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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | 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
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