openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Counting /DeficiencyVariance.lean
| 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 | |