| import OAI.Probability.ProjectionMoments.Frames |
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| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace ProjectionMoments |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| @[fun_prop] lemma measurable_coordinateMap (m : ℕ) : |
| Measurable (fun z : (Fin m → E) × E => coordinateMap z.1 z.2) := by |
| apply Measurable.of_eval |
| intro i |
| simp only [coordinateMap_apply] |
| fun_prop |
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| def projectionKernel (ν : Measure E) [SFinite ν] (m : ℕ) : Kernel (Fin m → E) (Fin m → ℝ) where |
| toFun A := ν.map (coordinateMap A) |
| measurable' := by |
| apply Measure.measurable_of_measurable_coe |
| intro s hs |
| have he (A : Fin m → E) : (ν.map (coordinateMap A)) s = |
| ∫⁻ x, s.indicator (fun _ => (1 : ℝ≥0∞)) (coordinateMap A x) ∂ν := by |
| rw [← lintegral_map (measurable_const.indicator hs) (by fun_prop)] |
| simp [lintegral_indicator hs] |
| simp_rw [he] |
| apply Measurable.lintegral_prod_right |
| exact (measurable_const.indicator hs).comp (measurable_coordinateMap m) |
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| instance projectionKernel_finite (ν : Measure E) [IsFiniteMeasure ν] (m : ℕ) : |
| IsFiniteKernel (projectionKernel ν m) := by |
| refine ⟨ν Set.univ, measure_lt_top _ _, ?_⟩ |
| intro A |
| change (ν.map (coordinateMap A)) Set.univ ≤ _ |
| rw [Measure.map_apply (by fun_prop) MeasurableSet.univ, Set.preimage_univ] |
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| lemma projectionKernel_compProd (ν : Measure E) [IsFiniteMeasure ν] (m : ℕ) : |
| (Measure.pi (fun _ : Fin m => stdGaussian E)) ⊗ₘ projectionKernel ν m = |
| NoiselessRegression.projectionLaw ν m := by |
| apply Measure.ext_of_lintegral |
| intro f hf |
| rw [Measure.lintegral_compProd hf, NoiselessRegression.projectionLaw, |
| lintegral_map hf (by fun_prop), lintegral_prod _ (by fun_prop)] |
| apply lintegral_congr |
| intro A |
| change (∫⁻ y, f (A,y) ∂ν.map (coordinateMap A)) = _ |
| rw [lintegral_map (by fun_prop) (by fun_prop)] |
| congr 1 |
| funext x |
| apply congrArg f |
| exact Prod.ext (by rfl) (funext fun i => coordinateMap_apply A x i) |
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| lemma ae_projection_ac (ν : Measure E) [IsFiniteMeasure ν] (m : ℕ) |
| (h : NoiselessRegression.projectionLaw ν m ≪ NoiselessRegression.matrixLabelReference (E := E) m) : |
| ∀ᵐ A : Fin m → E ∂Measure.pi (fun _ => stdGaussian E), |
| ν.map (coordinateMap A) ≪ (volume : Measure (Fin m → ℝ)) := by |
| let γ : Measure (Fin m → E) := Measure.pi (fun _ => stdGaussian E) |
| let η : Measure (Fin m → ℝ) := volume.toFinite |
| have hh : γ ⊗ₘ projectionKernel ν m ≪ γ ⊗ₘ Kernel.const (Fin m → E) η := by |
| rw [projectionKernel_compProd, Measure.compProd_const] |
| exact h.trans (Measure.AbsolutelyContinuous.refl γ |>.prod (absolutelyContinuous_toFinite volume)) |
| filter_upwards [hh.kernel_of_compProd] with A hA |
| exact hA.trans (toFinite_absolutelyContinuous volume) |
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| lemma sphere_ae_projection_ac [Nontrivial E] {m : ℕ} (hm : m+2 ≤ Module.finrank ℝ E) : |
| ∀ᵐ A : Fin m → E ∂Measure.pi (fun _ => stdGaussian E), |
| NoiselessRegression.ambientSphereLaw.map (coordinateMap A) ≪ (volume : Measure (Fin m → ℝ)) := by |
| apply ae_projection_ac |
| apply NoiselessRegression.projectionLaw_absolutelyContinuous (a := m+1) (n := 1) |
| (C := ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E)) |
| · intro z r hr |
| exact NoiselessRegression.ambientSphereLaw_closedBall z hr (m+1) (by omega) |
| · positivity |
| · exact ENNReal.ofReal_ne_top |
| · exact NoiselessRegression.ambientSphereLaw_support.trans Metric.sphere_subset_closedBall |
| · omega |
| · omega |
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| lemma linearEquiv_preserves_ac {m : ℕ} (T : (Fin m → ℝ) ≃L[ℝ] (Fin m → ℝ)) |
| {μ : Measure (Fin m → ℝ)} (hμ : μ ≪ volume) : μ.map T ≪ volume := by |
| exact (T.toHomeomorph.toMeasurableEquiv.measurableEmbedding.absolutelyContinuous_map hμ).trans |
| (Measure.LinearMap.quasiMeasurePreserving volume T.toLinearEquiv.toLinearMap |
| (LinearEquiv.isUnit_det' T.toLinearEquiv).ne_zero).absolutelyContinuous |
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| lemma sphere_projection_rotation [Nontrivial E] {m : ℕ} (O : Rotation E) (A : Fin m → E) : |
| NoiselessRegression.ambientSphereLaw.map (coordinateMap (rotateRows O A)) = |
| NoiselessRegression.ambientSphereLaw.map (coordinateMap A) := by |
| have hfun : (coordinateMap (rotateRows O A) : E → (Fin m → ℝ)) = |
| coordinateMap A ∘ (Unitary.linearIsometryEquiv O).symm := |
| funext (coordinateMap_rotateRows O A) |
| rw [hfun, ← Measure.map_map (by fun_prop) (by fun_prop), |
| (NoiselessRegression.ambientSphereLaw_isometry (Unitary.linearIsometryEquiv O).symm).map_eq] |
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| theorem sphere_frame_ac [Nontrivial E] {m : ℕ} (hm : m+2 ≤ Module.finrank ℝ E) |
| {u : Fin m → E} (hu : Orthonormal ℝ u) : |
| NoiselessRegression.ambientSphereLaw.map (coordinateMap u) ≪ (volume : Measure (Fin m → ℝ)) := by |
| obtain ⟨B,hB,hac⟩ := ((gaussian_rows_ae_linearIndependent m (by omega)).and |
| (sphere_ae_projection_ac hm)).exists |
| obtain ⟨v,hv,T,hT,hvol⟩ := rows_factor B hB |
| have he : (NoiselessRegression.ambientSphereLaw.map (coordinateMap B)).map T.symm = |
| NoiselessRegression.ambientSphereLaw.map (coordinateMap v) := by |
| rw [Measure.map_map (by fun_prop) (by fun_prop)] |
| congr 1 |
| funext x |
| simp only [Function.comp_apply, hT, T.symm_apply_apply] |
| have hvac := linearEquiv_preserves_ac T.symm hac |
| rw [he] at hvac |
| obtain ⟨O,hO⟩ := rotation_transitive hv hu |
| have heu : rotateRows O v = u := funext hO |
| rw [← heu, sphere_projection_rotation] |
| exact hvac |
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| theorem sphere_fullRank_ac [Nontrivial E] {m : ℕ} (hm : m+2 ≤ Module.finrank ℝ E) |
| {A : Fin m → E} (hA : LinearIndependent ℝ A) : |
| NoiselessRegression.ambientSphereLaw.map (coordinateMap A) ≪ (volume : Measure (Fin m → ℝ)) := by |
| obtain ⟨u,hu,T,hT,hvol⟩ := rows_factor A hA |
| have he : NoiselessRegression.ambientSphereLaw.map (coordinateMap A) = |
| (NoiselessRegression.ambientSphereLaw.map (coordinateMap u)).map T := by |
| rw [Measure.map_map (by fun_prop) (by fun_prop)] |
| congr 1 |
| exact funext hT |
| rw [he] |
| exact linearEquiv_preserves_ac T (sphere_frame_ac hm hu) |
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| end ProjectionMoments |
| end |
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| end |
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| end OAI |
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