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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import Mathlib
import OAI.Probability.LogConcave.Sampling.TransportMajorant
import OAI.Probability.LogConcave.JetEstimates.ArrayRelabel
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open Set Function
open scoped Classical NNReal BigOperators
open TensorEnergy
def composedJetMajorant (n : ℕ) : ℝ :=
∑c : OrderedFinpartition n,normalizedTensorMajorant (c.length+2) 1*∏i,transportMajorant (c.partSize i)
lemma composedJetMajorant_nonneg (n : ℕ) : 0≤composedJetMajorant n :=
Finset.sum_nonneg (fun _ _ => mul_nonneg (normalizedTensorMajorant_nonneg _ _)
(Finset.prod_nonneg (fun _ _ => transportMajorant_nonneg _)))
lemma backwardFirstJet_iterated_split {d n : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r R T : ℝ} (hr : 0<r)
(hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hR : 0<R) (hRT : R^2≤1-T^2)
(hT0 : 0≤T) (hT1 : T<1) (t : Icc (0:ℝ) T) (y : Point d) :
AllSplitBound (arrayTensor (iteratedFDeriv ℝ n
(fun z => matrixArray d (conditionalFirstJet F x r ((lam:ℝ)*r)
(t,terminalBackward hF x hr.le hl hT0 hT1 (t,z)))) y))
(((lam:ℝ)*r)*composedJetMajorant n/R^n) := by
let f := probabilityTransport hF x hr.le hl hT0 hT1 ⟨T,hT0,le_rfl⟩ t
let g := fun z => matrixArray d (conditionalFirstJet F x r ((lam:ℝ)*r) (t,z))
have hf : ContDiff ℝ (⊤:ℕ∞) f := probabilityTransport_smooth hF x hr.le hl hT0 hT1 _ _
have ht2 : (t:ℝ)^2<1 := by nlinarith [t.2.1,t.2.2]
have hg0 : ContDiff ℝ (⊤:ℕ∞) (fun z => conditionalFirstJet F x r ((lam:ℝ)*r) (t,z)) := by
apply contDiff_iff_contDiffAt.mpr
intro z
exact (conditionalFirstJet_smooth hF x hr hlam hl ((t:ℝ),z) ht2).comp z
(contDiffAt_const.prodMk contDiffAt_id)
have hg : ContDiff ℝ (⊤:ℕ∞) g := (matrixArray d).contDiff.comp hg0
have htR : R^2≤1-(t:ℝ)^2 := by nlinarith [t.2.1,t.2.2]
have hh := array_composition_scale (n:=n) y hf hg
(fun k => ((lam:ℝ)*r)*normalizedTensorMajorant (k+2) 1) transportMajorant
(fun k => mul_nonneg (mul_nonneg lam.2 hr.le) (normalizedTensorMajorant_nonneg _ _))
transportMajorant_nonneg R hR 0
(fun order _ => by simpa only [Nat.add_zero] using
conditionalFirstJet_iterated_split (n:=order) hF x hr hlam hl t.2.1 (t.2.2.trans_lt hT1) hR htR (f y))
(fun _ _ => transportMajorant_bound hF x hr hlam hl hR hRT hT0 hT1 _ _ y)
have he : (fun z => matrixArray d (conditionalFirstJet F x r ((lam:ℝ)*r)
(t,terminalBackward hF x hr.le hl hT0 hT1 (t,z))))=g ∘ f := by
funext z
rw [terminalBackward_eq]
rfl
rw [he]
simpa only [Nat.add_zero,composedJetMajorant,Finset.mul_sum,mul_assoc] using hh
def kernelMajorant (n : ℕ) : ℝ :=
∑s∈(Finset.univ : Finset (Fin n)).powerset,jacobianMajorant s.card*composedJetMajorant (n-s.card)
lemma kernelMajorant_nonneg (n : ℕ) : 0≤kernelMajorant n :=
Finset.sum_nonneg (fun _ _ =>
mul_nonneg (jacobianMajorant_nonneg _) (composedJetMajorant_nonneg _))
lemma steinIntegrand_iterated_split {d n : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r R T : ℝ} (hr : 0<r)
(hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hR : 0<R) (hRT : R^2≤1-T^2)
(hT0 : 0≤T) (hT1 : T<1) (t : Icc (0:ℝ) T) (y : Point d) :
AllSplitBound (arrayTensor (iteratedFDeriv ℝ n
(fun z => matrixArray d (steinIntegrand hF x hr hl hT0 hT1 (t,z))) y))
(((lam:ℝ)*r)*kernelMajorant n/R^n) := by
have hf : ContDiff ℝ (⊤:ℕ∞) (fun z => terminalJacobian hF x hr hl hT0 hT1 (t,z)) :=
(terminalJacobian_smooth hF x hr hl hT0 hT1).comp
(contDiff_const.prodMk (contDiff_id (E:=Point d)))
have hg : ContDiff ℝ (⊤:ℕ∞) (fun z => conditionalFirstJet F x r ((lam:ℝ)*r)
(t,terminalBackward hF x hr.le hl hT0 hT1 (t,z))) := by
apply contDiff_iff_contDiffAt.mpr
intro z
apply (conditionalFirstJet_smooth hF x hr hlam hl _ (by nlinarith [t.2.1,t.2.2])).comp z
exact contDiffAt_const.prodMk
(((terminalBackward_smooth hF x hr.le hl hT0 hT1).comp (contDiff_const.prodMk contDiff_id)).contDiffAt)
have hh := matrixComposition_split (n:=n) hf hg
(fun k => jacobianMajorant k/R^k) (fun k => ((lam:ℝ)*r)*composedJetMajorant k/R^k)
(fun k => div_nonneg (jacobianMajorant_nonneg _) (pow_nonneg hR.le _))
(fun k => div_nonneg (mul_nonneg (mul_nonneg lam.2 hr.le) (composedJetMajorant_nonneg _)) (pow_nonneg hR.le _)) y
(fun _ _ => terminalJacobian_iterated_split hF x hr hlam hl hR hRT hT0 hT1 t y)
(fun _ _ => backwardFirstJet_iterated_split hF x hr hlam hl hR hRT hT0 hT1 t y)
have he (s : Finset (Fin n)) :
(jacobianMajorant s.card/R^s.card)*(((lam:ℝ)*r)*composedJetMajorant (n-s.card)/R^(n-s.card))=
((lam:ℝ)*r)*(jacobianMajorant s.card*composedJetMajorant (n-s.card))/R^n := by
rw [div_mul_div_comm,←pow_add,Nat.add_sub_of_le ((Finset.card_le_univ s).trans_eq (Fintype.card_fin n))]
ring
simp_rw [he] at hh
simpa only [steinIntegrand,smoothTimeClip_eq hT0 hT1 t.2,kernelMajorant,
←Finset.sum_div,←Finset.mul_sum] using hh
end LogConcaveSampling
end
end
end
end OAI
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