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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 | import Mathlib
import OAI.Probability.LogConcave.JetEstimates.ProperSplit
import OAI.Probability.LogConcave.Sampling.NaturalMomentNorm
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.TensorEnergy
open MeasureTheory
open scoped Classical BigOperators Matrix.Norms.L2Operator ENNReal
universe u
variable {S : Type u} [Fintype S] {d : ℕ}
lemma properSplit_card_le : Fintype.card (ProperSplit S) ≤ 2^Fintype.card S := by
calc
_ ≤ Fintype.card (S → Bool) := Fintype.card_subtype_le _
_ = _ := by simp
lemma measurable_splitMatrix_norm {Ω : Type*} [MeasurableSpace Ω]
(T : Ω → (S → Fin d) → ℝ) (hT : ∀c,Measurable (fun x => T x c))
(p : ProperSplit S) : Measurable (fun x => ‖splitMatrix (T x) p‖) := by
let : MeasurableSpace (Matrix (OutSlots p → Fin d) (InSlots p → Fin d) ℝ) :=
inferInstanceAs (MeasurableSpace ((OutSlots p → Fin d) → (InSlots p → Fin d) → ℝ))
have : BorelSpace (Matrix (OutSlots p → Fin d) (InSlots p → Fin d) ℝ) :=
inferInstanceAs (BorelSpace ((OutSlots p → Fin d) → (InSlots p → Fin d) → ℝ))
apply Measurable.norm
exact Measurable.of_eval (fun _ => Measurable.of_eval (fun _ => hT _))
lemma moment_splitEnvelope {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
(T : Ω → (S → Fin d) → ℝ) (hT : ∀c,Measurable (fun x => T x c))
{n : ℕ} (hn : 0<n) (B : ℝ≥0∞)
(hB : ∀p : ProperSplit S,
(∫⁻x,ENNReal.ofReal ‖splitMatrix (T x) p‖^n ∂μ) ≤ B^n) :
(∫⁻x,ENNReal.ofReal (splitEnvelope (T x))^n ∂μ) ≤
((2:ℝ≥0∞)^Fintype.card S*B)^n := by
have hh := natural_moment_sum (μ:=μ) (fun p : ProperSplit S => fun x => ‖splitMatrix (T x) p‖)
(fun p => (measurable_splitMatrix_norm T hT p).aestronglyMeasurable)
(fun _ _ => norm_nonneg _) hn (fun _ => B) hB
apply hh.trans
apply pow_le_pow_left'
simpa using mul_le_mul'
(show (Fintype.card (ProperSplit S):ℝ≥0∞) ≤ 2^Fintype.card S by
exact_mod_cast properSplit_card_le (S:=S)) (le_refl B)
end LogConcaveSampling.TensorEnergy
end
end
end
end OAI
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