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Conditional independence of the two channel frames

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

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    Natural Language Statement

    Lemma

    At fixed primal matrices P,Q, the valid X and Y completions form a Cartesian product: their annihilator and injectivity conditions are separate. Uniform conditioning therefore gives independent uniform channel completions, with the actual finite normalizing factors.

    Concept map
    8 concepts
    100%
    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 channel_rank_failures proven

    2 conditional_point proven

    3 conditional_uniform proven

    4 raw_conditioning proven

    Lean source view on GitHub

    1import Lax342547.FrameSymmetry
    2import Lax342547.GramColumns
    3
    4/-!
    5---
    6title: Conditional independence of the two channel frames
    7type: lemma
    8---
    9At fixed primal matrices P,Q, the valid X and Y completions form a
    10Cartesian product: their annihilator and injectivity conditions are
    11separate. Uniform conditioning therefore gives independent uniform
    12channel completions, with the actual finite normalizing factors.
    13-/
    14
    15namespace Lax342547.ConditionalChannels
    16
    17open Lax342547.MomentSpace Lax342547.RawFrames
    18open scoped ENNReal
    19
    20variable {B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    21
    22abbrev PlusCompletion (P Q : Matrix N B Binary) :=
    23 {X : Matrix N H Binary // X.transpose * Q = 0 ∧ Function.Injective
    24 (fun z : (B → Binary) × (H → Binary) => P.mulVec z.1 + X.mulVec z.2)}
    25
    26abbrev MinusCompletion (P Q : Matrix N B Binary) :=
    27 {Y : Matrix N H Binary // P.transpose * Y = 0 ∧ Function.Injective
    28 (fun z : (B → Binary) × (H → Binary) => Q.mulVec z.1 + Y.mulVec z.2)}
    29
    30abbrev Fiber (E : Matrix B B Binary) (P Q : Matrix N B Binary) :=
    31 {F : Frame B H N E // F.P = P ∧ F.Q = Q}
    32
    33noncomputable instance (P Q : Matrix N B Binary) : Fintype (PlusCompletion (H := H) P Q) := by
    34 classical exact Subtype.fintype _
    35
    36noncomputable instance (P Q : Matrix N B Binary) : Fintype (MinusCompletion (H := H) P Q) := by
    37 classical exact Subtype.fintype _
    38
    39noncomputable instance (E : Matrix B B Binary) (P Q : Matrix N B Binary) :
    40 Fintype (Fiber (H := H) E P Q) := by classical exact Subtype.fintype _
    41
    42def channels {E : Matrix B B Binary} {P Q : Matrix N B Binary} (F : Fiber (H := H) E P Q) :
    43 PlusCompletion (H := H) P Q × MinusCompletion (H := H) P Q :=
    44 (⟨F.val.X, by
    45 simpa [← F.property.1, ← F.property.2] using
    46 And.intro F.val.plus_annihilator F.val.plus_injective⟩,
    47 ⟨F.val.Y, by
    48 simpa [← F.property.1, ← F.property.2] using
    49 And.intro F.val.minus_annihilator F.val.minus_injective⟩)
    50
    51def assemble {E : Matrix B B Binary} {P Q : Matrix N B Binary} (hgram : P.transpose * Q = E)
    52 (X : PlusCompletion (H := H) P Q) (Y : MinusCompletion (H := H) P Q) :
    53 Fiber (H := H) E P Q :=
    54 ⟨{ P := P, Q := Q, X := X.val, Y := Y.val,
    55 plus_injective := X.property.2, minus_injective := Y.property.2,
    56 gram := hgram, plus_annihilator := X.property.1, minus_annihilator := Y.property.1 },
    57 rfl, rfl⟩
    58
    59def channelEquiv {E : Matrix B B Binary} {P Q : Matrix N B Binary} (hgram : P.transpose * Q = E) :
    60 Fiber (H := H) E P Q ≃ (PlusCompletion (H := H) P Q × MinusCompletion (H := H) P Q) where
    61 toFun := channels
    62 invFun z := assemble hgram z.1 z.2
    63 left_inv F := by
    64 apply Subtype.ext
    65 apply frameData_injective
    66 exact Prod.ext F.property.1.symm (Prod.ext F.property.2.symm rfl)
    67 right_inv z := by cases z; rfl
    68
    69axiom conditional_uniform {E : Matrix B B Binary} {P Q : Matrix N B Binary}
    70 (hgram : P.transpose * Q = E) [Nonempty (Fiber (H := H) E P Q)]
    71 [Nonempty (PlusCompletion (H := H) P Q)] [Nonempty (MinusCompletion (H := H) P Q)] :
    72 (PMF.uniformOfFintype (Fiber (H := H) E P Q)).map channels =
    73 PMF.uniformOfFintype (PlusCompletion (H := H) P Q × MinusCompletion (H := H) P Q)
    74
    75axiom conditional_point {E : Matrix B B Binary} {P Q : Matrix N B Binary}
    76 (hgram : P.transpose * Q = E) [Nonempty (Fiber (H := H) E P Q)]
    77 [Nonempty (PlusCompletion (H := H) P Q)] [Nonempty (MinusCompletion (H := H) P Q)]
    78 (X : PlusCompletion (H := H) P Q) (Y : MinusCompletion (H := H) P Q) :
    79 (PMF.uniformOfFintype (Fiber (H := H) E P Q)).map channels (X, Y) =
    80 PMF.uniformOfFintype (PlusCompletion (H := H) P Q) X *
    81 PMF.uniformOfFintype (MinusCompletion (H := H) P Q) Y
    82
    83axiom raw_conditioning {E : Matrix B B Binary} {P Q : Matrix N B Binary}
    84 (hgram : P.transpose * Q = E) [Nonempty (Frame B H N E)]
    85 [Nonempty (Fiber (H := H) E P Q)]
    86 [Nonempty (PlusCompletion (H := H) P Q)] [Nonempty (MinusCompletion (H := H) P Q)]
    87 (h : ∃ F ∈ {F : Frame B H N E | F.P = P ∧ F.Q = Q},
    88 F ∈ (PMF.uniformOfFintype (Frame B H N E)).support) :
    89 ((PMF.uniformOfFintype (Frame B H N E)).filter {F | F.P = P ∧ F.Q = Q} h).map
    90 (fun F => (F.X, F.Y)) =
    91 (PMF.uniformOfFintype (PlusCompletion (H := H) P Q × MinusCompletion (H := H) P Q)).map
    92 (fun z => (z.1.val, z.2.val))
    93
    94axiom channel_rank_failures [DecidableEq B] [DecidableEq H] [DecidableEq N]
    95 (P Q : Matrix N B Binary) (hP : Function.Injective P.mulVec) (hQ : Function.Injective Q.mulVec)
    96 [Nonempty (GramColumns.GramFiber Q (0 : Matrix B H Binary))]
    97 [Nonempty (GramColumns.GramFiber P (0 : Matrix B H Binary))] :
    98 (PMF.uniformOfFintype (GramColumns.GramFiber Q (0 : Matrix B H Binary))).toOuterMeasure
    99 {X | ¬ Function.Injective (fun z : (B → Binary) × (H → Binary) =>
    100 P.mulVec z.1 + X.val.mulVec z.2)} ≤
    101 (2 : ℝ≥0∞) ^ (2 * Fintype.card B + Fintype.card H) / 2 ^ Fintype.card N ∧
    102 (PMF.uniformOfFintype (GramColumns.GramFiber P (0 : Matrix B H Binary))).toOuterMeasure
    103 {Y | ¬ Function.Injective (fun z : (B → Binary) × (H → Binary) =>
    104 Q.mulVec z.1 + Y.val.mulVec z.2)} ≤
    105 (2 : ℝ≥0∞) ^ (2 * Fintype.card B + Fintype.card H) / 2 ^ Fintype.card N
    106
    107end Lax342547.ConditionalChannels
    108
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