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b317718 | 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 | import OAI.Probability.GaussianRegression.Blocks
import OAI.Probability.GaussianRegression.Sphere
namespace OAI
/-!
Absolute continuity of noiseless spherical observation laws.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise
noncomputable section
universe u
namespace NoiselessRegression
section ObservationAC
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
lemma projectionLaw_absolutelyContinuous {ν : Measure E} [IsFiniteMeasure ν]
{a m n : ℕ} {C : ℝ≥0∞} (hν : PolynomialMass ν a C) (hC : C ≠ 0) (hCt : C ≠ ∞)
(hsupport : ν.support ⊆ Metric.closedBall 0 1) (hn : 1 ≤ n) (hmargin : n+m ≤ a) :
projectionLaw ν m ≪ matrixLabelReference (E := E) m := by
let ν' := C⁻¹ • ν
have hCi : C⁻¹ ≠ ∞ := ENNReal.inv_ne_top.mpr hC
let : IsFiniteMeasure ν' := ⟨by simp only [ν', Measure.smul_apply, smul_eq_mul]; finiteness⟩
have hlocal : HasLocalMassBound ν' (a : ℝ) := by
intro z _ r hr
have hh := mul_le_mul' (le_rfl : C⁻¹ ≤ C⁻¹) (hν z r hr)
simpa only [ν', Measure.smul_apply, smul_eq_mul, ← mul_assoc,
ENNReal.inv_mul_cancel hC hCt, one_mul, Real.rpow_natCast] using hh
have hs : ν'.support ⊆ Metric.closedBall 0 1 :=
(Measure.smul_absolutelyContinuous.support_mono).trans hsupport
have hbound (S : Set ((Fin m → E) × (Fin m → ℝ))) :=
projectionLaw_set_bound hlocal hs m n hn (by exact_mod_cast hmargin) S
have he : projectionLaw ν' m = C⁻¹ • projectionLaw ν m := by
simp only [projectionLaw, ν', Measure.prod_smul_right]
exact Measure.map_smul _ (by fun_prop)
intro S hS
have h := hbound S
rw [hS, ENNReal.zero_rpow_of_pos (by
have : (1 : ℝ) < n+1 := by exact_mod_cast (by omega : 1 < n+1)
exact sub_pos.mpr ((div_lt_one (by positivity)).mpr this)), mul_zero, he,
Measure.smul_apply, smul_eq_mul] at h
exact (mul_eq_zero.mp (le_antisymm h bot_le)).resolve_left (ENNReal.inv_ne_zero.mpr hCt)
lemma sphere_observation_quasi [Nontrivial E] (hd : 3 ≤ Module.finrank ℝ E) :
Measure.QuasiMeasurePreserving
(fun z : (Metric.sphere (0 : E) 1) × E => (z.2, ⟪z.1.val,z.2⟫))
((sphereLaw (E := E)).prod (stdGaussian E)) ((stdGaussian E).prod volume) := by
have hac : projectionLaw (ambientSphereLaw (E := E)) 1 ≪ matrixLabelReference (E := E) 1 := by
apply projectionLaw_absolutelyContinuous (a := 2) (n := 1)
(C := ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E))
· intro z r hr
exact ambientSphereLaw_closedBall z hr 2 (by omega)
· positivity
· exact ENNReal.ofReal_ne_top
· exact ambientSphereLaw_support.trans Metric.sphere_subset_closedBall
· omega
· omega
have hp : Measure.QuasiMeasurePreserving
(fun z : (Fin 1 → E) × E => (z.1, fun j => ⟪z.2,z.1 j⟫))
((Measure.pi (fun _ : Fin 1 => stdGaussian E)).prod (ambientSphereLaw (E := E)))
(matrixLabelReference (E := E) 1) := ⟨by fun_prop, hac⟩
have h1 := ((measurePreserving_funUnique (stdGaussian E) (Fin 1)).symm
(MeasurableEquiv.funUnique (Fin 1) E)).prod
(measurable_subtype_coe.measurePreserving (sphereLaw (E := E)))
have h2 := (measurePreserving_funUnique (stdGaussian E) (Fin 1)).prod
(volume_preserving_funUnique (Fin 1) ℝ)
have h3 := h2.quasiMeasurePreserving.comp (hp.comp (h1.quasiMeasurePreserving.comp
Measure.measurePreserving_swap.quasiMeasurePreserving))
convert h3 using 1
ext z <;> simp [MeasurableEquiv.funUnique]
end ObservationAC
end NoiselessRegression
end
end OAI
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