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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import OAI.Combinatorics.BooleanFunctions.Grouping
namespace OAI
open scoped BigOperators Topology NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter Real
noncomputable section
namespace SquareRootDegree
namespace Amplification
open Observations Switching GaussianGain MomentTransfer
attribute [local instance] Classical.propDecidable
structure Observation where
Coord : Type
[coordFintype : Fintype Coord]
[coordDecidableEq : DecidableEq Coord]
Label : Type
[labelFintype : Fintype Label]
map : Cube Coord → Label
cost : ℕ
cost_pos : 0 < cost
cell_le : ∀ b, fourierDegree (cell map b) ≤ cost
variance_pos : 0 < variance map
attribute [instance] Observation.coordFintype Observation.coordDecidableEq Observation.labelFintype
def ratio (F : Observation) : ℝ := variance F.map/F.cost
def costFactor : ℕ := 2*gridN+2
def factor : ℝ := 1+eta/costFactor
theorem costFactor_eq : Fintype.card Leaves+1 = costFactor := by
simp only [Leaves,Fintype.card_fin,costFactor]
theorem costFactor_pos : 0 < costFactor := by
unfold costFactor
omega
theorem factor_gt_one : 1 < factor := by
unfold factor
have : 0 < eta/(costFactor:ℝ) := div_pos eta_pos (Nat.cast_pos.mpr costFactor_pos)
linarith
theorem ratio_improvement {q n D : ℕ} (hq : 0 < q) (hn : 0 < n) (hD : 0 < D)
{v v' η : ℝ} (hv : (n:ℝ)*v*((q:ℝ)+η) < v') :
(1+η/q)*(v/D) < v'/(q*(n*D):ℕ) := by
rw [lt_div_iff₀ (Nat.cast_pos.mpr (Nat.mul_pos hq (Nat.mul_pos hn hD)))]
calc (1+η/q)*(v/D)*(q*(n*D):ℕ) = (n:ℝ)*v*((q:ℝ)+η) := by
have hD' : (D:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hD)
have hq' : (q:ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hq)
simp only [Nat.cast_mul]
field_simp
_ < _ := hv
theorem exists_step (F : Observation) : ∃ G : Observation, factor*ratio F < ratio G := by
obtain ⟨n,hn,S,L,hv⟩ := exists_amplification F.map F.variance_pos
have he : (Fintype.card Leaves:ℝ)+1 = (costFactor:ℝ) := by
rw [← costFactor_eq,Nat.cast_add,Nat.cast_one]
rw [he] at hv
let G : Observation := {
Coord := Slots Leaves × (Fin n × F.Coord)
Label := EventSpace (Fin n → F.Label) Leaves ⊕ (Fin n → F.Label)
map := switched (grouped (κ := Fin n) F.map) S L
cost := costFactor*(n*F.cost)
cost_pos := Nat.mul_pos costFactor_pos (Nat.mul_pos hn F.cost_pos)
cell_le := fun b => by
rw [← costFactor_eq]
exact switched_degree _ _ _ _
(fun a => by simpa only [Fintype.card_fin] using grouped_degree F.map F.cost F.cell_le a) b
variance_pos := lt_trans (mul_pos (mul_pos (Nat.cast_pos.mpr hn) F.variance_pos)
(add_pos (Nat.cast_pos.mpr costFactor_pos) eta_pos)) hv }
refine ⟨G,?_⟩
exact ratio_improvement costFactor_pos hn F.cost_pos hv
theorem identity_variance : variance (id : Cube (Fin 1) → Cube (Fin 1)) = 1 := by
have hs : score (id : Cube (Fin 1) → Cube (Fin 1)) = cubeSum :=
Projection.conditional_comp id cubeSum
unfold Observations.variance
rw [hs]
simp only [cubeSum,Fin.sum_univ_one,sign_sq]
exact average_const 1
def base : Observation where
Coord := Fin 1
Label := Cube (Fin 1)
map := id
cost := 1
cost_pos := Nat.zero_lt_one
cell_le := fun b => by
have hd : fourierDegree (cell id b) ≤ (Finset.univ : Finset (Fin 1)).card := by
apply fourierDegree_le_card_of_dependsOn
intro x y h
have he : x = y := funext fun i => h i (Finset.mem_univ i)
rw [he]
simpa using hd
variance_pos := by rw [identity_variance]; exact zero_lt_one
theorem base_ratio : ratio base = 1 := by
unfold ratio base
rw [identity_variance]
norm_num
theorem exists_iterate (k : ℕ) : ∃ F : Observation, factor^k ≤ ratio F := by
induction k with
| zero => exact ⟨base,by simp [base_ratio]⟩
| succ k ih =>
obtain ⟨F,hF⟩ := ih
obtain ⟨G,hG⟩ := exists_step F
refine ⟨G,?_⟩
calc factor^(k+1) = factor*factor^k := pow_succ' factor k
_ ≤ factor*ratio F := mul_le_mul_of_nonneg_left hF (le_trans zero_le_one factor_gt_one.le)
_ ≤ ratio G := hG.le
theorem exists_large_ratio (R : ℝ) : ∃ F : Observation, R < ratio F := by
obtain ⟨k,hk⟩ := pow_unbounded_of_one_lt R factor_gt_one
obtain ⟨F,hF⟩ := exists_iterate k
exact ⟨F,hk.trans_le hF⟩
end Amplification
end SquareRootDegree
end
end OAI
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