| import OAI.Combinatorics.BooleanFunctions.SmoothGaussian |
| import OAI.Combinatorics.BooleanFunctions.SmoothSelector |
|
|
| namespace OAI |
|
|
| open scoped BigOperators Topology NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Real |
|
|
| noncomputable section |
|
|
| namespace SquareRootDegree |
| namespace GaussianGain |
| open SmoothGaussian SmoothSelector |
|
|
| theorem gaussian_sq_integral (v : ℝ≥0) : |
| (∫ x, x^2 ∂gaussianReal 0 v) = v := by |
| have h := variance_fun_id_gaussianReal (μ := 0) (v := v) |
| rw [variance_eq_integral (show AEMeasurable (fun x : ℝ => x) (gaussianReal 0 v) from measurable_id.aemeasurable)] at h |
| simpa using h |
|
|
| theorem gaussian_sq_integrable (v : ℝ≥0) : Integrable (fun x : ℝ => x^2) (gaussianReal 0 v) := by |
| simpa using (memLp_id_gaussianReal (μ := 0) (v := v) 2).integrable_sq |
|
|
| theorem gaussian_poly_integrable (v : ℝ≥0) : |
| Integrable (fun x : ℝ => 1+x^2) (gaussianReal 0 v) := |
| (integrable_const 1).add (gaussian_sq_integrable v) |
|
|
| theorem density_div_mass_bound {a h δ R t : ℝ} (hh : 0 < h) |
| (hδh : δ+h ≤ 1) (ha : |a-(1+t)| ≤ δ) (ht : |t| ≤ R+1) : |
| gaussianPDFReal 0 1 t / mass (gaussianReal 0 1) (tent a h) ≤ |
| 2/h * Real.exp (2*R+4) := by |
| have hb := tent_mass_pos (a := a) hh |
| have hA : |a|+h ≤ |t|+2 := by |
| have he : |a| ≤ |a-(1+t)|+|1+t| := by |
| simpa using abs_add_le (a-(1+t)) (1+t) |
| have he' : |1+t| ≤ 1+|t| := by simpa using abs_add_le (1:ℝ) t |
| linarith |
| have hsq : (|a|+h)^2 ≤ (|t|+2)^2 := by |
| nlinarith [abs_nonneg a, abs_nonneg t] |
| have he : Real.exp (-t^2/2) ≤ Real.exp (-((|a|+h)^2)/2) * Real.exp (2*R+4) := by |
| rw [← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| nlinarith [sq_abs t] |
| apply (div_le_iff₀ hb).mpr |
| have hl := beta_lower (a := a) hh |
| have hp : 0 ≤ 2/h * Real.exp (2*R+4) := by positivity |
| calc |
| gaussianPDFReal 0 1 t ≤ (Real.sqrt (2*Real.pi))⁻¹ * |
| (Real.exp (-((|a|+h)^2)/2) * Real.exp (2*R+4)) := by |
| simpa only [gaussianPDFReal, NNReal.coe_one, mul_one, sub_zero] using |
| mul_le_mul_of_nonneg_left he (inv_nonneg.mpr (Real.sqrt_nonneg _)) |
| _ = (2/h * Real.exp (2*R+4)) * |
| (h/2 * ((Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-((|a|+h)^2)/2))) := by |
| field_simp |
| _ ≤ _ := mul_le_mul_of_nonneg_left hl hp |
|
|
| theorem density_tail_bound {R t : ℝ} (ht : R^2 ≤ t^2) : |
| gaussianPDFReal 0 1 t ≤ 2 * Real.exp (-(R^2/4)) * gaussianPDFReal 0 2 t := by |
| have he : Real.exp (-t^2/2) ≤ Real.exp (-(R^2/4)) * Real.exp (-t^2/4) := by |
| rw [← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| linarith |
| have hs : Real.sqrt (2*Real.pi*2) = Real.sqrt 2 * Real.sqrt (2*Real.pi) := by |
| rw [← Real.sqrt_mul (by norm_num : (0:ℝ) ≤ 2)] |
| congr 1; ring_nf |
| have hs2 : Real.sqrt 2 ≤ (2:ℝ) := by |
| nlinarith [Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2),Real.sqrt_nonneg 2] |
| have hs0 : 0 < Real.sqrt (2*Real.pi) := Real.sqrt_pos.mpr (by positivity) |
| have hc : (Real.sqrt (2*Real.pi))⁻¹ ≤ 2 * (Real.sqrt (2*Real.pi*2))⁻¹ := by |
| rw [hs, mul_inv_rev] |
| have h : (1:ℝ) ≤ 2 * (Real.sqrt 2)⁻¹ := by |
| rw [← div_eq_mul_inv, le_div_iff₀ (Real.sqrt_pos.mpr (by norm_num : (0:ℝ)<2))] |
| simpa only [one_mul] using hs2 |
| nlinarith [mul_nonneg (inv_nonneg.mpr hs0.le) (sub_nonneg.mpr h)] |
| simp only [gaussianPDFReal, NNReal.coe_one, NNReal.coe_ofNat, mul_one, sub_zero] |
| calc |
| _ ≤ (2*(Real.sqrt (2*Real.pi*2))⁻¹) * |
| (Real.exp (-(R^2/4)) * Real.exp (-t^2/4)) := |
| mul_le_mul hc he (by positivity) (by positivity) |
| _ = _ := by ring_nf |
|
|
| def centralKernel (N : ℕ) (δ : ℝ) (μ v β : Fin (2*N+1) → ℝ) (t : ℝ) : ℝ := |
| ∑ i, hat N δ i t * ((1+t-μ i)^2+v i) / β i |
|
|
| @[fun_prop] theorem centralKernel_continuous (N : ℕ) (δ : ℝ) |
| (μ v β : Fin (2*N+1) → ℝ) : Continuous (centralKernel N δ μ v β) := by |
| unfold centralKernel |
| fun_prop |
|
|
| theorem centralKernel_support {N : ℕ} {δ : ℝ} (hδ : 0 < δ) |
| (μ v β : Fin (2*N+1) → ℝ) {t : ℝ} (ht : (N+1)*δ < |t|) : |
| centralKernel N δ μ v β t = 0 := by |
| apply Finset.sum_eq_zero |
| intro i _ |
| have hi : hat N δ i t = 0 := by |
| by_contra hn |
| exact (not_le_of_gt ht) (hat_support hδ hn) |
| simp [hi] |
|
|
| theorem centralKernel_compactSupport {N : ℕ} {δ : ℝ} (hδ : 0 < δ) |
| (μ v β : Fin (2*N+1) → ℝ) : HasCompactSupport (centralKernel N δ μ v β) := by |
| apply HasCompactSupport.intro (isCompact_Icc (a := -(N+1)*δ) (b := (N+1)*δ)) |
| intro t ht |
| apply centralKernel_support hδ |
| by_contra hn |
| have hh := abs_le.mp (le_of_not_gt hn) |
| apply ht |
| constructor <;> linarith [hh.1,hh.2] |
|
|
| theorem central_density_bound {N : ℕ} {δ h R M : ℝ} (hδ : 0 < δ) |
| (hh : 0 < h) (hsmall : δ+h ≤ 1) (hR : N*δ = R) |
| (μ v : Fin (2*N+1) → ℝ) |
| (hμ : ∀ i, |μ i-center N δ i| ≤ h) (hv0 : ∀ i, 0 ≤ v i) (hv : ∀ i, v i ≤ M) |
| (t : ℝ) : |
| gaussianPDFReal 0 1 t * |
| centralKernel N δ μ v (fun i => mass (gaussianReal 0 1) (tent (center N δ i) h)) t ≤ |
| (2/h * Real.exp (2*R+4)) * ((δ+h)^2+M) := by |
| let K := 2/h * Real.exp (2*R+4) |
| let B := (δ+h)^2+M |
| have hK : 0 ≤ K := by dsimp [K]; positivity |
| have hM : 0 ≤ M := (hv0 ⟨0,by omega⟩).trans (hv _) |
| have hB : 0 ≤ B := add_nonneg (sq_nonneg _) hM |
| unfold centralKernel |
| rw [Finset.mul_sum] |
| calc |
| ∑ i, gaussianPDFReal 0 1 t * |
| (hat N δ i t * ((1+t-μ i)^2+v i) / |
| mass (gaussianReal 0 1) (tent (center N δ i) h)) ≤ |
| ∑ i, hat N δ i t * (K*B) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases hi : hat N δ i t = 0 |
| · simp [hi] |
| have ha := hat_local hδ hi |
| have ht := hat_support hδ hi |
| have ht' : |t| ≤ R+1 := by rw [add_mul, hR] at ht; nlinarith |
| have he := density_div_mass_bound hh hsmall ha.le ht' |
| have herr : |1+t-μ i| ≤ δ+h := by |
| have hx : |1+t-μ i| ≤ |center N δ i-(1+t)|+|μ i-center N δ i| := by |
| have := abs_add_le (center N δ i-(1+t)) (μ i-center N δ i) |
| simpa only [show center N δ i-(1+t)+(μ i-center N δ i) = -(1+t-μ i) by ring_nf, |
| abs_neg] using this |
| linarith [hμ i] |
| have hnum : (1+t-μ i)^2+v i ≤ B := by |
| dsimp [B] |
| nlinarith [sq_abs (1+t-μ i), abs_nonneg (1+t-μ i), hv i] |
| have hnum0 : 0 ≤ (1+t-μ i)^2+v i := add_nonneg (sq_nonneg _) (hv0 i) |
| calc |
| _ = hat N δ i t * |
| ((gaussianPDFReal 0 1 t / mass (gaussianReal 0 1) (tent (center N δ i) h)) * |
| ((1+t-μ i)^2+v i)) := by ring_nf |
| _ ≤ hat N δ i t * (K*B) := |
| mul_le_mul_of_nonneg_left (mul_le_mul he hnum hnum0 hK) (hat_nonneg ..) |
| _ = (∑ i, hat N δ i t) * (K*B) := (Finset.sum_mul ..).symm |
| _ ≤ K*B := by nlinarith [hats_sum_le_one N δ t, mul_nonneg hK hB] |
|
|
| theorem central_integral_bound {N : ℕ} {δ h R M : ℝ} (hδ : 0 < δ) |
| (hh : 0 < h) (hsmall : δ+h ≤ 1) (hR : N*δ = R) |
| (μ v : Fin (2*N+1) → ℝ) |
| (hμ : ∀ i, |μ i-center N δ i| ≤ h) (hv0 : ∀ i, 0 ≤ v i) (hv : ∀ i, v i ≤ M) : |
| (∫ t, centralKernel N δ μ v |
| (fun i => mass (gaussianReal 0 1) (tent (center N δ i) h)) t ∂gaussianReal 0 1) ≤ |
| 4*(R+δ)/h * Real.exp (2*R+4) * ((δ+h)^2+M) := by |
| let β := fun i => mass (gaussianReal 0 1) (tent (center N δ i) h) |
| let f := fun t => gaussianPDFReal 0 1 t * centralKernel N δ μ v β t |
| have hf : Integrable f volume := by |
| apply Continuous.integrable_of_hasCompactSupport |
| · dsimp [f]; unfold gaussianPDFReal; fun_prop |
| · exact (centralKernel_compactSupport hδ μ v β).mul_left |
| have he : ∫ t in Set.Icc (-(R+δ)) (R+δ), f t = ∫ t, f t := by |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro t ht |
| dsimp [f] |
| rw [centralKernel_support hδ μ v β] |
| · exact mul_zero _ |
| · by_contra hn |
| have hab := abs_le.mp (le_of_not_gt hn) |
| rw [add_mul, hR, one_mul] at hab |
| exact ht ⟨by linarith [hab.1],hab.2⟩ |
| have hbound := integral_mono_ae hf.integrableOn (integrable_const |
| ((2/h * Real.exp (2*R+4))*((δ+h)^2+M))) |
| (ae_of_all (volume.restrict (Set.Icc (-(R+δ)) (R+δ))) |
| (central_density_bound hδ hh hsmall hR μ v hμ hv0 hv)) |
| rw [he] at hbound |
| rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1:ℝ≥0) ≠ 0)] |
| change (∫ t, f t) ≤ _ |
| apply hbound.trans |
| simp only [integral_const, Measure.real, Measure.restrict_apply_univ, Real.volume_Icc, smul_eq_mul] |
| have hn : 0 ≤ R+δ-(-(R+δ)) := by |
| have : 0 ≤ (N:ℝ)*δ := mul_nonneg (Nat.cast_nonneg _) hδ.le |
| rw [hR] at this |
| linarith |
| rw [ENNReal.toReal_ofReal hn] |
| ring_nf |
|
|
| exact le_rfl |
|
|
| theorem density_integrable {v : ℝ≥0} (hv : v ≠ 0) {f : ℝ → ℝ} |
| (hf : Integrable f (gaussianReal 0 v)) : |
| Integrable (fun t => gaussianPDFReal 0 v t * f t) volume := by |
| rw [gaussianReal_of_var_ne_zero 0 hv] at hf |
| have he := (integrable_withDensity_iff_integrable_smul₀' |
| (show AEMeasurable (gaussianPDF 0 v) volume from |
| (show Measurable (gaussianPDF 0 v) from by fun_prop).aemeasurable) |
| (ae_of_all _ (fun _ => gaussianPDF_lt_top))).mp hf |
| simpa only [toReal_gaussianPDF, smul_eq_mul] using he |
|
|
| theorem middle_mass_bound {h : ℝ} (hh : 0 < h) (hh1 : h ≤ 1) : |
| (mass (gaussianReal 0 1) (tent 1 h))⁻¹ ≤ 54/h := by |
| have hb := tent_mass_pos (a := 1) hh |
| have hs : 0 < Real.sqrt (2*Real.pi) := Real.sqrt_pos.mpr (by positivity) |
| have hs3 : Real.sqrt (2*Real.pi) ≤ 3 := by |
| nlinarith [Real.sq_sqrt (by positivity : 0 ≤ 2*Real.pi),Real.pi_lt_four] |
| have hA : (|(1:ℝ)|+h)^2/2 ≤ 2 := by norm_num only [abs_one]; nlinarith |
| have he : Real.exp (-2) ≤ Real.exp (-((|(1:ℝ)|+h)^2)/2) := |
| Real.exp_le_exp.mpr (by linarith) |
| have hl : h/6*Real.exp (-2) ≤ mass (gaussianReal 0 1) (tent 1 h) := by |
| apply le_trans _ (beta_lower hh) |
| have hi : (1:ℝ)/3 ≤ (Real.sqrt (2*Real.pi))⁻¹ := by |
| rw [inv_eq_one_div] |
| exact one_div_le_one_div_of_le hs hs3 |
| have hm := mul_le_mul hi he (Real.exp_pos (-2)).le (inv_nonneg.mpr hs.le) |
| have hp := mul_le_mul_of_nonneg_left hm (show 0 ≤ h/2 by positivity) |
| nlinarith |
| have hE : Real.exp 2 ≤ 9 := by |
| calc Real.exp 2 = (Real.exp 1)^2 := by rw [← Real.exp_nat_mul]; norm_num |
| _ ≤ 3^2 := pow_le_pow_left₀ (Real.exp_pos _).le (le_trans Real.exp_one_lt_d9.le (by norm_num)) _ |
| _ = 9 := by norm_num |
| have hlim : (mass (gaussianReal 0 1) (tent 1 h))⁻¹ ≤ 6/h * Real.exp 2 := by |
| rw [inv_eq_one_div, div_le_iff₀ hb] |
| have heq : Real.exp (-2) * Real.exp 2 = 1 := by rw [← Real.exp_add]; norm_num |
| have hp := mul_le_mul_of_nonneg_right hl (show 0 ≤ 6/h*Real.exp 2 by positivity) |
| calc 1 = (h/6*Real.exp (-2)) * (6/h*Real.exp 2) := by |
| field_simp; nlinarith [heq] |
| _ ≤ _ := by simpa only [mul_comm] using hp |
| apply hlim.trans |
| calc 6/h * Real.exp 2 ≤ 6/h*9 := |
| mul_le_mul_of_nonneg_left hE (show 0 ≤ 6/h by positivity) |
| _ = 54/h := by ring_nf |
|
|
| def residual (N : ℕ) (δ t : ℝ) : ℝ := 1-∑ i, hat N δ i t |
|
|
| theorem residual_nonneg (N : ℕ) (δ t : ℝ) : 0 ≤ residual N δ t := |
| sub_nonneg.mpr (hats_sum_le_one ..) |
|
|
| theorem residual_le_one (N : ℕ) (δ t : ℝ) : residual N δ t ≤ 1 := by |
| have : 0 ≤ ∑ i, hat N δ i t := Finset.sum_nonneg (fun _ _ => hat_nonneg ..) |
| dsimp [residual]; linarith |
|
|
| theorem residual_eq_zero {N : ℕ} {δ t : ℝ} (hδ : 0 < δ) (ht : |t| ≤ N*δ) : |
| residual N δ t = 0 := by simp only [residual, hats_sum_eq_one hδ ht, sub_self] |
|
|
| theorem quadratic_integrable (c v : ℝ) (σ : ℝ≥0) : |
| Integrable (fun t : ℝ => (1+t-c)^2+v) (gaussianReal 0 σ) := by |
| have h0 : Integrable (fun _ : ℝ => (1-c)^2+v) (gaussianReal 0 σ) := integrable_const _ |
| have h1 : Integrable (fun t : ℝ => t) (gaussianReal 0 σ) := |
| (memLp_id_gaussianReal (μ := 0) (v := σ) 1).integrable (by norm_num) |
| have h2 := gaussian_sq_integrable σ |
| convert! (h2.add (h1.const_mul (2*(1-c)))).add h0 using 1 |
| funext t; simp only [Pi.add_apply]; ring_nf |
|
|
| theorem residual_quadratic_integrable {N : ℕ} {δ : ℝ} (hδ : 0 < δ) (c v : ℝ) : |
| Integrable (fun t => residual N δ t *((1+t-c)^2+v)) (gaussianReal 0 1) := by |
| have hi (i : Fin (2*N+1)) : Integrable (fun t => hat N δ i t *((1+t-c)^2+v)) (gaussianReal 0 1) := |
| (by fun_prop : Continuous (fun t => hat N δ i t *((1+t-c)^2+v))).integrable_of_hasCompactSupport |
| ((hat_compactSupport hδ i).mul_right) |
| convert! (quadratic_integrable c v 1).sub (integrable_finsetSum _ (fun i _ => hi i)) using 1 |
| funext t |
| simp only [Pi.sub_apply] |
| dsimp [residual] |
| rw [sub_mul,one_mul,Finset.sum_mul] |
|
|
| theorem tail_integral_bound {N : ℕ} {δ h R c v : ℝ} (hδ : 0 < δ) |
| (hh : 0 < h) (hh1 : h ≤ 1) (hR : N*δ = R) (hc : |c-1| ≤ h) |
| (hv0 : 0 ≤ v) (hv1 : v ≤ 1) : |
| (∫ t, residual N δ t * ((1+t-c)^2+v) / mass (gaussianReal 0 1) (tent 1 h) ∂gaussianReal 0 1) ≤ |
| 972/h * Real.exp (-(R^2/4)) := by |
| let β := mass (gaussianReal 0 1) (tent 1 h) |
| have hb : 0 < β := tent_mass_pos hh |
| have hβ : β⁻¹ ≤ 54/h := middle_mass_bound hh hh1 |
| have hi : Integrable (fun t => gaussianPDFReal 0 1 t * |
| (residual N δ t*((1+t-c)^2+v)/β)) volume := |
| density_integrable (by norm_num) ((residual_quadratic_integrable hδ c v).div_const β) |
| have hj : Integrable (fun t => (324/h*Real.exp (-(R^2/4))) * |
| (gaussianPDFReal 0 2 t*(1+t^2))) volume := |
| (density_integrable (by norm_num) (gaussian_poly_integrable 2)).const_mul _ |
| have hpoint (t : ℝ) : gaussianPDFReal 0 1 t * |
| (residual N δ t*((1+t-c)^2+v)/β) ≤ (324/h*Real.exp (-(R^2/4))) * |
| (gaussianPDFReal 0 2 t*(1+t^2)) := by |
| by_cases ht : |t| ≤ R |
| · rw [residual_eq_zero hδ (by rwa [hR])] |
| simp only [zero_mul,zero_div,mul_zero] |
| exact mul_nonneg (by positivity) (mul_nonneg (gaussianPDFReal_nonneg ..) (by positivity)) |
| have hR0 : 0 ≤ R := by rw [← hR]; positivity |
| have htsq : R^2 ≤ t^2 := by nlinarith [sq_abs t, abs_nonneg t,lt_of_not_ge ht] |
| have hd := density_tail_bound htsq |
| have hn : (1+t-c)^2+v ≤ 3*(1+t^2) := by |
| have hc' := abs_le.mp hc |
| nlinarith [sq_nonneg (t-(1-c)),sq_nonneg t] |
| have hn0 : 0 ≤ (1+t-c)^2+v := add_nonneg (sq_nonneg _) hv0 |
| have hrn : residual N δ t*((1+t-c)^2+v) ≤ 3*(1+t^2) := by |
| have := mul_le_mul (residual_le_one N δ t) hn hn0 (by norm_num : (0:ℝ)≤1) |
| simpa using this |
| have hq : residual N δ t*((1+t-c)^2+v)/β ≤ (54/h)*(3*(1+t^2)) := by |
| rw [div_eq_mul_inv] |
| calc _ ≤ (3*(1+t^2))*(54/h) := mul_le_mul hrn hβ (inv_nonneg.mpr hb.le) (by positivity) |
| _ = _ := by ring_nf |
| calc |
| _ ≤ (2*Real.exp (-(R^2/4))*gaussianPDFReal 0 2 t) * ((54/h)*(3*(1+t^2))) := |
| mul_le_mul hd hq (div_nonneg (mul_nonneg (residual_nonneg N δ t) hn0) hb.le) |
| (mul_nonneg (by positivity) (gaussianPDFReal_nonneg ..)) |
| _ = _ := by ring_nf |
| have hbound := integral_mono hi hj hpoint |
| rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1:ℝ≥0)≠0)] |
| change (∫ t, gaussianPDFReal 0 1 t*(residual N δ t*((1+t-c)^2+v)/β)) ≤ _ |
| apply hbound.trans |
| rw [integral_const_mul] |
| change _ * (∫ t, gaussianPDFReal 0 2 t • (1+t^2)) ≤ _ |
| rw [← integral_gaussianReal_eq_integral_smul (by norm_num : (2:ℝ≥0)≠0)] |
| rw [integral_add (integrable_const 1) (gaussian_sq_integrable 2), integral_const, |
| gaussian_sq_integral] |
| norm_num |
| ring_nf |
| exact le_rfl |
|
|
| def gridN : ℕ := (2^128)^8 |
| def width : ℝ := 1 / ((2:ℝ)^128)^12 |
| def radius : ℝ := 96 |
| def step : ℝ := radius / gridN |
|
|
| theorem gridN_pos : 0 < gridN := by norm_num [gridN] |
| theorem width_pos : 0 < width := by unfold width; positivity |
| theorem step_pos : 0 < step := by |
| exact div_pos (by norm_num [radius]) (by exact_mod_cast gridN_pos) |
| theorem step_small : step+width ≤ 1 := by norm_num [step,radius,gridN,width] |
| theorem width_le_one : width ≤ 1 := by linarith [step_small,step_pos] |
| theorem grid_radius : (gridN:ℝ)*step = radius := by |
| unfold step; field_simp [show (gridN:ℝ)≠0 by exact_mod_cast gridN_pos.ne'] |
| theorem variance_numeric : (2*gridN+1:ℝ)*width^2 < 1/4 := by |
| norm_num [gridN,width] |
|
|
| theorem central_numeric : |
| 4*(radius+step)/width * Real.exp (2*radius+4) * |
| ((step+width)^2+(2*gridN+1:ℝ)*width^2) < 1/4 := by |
| have he : Real.exp 196 < (3:ℝ)^196 := by |
| calc Real.exp 196 = (Real.exp 1)^196 := by rw [← Real.exp_nat_mul]; norm_num |
| _ < 3^196 := pow_lt_pow_left₀ (lt_trans Real.exp_one_lt_d9 (by norm_num)) |
| (Real.exp_pos _).le (by norm_num) |
| have hp : 0 < 4*(radius+step)/width*((step+width)^2+(2*gridN+1:ℝ)*width^2) := by |
| dsimp [radius,step,width,gridN]; positivity |
| calc _ = Real.exp 196 * (4*(radius+step)/width*((step+width)^2+(2*gridN+1:ℝ)*width^2)) := by |
| norm_num [radius]; ring_nf |
| _ < 3^196 * (4*(radius+step)/width*((step+width)^2+(2*gridN+1:ℝ)*width^2)) := |
| mul_lt_mul_of_pos_right he hp |
| _ < 1/4 := by norm_num [radius,step,width,gridN] |
|
|
| theorem tail_numeric : 972/width*Real.exp (-(radius^2/4)) < 1/4 := by |
| have he : (2:ℝ)^2304 < Real.exp 2304 := by |
| calc (2:ℝ)^2304 < (Real.exp 1)^2304 := |
| pow_lt_pow_left₀ (lt_trans (by norm_num) Real.exp_one_gt_d9) (by norm_num) (by norm_num) |
| _ = Real.exp 2304 := by rw [← Real.exp_nat_mul]; norm_num |
| have hi : Real.exp (-2304) < ((2:ℝ)^2304)⁻¹ := by |
| rw [Real.exp_neg] |
| exact (inv_lt_inv₀ (Real.exp_pos _) (by positivity)).mpr he |
| calc _ = 972/width*Real.exp (-2304) := by norm_num [radius] |
| _ < 972/width*((2:ℝ)^2304)⁻¹ := mul_lt_mul_of_pos_left hi (by unfold width; positivity) |
| _ < 1/4 := by |
| rw [show (2:ℝ)^2304 = ((2:ℝ)^128)^18 by rw [← pow_mul]] |
| norm_num [width] |
|
|
| abbrev Leaves := Fin (2*gridN+1) |
| def leafTest (i : Leaves) : ℝ → ℝ := tent (center gridN step i) width |
| def selectorTest (i : Leaves) : ℝ → ℝ := weight gridN step i |
| def beta (i : Leaves) : ℝ := mass (gaussianReal 0 1) (leafTest i) |
| def mu (i : Leaves) : ℝ := mean (gaussianReal 0 1) (leafTest i) |
| def sigma (i : Leaves) : ℝ := variance (gaussianReal 0 1) (leafTest i) |
| def totalVariance : ℝ := ∑ i, sigma i |
| def otherVariance (i : Leaves) : ℝ := ∑ j ∈ Finset.univ.erase i, sigma j |
|
|
| theorem beta_pos (i : Leaves) : 0 < beta i := tent_mass_pos width_pos |
| theorem mu_close (i : Leaves) : |mu i-center gridN step i| ≤ width := |
| (tent_moments width_pos).1 |
| theorem sigma_nonneg (i : Leaves) : 0 ≤ sigma i := (tent_moments width_pos).2.1 |
| theorem sigma_bound (i : Leaves) : sigma i ≤ width^2 := (tent_moments width_pos).2.2 |
| theorem otherVariance_nonneg (i : Leaves) : 0 ≤ otherVariance i := |
| Finset.sum_nonneg (fun _ _ => sigma_nonneg _) |
| theorem otherVariance_bound (i : Leaves) : otherVariance i ≤ (2*gridN+1:ℝ)*width^2 := by |
| calc otherVariance i ≤ ∑ j : Leaves, sigma j := |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.erase_subset _ _) (fun j _ _ => sigma_nonneg j) |
| _ ≤ ∑ _ : Leaves, width^2 := Finset.sum_le_sum (fun j _ => sigma_bound j) |
| _ = _ := by simp [Leaves] |
| theorem otherVariance_le_one (i : Leaves) : otherVariance i ≤ 1 := |
| (otherVariance_bound i).trans (le_trans variance_numeric.le (by norm_num)) |
|
|
| def kernel (t : ℝ) : ℝ := ∑ i, selectorTest i t *((1+t-mu i)^2+otherVariance i)/beta i |
|
|
| def J : ℝ := ∫ t, kernel t ∂gaussianReal 0 1 |
|
|
| theorem kernel_split_general (N : ℕ) (δ : ℝ) (μ v β : Fin (2*N+1) → ℝ) (t : ℝ) : |
| (∑ i, weight N δ i t *((1+t-μ i)^2+v i)/β i) = centralKernel N δ μ v β t + |
| residual N δ t *((1+t-μ (middle N))^2+v (middle N))/β (middle N) := by |
| unfold weight centralKernel residual |
| simp only [add_mul,add_div,Finset.sum_add_distrib] |
| congr 1 |
| simp only [ite_mul,zero_mul,ite_div,zero_div,Finset.sum_ite_eq',Finset.mem_univ,ite_true] |
|
|
| theorem kernel_split (t : ℝ) : kernel t = centralKernel gridN step mu otherVariance beta t + |
| residual gridN step t *((1+t-mu (middle gridN))^2+otherVariance (middle gridN))/beta (middle gridN) := |
| kernel_split_general gridN step mu otherVariance beta t |
|
|
| theorem kernel_integrable : Integrable kernel (gaussianReal 0 1) := by |
| rw [show kernel = fun t => centralKernel gridN step mu otherVariance beta t + |
| residual gridN step t *((1+t-mu (middle gridN))^2+otherVariance (middle gridN))/beta (middle gridN) |
| from funext kernel_split] |
| exact ((centralKernel_continuous ..).integrable_of_hasCompactSupport |
| (centralKernel_compactSupport step_pos ..)).add |
| ((residual_quadratic_integrable step_pos _ _).div_const _) |
|
|
| theorem J_lt_half : J < 1/2 := by |
| have hc := central_integral_bound step_pos width_pos step_small grid_radius mu otherVariance |
| mu_close otherVariance_nonneg otherVariance_bound |
| have ht := tail_integral_bound step_pos width_pos width_le_one grid_radius |
| (show |mu (middle gridN)-1| ≤ width by simpa using mu_close (middle gridN)) |
| (otherVariance_nonneg (middle gridN)) (otherVariance_le_one (middle gridN)) |
| have hb : beta (middle gridN) = mass (gaussianReal 0 1) (tent 1 width) := by |
| simp [beta,leafTest] |
| unfold J |
| simp_rw [kernel_split] |
| rw [integral_add ((centralKernel_continuous ..).integrable_of_hasCompactSupport |
| (centralKernel_compactSupport step_pos ..)) |
| ((residual_quadratic_integrable step_pos _ _).div_const _), hb] |
| change (∫ t, centralKernel gridN step mu otherVariance beta t ∂gaussianReal 0 1) + _ < _ |
| change (∫ t, centralKernel gridN step mu otherVariance beta t ∂gaussianReal 0 1) ≤ _ at hc |
| linarith [central_numeric,tail_numeric] |
|
|
| theorem J_nonneg : 0 ≤ J := by |
| apply integral_nonneg |
| intro t |
| apply Finset.sum_nonneg |
| intro i _ |
| exact div_nonneg (mul_nonneg (weight_nonneg ..) |
| (add_nonneg (sq_nonneg _) (otherVariance_nonneg _))) (beta_pos _).le |
|
|
| theorem positive_gaussian_deficit : |
| 0 < (∏ i : Leaves, beta i) * (1+totalVariance-J) := by |
| have hp : 0 < ∏ i : Leaves, beta i := Finset.prod_pos (fun _ _ => beta_pos _) |
| have hs : 0 ≤ totalVariance := Finset.sum_nonneg (fun _ _ => sigma_nonneg _) |
| exact mul_pos hp (by linarith [J_lt_half]) |
|
|
| end GaussianGain |
| end SquareRootDegree |
|
|
| end |
|
|
| end OAI |
|
|
