| import OAI.Combinatorics.Permanent.Perturbation |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace FourRow |
| open scoped BigOperators |
|
|
| def strictExponent : ℝ := 2 - 1 / 100000 |
|
|
| def robustRadius : ℝ := 1 / 100000 |
|
|
| theorem normalized_robust_permanent (ν : Law) (hp : IsProbability ν) |
| (hm : UniformMarginals ν) (htv : totalVariation ν < robustRadius) |
| (f : Functions) (hf : ∀ i j, 0 ≤ f i j) (hn : ∀ i, rowSq (f i) = 4) : |
| permanentExpectation ν f ≤ ∏ i, lpNorm strictExponent (f i) := by |
| have h1 := normalized_uniform_stability f hf hn |
| have h2 := normalized_perturbation ν hp hm f hf hn |
| have h3 := normalized_product_norm_lower strictExponent (by norm_num [strictExponent]) |
| (by norm_num [strictExponent]) f hf hn |
| have hd := totalDist_nonneg f |
| have ht := mul_le_mul_of_nonneg_right (le_of_lt htv) hd |
| dsimp [robustRadius] at ht |
| dsimp [strictExponent] at h3 ⊢ |
| linarith |
|
|
|
|
| /- Homogeneity extends the bound to unrestricted nonnegative rows. -/ |
| theorem lpNorm_scale (p : ℝ) (hp : p ≠ 0) (c : ℝ) (hc : 0 ≤ c) |
| (a : Site → ℝ) (ha : ∀ j, 0 ≤ a j) : |
| lpNorm p (fun j => c * a j) = c * lpNorm p a := by |
| unfold lpNorm |
| simp_rw [Real.mul_rpow hc (ha _)] |
| rw [← Finset.mul_sum, mul_div_assoc] |
| rw [Real.mul_rpow (Real.rpow_nonneg hc p) |
| (div_nonneg (Finset.sum_nonneg (fun j _ => Real.rpow_nonneg (ha j) p)) (by norm_num))] |
| rw [← Real.rpow_mul hc, one_div, mul_inv_cancel₀ hp, Real.rpow_one] |
|
|
| theorem expectation_scale (ν : Law) (c : Site → ℝ) (f : Functions) : |
| permanentExpectation ν (fun i j => c i * f i j) = |
| (∏ i, c i) * permanentExpectation ν f := by |
| unfold permanentExpectation |
| simp_rw [Finset.prod_mul_distrib] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro π _ |
| ring |
|
|
| theorem rowSq_zero_entries (a : Site → ℝ) (h : rowSq a = 0) : ∀ j, a j = 0 := by |
| intro j |
| have hs : a j ^ 2 ≤ rowSq a := |
| Finset.single_le_sum (fun k _ => sq_nonneg (a k)) (Finset.mem_univ j) |
| rw [h] at hs |
| nlinarith [sq_nonneg (a j)] |
|
|
| theorem robust_permanent_explicit (ν : Law) (hp : IsProbability ν) |
| (htv : totalVariation ν < robustRadius) (hm : UniformMarginals ν) |
| (f : Functions) (hf : ∀ i j, 0 ≤ f i j) : |
| permanentExpectation ν f ≤ ∏ i, lpNorm strictExponent (f i) := by |
| by_cases hz : ∃ i, rowSq (f i) = 0 |
| · obtain ⟨i, hi⟩ := hz |
| have hf0 := rowSq_zero_entries (f i) hi |
| have hprod (π : Perm) : ∏ k, f k (π k) = 0 := by |
| exact Finset.prod_eq_zero (Finset.mem_univ i) (hf0 (π i)) |
| have hexp : permanentExpectation ν f = 0 := by |
| simp only [permanentExpectation, hprod, mul_zero, Finset.sum_const_zero] |
| rw [hexp] |
| exact Finset.prod_nonneg (fun k _ => lpNorm_nonneg _ _ (hf k)) |
| · have hspos (i : Site) : 0 < rowSq (f i) := by |
| have := rowSq_nonneg (f i) |
| have : rowSq (f i) ≠ 0 := fun hi => hz ⟨i, hi⟩ |
| positivity |
| let c : Site → ℝ := fun i => Real.sqrt (rowSq (f i) / 4) |
| have hcpos (i : Site) : 0 < c i := Real.sqrt_pos.2 (div_pos (hspos i) (by norm_num)) |
| have hc0 (i : Site) : 0 ≤ c i := le_of_lt (hcpos i) |
| have hcne (i : Site) : c i ≠ 0 := ne_of_gt (hcpos i) |
| have hcsq (i : Site) : c i ^ 2 = rowSq (f i) / 4 := |
| Real.sq_sqrt (div_nonneg (le_of_lt (hspos i)) (by norm_num)) |
| let g : Functions := fun i j => f i j / c i |
| have hg0 (i j : Site) : 0 ≤ g i j := div_nonneg (hf i j) (hc0 i) |
| have hgn (i : Site) : rowSq (g i) = 4 := by |
| dsimp [rowSq, g] |
| simp_rw [div_pow, ← Finset.sum_div] |
| change rowSq (f i) / c i ^ 2 = 4 |
| rw [hcsq] |
| field_simp [ne_of_gt (hspos i)] |
| have heq : f = fun i j => c i * g i j := by |
| funext i j |
| dsimp [g] |
| field_simp [hcne i] |
| have h := normalized_robust_permanent ν hp hm htv g hg0 hgn |
| have hcprod : 0 ≤ ∏ i, c i := Finset.prod_nonneg (fun i _ => hc0 i) |
| have hscaled := mul_le_mul_of_nonneg_left h hcprod |
| rw [heq, expectation_scale] |
| simp_rw [lpNorm_scale strictExponent (by norm_num [strictExponent]) _ (hc0 _) _ (hg0 _)] |
| rw [Finset.prod_mul_distrib] |
| exact hscaled |
|
|
| /-- A robust permanent inequality with explicit absolute constants. -/ |
| theorem robust_permanent : |
| ∃ p₀ : ℝ, 4 / 3 < p₀ ∧ p₀ < 2 ∧ |
| ∃ ε : ℝ, 0 < ε ∧ ∀ ν : Law, |
| IsProbability ν → totalVariation ν < ε → UniformMarginals ν → |
| ∀ f : Functions, (∀ i j, 0 ≤ f i j) → |
| permanentExpectation ν f ≤ ∏ i, lpNorm p₀ (f i) := by |
| refine ⟨strictExponent, ?_, ?_, robustRadius, ?_, ?_⟩ |
| · norm_num [strictExponent] |
| · norm_num [strictExponent] |
| · norm_num [robustRadius] |
| · intro ν hp htv hm f hf |
| exact robust_permanent_explicit ν hp htv hm f hf |
|
|
| end FourRow |
| end |
|
|
| end OAI |
|
|
