| import OAI.Combinatorics.ContingencyTables.Sampling.ExactGateCost |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactPrintCost |
| import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperInputWordBudget |
|
|
| namespace OAI |
|
|
| /-! # Exact-sampler parameters in the public binary input size -/ |
|
|
| namespace ContingencyTables.SmallGraphProfiles |
| open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.LiteralCodec |
| open ProfilePrograms DensePrograms Algorithms Polynomial |
| open scoped BigOperators Classical |
| noncomputable section |
|
|
| variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) |
| local notation "S" => samplingSize n r 0 |
|
|
| lemma exact_public_length (htotal : (∑ i,r i)=∑ j,c j) : |
| (encodeMargins r c).length ≤ 5*S^2 := by |
| have h := encodeSamplingInput_length r c htotal 0 |
| have he : (encodeMargins r c).length ≤ (encodeSamplingInput r c 0).length := by |
| simp only [encodeSamplingInput,List.length_append] |
| omega |
| exact he.trans h |
|
|
| lemma exact_public_weight (htotal : (∑ i,r i)=∑ j,c j) : |
| weight (encodeMargins r c) ≤ 91*S^2 := by |
| have h := exact_public_length r c htotal |
| have hS : 1 ≤ S^2 := one_le_pow₀ (by unfold samplingSize; omega) |
| rw [weight_word] |
| omega |
|
|
| lemma exact_precision_bound : correctionPrecision r c ≤ 245^8*S^16 := by |
| let dim := CompletionCounts.dimensionAllowance (I := Fin m) (J := Fin n) |
| have hi := sampling_internal_size_bound r c 0 |
| have hd : dim ≤ 245*S^2 := by dsimp only [dim]; omega |
| have hb : smallBinaryLength r c ≤ 245*S^2 := by omega |
| change dim^6*(smallBinaryLength r c)^2 ≤ _ |
| calc |
| _ ≤ (245*S^2)^6*(245*S^2)^2 := |
| Nat.mul_le_mul (Nat.pow_le_pow_left hd _) (Nat.pow_le_pow_left hb _) |
| _ = 245^8*S^16 := by ring |
|
|
| lemma exact_margin_weight_polynomial : ∃ p : Polynomial ℕ, |
| ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), |
| (∑ i,r i)=∑ j,c j → weight (List.ofFn r,List.ofFn c) ≤ p.eval (samplingSize n r 0) := by |
| obtain ⟨p,hp⟩ := polynomial_exactParsedMargins |
| refine ⟨p.comp (91*X^2),?_⟩ |
| intro m n r c htotal |
| have h := (Nat.le_add_left _ _).trans (hp (encodeMargins r c)) |
| rw [exactParsedMargins_spec] at h |
| exact h.trans (by |
| rw [eval_comp] |
| simp only [eval_mul,eval_ofNat,eval_pow,eval_X] |
| exact polynomial_eval_monotone p (exact_public_weight r c htotal)) |
|
|
| lemma exact_parameter_polynomial : ∃ p : Polynomial ℕ, |
| ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), |
| (∑ i,r i)=∑ j,c j → exactParameterMeasure (List.ofFn r,List.ofFn c) ≤ |
| p.eval (samplingSize n r 0) := by |
| obtain ⟨p,hp⟩ := exact_margin_weight_polynomial |
| refine ⟨C (245^8)*X^16+p,?_⟩ |
| intro m n r c htotal |
| simp only [exactParameterMeasure,exactPrecision_ofFn,eval_add,eval_mul,eval_C,eval_pow,eval_X] |
| exact Nat.add_le_add (exact_precision_bound r c) (hp m n r c htotal) |
|
|
| lemma exact_total_weight : weight (∑ i,r i) ≤ 4*S+1 := by |
| have hs : (∑ i,r i).size ≤ Nat.clog 2 ((∑ i,r i)+1) := |
| Nat.size_le.mpr (Nat.lt_of_lt_of_le (Nat.lt_succ_self _) |
| (Nat.le_pow_clog (by decide) _)) |
| have hw := weight_nat_le (∑ i,r i) |
| unfold samplingSize |
| omega |
|
|
| lemma exact_matrix_size : ExactPrintCost.matrixSize (n := n) r ≤ 25*S^3 := by |
| have hS : 1 ≤ S := by unfold samplingSize; omega |
| have hm : m ≤ S := by unfold samplingSize; omega |
| have hn : n ≤ S := by unfold samplingSize; omega |
| have hw : 4*weight (∑ i,r i)+2 ≤ 22*S := by |
| have := exact_total_weight (n := n) r |
| omega |
| have hpow : S ≤ S^3 := Nat.le_self_pow (by decide) _ |
| calc |
| _ ≤ S*(S*(22*S)+2)+1 := by unfold ExactPrintCost.matrixSize; gcongr |
| _ ≤ 25*S^3 := by nlinarith |
|
|
| lemma exact_ordinary_bits_polynomial : ∃ p : Polynomial ℕ, |
| ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), |
| exactOrdinaryBits (List.ofFn r,List.ofFn c) ≤ p.eval (samplingSize n r 0) := by |
| let q : Polynomial ℕ := X+2*(C (245^8)*X^16) |
| refine ⟨C (344^644)*q^1288,?_⟩ |
| intro m n r c |
| have hD := exact_precision_bound r c |
| have hS : samplingSize n r (2*correctionPrecision r c) ≤ q.eval (samplingSize n r 0) := by |
| simp only [q,eval_add,eval_mul,eval_ofNat,eval_C,eval_pow,eval_X] |
| unfold samplingSize at * |
| omega |
| rw [exactOrdinaryBits_ofFn] |
| have hf : finiteSamplerBits r c (2*correctionPrecision r c) ≤ |
| 344^644*(samplingSize n r (2*correctionPrecision r c))^1288 := |
| sampling_word_budget_polynomial r c (2*correctionPrecision r c) |
| have hp (k : ℕ) : |
| 344^644*(samplingSize n r (2*correctionPrecision r c))^k ≤ |
| (C (344^644)*q^k).eval (samplingSize n r 0) := by |
| rw [eval_mul,eval_C,eval_pow] |
| exact Nat.mul_le_mul_left (344^644) (Nat.pow_le_pow_left hS k) |
| exact hf.trans (hp 1288) |
|
|
| lemma exact_gate_measure_polynomial : ∃ p : Polynomial ℕ, |
| ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), |
| (∑ i,r i)=∑ j,c j → ExactGateCost.measure (List.ofFn r,List.ofFn c) ≤ |
| p.eval (samplingSize n r 0) := by |
| obtain ⟨p,hp⟩ := exact_parameter_polynomial |
| obtain ⟨q,hq⟩ := exact_margin_weight_polynomial |
| obtain ⟨u,hu⟩ := exact_ordinary_bits_polynomial |
| refine ⟨p+q+u,?_⟩ |
| intro m n r c htotal |
| simpa only [ExactGateCost.measure,FiniteWordExecution.wordMeasure,eval_add,add_assoc] |
| using Nat.add_le_add (hp m n r c htotal) |
| (Nat.add_le_add (hq m n r c htotal) (hu m n r c)) |
|
|
| end |
| end ContingencyTables.SmallGraphProfiles |
|
|
| end OAI |
|
|
