| import OAI.Probability.ProjectionMoments.Graph |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| namespace ProjectionMoments |
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| section |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| lemma stdGaussian_le_volume : stdGaussian E ≤ volume := by |
| let b := stdOrthonormalBasis ℝ E |
| have hp : MeasurePreserving (fun x : Fin (Module.finrank ℝ E) → ℝ => |
| ∑ i, x i • b i) (volume : Measure (Fin (Module.finrank ℝ E) → ℝ)) volume := by |
| have he : (fun x : Fin (Module.finrank ℝ E) → ℝ => ∑ i, x i • b i) = |
| (fun x => b.repr.symm (WithLp.toLp 2 x)) := by |
| funext x |
| exact b.sum_repr_symm (WithLp.toLp 2 x) |
| rw [he] |
| exact b.measurePreserving_repr_symm.comp |
| (PiLp.volume_preserving_toLp (Fin (Module.finrank ℝ E))) |
| rw [stdGaussian_eq_map_pi_orthonormalBasis b, ← hp.map_eq] |
| exact Measure.map_mono (NoiselessRegression.pi_gaussianReal_le_volume _) hp.measurable |
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| lemma stdGaussian_submodule_null (S : Submodule ℝ E) (hS : S ≠ ⊤) : |
| stdGaussian E S = 0 := by |
| apply le_antisymm _ zero_le |
| exact (stdGaussian_le_volume (E := E) S).trans_eq (Measure.addHaar_submodule volume S hS) |
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| lemma gaussian_rows_ae_gramSchmidt_ne_zero (m : ℕ) (hm : m ≤ Module.finrank ℝ E) : |
| ∀ᵐ A : Fin m → E ∂Measure.pi (fun _ => stdGaussian E), ∀ i, gramSchmidt ℝ A i ≠ 0 := by |
| induction m with |
| | zero => exact Filter.Eventually.of_forall fun _ i => Fin.elim0 i |
| | succ m ih => |
| rw [NoiselessRegression.ae_pi_snoc _ (by |
| simpa only [sub_zero] using NoiselessRegression.measurableSet_gramSchmidt_ne_zero (0 : E) (m+1))] |
| filter_upwards [ih (by omega)] with v hv |
| let S := Submodule.span ℝ (Set.range v) |
| have hd : Module.finrank ℝ S ≤ m := by |
| simpa only [S, Fintype.card_fin, Set.finrank] using finrank_range_le_card (R := ℝ) v |
| have hs : S ≠ ⊤ := by |
| intro he |
| rw [he, finrank_top] at hd |
| omega |
| have ha : ∀ᵐ x ∂stdGaussian E, x ∉ S := by |
| rw [ae_iff] |
| convert stdGaussian_submodule_null S hs using 2 |
| ext x |
| simp |
| filter_upwards [ha] with x hx |
| intro i |
| induction i using Fin.lastCases with |
| | last => |
| apply norm_ne_zero_iff.mp |
| rw [NoiselessRegression.norm_gramSchmidt_eq_infDist, |
| NoiselessRegression.image_Iio_last] |
| simp only [Fin.snoc_castSucc, Fin.snoc_last] |
| exact ne_of_gt ((S.closed_of_finiteDimensional.notMem_iff_infDist_pos |
| ⟨0, S.zero_mem⟩).mp hx) |
| | cast j => simpa only [NoiselessRegression.gramSchmidt_castSucc, Fin.snoc_castSucc] using hv j |
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| lemma gaussian_rows_ae_linearIndependent (m : ℕ) (hm : m ≤ Module.finrank ℝ E) : |
| ∀ᵐ A : Fin m → E ∂Measure.pi (fun _ => stdGaussian E), LinearIndependent ℝ A := by |
| filter_upwards [gaussian_rows_ae_gramSchmidt_ne_zero m hm] with A hA |
| apply Matrix.linearIndependent_of_det_gram_ne_zero |
| rw [NoiselessRegression.det_gram_eq_prod_gramSchmidt] |
| exact Finset.prod_ne_zero_iff.mpr fun i _ => pow_ne_zero _ (norm_ne_zero_iff.mpr (hA i)) |
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| lemma lintegral_gaussianReal_sq : |
| ∫⁻ x : ℝ, ENNReal.ofReal (x^2) ∂gaussianReal 0 1 = 1 := by |
| have hi : Integrable (fun x : ℝ => x^2) (gaussianReal 0 1) := |
| (memLp_id_gaussianReal' 2 (by norm_num)).integrable_sq |
| rw [← ofReal_integral_eq_lintegral_ofReal hi (Filter.Eventually.of_forall (fun x => sq_nonneg x))] |
| have hv := variance_fun_id_gaussianReal (μ := 0) (v := 1) |
| rw [variance_eq_integral (X := fun x : ℝ => x) (by fun_prop)] at hv |
| simpa only [integral_id_gaussianReal, sub_zero, NNReal.coe_one, ENNReal.ofReal_one] using |
| congrArg ENNReal.ofReal hv |
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| lemma lintegral_stdGaussian_norm_sq : |
| ∫⁻ x : E, ENNReal.ofReal (‖x‖^2) ∂stdGaussian E = Module.finrank ℝ E := by |
| let b := stdOrthonormalBasis ℝ E |
| rw [stdGaussian_eq_map_pi_orthonormalBasis b, lintegral_map (by fun_prop) (by fun_prop)] |
| have he (x : Fin (Module.finrank ℝ E) → ℝ) : |
| ‖∑ i, x i • b i‖^2 = ∑ i, (x i)^2 := by |
| rw [b.sum_repr_symm (WithLp.toLp 2 x), LinearIsometryEquiv.norm_map] |
| exact EuclideanSpace.real_norm_sq_eq _ |
| simp_rw [he, ENNReal.ofReal_sum_of_nonneg (fun i _ => sq_nonneg _)] |
| rw [lintegral_finsetSum _ (fun i _ => by fun_prop)] |
| have hi (i : Fin (Module.finrank ℝ E)) : |
| ∫⁻ x : Fin (Module.finrank ℝ E) → ℝ, ENNReal.ofReal (x i ^ 2) |
| ∂Measure.pi (fun _ => gaussianReal 0 1) = 1 := by |
| rw [← lintegral_gaussianReal_sq] |
| exact (measurePreserving_eval (fun _ : Fin (Module.finrank ℝ E) => gaussianReal 0 1) i).lintegral_comp |
| (show Measurable (fun x : ℝ => ENNReal.ofReal (x^2)) by fun_prop) |
| simp_rw [hi] |
| simp |
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| lemma gaussian_norm_ball_mass (hd : 0 < Module.finrank ℝ E) : |
| (2 : ℝ≥0∞)⁻¹ ≤ stdGaussian E (Metric.closedBall 0 (Real.sqrt (2 * Module.finrank ℝ E))) := by |
| let d : ℝ := Module.finrank ℝ E |
| have hd' : 0 < d := by dsimp [d]; exact_mod_cast hd |
| have hneg : stdGaussian E {x | ENNReal.ofReal (2*d) ≤ ENNReal.ofReal (‖x‖^2)} ≤ (2 : ℝ≥0∞)⁻¹ := by |
| have h := meas_ge_le_lintegral_div (μ := stdGaussian E) |
| (f := fun x => ENNReal.ofReal (‖x‖^2)) (by fun_prop) |
| (ε := ENNReal.ofReal (2*d)) (by positivity) ENNReal.ofReal_ne_top |
| rw [lintegral_stdGaussian_norm_sq] at h |
| convert h using 1 |
| have hd0 : (Module.finrank ℝ E : ℝ≥0∞) ≠ 0 := by exact_mod_cast hd.ne' |
| simp only [ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2), ENNReal.ofReal_ofNat, d, |
| ENNReal.ofReal_natCast, div_eq_mul_inv] |
| rw [ENNReal.mul_inv (Or.inl (by norm_num)) (Or.inl (by norm_num)), mul_left_comm, ENNReal.mul_inv_cancel hd0 (by simp), mul_one] |
| have hsub : (Metric.closedBall (0 : E) (Real.sqrt (2*d)))ᶜ ⊆ |
| {x | ENNReal.ofReal (2*d) ≤ ENNReal.ofReal (‖x‖^2)} := by |
| intro x hx |
| have hx' : Real.sqrt (2*d) < ‖x‖ := by simpa only [mem_compl_iff, Metric.mem_closedBall, |
| dist_zero_right, not_le] using hx |
| apply ENNReal.ofReal_le_ofReal |
| have hsq := (sq_le_sq₀ (Real.sqrt_nonneg (2*d)) (norm_nonneg x)).mpr hx'.le |
| rwa [Real.sq_sqrt (by positivity)] at hsq |
| have hc := (measure_mono hsub).trans hneg |
| rw [measure_compl measurableSet_closedBall (measure_ne_top _ _), measure_univ] at hc |
| change (2 : ℝ≥0∞)⁻¹ ≤ stdGaussian E (Metric.closedBall 0 (Real.sqrt (2*d))) |
| have := tsub_le_iff_left.mp hc |
| have he : (1 : ℝ≥0∞) - (2 : ℝ≥0∞)⁻¹ = (2 : ℝ≥0∞)⁻¹ := by norm_num |
| rw [← he] |
| exact tsub_le_iff_right.mpr this |
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| end |
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| lemma map_withDensity_equiv {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) (e : X ≃ᵐ Y) (f : X → ℝ≥0∞) (hf : Measurable f) : |
| (μ.withDensity f).map e = (μ.map e).withDensity (fun y => f (e.symm y)) := by |
| ext s hs |
| rw [Measure.map_apply e.measurable hs, withDensity_apply _ (hs.preimage e.measurable), |
| withDensity_apply _ hs, ← lintegral_indicator hs, |
| lintegral_map (show Measurable (s.indicator (fun y => f (e.symm y))) from |
| (hf.comp e.symm.measurable).indicator hs) e.measurable] |
| have he (x : X) : s.indicator (fun y => f (e.symm y)) (e x) = |
| (e ⁻¹' s).indicator f x := by |
| by_cases hx : e x ∈ s <;> simp [hx] |
| simp_rw [he] |
| exact (lintegral_indicator (hs.preimage e.measurable) f).symm |
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| lemma density_moment_equiv {X : Type*} [MeasurableSpace X] (μ : Measure X) [SigmaFinite μ] |
| (e : X ≃ᵐ X) (c : ℝ≥0∞) (hc0 : c ≠ 0) (hct : c ≠ ∞) |
| (he : μ.map e = c • μ) {f g : X → ℝ≥0∞} |
| (hf : Measurable f) (hg : Measurable g) |
| (hfg : (μ.withDensity f).map e = μ.withDensity g) (n : ℕ) : |
| ∫⁻ y, g y ^ (n+1) ∂μ = c^n * ∫⁻ x, f x ^ (n+1) ∂μ := by |
| have hd : g =ᵐ[μ] (fun y => c * f (e.symm y)) := by |
| apply (withDensity_eq_iff_of_sigmaFinite hg.aemeasurable (by fun_prop)).mp |
| rw [← hfg, map_withDensity_equiv μ e f hf, he, withDensity_smul_measure] |
| exact (withDensity_smul c (hf.comp e.symm.measurable)).symm |
| have hd' : (fun y => g y ^ (n+1)) =ᵐ[μ] |
| (fun y => (c * f (e.symm y)) ^ (n+1)) := by |
| filter_upwards [hd] with y hy |
| rw [hy] |
| rw [lintegral_congr_ae hd'] |
| simp_rw [mul_pow] |
| rw [lintegral_const_mul _ (by fun_prop)] |
| have hchange : c * (∫⁻ y, f (e.symm y) ^ (n+1) ∂μ) = ∫⁻ x, f x ^ (n+1) ∂μ := by |
| calc |
| _ = ∫⁻ y, f (e.symm y) ^ (n+1) ∂(c • μ) := by |
| simp only [lintegral_smul_measure, smul_eq_mul] |
| _ = _ := by |
| rw [← he, lintegral_map_equiv] |
| simp |
| have hchange' : (∫⁻ y, f (e.symm y) ^ (n+1) ∂μ) = |
| c⁻¹ * ∫⁻ x, f x ^ (n+1) ∂μ := by |
| rw [← hchange, ← mul_assoc, ENNReal.inv_mul_cancel hc0 hct, one_mul] |
| rw [hchange', pow_succ, mul_assoc, ← mul_assoc c, ENNReal.mul_inv_cancel hc0 hct, |
| one_mul] |
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| end ProjectionMoments |
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| end |
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| end OAI |
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