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Lower concentration of the fibers of a uniform random map

Lax253009.RandomFibers · concepts/Lax253009/RandomFibers.lean · lax-253009

proven

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    Natural Language Statement

    Theorem

    For a uniformly random map from an NN-element set to an MM-element set, with N,M>0N,M>0, the probability that some fiber has fewer than N/(2M)N/(2M) elements is at most Me−N/(8M2)M e^{-N/(8M^2)}.

    This is a sufficient version of the concentration estimate in Lemma 4.3. For N=2wN=2^w and M=2sM=2^s, it tends to zero exponentially in 2w2^w for each fixed ss. The paper uses a sharper Chernoff bound; the threshold for ww in the soundness theorem can absorb the difference.

    Concept map
    4 concepts; 1 descendant hidden
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    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 any_small_fiber proven

    2 small_fiber proven

    Lean source view on GitHub

    1import Lax253009.ExponentialBounds
    2
    3/-!
    4---
    5title: Lower concentration of the fibers of a uniform random map
    6type: theorem
    7---
    8For a uniformly random map from an NN-element set to an MM-element set,
    9with N,M>0N,M>0, the probability that some fiber has fewer than N/(2M)N/(2M)
    10elements is at most Me−N/(8M2)M e^{-N/(8M^2)}.
    11
    12This is a sufficient version of the concentration estimate in Lemma 4.3.
    13For N=2wN=2^w and M=2sM=2^s, it tends to zero exponentially in 2w2^w for each
    14fixed ss. The paper uses a sharper Chernoff bound; the threshold for ww
    15in the soundness theorem can absorb the difference.
    16-/
    17
    18namespace Lax253009.RandomFibers
    19
    20open FiniteProbability
    21
    22def fiber {ι κ : Type} [Fintype ι] [DecidableEq κ] (f : ι → κ) (z : κ) : Finset ι :=
    23 Finset.univ.filter fun i ↦ f i = z
    24
    25axiom small_fiber {ι κ : Type} [Fintype ι] [DecidableEq ι]
    26 [Fintype κ] [DecidableEq κ] [Nonempty κ] (hN : 0 < Fintype.card ι) (z : κ) :
    27 probability (fun f : ι → κ ↦ (fiber f z).card <
    28 (Fintype.card ι : ℝ) / (2 * Fintype.card κ)) ≤
    29 Real.exp (-(Fintype.card ι : ℝ) / (8 * (Fintype.card κ : ℝ) ^ 2))
    30
    31axiom any_small_fiber {ι κ : Type} [Fintype ι] [DecidableEq ι]
    32 [Fintype κ] [DecidableEq κ] [Nonempty κ] (hN : 0 < Fintype.card ι) :
    33 probability (fun f : ι → κ ↦ ∃ z, (fiber f z).card <
    34 (Fintype.card ι : ℝ) / (2 * Fintype.card κ)) ≤
    35 (Fintype.card κ : ℝ) *
    36 Real.exp (-(Fintype.card ι : ℝ) / (8 * (Fintype.card κ : ℝ) ^ 2))
    37
    38end Lax253009.RandomFibers
    39
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