Stability of Stein kernels, moment maps and invariant measures Dan Mikulincer Weizmann Institute of Science Joint work with Max Fathi 1
What This Talk Is About Let X t , Y t be two diffusions in R d which satisfy dX t = a ( X t ) dt + τ ( X t ) dB t , dY t = b ( Y t ) dt + σ ( Y t ) dB t . Assume ν, µ to be their respective (unique) invariant measures. Question (Stability of invariant measures) Suppose that � a − b � · + � τ − σ � · is small, is µ close to ν ? 2
What This Talk Is About Let X t , Y t be two diffusions in R d which satisfy dX t = a ( X t ) dt + τ ( X t ) dB t , dY t = b ( Y t ) dt + σ ( Y t ) dB t . Assume ν, µ to be their respective (unique) invariant measures. Question (Stability of invariant measures) Suppose that � a − b � · + � τ − σ � · is small, is µ close to ν ? 2
The Motivation (What This Talk is Really About) If µ is a measure on R d we will associate to it the following objects: ❼ A matrix valued map τ µ : R d → M d ( R ), called a Stein kernel. ❼ A convex function ϕ µ : R d → R , called the moment map. Question (Stability of Stein kernels) Suppose that � τ µ − τ ν � · is small, is µ close to ν ? Question (Stability of moment maps) Suppose that � ϕ µ − ϕ ν � · is small, is µ close to ν ? 3
The Motivation (What This Talk is Really About) If µ is a measure on R d we will associate to it the following objects: ❼ A matrix valued map τ µ : R d → M d ( R ), called a Stein kernel. ❼ A convex function ϕ µ : R d → R , called the moment map. Question (Stability of Stein kernels) Suppose that � τ µ − τ ν � · is small, is µ close to ν ? Question (Stability of moment maps) Suppose that � ϕ µ − ϕ ν � · is small, is µ close to ν ? 3
The Motivation (What This Talk is Really About) If µ is a measure on R d we will associate to it the following objects: ❼ A matrix valued map τ µ : R d → M d ( R ), called a Stein kernel. ❼ A convex function ϕ µ : R d → R , called the moment map. Question (Stability of Stein kernels) Suppose that � τ µ − τ ν � · is small, is µ close to ν ? Question (Stability of moment maps) Suppose that � ϕ µ − ϕ ν � · is small, is µ close to ν ? 3
The Motivation (What This Talk is Really About) If µ is a measure on R d we will associate to it the following objects: ❼ A matrix valued map τ µ : R d → M d ( R ), called a Stein kernel. ❼ A convex function ϕ µ : R d → R , called the moment map. Question (Stability of Stein kernels) Suppose that � τ µ − τ ν � · is small, is µ close to ν ? Question (Stability of moment maps) Suppose that � ϕ µ − ϕ ν � · is small, is µ close to ν ? 3
The Motivation (What This Talk is Really About) If µ is a measure on R d we will associate to it the following objects: ❼ A matrix valued map τ µ : R d → M d ( R ), called a Stein kernel. ❼ A convex function ϕ µ : R d → R , called the moment map. Question (Stability of Stein kernels) Suppose that � τ µ − τ ν � · is small, is µ close to ν ? Question (Stability of moment maps) Suppose that � ϕ µ − ϕ ν � · is small, is µ close to ν ? 3
Stein’s method Basic observation: If G ∼ γ is the standard Gaussian on R d . Then, E [ � G , ∇ f ( G ) � ] = E [∆ f ( G )] , for any test function f : R d → R . Moreover, the Gaussian is the only measure which satisfies this relation. Stein’s idea: This property is stable. If X is any other random vector in R d . E [ � X , ∇ f ( X ) � ] ≃ E [∆ f ( X )] = ⇒ X ≃ G , 4
Stein’s method Basic observation: If G ∼ γ is the standard Gaussian on R d . Then, E [ � G , ∇ f ( G ) � ] = E [∆ f ( G )] , for any test function f : R d → R . Moreover, the Gaussian is the only measure which satisfies this relation. Stein’s idea: This property is stable. If X is any other random vector in R d . E [ � X , ∇ f ( X ) � ] ≃ E [∆ f ( X )] = ⇒ X ≃ G , 4
Stein Kernels A Stein kernel of X ∼ µ is a matrix valued map τ : R d → M d ( R ), such that � τ ( X ) , ∇ 2 f ( X ) � HS � � E [ � X , ∇ f ( X ) � ] = E . We have that τ ≡ Id iff µ = γ . The discrepancy is then defined as S 2 ( µ || γ ) = E µ � τ − Id � 2 � � . HS 5
Stein Kernels A Stein kernel of X ∼ µ is a matrix valued map τ : R d → M d ( R ), such that � τ ( X ) , ∇ 2 f ( X ) � HS � � E [ � X , ∇ f ( X ) � ] = E . We have that τ ≡ Id iff µ = γ . The discrepancy is then defined as S 2 ( µ || γ ) = E µ � τ − Id � 2 � � . HS 5
Stein Kernels - Example If X ∼ µ is a ’nice’ centered random variable on R , with density ρ its unique Stein kernel is given by ∞ � y ρ ( y ) dy x τ ( x ) := . ρ ( x ) Indeed, we can integrate by parts, ∞ ∞ ∞ � � � Xf ′ ( X ) f ′ ( x ) x ρ ( x ) dx = f ′′ ( x ) dx � � = y ρ ( y ) dy E −∞ −∞ x � ∞ � � y ρ ( y ) dy ∞ � f ′′ ( x ) x τ ( X ) f ′′ ( X ) � � = ρ ( x ) dx = E . ρ ( x ) −∞ 6
Stein Kernels - Example If X ∼ µ is a ’nice’ centered random variable on R , with density ρ its unique Stein kernel is given by ∞ � y ρ ( y ) dy x τ ( x ) := . ρ ( x ) Indeed, we can integrate by parts, ∞ ∞ ∞ � � � Xf ′ ( X ) f ′ ( x ) x ρ ( x ) dx = f ′′ ( x ) dx � � = y ρ ( y ) dy E −∞ −∞ x � ∞ � � y ρ ( y ) dy ∞ � f ′′ ( x ) x τ ( X ) f ′′ ( X ) � � = ρ ( x ) dx = E . ρ ( x ) −∞ 6
Stein Kernels ∞ � Suppose now that | τ ( x ) − 1 | is small. So, ρ ( x ) ≃ y ρ ( y ) dy . In x this case, one can use Gronwall’s inequality to show ρ ( x ) ≃ e − x 2 / 2 . yo yo In higher dimension, many different constructions for Stein kernels are known. The known constructions do not have explicit tractable expressions in general. 7
Stein Kernels ∞ � Suppose now that | τ ( x ) − 1 | is small. So, ρ ( x ) ≃ y ρ ( y ) dy . In x this case, one can use Gronwall’s inequality to show ρ ( x ) ≃ e − x 2 / 2 . yo yo In higher dimension, many different constructions for Stein kernels are known. The known constructions do not have explicit tractable expressions in general. 7
Stein Discrepancy Recall that S 2 ( µ || γ ) = E µ � τ − Id � 2 � � . It’s an exercise to show, HS W 1 ( µ, γ ) ≤ S ( µ || γ ) . What is more impressive is that, W 2 ( µ, γ ) ≤ S ( µ || γ ) , as well, as shown in (Ledoux, Nourdin, Pecatti 14’). In fact, Ent ( µ || γ ) ≤ 1 � 1 + I ( µ || γ ) � 2 S 2 ( µ || γ ) ln . S 2 ( µ || γ ) 8
Stein Discrepancy Recall that S 2 ( µ || γ ) = E µ � τ − Id � 2 � � . It’s an exercise to show, HS W 1 ( µ, γ ) ≤ S ( µ || γ ) . What is more impressive is that, W 2 ( µ, γ ) ≤ S ( µ || γ ) , as well, as shown in (Ledoux, Nourdin, Pecatti 14’). In fact, Ent ( µ || γ ) ≤ 1 � 1 + I ( µ || γ ) � 2 S 2 ( µ || γ ) ln . S 2 ( µ || γ ) 8
Stein Discrepancy Recall that S 2 ( µ || γ ) = E µ � τ − Id � 2 � � . It’s an exercise to show, HS W 1 ( µ, γ ) ≤ S ( µ || γ ) . What is more impressive is that, W 2 ( µ, γ ) ≤ S ( µ || γ ) , as well, as shown in (Ledoux, Nourdin, Pecatti 14’). In fact, Ent ( µ || γ ) ≤ 1 � 1 + I ( µ || γ ) � 2 S 2 ( µ || γ ) ln . S 2 ( µ || γ ) 8
Stein Discrepancy - Rough Sketch √ Consider the OU process dX t = − X t dt + 2 dB t , with X 0 ∼ µ . γ is the unique invariant measure of the process and we wish to bound: ∞ � d W 2 ( X 0 , X ∞ ) = dt W 2 ( X 0 , X t ) dt . 0 A result of Otto-Villani allows to bound d dt W 2 ( X 0 , X t ) by I ( X t || γ ) . yo Integration by parts is then used to bound I ( X t || γ ) by S 2 ( µ || γ ). 9
Stein Discrepancy - Rough Sketch √ Consider the OU process dX t = − X t dt + 2 dB t , with X 0 ∼ µ . γ is the unique invariant measure of the process and we wish to bound: ∞ � d W 2 ( X 0 , X ∞ ) = dt W 2 ( X 0 , X t ) dt . 0 A result of Otto-Villani allows to bound d dt W 2 ( X 0 , X t ) by I ( X t || γ ) . yo Integration by parts is then used to bound I ( X t || γ ) by S 2 ( µ || γ ). 9
Stein Discrepancy - Rough Sketch √ Consider the OU process dX t = − X t dt + 2 dB t , with X 0 ∼ µ . γ is the unique invariant measure of the process and we wish to bound: ∞ � d W 2 ( X 0 , X ∞ ) = dt W 2 ( X 0 , X t ) dt . 0 A result of Otto-Villani allows to bound d dt W 2 ( X 0 , X t ) by I ( X t || γ ) . yo Integration by parts is then used to bound I ( X t || γ ) by S 2 ( µ || γ ). 9
Stein Discrepancy with Respect to Other Measures Stein kernels and discrepancy have found numerous applications for normal approximations: ❼ Central limit theorems. ❼ Stability of functional inequalities. ❼ Second order Poincar´ e inequalities. Can we extend the theory by bounding dist ( µ, ν ) with � τ µ − τ ν � ? 10
Stein Discrepancy with Respect to Other Measures Stein kernels and discrepancy have found numerous applications for normal approximations: ❼ Central limit theorems. ❼ Stability of functional inequalities. ❼ Second order Poincar´ e inequalities. Can we extend the theory by bounding dist ( µ, ν ) with � τ µ − τ ν � ? 10
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