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c270adf | 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 | import Mathlib
import OAI.Combinatorics.IndependentSets.Repetition.ScalarCell
namespace OAI
namespace LargeIndependentSets.BooleanJunta
open MeasureTheory Set
open scoped BigOperators Classical NNReal ENNReal
noncomputable def scalarSampleInfluence {m : ℕ} (f : ℝ → ℝ) (i : Fin m) : ℝ :=
mean (fun q : Cube m => ∫ y in (0:ℝ)..1,
|(f (scalarCell q y) - f (scalarCell (flip i q) y))/2|)
lemma scalarSampleInfluence_succ {m : ℕ} (f : ℝ → ℝ) (i : Fin m) :
scalarSampleInfluence (m:=m+1) f i.succ =
(scalarSampleInfluence (m:=m) (f ∘ halfInput false) i +
scalarSampleInfluence (m:=m) (f ∘ halfInput true) i)/2 := by
rw [scalarSampleInfluence, mean_succ]
rfl
lemma scalarCell_difference {m : ℕ} {L : ℝ≥0} {f : ℝ → ℝ}
(hf : LipschitzWith L f) (q r : Cube m) {y : ℝ} (hy : y ∈ Icc (0:ℝ) 1) :
|f (scalarCell q y) - f (scalarCell r y)| ≤ variation f := by
have hq := scalarCell_mem q hy
have hr := scalarCell_mem r hy
rcases le_total (scalarCell q y) (scalarCell r y) with h | h
· rw [abs_sub_comm]
exact CubeGradient.slice_difference_le_variation hf hq.1 h hr.2
· exact CubeGradient.slice_difference_le_variation hf hr.1 h hq.2
theorem dyadic_sample_total_influence {m : ℕ} {L : ℝ≥0} {f : ℝ → ℝ}
(hf : LipschitzWith L f) :
(∑ i, scalarSampleInfluence (m:=m) f i) ≤ variation f := by
induction m generalizing L f with
| zero => simpa using variation_nonneg f
| succ m ih =>
rw [Fin.sum_univ_succ]
have hhead : scalarSampleInfluence (m:=m+1) f 0 ≤ variation f/2 := by
apply (mean_mono (fun q => ?_)).trans_eq
(show mean (fun _ : Cube (m+1) => variation f/2) = variation f/2 by simp [mean])
have hc : Continuous (fun y => |(f (scalarCell q y) - f (scalarCell (flip 0 q) y))/2|) := by
exact (((hf.continuous.comp (scalarCell_lipschitz q).continuous).sub
(hf.continuous.comp (scalarCell_lipschitz (flip 0 q)).continuous)).div_const 2).abs
have hh : (∫ y in (0:ℝ)..1, |(f (scalarCell q y)-f (scalarCell (flip 0 q) y))/2|) ≤
∫ _y in (0:ℝ)..1, variation f/2 := by
apply intervalIntegral.integral_mono_on (by norm_num) (hc.intervalIntegrable _ _)
(continuous_const.intervalIntegrable _ _)
intro y hy
rw [abs_div, abs_of_pos (by norm_num : (0:ℝ)<2)]
exact div_le_div_of_nonneg_right (scalarCell_difference hf _ _ hy) (by norm_num)
simpa using hh
have ht : (∑ i : Fin m, scalarSampleInfluence (m:=m+1) f i.succ) =
((∑ i : Fin m, scalarSampleInfluence (m:=m) (f ∘ halfInput false) i) +
(∑ i : Fin m, scalarSampleInfluence (m:=m) (f ∘ halfInput true) i))/2 := by
simp only [scalarSampleInfluence_succ]
rw [← Finset.sum_div, Finset.sum_add_distrib]
rw [ht]
have h0 := ih (hf.comp (halfInput_lipschitz false))
have h1 := ih (hf.comp (halfInput_lipschitz true))
have hv := variation_half_sum hf
linarith
end LargeIndependentSets.BooleanJunta
end OAI
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