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Actual conditioned affine coefficient dependence bounds

Lax342547.ConditionedAffineColumns · concepts/Lax342547/ConditionedAffineColumns.lean · lax-342547

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    Natural Language Statement

    Lemma

    The free parts of two independent full affine coefficient maps remain a uniform matrix pair. This pays nominal dependence before conditioning; conditioning each coefficient draw on an event of mass at least beta increases that failure bound by at most beta to the power minus two.

    Concept map
    17 concepts; 7 descendants hidden
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 conditioned_nominal_pair_failure proven

    2 conditioning_pair_bound proven

    3 freePair_uniform proven

    4 nominal_pair_failure proven

    Lean source view on GitHub

    1import Lax342547.AffineSliceIndependence
    2import Lax342547.RealCellLaws
    3
    4/-!
    5---
    6title: Actual conditioned affine coefficient dependence bounds
    7type: lemma
    8---
    9The free parts of two independent full affine coefficient maps remain a
    10uniform matrix pair. This pays nominal dependence before conditioning;
    11conditioning each coefficient draw on an event of mass at least beta
    12increases that failure bound by at most beta to the power minus two.
    13-/
    14
    15namespace Lax342547.ConditionedAffineColumns
    16noncomputable section
    17open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    18open Lax342547.AffineSliceIndependence Lax342547.RetainedImages Lax342547.RealCellLaws
    19open scoped BigOperators ENNReal
    20
    21def freePairMap {k n : ℕ} {D : Type} :
    22 (Matrix (Base k n) (Option D) Binary × Matrix (Base k n) (Option D) Binary) →ₗ[Binary]
    23 (Matrix (Fin n) (Option D) Binary × Matrix (Fin n) (Option D) Binary) where
    24 toFun V := (freePart V.1,freePart V.2)
    25 map_add' _ _ := rfl
    26 map_smul' _ _ := rfl
    27
    28axiom freePair_uniform {k n : ℕ} {D : Type} [Fintype D] [DecidableEq D] :
    29 (PMF.uniformOfFintype (Matrix (Base k n) (Option D) Binary × Matrix (Base k n) (Option D) Binary)).map
    30 (freePairMap (k := k) (n := n) (D := D)) =
    31 PMF.uniformOfFintype (Matrix (Fin n) (Option D) Binary × Matrix (Fin n) (Option D) Binary)
    32
    33axiom nominal_pair_failure {k n b degree : ℕ} {D : Type} [Fintype D] [DecidableEq D]
    34 (s : Fin b → Binary) :
    35 (PMF.uniformOfFintype (Matrix (Base k n) (Option D) Binary × Matrix (Base k n) (Option D) Binary)).toOuterMeasure
    36 {V | ¬ Function.Injective (fun uv : (Option D → Binary) × (Option D → Binary) =>
    37 (pointColumns (degree := degree) s V.1).mulVec uv.1+
    38 (pointColumns (degree := degree) s V.2).mulVec uv.2)} ≤
    39 (2 : ℝ≥0∞)^(2*Fintype.card D+1)/(2 : ℝ≥0∞)^n
    40
    41axiom conditioning_pair_bound {A : Type} [Fintype A] (ρ : A → ℝ) (C : A → Prop)
    42 (bad : A × A → Prop) (β δ : ℝ) (hρ : ∀ a,0 ≤ ρ a) (hβ : 0 < β)
    43 (hC : β ≤ cellMass ρ C)
    44 (hbad : cellMass (fun z : A × A => ρ z.1*ρ z.2) bad ≤ δ) :
    45 cellMass (fun z : A × A => conditionalLaw ρ C z.1*conditionalLaw ρ C z.2) bad ≤ δ/β^2
    46
    47axiom conditioned_nominal_pair_failure {k n b degree : ℕ} {D : Type} [Fintype D] [DecidableEq D]
    48 (s : Fin b → Binary) (C : Matrix (Base k n) (Option D) Binary → Prop) (β : ℝ)
    49 (hβ : 0 < β) (hC : β ≤ cellMass
    50 (weights (PMF.uniformOfFintype (Matrix (Base k n) (Option D) Binary))) C) :
    51 cellMass (fun V : Matrix (Base k n) (Option D) Binary × Matrix (Base k n) (Option D) Binary =>
    52 conditionalLaw (weights (PMF.uniformOfFintype _)) C V.1*
    53 conditionalLaw (weights (PMF.uniformOfFintype _)) C V.2)
    54 (fun V => ¬ Function.Injective (fun uv : (Option D → Binary) × (Option D → Binary) =>
    55 (pointColumns (degree := degree) s V.1).mulVec uv.1+
    56 (pointColumns (degree := degree) s V.2).mulVec uv.2)) ≤
    57 ((2 : ℝ)^(2*Fintype.card D+1)/(2 : ℝ)^n)/β^2
    58
    59end
    60end Lax342547.ConditionedAffineColumns
    61
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