File size: 3,078 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import Mathlib
import OAI.Probability.LogConcave.Sampling.ProbabilityNodes
import OAI.Probability.LogConcave.Complexity.CenteringStateBudget
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory TensorEnergy Quadrature
open scoped Classical BigOperators NNReal RealInnerProductSpace
local instance identityOutputEmbeddingDecidableEqUnit : DecidableEq Unit :=
Classical.decEq _
def identityOutputEmbedding (d : ℕ) : Fin d ↪ (Unit → Fin d) :=
⟨fun i _ => i,fun _ _ h => congrFun h ()⟩
def identityOutputProjection (d : ℕ) : EuclideanSpace ℝ (Unit → Fin d) →L[ℝ] Point d :=
finiteCoordinateProjection (identityOutputEmbedding d)
lemma identityOutputProjection_vector {d : ℕ} (f : Point d → Point d) (y : Point d) :
identityOutputProjection d (tensorVector (vectorArray f) y)=f y := by
ext i
change inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) (f y)=_
simp only [EuclideanSpace.basisFun_inner]
lemma identityOutputProjection_lie {d : ℕ} (H : Point d → ℝ)
(A : (Unit ⊕ Unit → Fin d) → Point d → ℝ)
(hA : ∀c,PolySmooth (A c))
(hsk : ∀i z y,A (Sum.elim (fun _ => i) (fun _ => z)) y=
-A (Sum.elim (fun _ => z) (fun _ => i)) y) (y : Point d) :
identityOutputProjection d (tensorVector
(iterTensorLie H A (vectorArray (id : Point d → Point d)) 1) y)=skewLieField H A y := by
ext i
change tensorLie H A (vectorArray (id : Point d → Point d)) (fun _ => i) y=_
rw [tensorLie_eq_directional H A _ (fun c => (hA c).smooth)
(fun c => (identityArray_polySmooth d c).smooth) hsk]
change fderiv ℝ ((innerSL ℝ (EuclideanSpace.basisFun (Fin d) ℝ i)) : Point d → ℝ) y
(skewLieField H A y)=_
rw [ContinuousLinearMap.fderiv]
simp only [innerSL_apply_apply, EuclideanSpace.basisFun_inner]
lemma projection_chainTaylor {E G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup G] [NormedSpace ℝ G]
(L : E →L[ℝ] G) (J : ℕ → ℝ → E) (n : ℕ) (a b : ℝ) :
L (chainTaylor J n a b)=chainTaylor (fun k t => L (J k t)) n a b := by
simp only [chainTaylor,map_sum,map_smul]
lemma projection_rms {Ω I J : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
[Fintype I] [Fintype J] (e : I ↪ J)
{f : Ω → EuclideanSpace ℝ J} (hm : Measurable f)
{B : ℝ} (hi : Integrable (fun y => ‖f y‖^2) μ) (hb : (∫y,‖f y‖^2 ∂μ)≤B) :
Integrable (fun y => ‖finiteCoordinateProjection e (f y)‖^2) μ ∧
(∫y,‖finiteCoordinateProjection e (f y)‖^2 ∂μ)≤B := by
have ha (y : Ω) : ‖finiteCoordinateProjection e (f y)‖^2≤‖f y‖^2 :=
pow_le_pow_left₀ (norm_nonneg _) (finiteCoordinateProjection_norm_le _ _) 2
have h := hi.mono' (((finiteCoordinateProjection e).continuous.measurable.comp hm).norm.pow_const 2).aestronglyMeasurable
(Filter.Eventually.of_forall (fun y => by simpa only [Real.norm_eq_abs,abs_sq,Function.comp_def] using ha y))
exact ⟨h,(integral_mono h hi ha).trans hb⟩
end LogConcaveSampling
end
end
end OAI
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