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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 | import OAI.Probability.PosteriorReplicas.Gaussian.GaussianLogMoment
import OAI.Probability.PosteriorReplicas.Geometry.BaseFunProp
namespace OAI
open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace PosteriorReplicas
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
@[fun_prop] lemma measurable_gramSchmidtNormed {n : ℕ} (i : Fin n) :
Measurable (fun v : Fin n → E => gramSchmidtNormed ℝ v i) := by
unfold gramSchmidtNormed
exact ((PosteriorRegression.measurable_gramSchmidt i).norm.inv).smul
(PosteriorRegression.measurable_gramSchmidt i)
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma gramSchmidtNormed_mem [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
{n : ℕ} (U : Submodule ℝ E) (v : Fin n → E)
(hv : ∀ i, v i ∈ U) (i : Fin n) : gramSchmidtNormed ℝ v i ∈ U := by
have hs : Submodule.span ℝ (range v) ≤ U := Submodule.span_le.mpr (by
rintro _ ⟨j,rfl⟩
exact hv j)
apply hs
rw [← span_gramSchmidt ℝ v, ← span_gramSchmidtNormed_range]
exact Submodule.subset_span (mem_range_self i)
lemma ae_gaussian_projected_columns_independent (U : Submodule ℝ E) (n : ℕ)
(hn : n ≤ Module.finrank ℝ U) :
∀ᵐ v : Fin n → E ∂Measure.pi (fun _ => stdGaussian E),
LinearIndependent ℝ (fun i => U.starProjection (v i)) := by
have hmap : (Measure.pi (fun _ : Fin n => stdGaussian E)).map
(fun v i => U.orthogonalProjectionOnto (v i)) = Measure.pi (fun _ : Fin n => stdGaussian U) := by
rw [Measure.pi_map_pi (fun _ => by fun_prop)]
simp_rw [gaussian_orthogonalProjectionOnto]
have h := ae_gaussian_columns_independent (E := U) n hn
rw [← hmap] at h
have hz : ∀ᵐ v : Fin n → E ∂Measure.pi (fun _ => stdGaussian E),
LinearIndependent ℝ (fun i => U.orthogonalProjectionOnto (v i)) :=
(ae_map_iff (f := fun v : Fin n → E => fun i => U.orthogonalProjectionOnto (v i))
((Measurable.of_eval (fun i => U.orthogonalProjectionOnto.continuous.measurable.comp
(measurable_pi_apply i))).aemeasurable) isOpen_setOfPred_linearIndependent.measurableSet).mp h
filter_upwards [hz] with v hv
exact hv.map' U.subtype (LinearMap.ker_eq_bot.mpr U.injective_subtype)
def randomComplement {ℓ m k : ℕ} (V : Rows ℓ m) (e : Rows ℓ k) : Rows ℓ k :=
gramSchmidtNormed ℝ (fun i => kernelComponent V (e i))
@[fun_prop] lemma measurable_randomComplement {ℓ m k : ℕ} :
Measurable (fun z : Rows ℓ m × Rows ℓ k => randomComplement z.1 z.2) := by
unfold randomComplement
apply Measurable.of_eval
intro i
exact (measurable_gramSchmidtNormed i).comp (by fun_prop)
lemma randomComplement_orthogonal {ℓ m k : ℕ} {V : Rows ℓ m}
(hV : LinearIndependent ℝ V) (e : Rows ℓ k) (i : Fin k) (j : Fin m) :
⟪randomComplement V e i, V j⟫ = 0 := by
apply (Submodule.mem_orthogonal' (Submodule.span ℝ (range V))
(randomComplement V e i)).mp ?_ (V j) (Submodule.subset_span (mem_range_self j))
unfold randomComplement
apply gramSchmidtNormed_mem
intro a
rw [kernelComponent_eq_starProjection hV]
exact Submodule.starProjection_apply_mem _ _
lemma ae_randomComplement_orthonormal {ℓ m k : ℕ} {V : Rows ℓ m}
(hV : LinearIndependent ℝ V) (hkm : k+m ≤ ℓ) :
∀ᵐ e ∂gaussianRows ℓ k, Orthonormal ℝ (randomComplement V e) := by
let U := (Submodule.span ℝ (range V))ᗮ
have hdim : k ≤ Module.finrank ℝ U := by
have h := (Submodule.span ℝ (range V)).finrank_add_finrank_orthogonal
rw [finrank_euclideanSpace, finrank_span_eq_card hV, Fintype.card_fin] at h
simp only [Fintype.card_fin] at h
dsimp [U]
omega
filter_upwards [ae_gaussian_projected_columns_independent U k hdim] with e he
apply gramSchmidtNormed_orthonormal
simpa only [kernelComponent_eq_starProjection hV] using he
end PosteriorReplicas
end
end OAI
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