File size: 2,866 Bytes
135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Parisi.QuantileTransport
import OAI.Probability.SKBarriers.Parisi.CDFOverlapConvergence
namespace OAI
section
noncomputable section
open scoped NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem scalarOverlapModulus_continuous (β h : ℝ) :
Continuous (fun D : ℝ => scalarOverlapModulus β D h) := by
unfold scalarOverlapModulus
fun_prop
theorem scalarOverlapModulus_mono (β : ℝ) {h : ℝ} (hh : 0≤h) :
Monotone (fun D : ℝ => scalarOverlapModulus β D h) := by
intro D E hDE
unfold scalarOverlapModulus
have hc : 0≤4*scalarTimeMassConstant β+2*scalarTimeMassConstantK β 3 :=
add_nonneg (mul_nonneg (by norm_num) (scalarTimeMassConstant_nonneg β))
(mul_nonneg (by norm_num) (scalarTimeMassConstantK_nonneg β 3))
apply add_le_add
· exact add_le_add (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hDE hc) hh) le_rfl
· apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply sub_le_sub_right
apply Real.exp_le_exp.mpr
exact mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hDE (scalarTimeMassConstant_nonneg β)) (by norm_num)
theorem scalarOverlapModulus_nonneg (β : ℝ) {D h : ℝ} (hD : 0≤D) (hh : 0≤h) :
0 ≤ scalarOverlapModulus β D h := by
have H := scalarOverlapModulus_mono β hh hD
simp only [scalarOverlapModulus,mul_zero,zero_div,Real.exp_zero,sub_self,add_zero,zero_add] at H
exact (mul_nonneg (by norm_num : (0:ℝ)≤12) hh).trans H
theorem quantileCDF_segment_distance {k : ℕ} (A B : Fin (k+1) → ℝ)
(hA : A∈admissibleQuantiles k) (hB : B∈admissibleQuantiles k)
{t : ℝ} (ht : t∈Icc (0:ℝ) 1)
(hQ : (1-t) • A+t • B∈admissibleQuantiles k) :
cdfDistance (quantileCDF k ((1-t) • A+t • B)) (quantileCDF k A)=
t*cdfDistance (quantileCDF k B) (quantileCDF k A) := by
rw [cdfDistance,cdfDistance,quantileCDF_L1_eq _ _ hQ hA,quantileCDF_L1_eq _ _ hB hA,Finset.mul_sum]
apply Finset.sum_congr rfl
intro j _
have he : ((1-t) • A+t • B) j-A j=t*(B j-A j) := by
simp only [Pi.add_apply,Pi.smul_apply,smul_eq_mul]; ring
rw [he,abs_mul,abs_of_nonneg ht.1]
ring
def cdfGradientTest (β : ℝ) (α : ℝ → ℝ) (s : ℝ) : ℝ :=
s-scalarCDFOverlap β α (projIcc 0 1 (by norm_num) s)
theorem cdfGradientTest_continuous {β : ℝ} {α : ℝ → ℝ}
(hΓ : ContinuousOn (scalarCDFOverlap β α) (Icc (0:ℝ) 1)) :
Continuous (cdfGradientTest β α) :=
continuous_id.sub ((continuousOn_iff_continuous_domRestrict.mp hΓ).comp continuous_projIcc)
theorem cdfGradientTest_eq {β : ℝ} {α : ℝ → ℝ} {s : ℝ} (hs : s∈Icc (0:ℝ) 1) :
cdfGradientTest β α s=s-scalarCDFOverlap β α s := by
simp only [cdfGradientTest,projIcc_of_mem _ hs]
end SK.Analytic
end
end
end OAI
|