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f0f05de | 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 99 100 101 102 103 104 105 106 107 | import OAI.Combinatorics.BooleanFunctions.CellDegree
import OAI.Combinatorics.BooleanFunctions.Producer
namespace OAI
open scoped BigOperators Topology NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter Real
noncomputable section
namespace SquareRootDegree
namespace Observations
open Switching FiniteCLT UniformFinite GaussianGain MomentTransfer
attribute [local instance] Classical.propDecidable
variable {α δ ι : Type*} [Fintype α] [Fintype δ] [Fintype ι]
def eventCongr (e : α ≃ δ) : EventSpace α ι ≃ EventSpace δ ι :=
Equiv.prodCongr e (Equiv.piCongrRight (fun _ : ι => e))
theorem deficit_equiv (e : α ≃ δ) (T : δ → ℝ) (b : ℝ)
(a : EventSpace δ ι → ℝ) :
SoftEvent.deficit (T ∘ e) b (a ∘ eventCongr e) = SoftEvent.deficit T b a := by
unfold SoftEvent.deficit
convert average_equiv (eventCongr (ι := ι) e)
(fun x => a x*(1-(T x.1+∑ j, T (x.2 j)-b)^2)) using 1
congr 1
variable {γ β : Type*} [Fintype γ] [Fintype β] [DecidableEq γ]
def groupScale (F : Cube γ → β) (n : ℕ) : ℝ := Real.sqrt n * Real.sqrt (variance F)
def groupStandard (F : Cube γ → β) (n : ℕ) : Cube (Fin n × γ) → ℝ :=
normalizedSum (standard F) n ∘ blockCurry (Fin n) γ Bool
omit [Fintype α] [Fintype δ] [Fintype ι] [Fintype β] in
theorem groupScale_sq (F : Cube γ → β) (n : ℕ) :
(groupScale F n)^2 = n*variance F := by
rw [groupScale,mul_pow,Real.sq_sqrt (Nat.cast_nonneg n),Real.sq_sqrt (variance_nonneg F)]
omit [Fintype α] [Fintype δ] [Fintype ι] in
theorem grouped_score_scaled (F : Cube γ → β) (hv : 0 < variance F) {n : ℕ} (hn : 0 < n) :
score (grouped (κ := Fin n) F) = fun x => groupScale F n * groupStandard F n x := by
rw [grouped_score]
funext x
simp only [groupScale,groupStandard,normalizedSum,Function.comp_apply,blockCurry,Equiv.coe_fn_mk,
standard,standardValue,← Finset.mul_sum]
have hv' := (Real.sqrt_pos.mpr hv).ne'
have hn' := (Real.sqrt_pos.mpr (Nat.cast_pos.mpr hn)).ne'
unfold score conditionalAverage
field_simp
omit [Fintype α] [Fintype δ] [Fintype ι] in
theorem groupStandard_zero (F : Cube γ → β) (n : ℕ) : average (groupStandard F n) = 0 := by
rw [groupStandard]
change average (fun x => normalizedSum (standard F) n (blockCurry (Fin n) γ Bool x)) = 0
rw [average_equiv]
exact average_normalizedSum (standard F) (standard_zero F) n
omit [Fintype α] [Fintype δ] [Fintype ι] in
theorem groupStandard_sq (F : Cube γ → β) (hv : 0 < variance F) {n : ℕ} (hn : 0 < n) :
average (fun x => (groupStandard F n x)^2) = 1 := by
unfold groupStandard
rw [show (fun x => (normalizedSum (standard F) n ∘ blockCurry (Fin n) γ Bool) x ^ 2) =
(fun x => (normalizedSum (standard F) n x)^2) ∘ blockCurry (Fin n) γ Bool from rfl]
change average (fun x => (normalizedSum (standard F) n (blockCurry (Fin n) γ Bool x))^2) = 1
exact (average_equiv (blockCurry (Fin n) γ Bool)
(fun x => (normalizedSum (standard F) n x)^2)).trans
(average_normalizedSum_sq (standard F) (standard_zero F) (standard_sq F hv) hn)
omit [Fintype α] [Fintype δ] [Fintype ι] in
theorem exists_amplification (F : Cube γ → β) (hv : 0 < variance F) :
∃ n : ℕ, 0 < n ∧ ∃ S : (Fin n → β) → Leaves, ∃ L : Leaves → (Fin n → β) → Bool,
(n:ℝ)*variance F*(Fintype.card Leaves+1+eta) <
variance (switched (grouped (κ := Fin n) F) S L) := by
let : MeasurableSpace (Cube γ) := ⊤
obtain ⟨n,hn,b,S,L,hg⟩ := Producer.exists_table_gain F (standardValue F)
(standard_zero F) (standard_sq F hv)
refine ⟨n,hn,S,L,?_⟩
let G := grouped (κ := Fin n) F
let T := groupStandard F n
let A := TableRounding.tableEvent S L
have hd : eta < SoftEvent.deficit T b
(fun x => if A (oldLabels G x) then 1 else 0) := by
have he := deficit_equiv (blockCurry (Fin n) γ Bool)
(normalizedSum (standard F) n) b
(fun x => if A ((fun j => F (x.1 j)),fun i j => F (x.2 i j)) then 1 else 0)
have hA : (fun x : EventSpace (Cube (Fin n × γ)) Leaves => if A (oldLabels G x) then (1:ℝ) else 0) =
(fun x => if A ((fun j => F (x.1 j)),fun i j => F (x.2 i j)) then 1 else 0) ∘
eventCongr (blockCurry (Fin n) γ Bool) := rfl
change eta < SoftEvent.deficit (normalizedSum (standard F) n ∘ blockCurry (Fin n) γ Bool) b _
rw [hA,he]
exact hg
have hr := retained_gain G A T (normalizedSum (standardValue F) n) (fun _ => rfl) b eta
(groupStandard_zero F n) (groupStandard_sq F hv hn) hd
have hs := switched_variance_scaled G S L T (groupScale F n) (grouped_score_scaled F hv hn)
rw [groupScale_sq] at hs
rw [hs]
have hnV : 0 < (n:ℝ)*variance F := mul_pos (Nat.cast_pos.mpr hn) hv
exact mul_lt_mul_of_pos_left hr hnV
end Observations
end SquareRootDegree
end
end OAI
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