| import OAI.Combinatorics.EditDistance.HashCuts |
|
|
| namespace OAI |
|
|
| /-! |
| # Hash separation and finite histogram inequalities |
| -/ |
|
|
| namespace EditCutHash |
| attribute [local instance] Classical.propDecidable |
| open scoped BigOperators |
| open Finset |
|
|
| section Histogram |
| variable {J K : Type*} [Fintype J] [Fintype K] |
|
|
| noncomputable def hit (a b : K) : ℝ := if a = b then 1 else 0 |
| omit [Fintype K] in |
| lemma hit_nonneg (a b : K) : 0 ≤ hit a b := by unfold hit; split <;> norm_num |
| omit [Fintype K] in |
| lemma hit_le_one (a b : K) : hit a b ≤ 1 := by unfold hit; split <;> norm_num |
| omit [Fintype K] in |
| @[simp] lemma hit_self (a : K) : hit a a = 1 := by simp [hit] |
|
|
| noncomputable def count (f : J → K) (z : K) : ℝ := ∑ j, hit (f j) z |
| noncomputable def histogramDistance (f g : J → K) : ℝ := ∑ z, |count f z-count g z| |
|
|
| omit [Fintype K] in |
| lemma count_nonneg (f : J → K) (z : K) : 0 ≤ count f z := sum_nonneg (fun _ _ => hit_nonneg _ _) |
| lemma histogramDistance_nonneg (f g : J → K) : 0 ≤ histogramDistance f g := |
| sum_nonneg (fun _ _ => abs_nonneg _) |
|
|
| lemma hit_sum (a : K) : ∑ z, hit a z = 1 := by |
| classical |
| simp [hit] |
|
|
| lemma count_sum (f : J → K) : ∑ z, count f z = Fintype.card J := by |
| unfold count |
| rw [sum_comm] |
| simp only [hit_sum,sum_const,card_univ,nsmul_eq_mul,mul_one] |
|
|
| lemma hit_distance (a b : K) : ∑ z, |hit a z-hit b z| = 2*sep a b := by |
| classical |
| by_cases hab : a = b |
| · subst b |
| simp [sep] |
| · have he (z : K) : |hit a z-hit b z| = hit a z+hit b z := by |
| by_cases haz : a = z |
| · subst z |
| simp [hit,Ne.symm hab] |
| · by_cases hbz : b = z |
| · subst z |
| simp [hit,hab] |
| · simp [hit,haz,hbz] |
| simp_rw [he] |
| rw [sum_add_distrib,hit_sum,hit_sum,sep_eq_one hab] |
| norm_num |
|
|
| lemma histogramDistance_le (f g : J → K) : histogramDistance f g ≤ 2*∑ j, sep (f j) (g j) := by |
| unfold histogramDistance count |
| simp_rw [←sum_sub_distrib] |
| calc |
| _ ≤ ∑ z : K, ∑ j : J, |hit (f j) z-hit (g j) z| := |
| sum_le_sum (fun _ _ => abs_sum_le_sum_abs _ _) |
| _ = 2*∑ j, sep (f j) (g j) := by |
| rw [sum_comm] |
| simp_rw [hit_distance] |
| rw [mul_sum] |
|
|
| lemma histogramDistance_disjoint (f g : J → K) (h : ∀ i j, f i ≠ g j) : |
| histogramDistance f g = 2*(Fintype.card J : ℝ) := by |
| have hzero (z : K) : count f z = 0 ∨ count g z = 0 := by |
| by_cases hf : ∃ i, f i = z |
| · obtain ⟨i,rfl⟩ := hf |
| right |
| unfold count |
| apply sum_eq_zero |
| intro j _ |
| exact ite_eq_right (Ne.symm (h i j)) |
| · left |
| unfold count |
| apply sum_eq_zero |
| intro i _ |
| exact ite_eq_right (fun he => hf ⟨i,he⟩) |
| have he (z : K) : |count f z-count g z| = count f z+count g z := by |
| rcases hzero z with hz|hz |
| · rw [hz,zero_sub,abs_neg,abs_of_nonneg (count_nonneg g z),zero_add] |
| · rw [hz,sub_zero,abs_of_nonneg (count_nonneg f z),add_zero] |
| unfold histogramDistance |
| simp_rw [he] |
| rw [sum_add_distrib,count_sum,count_sum] |
| ring |
|
|
| lemma histogramDistance_lower (f g : J → K) : |
| 2*(Fintype.card J : ℝ)*(1-∑ i : J, ∑ j : J, hit (f i) (g j)) ≤ |
| histogramDistance f g := by |
| classical |
| by_cases h : ∀ i j, f i ≠ g j |
| · rw [histogramDistance_disjoint f g h] |
| have hz : (∑ i : J, ∑ j : J, hit (f i) (g j)) = 0 := by |
| apply sum_eq_zero |
| intro i _ |
| apply sum_eq_zero |
| intro j _ |
| exact ite_eq_right (h i j) |
| rw [hz] |
| ring_nf |
| rfl |
| · push Not at h |
| obtain ⟨i,j,hij⟩ := h |
| have h1 : 1 ≤ ∑ k : J, hit (f i) (g k) := by |
| have hh := single_le_sum (fun k _ => hit_nonneg (f i) (g k)) (mem_univ j) |
| simpa only [hij,hit_self] using hh |
| have h2 : (∑ k : J, hit (f i) (g k)) ≤ ∑ l : J, ∑ k : J, hit (f l) (g k) := |
| single_le_sum (fun l _ => sum_nonneg (fun k _ => hit_nonneg (f l) (g k))) (mem_univ i) |
| have hh : 1-∑ l : J, ∑ k : J, hit (f l) (g k) ≤ 0 := by linarith |
| exact (mul_nonpos_of_nonneg_of_nonpos (by positivity) hh).trans |
| (histogramDistance_nonneg f g) |
|
|
| end Histogram |
| end EditCutHash |
|
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| end OAI |
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|
