| import OAI.Combinatorics.ContingencyTables.Sampling.Padding |
| import OAI.Combinatorics.ContingencyTables.Sampling.ResidualMixture |
| import Mathlib.Tactic.NormNum |
|
|
| namespace OAI |
|
|
| /-! |
| # Parameter arithmetic for the samplers |
|
|
| The manuscript's padding and rare tabulation estimates give explicit |
| inequalities for successful unpadding and expected correction cost. |
| -/ |
|
|
| namespace ContingencyTables |
|
|
| open scoped BigOperators |
|
|
| private theorem padding_error_le_half (d s : ℕ) (hd : 14 ≤ d) (hs : s ≤ d) : |
| (s : ℝ) * (4 * (d : ℝ) ^ 12 * (d : ℝ) ^ 3 / (d : ℝ) ^ 20) ≤ 1 / 2 := by |
| have hd₀ : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega) |
| have hd₁₄ : (14 : ℝ) ≤ d := by exact_mod_cast hd |
| have hs' : (s : ℝ) ≤ d := by exact_mod_cast hs |
| have hfour : (8 : ℝ) ≤ (d : ℝ) ^ 4 := by |
| have h := pow_le_pow_left₀ (by norm_num : (0 : ℝ) ≤ 14) hd₁₄ 4 |
| norm_num at h |
| linarith |
| have htwenty : (d : ℝ) ^ 20 = (d : ℝ) ^ 16 * (d : ℝ) ^ 4 := by ring |
| have hpow : (0 : ℝ) ≤ (d : ℝ) ^ 16 := by positivity |
| have hlarge := mul_le_mul_of_nonneg_left hfour hpow |
| have hnum : 8 * (s : ℝ) * (d : ℝ) ^ 12 * (d : ℝ) ^ 3 ≤ (d : ℝ) ^ 20 := by |
| calc |
| _ ≤ 8 * (d : ℝ) * (d : ℝ) ^ 12 * (d : ℝ) ^ 3 := by gcongr |
| _ = 8 * (d : ℝ) ^ 16 := by ring |
| _ ≤ (d : ℝ) ^ 20 := by rw [htwenty]; linarith |
| rw [← mul_div_assoc, div_le_iff₀ (pow_pos hd₀ 20)] |
| nlinarith |
|
|
| / |
| unpads successfully with probability at least one half. -/ |
| theorem padding_success_half {I J : Type*} [Fintype I] [Fintype J] |
| [DecidableEq I] [DecidableEq J] |
| (r : I → ℕ) (c : J → ℕ) (htotal : ∑ i, r i = ∑ j, c j) |
| (K : Finset (I × J)) (d : ℕ) (hd : 14 ≤ d) |
| (hrows : Fintype.card I ≤ d) (hcols : Fintype.card J ≤ d) |
| (hK : K.card ≤ d) (hrow : ∀ p ∈ K, d ^ 20 ≤ r p.1) |
| (hcol : ∀ p ∈ K, d ^ 20 ≤ c p.2) : |
| (1 / 2 : ℝ) ≤ (Fintype.card (Table r c) : ℝ) / |
| Fintype.card (Table (paddedRows r (largePadding K (d ^ 12))) |
| (paddedColumns c (largePadding K (d ^ 12)))) := by |
| have hd₀ : 0 < d := by omega |
| have hU : 4 * d ≤ d ^ 20 := by |
| have hd₄ : 4 ≤ d := by omega |
| have hpow : d ≤ d ^ 19 := Nat.le_self_pow (by decide) d |
| calc |
| 4 * d ≤ d * d ^ 19 := Nat.mul_le_mul hd₄ hpow |
| _ = d ^ 20 := by ring |
| have hL : 2 * d ^ 12 ≤ d ^ 20 := by |
| have hpow : d ≤ d ^ 8 := Nat.le_self_pow (by decide) d |
| have hd₂ : 2 ≤ d ^ 8 := by omega |
| calc |
| 2 * d ^ 12 ≤ d ^ 8 * d ^ 12 := Nat.mul_le_mul_right _ hd₂ |
| _ = d ^ 20 := by ring |
| have h := padding_success_fraction r c htotal K (d ^ 12) (d ^ 20) d |
| hd₀ hrows hcols hrow hcol hU hL |
| have he := padding_error_le_half d K.card hd hK |
| push_cast at h |
| linarith |
|
|
| / |
| smaller probability for the manuscript's `D=d⁶b²`. -/ |
| theorem rare_tabulation_exponent (d b : ℕ) (hd : 14 ≤ d) (_hb : 1 ≤ b) : |
| 500 * d * b ≤ d ^ 6 * b ^ 2 := by |
| have hfive : 500 ≤ d ^ 5 := by |
| have h := Nat.pow_le_pow_left hd 5 |
| norm_num at h |
| omega |
| have hb₂ : b ≤ b ^ 2 := Nat.le_self_pow (by decide) b |
| calc |
| 500 * d * b ≤ d ^ 5 * d * b := Nat.mul_le_mul_right b (Nat.mul_le_mul_right d hfive) |
| _ = d ^ 6 * b := by ring |
| _ ≤ d ^ 6 * b ^ 2 := Nat.mul_le_mul_left _ hb₂ |
|
|
| theorem rare_tabulation_cost (d b : ℕ) (hd : 14 ≤ d) (hb : 1 ≤ b) : |
| (2 : ℚ) ^ (500 * d * b) / (2 : ℚ) ^ (d ^ 6 * b ^ 2) ≤ 1 := by |
| apply (div_le_one (by positivity)).mpr |
| exact pow_le_pow_right₀ (by norm_num) (rare_tabulation_exponent d b hd hb) |
|
|
| end ContingencyTables |
|
|
| end OAI |
|
|
