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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Combinatorics.FiniteCircle.Separation
namespace OAI
namespace FiniteCircle
open scoped BigOperators List
open scoped ComplexConjugate
open RCLike Module
open Classical
theorem histogram_embedding_step {α : Type*} [Fintype α] (a : α) {m M : ℕ}
(hm : 0 < m) (b : ℕ) (F : List.Vector α m → Fin M → ℝ) (Λ : ℝ) (hΛ : 0 ≤ Λ)
(hFlo : ∀ u v, (deficit u.val v.val : ℝ) ≤ coordDist F u v)
(hFhi : ∀ u v, coordDist F u v ≤ Λ * deficit u.val v.val) :
∃ M' : ℕ, ∃ F' : List.Vector α (b*m) → Fin M' → ℝ,
∀ x y, (deficit x.val y.val : ℝ) ≤ coordDist F' x y ∧
coordDist F' x y ≤ (16 * (Nat.log 2 m + 1) *
(b + 16 * Real.log ((m:ℝ)+1) * Λ)) * deficit x.val y.val := by
classical
let X := List.Vector α m
let : DecidableEq X := Classical.decEq X
let Ω := PartitionSpace X
let K := Nat.log 2 m + 1
have hav (k : Fin K) : 1 ≤ 2^k.val ∧ 2^k.val ≤ m := by
refine ⟨?_, Nat.pow_le_of_le_log hm.ne' (by have := k.isLt; dsimp [K] at *; omega)⟩
have hp : 0 < 2^k.val := pow_pos (by norm_num) _
omega
have hp (k : Fin K) := finite_partition (finite_l1_cuts F) (partitionRate (2^k.val))
(partitionRate_nonneg (2^k.val))
choose μ label he using hp
let I := (Fin b × Fin K) × (Ω × Ω)
let Φ : List α → I → ℝ := fun x i =>
4 * histMap (μ i.1.2) (label i.1.2) (windowWords a x (i.1.1*m) m (2^i.1.2.val)) i.2
have hΦ (x y : List α) : coordDist Φ x y =
4 * ∑ j : Fin b, ∑ k : Fin K, (μ k).expect (fun ω => histDist (label k ω)
(windowWords a x (j*m) m (2^k.val)) (windowWords a y (j*m) m (2^k.val))) := by
unfold coordDist
rw [Fintype.sum_prod_type]
simp only [Φ, ← mul_sub, abs_mul, show |(4:ℝ)| = 4 by norm_num]
simp_rw [← Finset.mul_sum, histMap_distance]
rw [Fintype.sum_prod_type]
have hnonneg (x y : List α) (j : Fin b) (k : Fin K) :
0 ≤ (μ k).expect (fun ω => histDist (label k ω)
(windowWords a x (j*m) m (2^k.val)) (windowWords a y (j*m) m (2^k.val))) :=
(μ k).expect_nonneg (fun _ => histDist_nonneg _ _ _)
have hlo (x y : List.Vector α (b*m)) : (deficit x.val y.val : ℝ) ≤ coordDist Φ x.val y.val := by
have hx (j : Fin b) : (segment x.val (j*m) m).length = m := by
apply length_segment
rw [x.property]
have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt j.isLt)
simpa only [Nat.succ_mul] using h
have hy (j : Fin b) : (segment y.val (j*m) m).length = m := by
apply length_segment
rw [y.property]
have h := Nat.mul_le_mul_right m (Nat.succ_le_of_lt j.isLt)
simpa only [Nat.succ_mul] using h
have hblock (j : Fin b) : (deficit (segment x.val (j*m) m) (segment y.val (j*m) m) : ℝ) ≤
4 * ∑ k : Fin K, (μ k).expect (fun ω => histDist (label k ω)
(windowWords a x.val (j*m) m (2^k.val)) (windowWords a y.val (j*m) m (2^k.val))) := by
let q := deficit (segment x.val (j*m) m) (segment y.val (j*m) m)
by_cases hq : q = 0
· change (q:ℝ) ≤ _
rw [hq, Nat.cast_zero]
exact mul_nonneg (by norm_num) (Finset.sum_nonneg (fun k _ => hnonneg x.val y.val j k))
· have hqm : q ≤ m := by dsimp [q, deficit]; rw [hx]; omega
obtain ⟨k, hs, hsm, hqs, hsq⟩ := exists_separating_scale (Nat.pos_of_ne_zero hq) hqm
have hsep := block_histogram_lower a F hFlo (μ k) (label k) (he k)
(hx j) (hy j) hs hsm hsq
rw [windowWords_segment a x.val hs hsm, windowWords_segment a y.val hs hsm] at hsep
calc
_ = (q:ℝ) := rfl
_ ≤ 4 * (2^k.val : ℕ) := by exact_mod_cast hqs
_ ≤ 4 * (μ k).expect (fun ω => histDist (label k ω)
(windowWords a x.val (j*m) m (2^k.val)) (windowWords a y.val (j*m) m (2^k.val))) := by linarith
_ ≤ _ := mul_le_mul_of_nonneg_left
(Finset.single_le_sum (fun k _ => hnonneg x.val y.val j k) (Finset.mem_univ _)) (by norm_num)
calc
_ ≤ ∑ j : Fin b, (deficit (segment x.val (j*m) m) (segment y.val (j*m) m) : ℝ) := by
exact_mod_cast deficit_le_sum_blocks x.property y.property
_ ≤ ∑ j : Fin b, 4 * ∑ k : Fin K, (μ k).expect (fun ω => histDist (label k ω)
(windowWords a x.val (j*m) m (2^k.val)) (windowWords a y.val (j*m) m (2^k.val))) :=
Finset.sum_le_sum (fun j _ => hblock j)
_ = _ := by rw [hΦ, Finset.mul_sum]
have hmove (x y : List α) (hxy : Slide x y) : coordDist Φ x y ≤
4*K * (2*b + 32 * Real.log ((m:ℝ)+1) * Λ) := by
rw [hΦ, Finset.sum_comm]
calc
_ ≤ 4 * ∑ k : Fin K, (2*b + 32 * Real.log ((m:ℝ)+1) * Λ) := by
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply Finset.sum_le_sum
intro k _
refine (histogram_slide_bound a F Λ hΛ hFhi (μ k) (label k) (he k) hxy b
(hav k).1 (hav k).2).trans ?_
have hlog : Real.log (((2^k.val : ℕ):ℝ)+1) ≤ Real.log ((m:ℝ)+1) :=
Real.log_le_log (by positivity) (by exact_mod_cast Nat.add_le_add_right (hav k).2 1)
gcongr
_ = _ := by simp; ring
let e := Fintype.equivFin I
let F' : List.Vector α (b*m) → Fin (Fintype.card I) → ℝ := fun x i => Φ x.val (e.symm i)
have hdist (x y : List.Vector α (b*m)) : coordDist F' x y = coordDist Φ x.val y.val :=
e.symm.sum_comp (fun i => |Φ x.val i - Φ y.val i|)
refine ⟨Fintype.card I, F', fun x y => ⟨?_, ?_⟩⟩
· rw [hdist]
exact hlo x y
· rw [hdist]
have hp := (exists_slidePath a (x.property.trans y.property.symm)).coordDist_le Φ
(4*K * (2*b + 32 * Real.log ((m:ℝ)+1) * Λ)) hmove
refine hp.trans_eq ?_
push_cast
dsimp [K]
push_cast
ring
end FiniteCircle
end OAI
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