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5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import OAI.Combinatorics.LowSpaceSubsetSum.Counting.Deficiency
import OAI.Combinatorics.BalancedRyser.Probability.FiniteUniformVariance
namespace OAI
/-!
# Deficiency under sampling without replacement
The variance estimate in the paper's deficiency lemma is a specialization
of the finite completion-coupling theorem
`Balanced.WithoutReplacement.finiteVariance_powersetCard_le`.
Its probability space is exactly the uniform law on all size-`v` subsets
of the supplied domain `U`, including the cases `v = 0` and `v = |U|`.
-/
namespace LowSpaceSubsetSum
open scoped BigOperators
open Balanced.WithoutReplacement
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
/-- Mean deficiency of a uniform size-`v` subset of `U`. -/
noncomputable def meanDeficiency (a : ι → ℤ) (U : Finset ι) (v : ℕ) : ℝ :=
𝔼 V ∈ U.powersetCard v, deficiency a V
/-- Variance for uniform sampling of exactly `v` indices, without replacement. -/
noncomputable def deficiencyVariance (a : ι → ℤ) (U : Finset ι) (v : ℕ) : ℝ :=
finiteVariance (U.powersetCard v) (deficiency a)
theorem deficiency_exchange_bound (a : ι → ℤ) (S : Finset ι) {i j : ι}
(hi : i ∈ S) (hj : j ∉ S) :
|deficiency a S - deficiency a (insert j (S.erase i))| ≤ 1 := by
have hj' : j ∉ S.erase i := fun h => hj (Finset.mem_erase.mp h).2
simpa only [Finset.insert_erase hi] using
(deficiency_replacement_lipschitz a (Finset.notMem_erase i S) hj')
/-- The paper's exact variance bound `Var(d(V)) ≤ v`. -/
theorem deficiencyVariance_le (a : ι → ℤ) (U : Finset ι) (v : ℕ)
(hv : v ≤ U.card) : deficiencyVariance a U v ≤ v := by
have h := finiteVariance_powersetCard_le U v (deficiency a)
(c := 1) zero_le_one hv (by
intro S _ _ i hi j _ hj
exact deficiency_exchange_bound a S hi hj)
simpa only [deficiencyVariance, one_pow, mul_one] using h
theorem deficiency_centered_secondMoment_le (a : ι → ℤ) (U : Finset ι) (v : ℕ)
(hv : v ≤ U.card) :
(𝔼 V ∈ U.powersetCard v, (deficiency a V - meanDeficiency a U v) ^ 2) ≤ v :=
deficiencyVariance_le a U v hv
/-- A direct finite Chebyshev bound for deviations from the mean deficiency.
The left side is the exact probability under uniform size-`v` sampling. -/
theorem deficiency_tail_fraction_le (a : ι → ℤ) (U : Finset ι) (v : ℕ)
(hv : v ≤ U.card) {t : ℝ} (ht : 0 < t) :
(((U.powersetCard v).filter
(fun V => t ≤ |deficiency a V - meanDeficiency a U v|)).card : ℝ) /
(U.powersetCard v).card ≤ (v : ℝ) / t ^ 2 := by
classical
let D := U.powersetCard v
let B := D.filter fun V => t ≤ |deficiency a V - meanDeficiency a U v|
have hD : (0 : ℝ) < D.card := by
exact_mod_cast (Finset.powersetCard_nonempty.mpr hv).card_pos
have hpoint : ∀ V ∈ D,
(if t ≤ |deficiency a V - meanDeficiency a U v| then t ^ 2 else 0) ≤
(deficiency a V - meanDeficiency a U v) ^ 2 := by
intro V _
split_ifs with h
· have hsq := (sq_le_sq₀ ht.le (abs_nonneg _)).mpr h
simpa only [sq_abs] using hsq
· exact sq_nonneg _
have htail : (B.card : ℝ) * t ^ 2 ≤
∑ V ∈ D, (deficiency a V - meanDeficiency a U v) ^ 2 := by
calc
_ = ∑ V ∈ D,
if t ≤ |deficiency a V - meanDeficiency a U v| then t ^ 2 else 0 := by
rw [← Finset.sum_filter]
simp [B]
_ ≤ _ := Finset.sum_le_sum hpoint
have hvar : (∑ V ∈ D, (deficiency a V - meanDeficiency a U v) ^ 2) ≤
(v : ℝ) * D.card := by
have h := deficiency_centered_secondMoment_le a U v hv
rw [Finset.expect_eq_sum_div_card] at h
exact (div_le_iff₀ hD).mp h
change (B.card : ℝ) / D.card ≤ (v : ℝ) / t ^ 2
exact (div_le_div_iff₀ hD (sq_pos_of_pos ht)).mpr (htail.trans hvar)
/-- Measure-theoretic form for the actual uniform fixed-cardinality law. -/
theorem deficiency_uniform_variance_le
[MeasurableSpace (Finset ι)] [MeasurableSingletonClass (Finset ι)]
(a : ι → ℤ) (U : Finset ι) (v : ℕ) (hv : v ≤ U.card) :
ProbabilityTheory.variance (deficiency a)
(PMF.uniformOfFinset (U.powersetCard v)
(Finset.powersetCard_nonempty.mpr hv)).toMeasure ≤ v := by
rw [variance_uniformOfFinset]
exact deficiencyVariance_le a U v hv
end LowSpaceSubsetSum
end OAI
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