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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 | import Mathlib
import OAI.Probability.LogConcave.Dynamics.StationaryPath
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory Quadrature RMSIntegral
open scoped Classical BigOperators NNReal
variable {E H : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[MeasurableSpace E] [BorelSpace E]
[NormedAddCommGroup H] [InnerProductSpace ℝ H]
[MeasurableSpace H] [BorelSpace H] [SecondCountableTopology H]
{μ : Measure E} {V : E → E}
lemma StationaryPath.observableIntegral_measurable (X : StationaryPath μ V)
{M : E → H} (hM : Continuous M) :
Measurable (fun y => ∫t in (0:ℝ)..1,M (X.path y t)) := by
simp only [intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1)]
exact (hM.measurable.comp X.measurable).stronglyMeasurable.integral_prod_left'.measurable
theorem stationaryOutput_rms (X : StationaryPath μ V)
{M : E → H} {L : ℝ≥0} (hM : LipschitzWith L M)
(n : ℕ) (Z : Fin (n+1) → E → E) (hmZ : ∀j,Measurable (Z j))
{Bz Bq : ℝ} (hBz : 0 ≤ Bz)
(hZi : ∀j,Integrable (fun y => ‖Z j y-X.path y (probabilityNodes n j)‖^2) μ)
(hZb : ∀j,(∫y,‖Z j y-X.path y (probabilityNodes n j)‖^2 ∂μ) ≤ Bz)
(hqi : Integrable (fun y => ‖(∫t in (0:ℝ)..1,M (X.path y t))-
∑j,weight (probabilityNodes n) j 0 1 • M (X.path y (probabilityNodes n j))‖^2) μ)
(hqb : (∫y,‖(∫t in (0:ℝ)..1,M (X.path y t))-
∑j,weight (probabilityNodes n) j 0 1 • M (X.path y (probabilityNodes n j))‖^2 ∂μ) ≤ Bq)
(anchor : E → H) :
Integrable (fun y => ‖(anchor y+∑j,weight (probabilityNodes n) j 0 1 • M (Z j y))-
(anchor y+∫t in (0:ℝ)..1,M (X.path y t))‖^2) μ ∧
(∫y,‖(anchor y+∑j,weight (probabilityNodes n) j 0 1 • M (Z j y))-
(anchor y+∫t in (0:ℝ)..1,M (X.path y t))‖^2 ∂μ) ≤
2*(centeringRowBudget n:ℝ)^2*(L:ℝ)^2*Bz+2*Bq := by
let w := fun j => weight (probabilityNodes n) j 0 1
let e := fun j y => M (Z j y)-M (X.path y (probabilityNodes n j))
have hem (j : Fin (n+1)) : Measurable (e j) :=
(hM.continuous.measurable.comp (hmZ j)).sub
(hM.continuous.measurable.comp (X.measurable_slice _))
have heb (j : Fin (n+1)) := lipschitz_error_sq (hmZ j) (X.measurable_slice _)
hM (hZi j) (hZb j)
have hsum := weighted_sum_sq w e (fun j => (hem j).aestronglyMeasurable)
(fun j => (heb j).1) (fun j => (heb j).2)
let q := fun y => (∫t in (0:ℝ)..1,M (X.path y t))-
∑j,w j • M (X.path y (probabilityNodes n j))
have hqm : Measurable q := (X.observableIntegral_measurable hM.continuous).sub
(Finset.measurable_sum _ (fun j _ =>
(hM.continuous.measurable.comp (X.measurable_slice _)).const_smul _))
have hsm : Measurable (fun y => ∑j,w j • e j y) :=
Finset.measurable_sum _ (fun j _ => (hem j).const_smul _)
have hh := sub_sq_bound hsm.aestronglyMeasurable hqm.aestronglyMeasurable
hsum.1 hqi hsum.2 hqb
have hid (y : E) : (anchor y+∑j,w j • M (Z j y))-
(anchor y+∫t in (0:ℝ)..1,M (X.path y t))=(∑j,w j • e j y)-q y := by
simp only [e,q,smul_sub,Finset.sum_sub_distrib]
abel
dsimp only [w] at hid
simp_rw [hid]
refine ⟨hh.1,hh.2.trans ?_⟩
have hw := pow_le_pow_left₀ (Finset.sum_nonneg (fun _ _ => abs_nonneg _))
(centeringFullWeight_bound n) 2
have hv := mul_le_mul_of_nonneg_right hw (show 0 ≤ (L:ℝ)^2*Bz by positivity)
change (∑j,|w j|)^2*((L:ℝ)^2*Bz) ≤ _ at hv
nlinarith
end LogConcaveSampling
end
end
end OAI
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