| import OAI.Combinatorics.EditDistance.Histograms |
|
|
| namespace OAI |
|
|
| /-! |
| # Mean histograms as finite cut distances |
| -/ |
|
|
| namespace EditCutHash |
| attribute [local instance] Classical.propDecidable |
| open scoped BigOperators |
| open Finset |
|
|
| section MeanHistogram |
| variable {I J X : Type*} [Fintype I] [Fintype J] |
|
|
| noncomputable def meanHistogram (w : I → ℝ) (a : ℝ) (c : I → X → Bool) (f g : J → X) : ℝ := |
| ∑ h : I → Bool, probability w a h * histogramDistance (fun j => key c h (f j)) (fun j => key c h (g j)) |
|
|
| lemma meanHistogram_nonneg {w : I → ℝ} (hw : ∀ i, 0 ≤ w i) {a : ℝ} (ha : 0 ≤ a) |
| (c : I → X → Bool) (f g : J → X) : 0 ≤ meanHistogram w a c f g := |
| sum_nonneg (fun h _ => mul_nonneg (probability_nonneg hw ha h) (histogramDistance_nonneg _ _)) |
|
|
| lemma meanHistogram_upper {w : I → ℝ} (hw : ∀ i, 0 ≤ w i) {a : ℝ} (ha : 0 ≤ a) |
| (c : I → X → Bool) (f g : J → X) : |
| meanHistogram w a c f g ≤ 2*a*∑ j, distance w c (f j) (g j) := by |
| calc |
| _ ≤ ∑ h : I → Bool, probability w a h * (2*∑ j, sep (key c h (f j)) (key c h (g j))) := |
| sum_le_sum (fun h _ => mul_le_mul_of_nonneg_left (histogramDistance_le _ _) (probability_nonneg hw ha h)) |
| _ = 2*∑ j : J, ∑ h : I → Bool, probability w a h * sep (key c h (f j)) (key c h (g j)) := by |
| simp_rw [mul_left_comm (probability w a _ : ℝ) 2,mul_sum] |
| rw [sum_comm] |
| _ ≤ 2*∑ j : J, a*distance w c (f j) (g j) := by |
| apply mul_le_mul_of_nonneg_left ?_ (by norm_num) |
| exact sum_le_sum (fun j _ => separation_bound w a c (f j) (g j)) |
| _ = _ := by rw [←mul_sum]; ring |
|
|
| lemma mean_cross_collisions (w : I → ℝ) (a : ℝ) (c : I → X → Bool) (f g : J → X) : |
| (∑ h : I → Bool, probability w a h * |
| (∑ i : J, ∑ j : J, hit (key c h (f i)) (key c h (g j)))) = |
| ∑ i : J, ∑ j : J, Real.exp (-a*distance w c (f i) (g j)) := by |
| simp_rw [mul_sum] |
| rw [sum_comm] |
| apply sum_congr rfl |
| intro i _ |
| rw [sum_comm] |
| apply sum_congr rfl |
| intro j _ |
| apply Eq.trans ?_ (collision_probability w a c (f i) (g j)) |
| apply sum_congr rfl |
| intro h _ |
| congr 1 |
| by_cases he : key c h (f i) = key c h (g j) <;> simp [hit,he] |
|
|
| theorem meanHistogram_lower {w : I → ℝ} (hw : ∀ i, 0 ≤ w i) {a : ℝ} (ha : 0 ≤ a) |
| (c : I → X → Bool) (f g : J → X) {R : ℝ} (hR : ∀ i j, R ≤ distance w c (f i) (g j)) : |
| 2*(Fintype.card J : ℝ)*(1-(Fintype.card J : ℝ)^2*Real.exp (-a*R)) ≤ |
| meanHistogram w a c f g := by |
| have hc : (∑ i : J, ∑ j : J, Real.exp (-a*distance w c (f i) (g j))) ≤ |
| (Fintype.card J : ℝ)^2*Real.exp (-a*R) := by |
| calc |
| _ ≤ ∑ _i : J, ∑ _j : J, Real.exp (-a*R) := by |
| apply sum_le_sum |
| intro i _ |
| apply sum_le_sum |
| intro j _ |
| apply Real.exp_le_exp.mpr |
| nlinarith [mul_le_mul_of_nonneg_left (hR i j) ha] |
| _ = _ := by simp only [sum_const,card_univ,nsmul_eq_mul]; ring |
| have hp : (∑ h : I → Bool, probability w a h * |
| (2*(Fintype.card J : ℝ)*(1-∑ i : J, ∑ j : J, hit (key c h (f i)) (key c h (g j))))) ≤ |
| meanHistogram w a c f g := |
| sum_le_sum (fun h _ => mul_le_mul_of_nonneg_left (histogramDistance_lower _ _) (probability_nonneg hw ha h)) |
| have he : (∑ h : I → Bool, probability w a h * |
| (2*(Fintype.card J : ℝ)*(1-∑ i : J, ∑ j : J, hit (key c h (f i)) (key c h (g j))))) = |
| 2*(Fintype.card J : ℝ)*(1-∑ i : J, ∑ j : J, Real.exp (-a*distance w c (f i) (g j))) := by |
| simp_rw [mul_left_comm (probability w a _) (2*(Fintype.card J : ℝ))] |
| rw [←mul_sum] |
| congr 1 |
| simp_rw [mul_sub,mul_one] |
| rw [sum_sub_distrib,probability_sum,mean_cross_collisions] |
| rw [he] at hp |
| apply le_trans ?_ hp |
| gcongr |
|
|
| noncomputable def rate (s : ℕ) : ℝ := Real.log (2*(s : ℝ)^2)/(s : ℝ) |
|
|
| lemma rate_nonneg {s : ℕ} (hs : 0 < s) : 0 ≤ rate s := by |
| have hs1 : (1 : ℝ) ≤ s := by exact_mod_cast hs |
| apply div_nonneg (Real.log_nonneg (by nlinarith)) (by positivity) |
|
|
| lemma rate_exp {s : ℕ} (hs : 0 < s) : Real.exp (-rate s*(s : ℝ)) = 1/(2*(s : ℝ)^2) := by |
| have sp : (0 : ℝ) < s := by exact_mod_cast hs |
| rw [rate,neg_mul,div_mul_cancel₀ _ sp.ne',Real.exp_neg,Real.exp_log (by positivity)] |
| simp only [one_div] |
| |
| lemma meanHistogram_separated {s : ℕ} (hs : 0 < s) {w : I → ℝ} (hw : ∀ i, 0 ≤ w i) |
| (c : I → X → Bool) (f g : Fin s → X) |
| (hR : ∀ i j, (s : ℝ) ≤ distance w c (f i) (g j)) : |
| (s : ℝ) ≤ meanHistogram w (rate s) c f g := by |
| have hh := meanHistogram_lower hw (rate_nonneg hs) c f g hR |
| rw [Fintype.card_fin,rate_exp hs] at hh |
| have sp : (0 : ℝ) < s := by exact_mod_cast hs |
| have he : 2*(s : ℝ)*(1-(s : ℝ)^2*(1/(2*(s : ℝ)^2))) = (s : ℝ) := by |
| field_simp |
| ring |
| simpa only [he] using hh |
| |
| end MeanHistogram |
| end EditCutHash |
| |
| namespace EditCutHash |
| attribute [local instance] Classical.propDecidable |
| open scoped BigOperators |
| open Finset |
| |
| lemma histogramDistance_shift {K : Type*} [Fintype K] {s : ℕ} |
| (f g : Fin (s+1) → K) (h : ∀ i : Fin s, f i.castSucc = g i.succ) : |
| histogramDistance f g ≤ 2 := by |
| have hc (z : K) : count f z-count g z = hit (f (Fin.last s)) z-hit (g 0) z := by |
| unfold count |
| rw [Fin.sum_univ_castSucc,Fin.sum_univ_succ] |
| simp_rw [h] |
| ring |
| unfold histogramDistance |
| simp_rw [hc] |
| rw [hit_distance] |
| have hh := sep_le_one (f (Fin.last s)) (g 0) |
| linarith |
| |
| lemma meanHistogram_shift {I X : Type*} [Fintype I] {s : ℕ} (hs : 0 < s) |
| {w : I → ℝ} (hw : ∀ i, 0 ≤ w i) {a : ℝ} (ha : 0 ≤ a) |
| (c : I → X → Bool) (f g : Fin s → X) |
| (h : ∀ i : Fin (s-1), f ⟨i.val,by omega⟩ = g ⟨i.val+1,by omega⟩) : |
| meanHistogram w a c f g ≤ 2 := by |
| obtain ⟨t,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hs) |
| have hd (r : I → Bool) : histogramDistance (fun i => key c r (f i)) (fun i => key c r (g i)) ≤ 2 := by |
| apply histogramDistance_shift |
| intro i |
| congr 1 |
| exact h ⟨i.val,by omega⟩ |
| calc |
| _ ≤ ∑ r : I → Bool, probability w a r * 2 := |
| sum_le_sum (fun r _ => mul_le_mul_of_nonneg_left (hd r) (probability_nonneg hw ha r)) |
| _ = 2 := by rw [←sum_mul,probability_sum,one_mul] |
| |
| @[simp] lemma meanHistogram_self {I J X : Type*} [Fintype I] [Fintype J] |
| (w : I → ℝ) (a : ℝ) (c : I → X → Bool) (f : J → X) : |
| meanHistogram w a c f f = 0 := by |
| simp [meanHistogram,histogramDistance] |
| |
| end EditCutHash |
| |
| end OAI |
| |
