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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 | import Mathlib
import OAI.Probability.LogConcave.JetEstimates.AssignmentsFinset
import OAI.Probability.LogConcave.JetEstimates.MixedJetAdd
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology
section UpperProof
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open Function MeasureTheory Set Filter
open scoped Topology NNReal
namespace LogConcaveSampling.Appell
open MeasureTheory
variable {E : Type} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
variable {ι ϑ κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype ϑ] [DecidableEq ϑ]
[Fintype κ]
theorem actual_jet_cumulant_energy {μ : Measure E} [IsProbabilityMeasure μ]
(hμ : HasExpMoments μ) {β : ℝ} (hβ : 0≤β) (hP : HasPoincare μ β)
(b : OrthonormalBasis κ ℝ E) (l : List ι) (k : List ϑ)
(hl : l.Nodup) (hlall : l.toFinset=Finset.univ)
(hk : k.Nodup) (hkall : k.toFinset=Finset.univ)
(hlne : l≠[]) (hkne : k≠[]) (o : List (ι ⊕ ϑ))
(ho : o.Perm (l.map Sum.inl++k.map Sum.inr)) :
TensorEnergy.Bound (fun (a : ι → κ) (c : ϑ → κ) =>
JetCalculus.jet (Sum.elim (fun i => b (a i)) (fun j => b (c j))) o
(fun θ => Real.log (laplace μ (fun _ => (1:ℝ)) θ)) 0)
(((Fintype.card ι+Fintype.card ϑ+1:ℕ):ℝ)^(2*(Fintype.card ι+Fintype.card ϑ+1)) *
splitConstant β (Fintype.card ι) (Fintype.card ϑ)) := by
classical
have hs := (contDiff_laplace hμ continuous_const (HasGrowth.const (1:ℝ))).log
(fun θ => (laplace_one_pos hμ θ).ne')
simp_rw [JetCalculus.jet_perm hs _ ho, ← JetCalculus.mixedJet_add hs]
exact actual_cumulant_energy hμ hβ hP b l k hl hlall hk hkall hlne hkne
end LogConcaveSampling.Appell
namespace LogConcaveSampling.TensorEnergy
open scoped BigOperators
variable {I J K L : Type*} [Fintype I] [Fintype J] [Fintype K] [Fintype L]
noncomputable def zeroExtension (e : K ↪ I) (v : K → ℝ) (i : I) : ℝ := by
classical
exact ∑ k, if e k=i then v k else 0
omit [Fintype I] in
lemma zeroExtension_apply (e : K ↪ I) (v : K → ℝ) (k : K) :
zeroExtension e v (e k)=v k := by
classical
simp [zeroExtension,e.injective.eq_iff]
lemma dot_zeroExtension (e : K ↪ I) (v : K → ℝ) (g : I → ℝ) :
(∑ i, g i*zeroExtension e v i)=∑ k,g (e k)*v k := by
classical
simp only [zeroExtension,Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro k hk
simp [mul_ite]
lemma energy_zeroExtension (e : K ↪ I) (v : K → ℝ) :
(∑ i, (zeroExtension e v i)^2)=∑ k,(v k)^2 := by
simp only [pow_two]
rw [dot_zeroExtension]
simp only [zeroExtension_apply]
lemma Bound.submatrix {A : I → J → ℝ} {C : ℝ} (h : Bound A C)
(e : K ↪ I) (f : L ↪ J) : Bound (fun k l => A (e k) (f l)) C := by
classical
intro v
have hh := h (zeroExtension f v)
simp only [dot_zeroExtension,energy_zeroExtension] at hh
apply le_trans _ hh
let g : I → ℝ := fun i => (∑ l,A i (f l)*v l)^2
change (∑ k,g (e k)) ≤ ∑ i,g i
rw [← Finset.sum_image (s := Finset.univ) (f := g) e.injective.injOn]
apply Finset.sum_le_univ_sum_of_nonneg
intro i
exact sq_nonneg _
end LogConcaveSampling.TensorEnergy
end UpperProof
end
end
end
end OAI
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