| 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] |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
|
|
