large β, νβ,δ concentrates around minimal values of Uδ .
νβ,δ is the invariant measure for the diffusion process associated to Lβ,δ
β
dXt = ”dBt ” − ∇Uδ (Xt ) dt.
2
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Back to general situation. Assume M is a compact manifold.
Z
U(x) =
κ(x, y )µ(dy ).
M
M set of minimizers of U.
PB: find Xt → M
Z
Need for δ > 0 small, κδ (x, y ) =
Z
Uδ (x) =
κδ (x, y )µ(dy ).
M
p(δ, x, z)κ(z, y )λ(dz)
M
1
exp (−βUδ (x)) λ(dx)
Zβ,δ
1
1
= ∆ − β∇Uδ
2
2
Gibbs measure νβ,δ =
Generator Lβ,δ
Facts
For large β, νβ,δ concentrates around minimal values of Uδ .
νβ,δ is the invariant measure for the diffusion process associated to Lβ,δ
β
dXt = ”dBt ” − ∇Uδ (Xt ) dt.
2
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Back to general situation. Assume M is a compact manifold.
Z
U(x) =
κ(x, y )µ(dy ).
M
M set of minimizers of U.
PB: find Xt → M
Z
Need for δ > 0 small, κδ (x, y ) =
Z
Uδ (x) =
κδ (x, y )µ(dy ).
M
p(δ, x, z)κ(z, y )λ(dz)
M
1
exp (−βUδ (x)) λ(dx)
Zβ,δ
1
1
= ∆ − β∇Uδ
2
2
Gibbs measure νβ,δ =
Generator Lβ,δ
Facts
For large β, νβ,δ concentrates around minimal values of Uδ .
νβ,δ is the invariant measure for the diffusion process associated to Lβ,δ
β
dXt = ”dBt ” − ∇Uδ (Xt ) dt.
2
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Back to general situation. Assume M is a compact manifold.
Z
U(x) =
κ(x, y )µ(dy ).
M
M set of minimizers of U.
PB: find Xt → M
Z
Need for δ > 0 small, κδ (x, y ) =
Z
Uδ (x) =
κδ (x, y )µ(dy ).
M
p(δ, x, z)κ(z, y )λ(dz)
M
1
exp (−βUδ (x)) λ(dx)
Zβ,δ
1
1
= ∆ − β∇Uδ
2
2
Gibbs measure νβ,δ =
Generator Lβ,δ
Facts
For large β, νβ,δ concentrates around minimal values of Uδ .
νβ,δ is the invariant measure for the diffusion process associated to Lβ,δ
β
dXt = ”dBt ” − ∇Uδ (Xt ) dt.
2
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Back to general situation. Assume M is a compact manifold.
Z
U(x) =
κ(x, y )µ(dy ).
M
M set of minimizers of U.
PB: find Xt → M
Z
Need for δ > 0 small, κδ (x, y ) =
Z
Uδ (x) =
κδ (x, y )µ(dy ).
M
p(δ, x, z)κ(z, y )λ(dz)
M
1
exp (−βUδ (x)) λ(dx)
Zβ,δ
1
1
= ∆ − β∇Uδ
2
2
Gibbs measure νβ,δ =
Generator Lβ,δ
Facts
For large β, νβ,δ concentrates around minimal values of Uδ .
νβ,δ is the invariant measure for the diffusion process associated to Lβ,δ
β
dXt = ”dBt ” − ∇Uδ (Xt ) dt.
2
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Let nt the density of Xt with respect to νβ,δ , ft =
Z
Jt = varνβ,δ (ft ) =
nt
and
νβ,δ
(ft − 1)2 dνβ,δ .
M
Then Jt0 = −νβ,δ (|∇ft |2 )
Poincaré inequality (Holley-Kusuoka-Stroock)
varνβ,δ (ft ) ≤ CM [1 ∧ βk∇Uδ k∞ ]5m−2 exp(b(Uδ )β)νβ,δ (|∇ft |2 )
Consequence
Jt0 ≤ −C(βδ −1 )2−5m exp(b(Uδ )β)Jt
Let βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 . With this choice Jt → 0 as t → ∞. So
Theorem 7
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Pb: difficult to compute ∇Uδ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Let nt the density of Xt with respect to νβ,δ , ft =
Z
Jt = varνβ,δ (ft ) =
nt
and
νβ,δ
(ft − 1)2 dνβ,δ .
M
Then Jt0 = −νβ,δ (|∇ft |2 )
Poincaré inequality (Holley-Kusuoka-Stroock)
varνβ,δ (ft ) ≤ CM [1 ∧ βk∇Uδ k∞ ]5m−2 exp(b(Uδ )β)νβ,δ (|∇ft |2 )
Consequence
Jt0 ≤ −C(βδ −1 )2−5m exp(b(Uδ )β)Jt
Let βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 . With this choice Jt → 0 as t → ∞. So
Theorem 7
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Pb: difficult to compute ∇Uδ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Let nt the density of Xt with respect to νβ,δ , ft =
Z
Jt = varνβ,δ (ft ) =
nt
and
νβ,δ
(ft − 1)2 dνβ,δ .
M
Then Jt0 = −νβ,δ (|∇ft |2 )
Poincaré inequality (Holley-Kusuoka-Stroock)
varνβ,δ (ft ) ≤ CM [1 ∧ βk∇Uδ k∞ ]5m−2 exp(b(Uδ )β)νβ,δ (|∇ft |2 )
Consequence
Jt0 ≤ −C(βδ −1 )2−5m exp(b(Uδ )β)Jt
Let βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 . With this choice Jt → 0 as t → ∞. So
Theorem 7
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Pb: difficult to compute ∇Uδ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Let nt the density of Xt with respect to νβ,δ , ft =
Z
Jt = varνβ,δ (ft ) =
nt
and
νβ,δ
(ft − 1)2 dνβ,δ .
M
Then Jt0 = −νβ,δ (|∇ft |2 )
Poincaré inequality (Holley-Kusuoka-Stroock)
varνβ,δ (ft ) ≤ CM [1 ∧ βk∇Uδ k∞ ]5m−2 exp(b(Uδ )β)νβ,δ (|∇ft |2 )
Consequence
Jt0 ≤ −C(βδ −1 )2−5m exp(b(Uδ )β)Jt
Let βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 . With this choice Jt → 0 as t → ∞. So
Theorem 7
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Pb: difficult to compute ∇Uδ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Let nt the density of Xt with respect to νβ,δ , ft =
Z
Jt = varνβ,δ (ft ) =
nt
and
νβ,δ
(ft − 1)2 dνβ,δ .
M
Then Jt0 = −νβ,δ (|∇ft |2 )
Poincaré inequality (Holley-Kusuoka-Stroock)
varνβ,δ (ft ) ≤ CM [1 ∧ βk∇Uδ k∞ ]5m−2 exp(b(Uδ )β)νβ,δ (|∇ft |2 )
Consequence
Jt0 ≤ −C(βδ −1 )2−5m exp(b(Uδ )β)Jt
Let βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 . With this choice Jt → 0 as t → ∞. So
Theorem 7
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Pb: difficult to compute ∇Uδ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Let nt the density of Xt with respect to νβ,δ , ft =
Z
Jt = varνβ,δ (ft ) =
nt
and
νβ,δ
(ft − 1)2 dνβ,δ .
M
Then Jt0 = −νβ,δ (|∇ft |2 )
Poincaré inequality (Holley-Kusuoka-Stroock)
varνβ,δ (ft ) ≤ CM [1 ∧ βk∇Uδ k∞ ]5m−2 exp(b(Uδ )β)νβ,δ (|∇ft |2 )
Consequence
Jt0 ≤ −C(βδ −1 )2−5m exp(b(Uδ )β)Jt
Let βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 . With this choice Jt → 0 as t → ∞. So
Theorem 7
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Pb: difficult to compute ∇Uδ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
Let nt the density of Xt with respect to νβ,δ , ft =
Z
Jt = varνβ,δ (ft ) =
nt
and
νβ,δ
(ft − 1)2 dνβ,δ .
M
Then Jt0 = −νβ,δ (|∇ft |2 )
Poincaré inequality (Holley-Kusuoka-Stroock)
varνβ,δ (ft ) ≤ CM [1 ∧ βk∇Uδ k∞ ]5m−2 exp(b(Uδ )β)νβ,δ (|∇ft |2 )
Consequence
Jt0 ≤ −C(βδ −1 )2−5m exp(b(Uδ )β)Jt
Let βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 . With this choice Jt → 0 as t → ∞. So
Theorem 7
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Pb: difficult to compute ∇Uδ .
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith
Introduction
Manifolds with convex geometry
Robustness of medians
Almost everywhere uniqueness for generic data points
Finding general means in compact manifolds with simulated annealing
1
Let s 7→ φδ (s, x, y ) the value at time s of the flow started at x of z 7→ − ∇z κδ (·, y )
2
Define the generator
Z
1
1
Lα,β,δ (f )(x) = ∆f (x) +
[f (φδ (βα, x, y )) − f (x)] µ(dy )
2
α M
and Xt an inhomogeneous diffusion process with generator Lαt ,βt ,δt ,
αt = (1 + t)−1 , βt = ln t/k , k > b(U), δt = (ln(2 + t))−1 :
(α)
Let (Nt )t≥0 be a standard Poisson process and Nt = NR t α−1 ds .
0
s
Let (Pk )k ≥1 an independent sequence of random variables with law µ.
Between jump times of Ntα , dXt = ”dBt ”.
If T is a jump time, XT = φδT αT βT , XT − , P (α)
NT
Theorem 8
For any neighbourhood N of M, P(Xt ∈ N ) → 1 as t → ∞.
Marc Arnaudon
Genralized means in manifolds Existence, uniqueness, robustness and algorith