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The distribution of mixer rows on free directions

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

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    Natural Language Statement

    Lemma

    For a uniform matrix L, the joint observations ZᵀLD and DᵀLW have density at most 2^(rank(Z) rank(W)) relative to the completely uniform list, when Z,W,D have independent columns. This is the single-matrix domination bound in the proof of Lemma 5.5.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    Lean source view on GitHub

    1import Lax342547.MixerCompatibility
    2import Lax342547.FiniteLinearLaw
    3
    4/-!
    5---
    6title: The distribution of mixer rows on free directions
    7type: lemma
    8---
    9For a uniform matrix L, the joint observations ZᵀLD and DᵀLW have
    10density at most 2^(rank(Z) rank(W)) relative to the completely uniform
    11list, when Z,W,D have independent columns. This is the single-matrix
    12domination bound in the proof of Lemma 5.5.
    13-/
    14
    15namespace Lax342547.MixerRowLaw
    16
    17variable {K I E F N : Type} [Field K] [Fintype I]
    18
    19def directions (D : Matrix I N K) :
    20 (Matrix E I K × Matrix I F K) →ₗ[K] (Matrix E N K × Matrix N F K) where
    21 toFun p := (p.1 * D, D.transpose * p.2)
    22 map_add' x y := by simp [Matrix.add_mul, Matrix.mul_add]
    23 map_smul' c x := by simp [Matrix.smul_mul, Matrix.mul_smul]
    24
    25def observations (Z : Matrix I E K) (W : Matrix I F K) (D : Matrix I N K) :
    26 Matrix I I K →ₗ[K] (Matrix E N K × Matrix N F K) :=
    27 (directions D).comp (MixerCompatibility.restrictions Z W)
    28
    29open Lax342547.MomentSpace
    30open scoped ENNReal
    31
    32axiom row_law_bound {I E F N : Type} [Fintype I] [Fintype E] [Fintype F] [Fintype N]
    33 [DecidableEq I] [DecidableEq E] [DecidableEq F] [DecidableEq N]
    34 (Z : Matrix I E Binary) (W : Matrix I F Binary) (D : Matrix I N Binary)
    35 (hZ : Function.Injective Z.mulVec) (hW : Function.Injective W.mulVec) (hD : Function.Injective D.mulVec)
    36 (y : Matrix E N Binary × Matrix N F Binary) :
    37 (PMF.uniformOfFintype (Matrix I I Binary)).map (observations Z W D) y ≤
    38 2 ^ (Fintype.card E * Fintype.card F) * PMF.uniformOfFintype (Matrix E N Binary × Matrix N F Binary) y
    39
    40end Lax342547.MixerRowLaw
    41
    Show Proof
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