File size: 2,528 Bytes
f0f05de | 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 | import OAI.Combinatorics.BinaryEdit.FiniteSpectralTree
import OAI.Combinatorics.BinaryEdit.FiniteDisplacement
import OAI.Combinatorics.BinaryEdit.FiniteNumeric
namespace OAI
/-!
# Actual ℓ₁ maps and the finite-tree obstruction
The finite-group displacement identity yields the finite-tree root estimate
from geometric bounds on the concrete translations.
-/
noncomputable section
namespace TreeEdit.BinaryLower.FiniteActivity
open Classical
open scoped BigOperators
/-- Leaf periods, sparse-character detection and translation-cost bounds
imply the root averaged-distance estimate for arbitrary maps into ℓ₁. -/
theorem finite_tree_l1_mean_bound {A σ : Type*}
[AddCommGroup A] [Fintype A] [Fintype σ] [Nonempty σ]
(m b h : ℕ) (hm : m ≠ 0) (hb : 0 < b)
(w : List (Fin m) → A)
(Q : (Fin h → Fin m) → ℕ) [∀ f, NeZero (Q f)]
(φ : ∀ f, ZMod (Q f) →+ A)
(hgen : ∀ f : Fin h → Fin m, w (List.ofFn f) = φ f 1)
(F : A → RealL1)
(shift : InternalNode (Fin m) h → σ → A) (M C : ℝ)
(hleaf : ∀ f (z : ZMod (Q f)) x,
‖F (x + φ f z) - F x‖ ≤ M / (m : ℝ) ^ h)
(hdetect : ∀ v χ, characterSparse w b v χ →
(Fintype.card σ : ℝ) ≤ ∑ s, ‖1 - χ (shift v s)‖ ^ 2)
(hcost : ∀ v s x,
‖F (x + shift v s) - F x‖ ≤ C / (m : ℝ) ^ (v.1 : ℕ)) :
(𝔼 x : A, ‖F (x + w []) - F x‖) ≤
4 * ((h : ℝ) * C + M / (2 * (b : ℝ) ^ h)) := by
classical
obtain ⟨H, hH, hFourier⟩ := finite_group_displacement F
rw [hFourier]
apply finite_tree_spectral_bound m b h hm hb w Q φ hgen H hH shift M C
· intro f z
rw [← hFourier]
exact Finset.expect_le Finset.univ_nonempty (fun x _ => hleaf f z x)
· exact hdetect
· intro v s
rw [← hFourier]
exact Finset.expect_le Finset.univ_nonempty (fun x _ => hcost v s x)
/-- Parameter normalization gives the root estimate with the absolute
constant 34. -/
theorem source_mean_bound34 {A : Type*} [AddCommGroup A] [Fintype A]
(F : A → RealL1) (root : A) {k : ℕ} {M : ℝ}
(hk : 1 ≤ k) (hM : 0 ≤ M)
(hroot : (𝔼 x : A, ‖F (x + root) - F x‖) ≤
4 * ((k : ℝ) * ((2 * (4 * (k : ℝ) ^ (10 * k)) /
(k : ℝ) ^ (20 * k)) * M) + M / (2 * (k : ℝ) ^ k))) :
(𝔼 x : A, ‖F (x + root) - F x‖) ≤ 34 * M / (k : ℝ) ^ k := by
apply TreeEdit.BinaryLower.FiniteNumeric.finite_root_bound hk hM
convert hroot using 1
ring
end TreeEdit.BinaryLower.FiniteActivity
end
end OAI
|