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Uniform relative Gram normalization errors

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

proven

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    Natural Language Statement

    Lemma

    The actual injective Gram event differs from its independent-bit factor by at most twice the column-rank exception. Independent groups multiply with an additive relative error budget, including growing slot batches.

    Concept map
    6 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 half_ambient_error proven

    2 independent_groups_error proven

    3 normalized_bounds proven

    4 normalized_error proven

    5 product_error proven

    Lean source view on GitHub

    1import Lax342547.GramNormalization
    2
    3/-!
    4---
    5title: Uniform relative Gram normalization errors
    6type: lemma
    7---
    8The actual injective Gram event differs from its independent-bit factor
    9by at most twice the column-rank exception. Independent groups multiply
    10with an additive relative error budget, including growing slot batches.
    11-/
    12
    13namespace Lax342547.HistogramGram
    14
    15open Lax342547.MomentSpace
    16open scoped BigOperators
    17
    18noncomputable def normalized {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    19 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary) : ℝ :=
    20 (2 : ℝ)^(Fintype.card I*Fintype.card J)*(Lax342547.GramNormalization.probability (N := N) G).toReal
    21
    22axiom normalized_bounds {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    23 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary)
    24 (hN : Fintype.card I+Fintype.card J ≤ Fintype.card N) :
    25 0 ≤ normalized (N := N) G ∧ normalized (N := N) G ≤ 1 ∧
    26 1-2*((2 : ℝ)^(Fintype.card I+Fintype.card J)/(2 : ℝ)^Fintype.card N) ≤ normalized (N := N) G
    27
    28axiom normalized_error {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    29 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary)
    30 (hN : Fintype.card I+Fintype.card J ≤ Fintype.card N) :
    31 |normalized (N := N) G-1| ≤
    32 2*((2 : ℝ)^(Fintype.card I+Fintype.card J)/(2 : ℝ)^Fintype.card N)
    33
    34axiom product_error {D : Type} (S : Finset D) (q ε : D → ℝ)
    35 (hq : ∀ d ∈ S,0 ≤ q d ∧ q d ≤ 1) (hε : ∀ d ∈ S,0 ≤ ε d)
    36 (herr : ∀ d ∈ S,1-ε d ≤ q d) :
    37 |(∏ d ∈ S,q d)-1| ≤ ∑ d ∈ S,ε d
    38
    39axiom independent_groups_error {S N : Type} [Fintype S] [Fintype N] [DecidableEq N]
    40 {I J : S → Type} [∀ s,Fintype (I s)] [∀ s,Fintype (J s)]
    41 [∀ s,DecidableEq (I s)] [∀ s,DecidableEq (J s)]
    42 (G : ∀ s,Matrix (I s) (J s) Binary)
    43 (hN : ∀ s,Fintype.card (I s)+Fintype.card (J s) ≤ Fintype.card N) :
    44 |(∏ s,normalized (N := N) (G s))-1| ≤
    45 ∑ s,2*((2 : ℝ)^(Fintype.card (I s)+Fintype.card (J s))/(2 : ℝ)^Fintype.card N)
    46
    47axiom half_ambient_error {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    48 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary)
    49 (hN : Fintype.card I+Fintype.card J ≤ Fintype.card N/2) :
    50 |normalized (N := N) G-1| ≤ 2/(2 : ℝ)^(Fintype.card N/2)
    51
    52end Lax342547.HistogramGram
    53
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