File size: 7,977 Bytes
d95525c | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import OAI.Analysis.FourierLLogL.Compress.Prefix
/-! # Selecting conditional iid comparator -/
open Set Function Filter
open scoped BigOperators
noncomputable section
namespace OAI
namespace FourierLLogL.Compress
open Real Law
private lemma choose_slow {a : ℕ → ℝ} (ha : Monotone a)
(h0 : ∀ j, 0 ≤ a j) {E : ℝ} (hE : 0 ≤ E) (T N m : ℕ)
(hN : 0 < N) (hm : m ≤ N) (hT : 3*N ≤ T)
(hh : ∑ j ∈ Finset.range T, a j ≤ E) :
∃ j, j < N ∧ a j ≤ E/N ∧
∑ i ∈ Finset.range m, (a (j+i)-a j) ≤ m^2*E/(N:ℝ)^2 := by
have hp : 0 < (N:ℝ) := by exact_mod_cast hN
let D := E/N
have hs (l : ℕ) (hl : l ≤ 2*N) : a l ≤ D := by
have Hlt : ∑ j ∈ Finset.range (l+N), a j ≤ E :=
le_trans (Finset.sum_le_sum_of_subset_of_nonneg
(Finset.range_mono (by omega)) (fun i _ _ => h0 i)) hh
rw [Finset.sum_range_add] at Hlt
have H : ∑ _ ∈ Finset.range N, a l ≤ ∑ i ∈ Finset.range N, a (l+i) :=
Finset.sum_le_sum fun i _ => ha (by omega : l ≤ l+i)
simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul] at H
apply (le_div_iff₀ hp).mpr
have : 0 ≤ ∑ j ∈ Finset.range l, a j := Finset.sum_nonneg fun j _ => h0 j
linarith
have HD : 0 ≤ D := div_nonneg hE hp.le
have hr (i : ℕ) (hi : i ≤ m) :
∑ j ∈ Finset.range N, (a (j+i)-a j) ≤ m*D := by
have h1 := Finset.sum_range_add a N i
have h2 := Finset.sum_range_add a i N
have hcmp : ∑ j ∈ Finset.range i, a (N+j) ≤ ∑ j ∈ Finset.range i, D := by
apply Finset.sum_le_sum
intro j hj
apply hs; rw [Finset.mem_range] at hj; omega
simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul] at hcmp
have hl : 0 ≤ ∑ j ∈ Finset.range i, a j := Finset.sum_nonneg fun j _ => h0 j
have hle : (i:ℝ)*D ≤ (m:ℝ)*D := by gcongr
rw [show i+N=N+i by omega] at h2
simp_rw [show ∀ j, j+i=i+j from fun j => add_comm _ _, Finset.sum_sub_distrib]
linarith
let f j := ∑ i ∈ Finset.range m, (a (j+i)-a j)
have hAvg : ∑ j ∈ Finset.range N, f j ≤ ∑ i ∈ Finset.range N, ((m:ℝ)^2*E/(N:ℝ)^2) := by
unfold f
rw [Finset.sum_comm]
calc
_ ≤ ∑ i ∈ Finset.range m, ((m:ℝ)*D) :=
Finset.sum_le_sum fun i hi => hr i (Finset.mem_range.mp hi).le
_ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul,D]; field_simp
by_contra! hy
apply (not_lt_of_ge hAvg)
refine Finset.sum_lt_sum_of_nonempty ⟨0,Finset.mem_range.mpr hN⟩ (fun j hj => ?_)
have hj := Finset.mem_range.mp hj
exact hy _ hj (hs _ (by omega))
variable {α G Ω : Type} [Fintype α] [Fintype G] [Fintype Ω] [AddCommGroup G]
omit [AddCommGroup G] in
private lemma choose3 (P : Law Ω) (a b c : Ω → ℝ)
(ha : ∀ x, 0 ≤ a x) (hb : ∀ x, 0 ≤ b x) (hc : ∀ x, 0 ≤ c x)
(A B C : ℝ) (hA : 0 < A) (hB : 0 < B) (hC : 0 < C)
(h1 : P.Ex a ≤ A) (h2 : P.Ex b ≤ B) (h3 : P.Ex c ≤ C) :
∃ x, 0 < P x ∧ a x ≤ 3*A ∧ b x ≤ 3*B ∧ c x ≤ 3*C := by
obtain ⟨x,hx,hh⟩ := P.exists_le_Ex fun x => a x*A⁻¹+b x*B⁻¹+c x*C⁻¹
have H : a x/A+b x/B+c x/C ≤ 3 := by
simp_rw [Ex_add,Ex_mul, ← div_eq_mul_inv] at hh
linarith [(div_le_one hA).mpr h1,(div_le_one hB).mpr h2,(div_le_one hC).mpr h3]
have hI := div_nonneg (ha x) hA.le
have hJ := div_nonneg (hb x) hB.le
have hK := div_nonneg (hc x) hC.le
exact ⟨x,hx,(div_le_iff₀ hA).mp (by linarith),(div_le_iff₀ hB).mp
(by linarith),(div_le_iff₀ hC).mp (by linarith)⟩
lemma kl_covered {ι : Type} [Fintype ι] [DecidableEq ι]
(P : Law Ω) (X : Ω → ι → α) (q : Law α) (h : ∀ i, P.send (fun a => X a i) = q) :
covered P X q := by
intro a ha i
rw [← h i]
exact P.send_pos (fun a => X a i) ha
def iid (p : Law α) (n : ℕ) := pack (fun _ : Fin n => p)
theorem select_context (tag : α → G) (c : G → ℝ) (hc : ∀ x, 0 ≤ c x)
(T N m : ℕ) (hm0 : 0 < m) (hm : m ≤ N) (hT : 3*N ≤ T)
(E : ℝ) (hE : 0 < E) [Nonempty α]
(h : ∀ p : Law α, exp (-E) ≤ heatMoment (Fin T) p tag c) :
∃ (p : Law α) (Ψ : Law (WordSample N α G)), exch Ψ ∧
(∀ i : Fin N, Ψ.send (fun a => a.1 i) = p) ∧
log (Fintype.card α) - p.entropy ≤ 3*(E/N) ∧
bill tag c p Ψ Prod.fst Prod.snd ≤ 3*E ∧
tv (Ψ.send (prefixB Prod.fst m hm)) (iid p m) ≤ 2*m*√E/N := by
obtain ⟨P,hp,hx,hu⟩ := noisy_symm tag c E h
let X : WordSample T α G → Letters T α := Prod.fst
let L := log (Fintype.card α)
let a j := L-rateH P X j
have h0 : 0 < N := by omega
have h1 (j : ℕ) : 0 ≤ a j := sub_nonneg.mpr (rate_bdd P X j)
let u := uniform α
have hcov : covered P X u := kl_covered P X u hu
have heq : ∑ j ∈ Finset.range T, a j ≤ E := by
have H := kl_le_bill hcov tag c hc Prod.snd
dsimp only [a,L]
rw [sum_deficit]
exact le_trans (le_trans
(le_add_of_nonneg_right (P.Ex_nonneg fun x => hc (x.2))) H) hp
obtain ⟨j,hj,hb,hj'⟩ := choose_slow (a:=a) (fun x y hh =>
sub_le_sub_left (rate_antitone hx hh) _) h1 hE.le T N m h0 hm hT heq
let W := prefixB X j (by omega)
let V := readBlock X j N (by omega)
let Y := readBlock X j m (by omega)
let R (w) := P.after W w
let H := readLetter X j (by omega)
let ν (w) := (R w).send H
let Pw := P.send W
let b (w) := L-(ν w).entropy
let k (w) := bill tag c u (R w) V Prod.snd
let d (w) := kl ((R w).send Y) (pack (fun _ : Fin m => ν w))
have Hb (w : Letters j α) : 0 ≤ b w :=
sub_nonneg.mpr (entropy_le_log_card _)
have Hk (w : Letters j α) : 0 ≤ k w := by
apply bill_nonneg _ tag c hc
intro a _ _
rw [uniform_apply]
positivity
have hv (w) (i : Fin m) :
(R w).send (fun x => Y x i) = ν w := conditional_block_marg hx (j:=j) (m:=m) (by omega) w i
have hv' (w) (i : Fin N) :
(R w).send (fun x => V x i) = ν w := conditional_block_marg hx (j:=j) (m:=N) (by omega) w i
have Hd (w : Letters j α) : 0 ≤ d w := kl_nonneg
(covered_pack (kl_covered _ _ _ (hv w)))
have Hm' : (0:ℝ) < m := by exact_mod_cast hm0
have HN : (0:ℝ) < N := by exact_mod_cast h0
have H1 : Pw.Ex b ≤ E/N := by
unfold b ν Pw
rw [Ex_sub,Ex_const]
exact le_trans (le_of_eq (by
dsimp only [a]
rw [rate_lt P X (show j<T by omega),Hcond_after]; rfl)) hb
have H2 : Pw.Ex k ≤ E := le_trans
(prefix_bill tag c u P X _ hcov j N (by omega)) hp
have H3 : Pw.Ex d ≤ (m:ℝ)^2*E/N^2 := by
unfold Pw d
have HH := avg_block_kl hx (j:=j) (m:=m) (by omega) (by omega)
apply le_trans (b:=∑ i ∈ Finset.range m, (rateH P X j - rateH P X (j+i))) HH.le
apply le_trans (b:=∑ i ∈ Finset.range m, (a (j+i)-a j)) ?_ hj'
apply Finset.sum_le_sum; intro i _; dsimp only [a]; linarith
obtain ⟨w,hw,h1,h2,h3⟩ := choose3 Pw b k d Hb Hk Hd (E/N) E ((m:ℝ)^2*E/N^2)
(by positivity) hE (by positivity) H1 H2 H3
let Ψ := (R w).send (fun x => (V x,x.2))
have hmx := conditional_block_exch hx (j:=j) (m:=N) (by omega) w
refine ⟨ν w,Ψ,hmx,fun i => ?_,h1,?_,?_⟩
· dsimp only [Ψ]
rw [send_send]; exact hv' w i
· dsimp only [Ψ]
rw [bill_send]
exact (bill_change (covered_subfamily (covered_after hcov W w hw) (offset j N (by omega)))
tag c Prod.snd (ν w) (hv' w)).trans h2
· have hh : Ψ.send (prefixB Prod.fst m hm) = (R w).send Y := by
unfold Ψ; rw [send_send, show Y = _ from block_cut X j N m (by omega) hm]; rfl
rw [hh]
let v := tv ((R w).send Y) (iid (ν w) m)
have ht : v^2 ≤ 3*((m:ℝ)^2*E/N^2) :=
(tv_sq_le_kl (covered_pack (kl_covered _ _ _ (hv w)))).trans h3
have h4 : 3*((m:ℝ)^2*E/N^2) ≤ (2*m*√E/N)^2 := by
rw [div_pow,mul_pow,mul_pow, sq_sqrt hE.le]
have H := pow_pos HN 2
field_simp; nlinarith
change v ≤ _
nlinarith [ht.trans h4,sqrt_nonneg E,show 0 ≤ 2*m*√E/N by positivity]
end FourierLLogL.Compress
end OAI
|