| import OAI.Analysis.FourierLLogL.Compress.Schedule |
| import OAI.Analysis.FourierLLogL.Compress.Phase |
| import OAI.Analysis.FourierLLogL.Compress.Low |
|
|
| open Function Filter Set |
| open scoped BigOperators |
| noncomputable section |
| namespace OAI |
|
|
| namespace FourierLLogL.Compress |
| open Law Real |
| variable {A G Θ : Type} [Fintype A] [Fintype G] [DecidableEq G] |
| [AddCommGroup G] [Nonempty A] [Fintype Θ] |
|
|
| omit [Nonempty A] in |
| lemma init_gap (tag : A → G) (hi : Injective tag) (P : Law A) : |
| fundG tag (initial P) 1 = P.entropy := by |
| rw [G_initial] |
| let Q := iid P 1 |
| have h : Q.send (fun x => x 0) = P := marginalPack (fun _ => P) _ |
| change Q.H _ = _; rw [← h] |
| refine H_partition Q (X:=tot tag) (Y:=fun x => x 0) (fun _ _ => ?_) |
| simp only [tot,Fin.sum_univ_one,hi.eq_iff] |
|
|
| / |
| choice of scales in terms of height. -/ |
| lemma compress_arrays (tag : A → G) (hi : Injective tag) (q : Law G) |
| (hq : ∀ x, 0 < q x) (c : G → ℝ) (hc : ∀ x, 0 ≤ c x) (hc0 : c 0=0) |
| (Q : Law Θ) (χ : Θ → AddChar G ℂ) (hn : ∀ t x, ‖χ t x‖=1) |
| (hrq : ∀ g, (↑(exp (-c g)):ℂ)=Q.Cx fun t => χ t g) |
| (hlog : ∀ g, -log (q g) = -log (q 0) + c g) |
| (F L C : ℝ) (hF : 1 ≤ F) (hL : 0 ≤ L) |
| (T N m : ℕ) (hm : 0 < m) (hN : m ≤ N) (hT : 3*N ≤ T) (S : Schedule) |
| (l u r : ℕ → ℕ) |
| (hl : ∀ i, i < S.size → l i ≤ u i ∧ u i ≤ m ∧ 0 < u i ∧ 0 < r i ∧ |
| 2*r i*l i ≤ N ∧ L ≤ (r i:ℝ)*(u i:ℝ)) |
| (hp : ∀ i j, i < j → j < S.size → u j ≤ l i ∧ r i ≤ r j) |
| (hg : ∀ P : Law A, exp (-F) ≤ heatMoment (Fin T) P tag c) |
| (hC : 10*F+2*(-log (q 0))+1 ≤ C) |
| (hh : ∀ i, i < S.size → 0 < l i) |
| (hs : ∀ i, i < S.size → |
| 2*m*√F/N*lnS (Fin (u i)) tag ≤ F ∧ |
| log (Fintype.card (Ticket (r i) (2*r i))) + r i + |
| 2*(2*m*√F/N)*lnS (Fin (r i*l i)) tag ≤ 6*F) : |
| ∃ (a : A) (s : Finset A), s.Nonempty ∧ |
| log (Fintype.card A)-log (s.card:ℝ) ≤ |
| 8*(3*(F/N)+S.loss*C*C/F)+2 ∧ small_defects tag χ Q L s a (350*F) := by |
| have hFp : 0 < F := by linarith |
| have hCp : 0 < C := by |
| have hh := Real.log_nonpos (q.nonneg 0) (q.le_one 0) |
| linarith |
| have hNp : (0:ℝ)< N := by |
| exact_mod_cast (Nat.lt_of_lt_of_le hm hN) |
| obtain ⟨p,R,h1,h2,h3,h4,h5⟩ := select_context tag c hc T N m hm hN hT F hFp hg |
| let ε := 2*m*√F/N |
| let x : Starting tag c N m (3*F) ε := ⟨p,R,h1,h2,h4,hN,h5⟩ |
| let W := initial p |
| have ha (i) (hh : i < S.size) : fundG tag W (u i) ≤ C := by |
| have H := x.G_upper hc q hq (-log (q 0)) hlog _ (hl i hh).2.1 |
| apply H.trans |
| unfold ε |
| have hj := Real.log_le_sub_one_of_pos (show (0:ℝ)<2 by norm_num) |
| linarith [(hs i hh).1] |
| have hb (i) (hh : i < S.size) (hle : 1 ≤ l i) : fundI tag W (r i) ≤ C := by |
| refine (x.I_upper hc q hq (-log (q 0)) hlog _ ?_).trans ?_ |
| · have := (hl i hh).2.2.2.2.1 |
| nlinarith |
| linarith |
|
|
| obtain ⟨P,hv,hW,i,hi',hK⟩ := |
| reach tag hi c hc W C F hCp hFp S m l u r hm (fun i h => ⟨(hl i h).1,(hl i h).2.1⟩) |
| hp ha (fun i h => hb i h (hh i h)) |
| obtain ⟨hi1,hi2,hu',hr',hn',hL'⟩ := hl i hi' |
| have HK : fundK tag c P (u i) (r i) ≤ 10*F := by |
| have H := x.K_upper hc hc0 (l i) (r i) hr' (le_trans hi1 hi2) hn' |
| have HH := K_anti tag c hv (l i) (r i) |
| change fundK tag c W _ _ ≤ _ at H |
| dsimp only [ε] at H |
| linarith [(hs i hi').2] |
| let D := 3*(F/N)+S.loss*C*C/F |
| have hD : 0 < D := by unfold D; have := S.loss_pos; positivity |
| have Hd : log (Fintype.card A) - fundG tag P 1 ≤ D := by |
| rw [show fundG tag W 1 = _ from init_gap tag hi p] at hW |
| linarith |
| obtain ⟨p',z,hz,hz'⟩ := freeze tag hi c hc P (u i) (r i) hu' hr' (10*F) D |
| (by positivity) hD HK Hd |
| have : NeZero (r i) := ⟨hr'.ne'⟩ |
| have : NeZero (u i) := ⟨hu'.ne'⟩ |
| have H0 := alignment tag c Q χ hn hrq p' z (10*F) D L (by linarith) hD hL |
| (by simpa only [Fintype.card_prod,Fintype.card_fin,Nat.cast_mul]) hz (by |
| simpa only [Fintype.card_prod,Fintype.card_fin,Nat.cast_mul]) |
| rwa [show 350*F = 35*(10*F) by ring] |
| end FourierLLogL.Compress |
|
|
| end OAI |
|
|
