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b317718 | 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 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.ReferenceBounds
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace ReferenceGaussian
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open GaussianTools
lemma integrable_energy_joint {p d : ℕ} (τ : Measure (Vec d)) [IsProbabilityMeasure τ]
(s : Vec d) {R : ℝ} (hR : 0 ≤ R) (hs : ∀ᵐ u ∂τ,u ∈ closedBall s R)
(e : Fin p → ℝ) :
Integrable (fun t : Mat p d × Vec d => energy t.1 (s-t.2) e)
((Measure.pi (fun _ : Fin p => scalarRow d)).prod τ) := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
have hm : Measurable (fun t : Mat p d × Vec d => energy t.1 (s-t.2) e) := by
unfold energy applyRows; fun_prop
apply ((((integrable_matEnergy (p := p) (d := d)).comp_fst τ).const_mul (2*R^2)).add
(integrable_const (2*∑ i,e i^2))).mono' hm.aestronglyMeasurable
have hae : ∀ᵐ t : Mat p d × Vec d ∂(Measure.pi (fun _ : Fin p => scalarRow d)).prod τ,
t.2 ∈ closedBall s R := measurePreserving_snd.quasiMeasurePreserving.ae hs
filter_upwards [hae] with t ht
rw [Real.norm_eq_abs,abs_of_nonneg (energy_nonneg _ _ _)]
have ht' : ‖s-t.2‖ ≤ R := by simpa only [mem_closedBall,dist_eq_norm,norm_sub_rev] using ht
have hb := energy_le t.1 (s-t.2) e
have hsq : ‖s-t.2‖^2 ≤ R^2 := (sq_le_sq₀ (norm_nonneg _) hR).mpr ht'
change energy t.1 (s-t.2) e ≤ 2*R^2*matEnergy t.1+2*∑ i,e i^2
nlinarith [mul_le_mul_of_nonneg_left hsq (matEnergy_nonneg t.1)]
lemma energy_rows_local_le {p d : ℕ} (τ : Measure (Vec d)) [IsProbabilityMeasure τ]
(s : Vec d) {R : ℝ} (hR : 0 ≤ R) (hs : ∀ᵐ u ∂τ,u ∈ closedBall s R)
(e : Fin p → ℝ) :
(∫ X : Mat p d,∫ u,energy X (s-u) e ∂τ ∂Measure.pi (fun _ : Fin p => scalarRow d)) ≤
2*(p:ℝ)*R^2+2*∑ i,e i^2 := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
rw [integral_integral_swap (integrable_energy_joint τ s hR hs e)]
have hi := (integrable_energy_joint τ s hR hs e).integral_prod_right
calc
_ ≤ ∫ _u,2*(p:ℝ)*R^2+2*∑ i,e i^2 ∂τ := by
apply integral_mono_ae hi (integrable_const _)
filter_upwards [hs] with u hu
have hu' : ‖s-u‖ ≤ R := by simpa only [mem_closedBall,dist_eq_norm,norm_sub_rev] using hu
have hsq : ‖s-u‖^2 ≤ R^2 := pow_le_pow_left₀ (norm_nonneg _) hu' 2
exact (energy_rows_le (s-u) e).trans (by gcongr)
_ = _ := by simp
lemma integrable_log_shift_rows {p d : ℕ} (τ : Measure (Vec d)) [IsProbabilityMeasure τ]
(s : Vec d) {R : ℝ} (hR : 0 ≤ R) (hs : ∀ᵐ u ∂τ,u ∈ closedBall s R)
{r : ℝ} (hr : 0 < r) (e : Fin p → ℝ) :
Integrable (fun X : Mat p d => ∫ u,Real.log (atDensity r X (applyRows X s+e) u) ∂τ)
(Measure.pi (fun _ : Fin p => scalarRow d)) := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
have hi := (integrable_energy_joint τ s hR hs e).integral_prod_left
have he (X : Mat p d) :
(∫ u,Real.log (atDensity r X (applyRows X s+e) u) ∂τ) =
-(p:ℝ)*Real.log r-(p:ℝ)/2*Real.log (2*Real.pi)-(∫ u,energy X (s-u) e ∂τ)/(2*r^2) := by
simp_rw [log_atDensity_shift hr]
have hei : Integrable (fun u => energy X (s-u) e) τ := by
apply continuous_integrable_on_support τ _ s R hs
unfold energy applyRows; fun_prop
rw [integral_sub (integrable_const _) (hei.div_const _),integral_div]
simp
simp_rw [he]
exact (integrable_const _).sub (hi.div_const _)
lemma log_shift_rows_local_ge {p d : ℕ} (τ : Measure (Vec d)) [IsProbabilityMeasure τ]
(s : Vec d) {R : ℝ} (hR : 0 ≤ R) (hs : ∀ᵐ u ∂τ,u ∈ closedBall s R)
{r : ℝ} (hr : 0 < r) (e : Fin p → ℝ) :
-(p:ℝ)*Real.log r-(p:ℝ)/2*Real.log (2*Real.pi)-
((p:ℝ)*R^2+∑ i,e i^2)/r^2 ≤
(∫ X : Mat p d,∫ u,Real.log (atDensity r X (applyRows X s+e) u) ∂τ
∂Measure.pi (fun _ : Fin p => scalarRow d)) := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
simp_rw [log_atDensity_shift hr]
have hi := integrable_energy_joint τ s hR hs e
have he (X : Mat p d) :
(∫ u,-(p:ℝ)*Real.log r-(p:ℝ)/2*Real.log (2*Real.pi)-energy X (s-u) e/(2*r^2) ∂τ) =
-(p:ℝ)*Real.log r-(p:ℝ)/2*Real.log (2*Real.pi)-(∫ u,energy X (s-u) e ∂τ)/(2*r^2) := by
have hei : Integrable (fun u => energy X (s-u) e) τ := by
apply continuous_integrable_on_support τ _ s R hs
unfold energy applyRows; fun_prop
rw [integral_sub (integrable_const _) (hei.div_const _),integral_div]; simp
simp_rw [he]
rw [integral_sub (integrable_const _) (hi.integral_prod_left.div_const _),integral_div]
simp only [integral_const,measureReal_def,measure_univ,ENNReal.toReal_one,one_smul]
have hb := energy_rows_local_le τ s hR hs e
have hrr : 0 < r^2 := sq_pos_of_pos hr
have hdiv := (div_le_div_of_nonneg_right hb (by positivity : 0 ≤ 2*r^2))
have heq : (2*(p:ℝ)*R^2+2*∑ i,e i^2)/(2*r^2)=((p:ℝ)*R^2+∑ i,e i^2)/r^2 := by ring
rw [heq] at hdiv
linarith
end ReferenceGaussian
end OAI
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