| import OAI.Probability.PosteriorReplicas.Gaussian.GaussianLogMoment |
| import OAI.Probability.PosteriorReplicas.Measures.ConstrainedDensityExpectation |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Filter Metric |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace PosteriorReplicas |
|
|
| lemma liftLabel_norm_coordinates {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (y : Labels k) : ‖liftLabel A y‖ = labelNorm ((pairingTriangular A).transpose⁻¹ *ᵥ y) := by |
| have h := pairingLift_energy hA y |
| have he : labelNorm ((pairingTriangular A).transpose⁻¹ *ᵥ y)^2 = |
| ∑ i, (((pairingTriangular A).transpose⁻¹ *ᵥ y) i)^2 := by |
| exact EuclideanSpace.real_norm_sq_eq _ |
| have hn : 0 ≤ labelNorm ((pairingTriangular A).transpose⁻¹ *ᵥ y) := norm_nonneg _ |
| change ‖liftLabel A y‖^2 = _ at h |
| nlinarith [norm_nonneg (liftLabel A y)] |
|
|
| lemma label_closedBall_volume {k : ℕ} (r : ℝ) : |
| (volume : Measure (Labels k)) {y | labelNorm y ≤ r} = |
| volume (closedBall (0 : Vec k) r) := by |
| rw [← (PiLp.volume_preserving_toLp (Fin k)).measure_preimage (by measurability)] |
| congr 1 |
| ext y |
| simp [labelNorm, mem_closedBall, dist_zero_right] |
|
|
| lemma label_closedBall_scale {k : ℕ} {r : ℝ} (hr : 0 ≤ r) : |
| (volume : Measure (Labels k)) {y | labelNorm y ≤ r} = |
| ENNReal.ofReal (r^k) * volume {y : Labels k | labelNorm y ≤ 1} := by |
| rw [label_closedBall_volume, label_closedBall_volume] |
| simpa using Measure.addHaar_closedBall_mul (volume : Measure (Vec k)) (0 : Vec k) hr (by norm_num : (0:ℝ) ≤ 1) |
|
|
| lemma label_ellipsoid_volume {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (r : ℝ) : |
| (volume : Measure (Labels k)) {y | ‖liftLabel A y‖ ≤ r} = |
| ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ r} := by |
| have hd := (pairingTriangular_det_pos hA).ne' |
| have hm := Real.map_matrix_volume_pi_eq_smul_volume_pi (M := (pairingTriangular A).transpose⁻¹) |
| (by simpa [Matrix.det_nonsing_inv] using inv_ne_zero hd) |
| have h := congrArg (fun μ : Measure (Labels k) => μ {y | labelNorm y ≤ r}) hm |
| rw [Measure.map_apply (by fun_prop) (measurableSet_le (by fun_prop) measurable_const), |
| Measure.smul_apply, smul_eq_mul] at h |
| simp only [Matrix.det_nonsing_inv, Ring.inverse_eq_inv, inv_inv, pairingTriangular_det, abs_of_nonneg (Real.sqrt_nonneg _)] at h |
| convert h using 1 |
| congr 1 |
| ext y |
| simp only [mem_ofPred_eq, mem_preimage, Matrix.toLin'_apply, liftLabel_norm_coordinates hA] |
| all_goals rfl |
|
|
| lemma label_ellipsoid_translate_volume {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (y : Labels k) (r : ℝ) : |
| (volume : Measure (Labels k)) {z | ‖liftLabel A (z-y)‖ ≤ r} = |
| ENNReal.ofReal (gramVolume A) * volume {z : Labels k | labelNorm z ≤ r} := by |
| have h := congrArg (fun μ : Measure (Labels k) => μ {z | ‖liftLabel A z‖ ≤ r}) |
| (map_add_right_eq_self (volume : Measure (Labels k)) (-y)) |
| rw [Measure.map_apply (by fun_prop) (measurableSet_le (by fun_prop) measurable_const)] at h |
| simpa only [preimage_ofPred_eq, sub_eq_add_neg, label_ellipsoid_volume hA r] using h |
|
|
| lemma projectedDensity_support_ellipsoid {d k : ℕ} (f : Vec d → ℝ≥0∞) |
| (A : Rows d k) (y : Labels k) (hy : projectedDensity .spherical f A y ≠ 0) : |
| ‖liftLabel A y‖ < 1 := by |
| unfold projectedDensity baseDensity at hy |
| split_ifs at hy with hA hy' <;> simp_all |
|
|
| lemma projectedDensity_support_volume {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (f : Vec d → ℝ≥0∞) : |
| (volume : Measure (Labels k)) {y | projectedDensity .spherical f A y ≠ 0} ≤ |
| ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ 1} := by |
| rw [← label_ellipsoid_volume hA 1] |
| exact measure_mono (fun y hy => (projectedDensity_support_ellipsoid f A y hy).le) |
|
|
| lemma gaussian_integrable_gramVolume {d k : ℕ} : |
| Integrable (gramVolume : Rows d k → ℝ) (gaussianRows d k) := by |
| have hp : Integrable (fun A : Rows d k => ∏ i, ‖A i‖) (gaussianRows d k) := |
| Integrable.fintype_prod (fun _ => (IsGaussian.integrable_id (μ := stdGaussian (Vec d))).norm) |
| apply hp.mono' (by fun_prop) |
| exact ae_of_all _ (fun A => by |
| rw [Real.norm_eq_abs, abs_of_nonneg (show 0 ≤ gramVolume A from Real.sqrt_nonneg _), gramVolume_eq_prod] |
| exact Finset.prod_le_prod₀ (fun _ _ => norm_nonneg _) |
| (fun index _ => norm_gramSchmidt_le_norm A index)) |
|
|
| lemma gaussian_label_support_volume_finite {d k : ℕ} (hd : 3 ≤ d) (hk : k < d) |
| {f : Vec d → ℝ≥0∞} (_hf : Measurable f) : |
| (∫⁻ A, (volume : Measure (Labels k)) {y | projectedDensity .spherical f A y ≠ 0} |
| ∂gaussianRows d k) < ∞ := by |
| apply lt_of_le_of_lt (lintegral_mono_ae (g := fun A : Rows d k => |
| ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ 1}) ?_) ?_ |
| · filter_upwards [ae_gaussian_rows_full_rank hd hk] with A hA |
| exact projectedDensity_support_volume hA f |
| · rw [lintegral_mul_const _ (by fun_prop)] |
| apply ENNReal.mul_lt_top |
| · exact (hasFiniteIntegral_iff_ofReal (ae_of_all _ (fun _ => Real.sqrt_nonneg _))).mp |
| gaussian_integrable_gramVolume.hasFiniteIntegral |
| · rw [label_closedBall_volume] |
| exact measure_closedBall_lt_top |
|
|
| end PosteriorReplicas |
| end |
|
|
| end OAI |
|
|
