| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.BallAverages |
|
|
| 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_translated_ball_mixture (m : ℕ) (hm : 1 ≤ m) {δ : ℝ} (hδ : 0 < δ) |
| (y x : Space m) : |
| ENNReal.ofReal (GaussianKernel m δ (y-x)) = |
| ∫⁻ r in Ioi 0, (Metric.ball y r).indicator |
| (fun _ => ENNReal.ofReal ((r/δ^2)*GaussianKernel m δ (r • CoordinateUnit m))) x := by |
| rw [gaussianKernel_ball_mixture m hm hδ] |
| apply lintegral_congr |
| intro r |
| have he : y-x ∈ Metric.ball (0 : Space m) r ↔ x ∈ Metric.ball y r := by |
| simp only [Metric.mem_ball, dist_eq_norm, sub_zero, norm_sub_rev] |
| by_cases hx : x ∈ Metric.ball y r |
| · rw [Set.indicator_of_mem (he.mpr hx), Set.indicator_of_mem hx] |
| · rw [Set.indicator_of_notMem (mt he.mp hx), Set.indicator_of_notMem hx] |
|
|
| def GaussianConvolution {m : ℕ} (ν : Measure (Space m)) (δ : ℝ) (y : Space m) : ℝ := |
| ∫ x, GaussianKernel m δ (y-x) ∂ν |
|
|
| lemma gaussianConvolution_shell {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) |
| [IsFiniteMeasure ν] {δ : ℝ} (hδ : 0 < δ) (y : Space m) : |
| GaussianConvolution ν δ y = ∫ r in Ioi 0, |
| ((r/δ^2)*GaussianKernel m δ (r • CoordinateUnit m)) * (ν (Metric.ball y r)).toReal := by |
| have hk : Measurable (GaussianKernel m δ) := (gaussianKernel_continuous m δ).measurable |
| have hmeas : Measurable (fun z : Space m × ℝ => |
| (Metric.ball y z.2).indicator |
| (fun _ => ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m))) z.1) := by |
| have he : (fun z : Space m × ℝ => (Metric.ball y z.2).indicator |
| (fun _ => ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m))) z.1) = |
| {z : Space m × ℝ | dist z.1 y < z.2}.indicator |
| (fun z => ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m))) := by |
| ext z |
| by_cases hz : z.1 ∈ Metric.ball y z.2 |
| · rw [Set.indicator_of_mem hz, Set.indicator_of_mem |
| (show z ∈ {z : Space m × ℝ | dist z.1 y < z.2} from hz)] |
| · rw [Set.indicator_of_notMem hz, Set.indicator_of_notMem |
| (show z ∉ {z : Space m × ℝ | dist z.1 y < z.2} from hz)] |
| rw [he] |
| exact (by fun_prop : Measurable (fun z : Space m × ℝ => |
| ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m)))).indicator |
| (measurableSet_lt (by fun_prop) (by fun_prop)) |
| unfold GaussianConvolution |
| rw [integral_eq_lintegral_of_nonneg_ae |
| (Eventually.of_forall (fun _ => gaussianKernel_nonneg m δ _)) |
| ((hk.comp (by fun_prop)).aestronglyMeasurable)] |
| simp_rw [gaussianKernel_translated_ball_mixture m hm hδ y] |
| rw [lintegral_lintegral_swap hmeas.aemeasurable] |
| simp_rw [lintegral_indicator Metric.isOpen_ball.measurableSet, setLIntegral_const] |
| have hmass : Measurable (fun r : ℝ => (ν (Metric.ball y r)).toReal) := |
| (show Monotone (fun r : ℝ => ν (Metric.ball y r)) from |
| fun a b h => measure_mono (Metric.ball_subset_ball h)).measurable.ennreal_toReal |
| rw [integral_eq_lintegral_of_nonneg_ae |
| ((ae_restrict_mem measurableSet_Ioi).mono (fun r hr => |
| mul_nonneg (mul_nonneg (div_nonneg hr.le (sq_nonneg δ)) (gaussianKernel_nonneg m δ _)) |
| ENNReal.toReal_nonneg)) (by fun_prop)] |
| congr 1 |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr |
| rw [ENNReal.ofReal_mul (mul_nonneg (div_nonneg hr.le (sq_nonneg δ)) |
| (gaussianKernel_nonneg m δ _)), ENNReal.ofReal_toReal (measure_ne_top _ _)] |
|
|
| lemma gaussianConvolution_mixture {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) |
| [IsFiniteMeasure ν] {δ : ℝ} (hδ : 0 < δ) (y : Space m) : |
| GaussianConvolution ν δ y = ∫ r in Ioi 0, RadialMixture m δ r * BallAverage ν y r := by |
| rw [gaussianConvolution_shell hm ν hδ y] |
| apply integral_congr_ae |
| filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr |
| have hr0 : 0 < r := hr |
| unfold RadialMixture BallAverage |
| field_simp [(unitBallVolume_pos m).ne', hr0.ne'] |
|
|
| lemma gaussianConvolution_scaled_mixture {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) |
| [IsFiniteMeasure ν] {δ : ℝ} (hδ : 0 < δ) (y : Space m) : |
| GaussianConvolution ν δ y = ∫ r in Ioi 0, RadialMixture m 1 r * BallAverage ν y (δ*r) := by |
| rw [gaussianConvolution_mixture hm ν hδ y] |
| have he := integral_comp_mul_left_Ioi' |
| (fun r => RadialMixture m δ r * BallAverage ν y r) 0 hδ |
| simp only [mul_zero, smul_eq_mul] at he |
| rw [← he, ← integral_const_mul] |
| apply integral_congr_ae |
| exact Eventually.of_forall (fun radius => by |
| dsimp only |
| rw [radialMixture_scale m hδ] |
| field_simp [hδ.ne']) |
|
|
| lemma ae_tendsto_gaussianConvolution {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) |
| [IsFiniteMeasure ν] (hν : ν ≪ volume) : |
| ∀ᵐ y ∂(volume : Measure (Space m)), Tendsto (fun δ : ℝ => GaussianConvolution ν δ y) |
| (𝓝[>] (0:ℝ)) (𝓝 ((ν.rnDeriv volume y).toReal)) := by |
| filter_upwards [ae_tendsto_ballAverage hm ν hν] with y hy |
| apply (tendsto_radial_average hm ν y ENNReal.toReal_nonneg hy).congr' |
| filter_upwards [self_mem_nhdsWithin] with δ hδ |
| exact (gaussianConvolution_scaled_mixture hm ν hδ y).symm |
| end MemoryPrecision |
| |
| |
| end |
| |
| end OAI |
| |
