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Factorization and counting of low-rank binary matrices

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

proven

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    Natural Language Statement

    Theorem

    A rank-at-most-r matrix factors through r coordinates. Counting the two factors gives the rectangular low-rank estimate used for binary mixer tests, including the alternating off-diagonal block argument.

    Concept map
    2 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 card_low_rank proven

    2 factorization proven

    3 uniform_low_rank_bound proven

    Lean source view on GitHub

    1import Lax342547.MomentSpace
    2import Mathlib.LinearAlgebra.Matrix.Rank
    3import Mathlib.Probability.Distributions.Uniform
    4
    5/-!
    6---
    7title: Factorization and counting of low-rank binary matrices
    8type: theorem
    9---
    10A rank-at-most-r matrix factors through r coordinates. Counting the two
    11factors gives the rectangular low-rank estimate used for binary mixer
    12tests, including the alternating off-diagonal block argument.
    13-/
    14
    15namespace Lax342547.LowRankCounting
    16
    17axiom factorization {K I J : Type} [Field K] [Fintype I] [Fintype J]
    18 (M : Matrix I J K) (r : ℕ) (hr : M.rank ≤ r) :
    19 ∃ X : Matrix I (Fin r) K, ∃ Y : Matrix (Fin r) J K, X * Y = M
    20
    21open Lax342547.MomentSpace
    22open scoped ENNReal
    23
    24axiom card_low_rank {I J : Type} [Fintype I] [Fintype J] (r : ℕ) :
    25 Nat.card {M : Matrix I J Binary // M.rank ≤ r} ≤ 2 ^ ((Fintype.card I + Fintype.card J) * r)
    26
    27axiom uniform_low_rank_bound {I J : Type} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
    28 (r : ℕ) : (PMF.uniformOfFintype (Matrix I J Binary)).toOuterMeasure {M | M.rank ≤ r} ≤
    29 (2 : ℝ≥0∞) ^ ((Fintype.card I + Fintype.card J) * r) / 2 ^ (Fintype.card I * Fintype.card J)
    30
    31end Lax342547.LowRankCounting
    32
    Show ProofShow ProofShow Proof
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