| import Mathlib |
| import OAI.Geometry.HyperbolicGroups.Moment |
| import OAI.Geometry.HyperbolicGroups.Random |
|
|
| namespace OAI |
|
|
| section |
| section |
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|
|
| noncomputable section |
| open scoped BigOperators |
| namespace HyperbolicObstruction |
| namespace FiniteLaw |
| variable {α : Type*} [Fintype α] |
|
|
| lemma prob_mono (μ : FiniteLaw α) {E F : α → Prop} [DecidablePred E] [DecidablePred F] |
| (h : ∀ a, E a → F a) : μ.prob E ≤ μ.prob F := by |
| apply μ.expect_mono |
| intro a |
| by_cases he : E a |
| · simp [he,h a he] |
| · simp [he] |
| split_ifs <;> norm_num |
|
|
| lemma prob_or_le (μ : FiniteLaw α) (E F : α → Prop) [DecidablePred E] [DecidablePred F] : |
| μ.prob (fun a => E a ∨ F a) ≤ μ.prob E + μ.prob F := by |
| rw [prob, prob, prob, ← μ.expect_add] |
| apply μ.expect_mono |
| intro a |
| by_cases he : E a <;> by_cases hf : F a <;> simp [he,hf] |
|
|
| lemma bernoulli_centered_exp_le {p t : ℝ} (hp : p ∈ Set.Icc (0 : ℝ) 1) (ht : |t| ≤ 1) : |
| (1-p)*Real.exp (t*(0-p)) + p*Real.exp (t*(1-p)) ≤ Real.exp (p*t^2) := by |
| have he : Real.exp t - 1 - t ≤ t^2 := |
| (le_abs_self _).trans (Real.abs_exp_sub_one_sub_id_le ht) |
| calc |
| _ = Real.exp (-t*p) * (1+p*(Real.exp t-1)) := by |
| rw [show t*(1-p) = -t*p+t by ring, show t*(0-p) = -t*p by ring, Real.exp_add] |
| ring |
| _ ≤ Real.exp (-t*p) * Real.exp (p*(Real.exp t-1)) := by |
| apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le |
| have hh := Real.add_one_le_exp (p*(Real.exp t-1)) |
| linarith |
| _ = Real.exp (p*(Real.exp t-1-t)) := by rw [← Real.exp_add]; congr 1; ring |
| _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left he hp.1) |
|
|
| / |
| other table entries are integrated out, not made dependent. -/ |
| def centeredCount {I : Type*} [Fintype I] (p : I → ℝ) (S : Finset I) (w : I → Bool) : ℝ := |
| ∑ i ∈ S, ((if w i then (1:ℝ) else 0) - p i) |
|
|
| lemma centeredCount_mgf {I : Type*} [Fintype I] [DecidableEq I] |
| (p : I → ℝ) (hp : ∀ i, p i ∈ Set.Icc (0 : ℝ) 1) (S : Finset I) {t : ℝ} (ht : |t| ≤ 1) : |
| (bernoulliProduct p hp).expect (fun w => Real.exp (t*centeredCount p S w)) ≤ |
| Real.exp ((∑ i ∈ S, p i)*t^2) := by |
| classical |
| have hx (w : I → Bool) : Real.exp (t*centeredCount p S w) = |
| ∏ i, Real.exp (t*(if i ∈ S then (if w i then (1:ℝ) else 0)-p i else 0)) := by |
| rw [← Real.exp_sum] |
| congr 1 |
| simp only [centeredCount,Finset.mul_sum,mul_ite,mul_zero] |
| simp only [← Finset.sum_filter,Finset.filter_mem_eq_inter,Finset.univ_inter] |
| simp_rw [hx] |
| rw [bernoulliProduct_expect_product p hp |
| (fun i (b : Bool) => Real.exp (t*(if i ∈ S then (if b then (1:ℝ) else 0)-p i else 0)))] |
| calc |
| _ ≤ ∏ i, Real.exp (if i ∈ S then p i*t^2 else 0) := by |
| apply Finset.prod_le_prod₀ |
| · intro i _ |
| exact add_nonneg (mul_nonneg (sub_nonneg.mpr (hp i).2) (Real.exp_pos _).le) |
| (mul_nonneg (hp i).1 (Real.exp_pos _).le) |
| · intro i _ |
| by_cases hi : i ∈ S |
| · simpa [hi] using bernoulli_centered_exp_le (hp i) ht |
| · simp [hi] |
| _ = _ := by |
| rw [← Real.exp_sum] |
| congr 1 |
| simp [Finset.sum_mul] |
|
|
| / |
| It suffices for the simultaneous six-letter pair counts. -/ |
| theorem centeredCount_chernoff {I : Type*} [Fintype I] [DecidableEq I] |
| (p : I → ℝ) (hp : ∀ i, p i ∈ Set.Icc (0 : ℝ) 1) (S : Finset I) |
| {ε : ℝ} (hε : 0 < ε) (hε2 : ε ≤ 2) : |
| (bernoulliProduct p hp).prob (fun w => ε*(∑ i ∈ S, p i) ≤ |centeredCount p S w|) ≤ |
| 2*Real.exp (-ε^2*(∑ i ∈ S, p i)/4) := by |
| classical |
| let μ := bernoulliProduct p hp |
| let m := ∑ i ∈ S, p i |
| let t := ε/2 |
| have ht : 0 < t := half_pos hε |
| have ht1 : t ≤ 1 := by dsimp [t]; linarith |
| have htail (s : ℝ) (hs : |s| = t) : |
| μ.prob (fun w => t*ε*m ≤ s*centeredCount p S w) ≤ Real.exp (-ε^2*m/4) := by |
| have hh := μ.markov (fun w => Real.exp (s*centeredCount p S w)) |
| (fun _ => (Real.exp_pos _).le) (Real.exp_pos (t*ε*m)) |
| have hm := centeredCount_mgf p hp S (show |s| ≤ 1 by rw [hs]; exact ht1) |
| have hsq : s^2 = t^2 := by rw [← sq_abs,hs] |
| have he : μ.prob (fun w => t*ε*m ≤ s*centeredCount p S w) = |
| μ.prob (fun w => Real.exp (t*ε*m) ≤ Real.exp (s*centeredCount p S w)) := by |
| congr 1; funext w; exact propext Real.exp_le_exp.symm |
| rw [he] |
| apply hh.trans |
| apply (div_le_div_of_nonneg_right hm (Real.exp_pos _).le).trans |
| rw [← Real.exp_sub,hsq] |
| apply le_of_eq |
| congr 1 |
| dsimp [t,m] |
| ring |
| have hu := htail t (abs_of_pos ht) |
| have hl := htail (-t) (by rw [abs_neg,abs_of_pos ht]) |
| have hh : μ.prob (fun w => ε*m ≤ |centeredCount p S w|) ≤ |
| μ.prob (fun w => t*ε*m ≤ t*centeredCount p S w ∨ t*ε*m ≤ -t*centeredCount p S w) := by |
| apply μ.prob_mono |
| intro w hw |
| rcases le_abs.mp hw with h | h |
| · left; nlinarith |
| · right; nlinarith |
| exact hh.trans ((μ.prob_or_le _ _).trans (by linarith)) |
|
|
| end FiniteLaw |
| end HyperbolicObstruction |
| end |
| end |
|
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| section |
|
|
| end |
| end |
|
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| end OAI |
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