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e320e82 | 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 | import OAI.Combinatorics.ExpectationThreshold.Basic
import OAI.Combinatorics.ExpectationThreshold.ProbabilityColors
import OAI.Combinatorics.ExpectationThreshold.SelectorTruncationDefinitions
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Finset.Lattice.Fold
namespace OAI
/-!
# Actual moves and profiles for the simultaneous selector
We retain all colorings of minimum total movement for at least one original
family member. This avoids an arbitrary tie-breaking order; each retained
coloring still has the member-specific minimality used in reconstruction.
-/
noncomputable section
open scoped BigOperators
namespace TalagrandThreshold.Selector
variable {α : Type*} [Fintype α] [DecidableEq α]
/-- A probability mass for each actual member of the family. -/
structure MassFamily (F : Family α) where
weight : ↥F → α → ℝ
nonneg : ∀ H x, 0 ≤ weight H x
total : ∀ H, ∑ x, weight H x = 1
supported : ∀ H x, x ∉ H.val → weight H x = 0
/-- Colors have zero-based indices; the actual color level is one larger. -/
def prefixSet {s : ℕ} (z : Coloring α s) (i : Fin s) : Finset α :=
Finset.univ.filter fun x => (z x).val ≤ i.val
def dscale {s : ℕ} (i : Fin s) : ℝ := ((2 : ℝ) ^ (i.val + 1))⁻¹
/-- `Earlier a z` means that the move from `a` to `z` only lowers colors. -/
def Earlier {s : ℕ} (a z : Coloring α s) : Prop := ∀ x, z x ≤ a x
def changes {s : ℕ} (a z : Coloring α s) : Finset α :=
Finset.univ.filter fun x => z x < a x
def crossings {s : ℕ} (a z : Coloring α s) (i : Fin s) : ℕ :=
(prefixSet z i \ prefixSet a i).card
def movement {s : ℕ} (a z : Coloring α s) : ℕ :=
∑ x, ((a x).val - (z x).val)
def movedRow {s : ℕ} (a z : Coloring α s) (i : Fin s) : Finset α :=
Finset.univ.filter fun x => (z x).val = i.val ∧ (z x).val < (a x).val
/-- The full profile retains both final and original colors. -/
abbrev Profile (s : ℕ) := Fin s → Color s → ℕ
def profile {s : ℕ} (a z : Coloring α s) : Profile s :=
fun i h => (Finset.univ.filter fun x =>
(z x).val = i.val ∧ a x = h ∧ i.val < h.val).card
def rowCount {s : ℕ} (n : Profile s) (i : Fin s) : ℕ := ∑ h, n i h
def profileSize {s : ℕ} (n : Profile s) : ℕ := ∑ i, rowCount n i
def profileCrossings {s : ℕ} (n : Profile s) (i : Fin s) : ℕ :=
∑ j : Fin s, ∑ h : Color s,
if j.val ≤ i.val ∧ i.val < h.val then n j h else 0
def FeasibleFor {F : Family α} (mu : MassFamily F) {s : ℕ}
(a z : Coloring α s) (H : ↥F) : Prop :=
∀ i : Fin s, ∃ ε : ℝ,
Admissible (mu.weight H) (prefixSet z i) (crossings a z i : ℝ) (dscale i) ε
def Feasible {F : Family α} (mu : MassFamily F) {s : ℕ}
(a z : Coloring α s) : Prop :=
Earlier a z ∧ ∃ H : ↥F, FeasibleFor mu a z H
def BoundedFeasible {F : Family α} (mu : MassFamily F) {s : ℕ}
(a : Coloring α s) (H₀ : ↥F) (z : Coloring α s) : Prop :=
Feasible mu a z ∧ changes a z ⊆ H₀.val
def IsMinimalMove {F : Family α} (mu : MassFamily F) {s : ℕ}
(a : Coloring α s) (H₀ : ↥F) (z : Coloring α s) : Prop :=
BoundedFeasible mu a H₀ z ∧
∀ w, BoundedFeasible mu a H₀ w → movement a z ≤ movement a w
def Selected {F : Family α} (mu : MassFamily F) {s : ℕ}
(a z : Coloring α s) : Prop := ∃ H₀ : ↥F, IsMinimalMove mu a H₀ z
def selectedMoves {F : Family α} (mu : MassFamily F) {s : ℕ}
(a : Coloring α s) : Finset (Coloring α s) := by
classical
exact Finset.univ.filter (Selected mu a)
/-- An actual bad original coloring, its selected endpoint, and its full profile. -/
def Captured {F : Family α} (mu : MassFamily F) {s : ℕ}
(a : Coloring α s) : Prop :=
∃ H : ↥F, ∀ i : Fin s,
1 - dscale i ≤ ∑ x ∈ prefixSet a i, mu.weight H x
def Bad {F : Family α} (mu : MassFamily F) {s : ℕ}
(a : Coloring α s) : Prop := ¬ Captured mu a
def groupOriginals {F : Family α} (mu : MassFamily F) {s : ℕ}
(z : Coloring α s) (n : Profile s) : Finset (Coloring α s) := by
classical
exact Finset.univ.filter fun a => Bad mu a ∧ Selected mu a z ∧ profile a z = n
def meanColor {F : Family α} (mu : MassFamily F) {s : ℕ}
(H : ↥F) (a : Coloring α s) : ℝ :=
∑ x, ((a x).val + 1 : ℝ) * mu.weight H x
end TalagrandThreshold.Selector
end
end OAI
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