| 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 |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
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|
