| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.RadialMixture |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
|
|
| noncomputable section |
| universe u v |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision |
| open NoiselessRegression |
|
|
| lemma gaussianKernel_scale (m : ℕ) {δ : ℝ} (hδ : 0 < δ) (y : Space m) : |
| GaussianKernel m δ (δ • y) = δ⁻¹ ^ m * GaussianKernel m 1 y := by |
| have he (a : ℝ) : gaussianPDFReal 0 ⟨δ^2,sq_nonneg δ⟩ (δ*a) = |
| δ⁻¹ * gaussianPDFReal 0 1 a := by |
| have h := gaussianPDFReal_inv_mul (μ := 0) (v := 1) hδ.ne' (δ*a) |
| simp only [inv_mul_cancel_left₀ hδ.ne', mul_zero, mul_one, abs_of_pos hδ] at h |
| rw [h] |
| field_simp |
| rfl |
| unfold GaussianKernel |
| simp only [PiLp.smul_apply, smul_eq_mul, one_pow] |
| simp_rw [he] |
| simp only [Finset.prod_mul_distrib, Finset.prod_const, Finset.card_univ, Fintype.card_fin] |
| rfl |
|
|
| lemma radialMixture_scale (m : ℕ) {δ : ℝ} (hδ : 0 < δ) (r : ℝ) : |
| RadialMixture m δ (δ*r) = δ⁻¹ * RadialMixture m 1 r := by |
| unfold RadialMixture |
| rw [mul_smul, gaussianKernel_scale m hδ] |
| simp only [one_pow, div_one, mul_pow, inv_pow] |
| field_simp |
|
|
| def BallAverage {m : ℕ} (ν : Measure (Space m)) (y : Space m) (r : ℝ) : ℝ := |
| (ν (Metric.ball y r)).toReal / (UnitBallVolume m * r^m) |
|
|
| lemma ballAverage_measurable {m : ℕ} (ν : Measure (Space m)) (y : Space m) : |
| Measurable (BallAverage ν y) := by |
| have h : Monotone (fun r : ℝ => ν (Metric.ball y r)) := fun r t h => |
| measure_mono (Metric.ball_subset_ball h) |
| exact h.measurable.ennreal_toReal.div (by fun_prop) |
|
|
| lemma ballAverage_nonneg {m : ℕ} (ν : Measure (Space m)) (y : Space m) {r : ℝ} |
| (hr : 0 < r) : 0 ≤ BallAverage ν y r := by |
| exact div_nonneg ENNReal.toReal_nonneg (mul_nonneg (unitBallVolume_pos m).le |
| (pow_nonneg hr.le _)) |
|
|
| lemma ac_ball_eq_closedBall {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) |
| (hν : ν ≪ volume) (y : Space m) (r : ℝ) : |
| ν (Metric.ball y r) = ν (Metric.closedBall y r) := by |
| have : NeZero m := ⟨by omega⟩ |
| apply measure_eq_measure_of_null_sdiff Metric.ball_subset_closedBall |
| rw [Metric.closedBall_sdiff_ball] |
| exact hν (Measure.addHaar_sphere volume y r) |
|
|
| lemma ae_tendsto_ballAverage {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) |
| [IsFiniteMeasure ν] (hν : ν ≪ volume) : |
| ∀ᵐ y ∂(volume : Measure (Space m)), Tendsto (BallAverage ν y) |
| (𝓝[>] (0:ℝ)) (𝓝 ((ν.rnDeriv volume y).toReal)) := by |
| have : NeZero m := ⟨by omega⟩ |
| filter_upwards [Besicovitch.ae_tendsto_rnDeriv ν volume, ν.rnDeriv_lt_top volume] with y hy hyfin |
| have ht := (ENNReal.continuousAt_toReal hyfin.ne).tendsto.comp hy |
| apply ht.congr' |
| filter_upwards [self_mem_nhdsWithin] with r hr |
| unfold BallAverage |
| simp only [Function.comp_apply] |
| rw [ENNReal.toReal_div, Measure.addHaar_closedBall_eq_addHaar_ball volume y r, |
| ← ac_ball_eq_closedBall hm ν hν y r, volume_ball_eq m hm y hr, |
| ENNReal.toReal_ofReal (mul_nonneg (unitBallVolume_pos m).le (pow_nonneg hr.le _))] |
| |
| lemma ballAverage_bounded {m : ℕ} (ν : Measure (Space m)) [IsFiniteMeasure ν] |
| (y : Space m) {f : ℝ} (hf : 0 ≤ f) |
| (ht : Tendsto (BallAverage ν y) (𝓝[>] (0:ℝ)) (𝓝 f)) : |
| ∃ B : ℝ, 0 ≤ B ∧ ∀ r : ℝ, 0 < r → ‖BallAverage ν y r‖ ≤ B := by |
| rcases Metric.tendsto_nhdsWithin_nhds.mp ht 1 zero_lt_one with ⟨b,hb,hnear⟩ |
| let C : ℝ := max (f+1) ((ν Set.univ).toReal / (UnitBallVolume m * b^m)) |
| refine ⟨C, le_trans (by linarith : 0 ≤ f+1) (le_max_left _ _), ?_⟩ |
| intro r hr |
| rw [Real.norm_eq_abs, abs_of_nonneg (ballAverage_nonneg ν y hr)] |
| by_cases h : r < b |
| · have he := hnear hr (by simpa [Real.dist_eq, abs_of_pos hr] using h) |
| have he' : BallAverage ν y r < f+1 := by |
| rw [Real.dist_eq] at he |
| have := (abs_lt.mp he).2 |
| linarith |
| exact he'.le.trans (le_max_left _ _) |
| · apply le_trans ?_ (le_max_right _ _) |
| unfold BallAverage |
| have hvol : 0 < UnitBallVolume m := unitBallVolume_pos m |
| apply div_le_div₀ (by positivity) ?_ (by positivity) ?_ |
| · exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) (measure_ne_top _ _)).mpr |
| (measure_mono (Set.subset_univ _)) |
| · exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hb.le (le_of_not_gt h) _) |
| (unitBallVolume_pos m).le |
| |
| lemma tendsto_radial_average {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) [IsFiniteMeasure ν] |
| (y : Space m) {f : ℝ} (hf : 0 ≤ f) |
| (ht : Tendsto (BallAverage ν y) (𝓝[>] (0:ℝ)) (𝓝 f)) : |
| Tendsto (fun δ : ℝ => ∫ r in Ioi 0, RadialMixture m 1 r * BallAverage ν y (δ*r)) |
| (𝓝[>] (0:ℝ)) (𝓝 f) := by |
| obtain ⟨B,hB,hbound⟩ := ballAverage_bounded ν y hf ht |
| have hw : IntegrableOn (RadialMixture m 1) (Ioi 0) := |
| integrable_of_integral_eq_one (radialMixture_integral m hm zero_lt_one) |
| have hl := tendsto_integral_filter_of_dominated_convergence |
| (μ := volume.restrict (Ioi 0)) (l := 𝓝[>] (0:ℝ)) |
| (F := fun δ r => RadialMixture m 1 r * BallAverage ν y (δ*r)) |
| (f := fun r => RadialMixture m 1 r * f) |
| (fun r => RadialMixture m 1 r * B) |
| (Eventually.of_forall (fun δ => |
| ((radialMixture_measurable m 1).mul ((ballAverage_measurable ν y).comp |
| (by fun_prop))).aestronglyMeasurable)) |
| (show ∀ᶠ δ in 𝓝[>] (0:ℝ), ∀ᵐ r ∂volume.restrict (Ioi 0), |
| ‖RadialMixture m 1 r * BallAverage ν y (δ*r)‖ ≤ RadialMixture m 1 r * B from by |
| filter_upwards [self_mem_nhdsWithin] with δ hδ |
| filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr |
| rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (radialMixture_nonneg m 1 hr)] |
| exact mul_le_mul_of_nonneg_left (hbound _ (mul_pos hδ hr)) (radialMixture_nonneg m 1 hr)) |
| (hw.mul_const B) (show ∀ᵐ r ∂volume.restrict (Ioi 0), |
| Tendsto (fun δ => RadialMixture m 1 r * BallAverage ν y (δ*r)) |
| (𝓝[>] (0:ℝ)) (𝓝 (RadialMixture m 1 r * f)) from by |
| filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr |
| apply Tendsto.const_mul |
| apply ht.comp |
| apply tendsto_nhdsWithin_iff.mpr |
| refine ⟨?_, ?_⟩ |
| · simpa using ((continuous_id.mul_const r).tendsto (0:ℝ)).mono_left nhdsWithin_le_nhds |
| · filter_upwards [self_mem_nhdsWithin] with δ hδ |
| change 0 < δ*r |
| exact mul_pos hδ hr) |
| simpa only [integral_mul_const, radialMixture_integral m hm zero_lt_one, one_mul] using hl |
| end MemoryPrecision |
| |
| |
| end |
| |
| end OAI |
| |
