| import OAI.Combinatorics.FiniteCircle.Separation |
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| namespace OAI |
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| namespace FiniteCircle |
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| open scoped BigOperators List |
| open scoped ComplexConjugate |
| open RCLike Module |
| open Classical |
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| 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 |
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| end FiniteCircle |
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| end OAI |
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