| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology |
| namespace HoneycombLocal |
|
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| lemma norm_iteratedDeriv_charFun_le {μ : Measure ℝ} [IsFiniteMeasure μ] |
| {n : ℕ} (h : MemLp id n μ) (t : ℝ) : |
| ‖iteratedDeriv n (charFun μ) t‖ ≤ ∫ x, ‖x‖ ^ n ∂μ := by |
| rw [iteratedDeriv_charFun h, norm_mul, norm_pow, Complex.norm_I, one_pow, one_mul] |
| refine (norm_integral_le_integral_norm _).trans_eq ?_ |
| congr 1 |
| ext x |
| simp only [norm_mul, norm_pow, Complex.norm_real] |
| rw [← Complex.ofReal_mul, Complex.norm_exp_ofReal_mul_I, mul_one] |
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| lemma taylorWithinEval_eq_univ {f : ℝ → ℂ} {t : ℝ} (ht : 0 < t) |
| (hf : ContDiff ℝ 3 f) : |
| taylorWithinEval f 2 (Icc 0 t) 0 t = taylorWithinEval f 2 univ 0 t := by |
| simp only [taylor_within_apply] |
| apply Finset.sum_congr rfl |
| intro k hk |
| have hk3 : k ≤ 3 := by have := Finset.mem_range.mp hk; omega |
| have hd : ContDiffAt ℝ k f 0 := hf.contDiffAt.of_le (by exact_mod_cast hk3) |
| rw [iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_Icc ht) hd ⟨le_rfl, ht.le⟩, |
| iteratedDerivWithin_eq_iteratedDeriv uniqueDiffOn_univ hd (mem_univ 0)] |
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| lemma taylor_charFun_bound_nonneg {μ : Measure ℝ} [IsProbabilityMeasure μ] |
| (h3 : MemLp id 3 μ) (hmean : ∫ x, x ∂μ = 0) {t : ℝ} (ht : 0 ≤ t) : |
| ‖charFun μ t - (1 - ((∫ x, x ^ 2 ∂μ : ℝ) : ℂ) * (t : ℂ) ^ 2 / 2)‖ ≤ |
| (∫ x, ‖x‖ ^ 3 ∂μ) * t ^ 3 / 2 := by |
| rcases ht.eq_or_lt with rfl | ht |
| · simp |
| have hc : ContDiff ℝ 3 (charFun μ) := contDiff_charFun h3 |
| have h2 : MemLp id 2 μ := h3.mono_exponent (by norm_num) |
| have h := taylor_mean_remainder_bound (n := 2) (a := 0) (b := t) |
| (C := ∫ x, ‖x‖ ^ 3 ∂μ) ht.le hc.contDiffOn ⟨ht.le, le_rfl⟩ (by |
| intro y hy |
| rw [iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_Icc ht) hc.contDiffAt hy] |
| exact norm_iteratedDeriv_charFun_le h3 y) |
| rw [taylorWithinEval_eq_univ ht hc] at h |
| have htaylor := taylorWithinEval_charFun_two_zero (P := μ) (X := id) |
| aemeasurable_id (by simpa using h2) t |
| simp only [Measure.map_id, id_eq, Pi.pow_apply, hmean, Complex.ofReal_zero, |
| zero_mul, add_zero] at htaylor |
| rw [htaylor] at h |
| simpa using h |
|
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| lemma taylor_charFun_bound {μ : Measure ℝ} [IsProbabilityMeasure μ] |
| (h3 : MemLp id 3 μ) (hmean : ∫ x, x ∂μ = 0) (t : ℝ) : |
| ‖charFun μ t - (1 - ((∫ x, x ^ 2 ∂μ : ℝ) : ℂ) * (t : ℂ) ^ 2 / 2)‖ ≤ |
| (∫ x, ‖x‖ ^ 3 ∂μ) * |t| ^ 3 / 2 := by |
| by_cases ht : 0 ≤ t |
| · simpa [abs_of_nonneg ht] using taylor_charFun_bound_nonneg h3 hmean ht |
| · have ht' : 0 ≤ -t := by linarith |
| have h := taylor_charFun_bound_nonneg h3 hmean ht' |
| have heq : charFun μ (-t) - |
| (1 - ((∫ x, x ^ 2 ∂μ : ℝ) : ℂ) * ((-t : ℝ) : ℂ) ^ 2 / 2) = |
| star (charFun μ t - (1 - ((∫ x, x ^ 2 ∂μ : ℝ) : ℂ) * (t : ℂ) ^ 2 / 2)) := by |
| simp [charFun_neg] |
| rw [heq, norm_star] at h |
| simpa [abs_of_neg (lt_of_not_ge ht)] using h |
|
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| lemma norm_pow_difference {z w : ℂ} (hz : ‖z‖ ≤ 1) (hw : ‖w‖ ≤ 1) (k : ℕ) : |
| ‖z ^ k - w ^ k‖ ≤ k * ‖z - w‖ := by |
| induction k with |
| | zero => simp |
| | succ k hk => |
| calc |
| ‖z ^ (k + 1) - w ^ (k + 1)‖ = |
| ‖z * (z ^ k - w ^ k) + (z - w) * w ^ k‖ := by congr 1; ring |
| _ ≤ ‖z * (z ^ k - w ^ k)‖ + ‖(z - w) * w ^ k‖ := norm_add_le _ _ |
| _ = ‖z‖ * ‖z ^ k - w ^ k‖ + ‖z - w‖ * ‖w‖ ^ k := by simp [norm_pow] |
| _ ≤ 1 * (k * ‖z - w‖) + ‖z - w‖ * 1 := by |
| gcongr |
| exact pow_le_one₀ (norm_nonneg _) hw |
| _ = (k + 1 : ℕ) * ‖z - w‖ := by push_cast; ring |
|
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| lemma exp_neg_error_le {y : ℝ} (hy : 0 ≤ y) (hy1 : y ≤ 1) : |
| ‖Complex.exp (-(y : ℂ)) - (1 - (y : ℂ))‖ ≤ y ^ 2 := by |
| have h := Complex.norm_exp_sub_one_sub_id_le |
| (x := -(y : ℂ)) (by simpa [abs_of_nonneg hy] using hy1) |
| convert h using 1 |
| · congr 1; ring |
| · simp [abs_of_nonneg hy] |
|
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| lemma gaussian_power_error {μ : Measure ℝ} [IsProbabilityMeasure μ] |
| (h3 : MemLp id 3 μ) (hmean : ∫ x, x ∂μ = 0) (k : ℕ) (t : ℝ) |
| (hsmall : (∫ x, x ^ 2 ∂μ) * t ^ 2 / 2 ≤ 1) : |
| ‖charFun μ t ^ k - Complex.exp (-((k : ℂ) * |
| ((∫ x, x ^ 2 ∂μ : ℝ) : ℂ) * (t : ℂ) ^ 2 / 2))‖ ≤ |
| k * ((∫ x, ‖x‖ ^ 3 ∂μ) * |t| ^ 3 / 2 + |
| ((∫ x, x ^ 2 ∂μ) * t ^ 2 / 2) ^ 2) := by |
| let v : ℝ := ∫ x, x ^ 2 ∂μ |
| let y : ℝ := v * t ^ 2 / 2 |
| have hv : 0 ≤ v := integral_nonneg (fun x => sq_nonneg x) |
| have hy : 0 ≤ y := by dsimp [y]; positivity |
| have hnorm : ‖Complex.exp (-(y : ℂ))‖ ≤ 1 := by |
| rw [← Complex.ofReal_neg, Complex.norm_exp_ofReal] |
| exact Real.exp_le_one_iff.mpr (neg_nonpos.mpr hy) |
| have hdiff : ‖charFun μ t - Complex.exp (-(y : ℂ))‖ ≤ |
| (∫ x, ‖x‖ ^ 3 ∂μ) * |t| ^ 3 / 2 + y ^ 2 := by |
| refine (norm_sub_le_norm_sub_add_norm_sub (charFun μ t) (1 - (y : ℂ)) |
| (Complex.exp (-(y : ℂ)))).trans ?_ |
| apply add_le_add |
| · simpa [y, v] using taylor_charFun_bound h3 hmean t |
| · rw [norm_sub_rev] |
| exact exp_neg_error_le hy hsmall |
| have hpow := (norm_pow_difference (norm_charFun_le_one t) hnorm k).trans |
| (mul_le_mul_of_nonneg_left hdiff (Nat.cast_nonneg k)) |
| have heq : Complex.exp (-(y : ℂ)) ^ k = |
| Complex.exp (-((k : ℂ) * (v : ℂ) * (t : ℂ) ^ 2 / 2)) := by |
| rw [← Complex.exp_nat_mul] |
| congr 1 |
| simp only [y, Complex.ofReal_div, Complex.ofReal_mul, Complex.ofReal_pow, |
| Complex.ofReal_ofNat] |
| ring |
| rw [heq] at hpow |
| exact hpow |
|
|
| lemma normalized_gaussian_power_error {μ : Measure ℝ} [IsProbabilityMeasure μ] |
| (h3 : MemLp id 3 μ) (hmean : ∫ x, x ∂μ = 0) {k : ℕ} (hk : 0 < k) |
| {σ : ℝ} (hσ : 0 < σ) (hvariance : σ ^ 2 = k * ∫ x, x ^ 2 ∂μ) |
| (t : ℝ) (hsmall : t ^ 2 ≤ 2 * k) : |
| ‖charFun μ (t / σ) ^ k - Complex.exp (-((t : ℂ) ^ 2 / 2))‖ ≤ |
| ((k : ℝ) * (∫ x, ‖x‖ ^ 3 ∂μ) / σ ^ 3) * |t| ^ 3 / 2 + |
| t ^ 4 / (4 * k) := by |
| have hk' : (0 : ℝ) < k := by exact_mod_cast hk |
| have heq : (∫ x, x ^ 2 ∂μ) * (t / σ) ^ 2 / 2 = t ^ 2 / (2 * k) := by |
| field_simp |
| nlinarith [hvariance] |
| have h := gaussian_power_error h3 hmean k (t / σ) (by |
| rw [heq]; exact (div_le_one (by positivity)).2 hsmall) |
| have hexp : -((k : ℂ) * ((∫ x, x ^ 2 ∂μ : ℝ) : ℂ) * |
| ((t / σ : ℝ) : ℂ) ^ 2 / 2) = -((t : ℂ) ^ 2 / 2) := by |
| have hvr : (∫ x, x ^ 2 ∂μ) = σ ^ 2 / k := by |
| exact (eq_div_iff (ne_of_gt hk')).2 (by simpa [mul_comm] using hvariance.symm) |
| rw [hvr] |
| push_cast |
| have hkC : (k : ℂ) ≠ 0 := by exact_mod_cast hk.ne' |
| have hσC : (σ : ℂ) ≠ 0 := by exact_mod_cast hσ.ne' |
| field_simp |
| rw [hexp, heq] at h |
| convert h using 1 |
| rw [abs_div, abs_of_pos hσ] |
| field_simp |
| ring |
|
|
| end HoneycombLocal |
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| end |
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| end OAI |
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