| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.CubeContinuation |
| import OAI.Probability.MemoryPrecision.JointL1 |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
|
|
| noncomputable section |
| universe u v |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision |
| open NoiselessRegression |
|
|
| section RadialCoordinates |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| def radialShrink (c : ℝ) (v : E) : E := |
| (Real.sqrt (‖v‖^2 + c^2))⁻¹ • v |
|
|
| def radialExpand (c : ℝ) (u : E) : E := |
| (c / Real.sqrt (1 - ⟪u,u⟫)) • u |
|
|
| lemma radialShrink_mem_ball {c : ℝ} (hc : 0 < c) (v : E) : |
| radialShrink c v ∈ Metric.ball 0 1 := by |
| have hp : 0 < ‖v‖^2 + c^2 := by positivity |
| have hs : 0 < Real.sqrt (‖v‖^2 + c^2) := Real.sqrt_pos.2 hp |
| have hsq := Real.sq_sqrt hp.le |
| rw [Metric.mem_ball, dist_zero_right, radialShrink, norm_smul, |
| Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hs)] |
| have hlt : ‖v‖ < Real.sqrt (‖v‖^2 + c^2) := by nlinarith [sq_pos_of_pos hc] |
| simpa [div_eq_mul_inv, mul_comm] using (div_lt_one hs).mpr hlt |
|
|
| lemma radialExpand_shrink {c : ℝ} (hc : 0 < c) (v : E) : |
| radialExpand c (radialShrink c v) = v := by |
| let t := Real.sqrt (‖v‖^2+c^2) |
| have ht : 0 < t := Real.sqrt_pos.2 (by positivity) |
| have ht2 : t^2 = ‖v‖^2+c^2 := Real.sq_sqrt (by positivity) |
| have he : 1 - ⟪radialShrink c v,radialShrink c v⟫ = (c/t)^2 := by |
| simp only [radialShrink, inner_smul_left, inner_smul_right, RCLike.conj_to_real, |
| real_inner_self_eq_norm_sq] |
| change 1 - t⁻¹ * (t⁻¹ * ‖v‖^2) = (c/t)^2 |
| field_simp |
| nlinarith [ht2] |
| rw [radialExpand, he, Real.sqrt_sq (le_of_lt (div_pos hc ht)), radialShrink, |
| smul_smul] |
| change (c / (c/t) * t⁻¹) • v = v |
| have he' : c / (c/t) * t⁻¹ = 1 := by field_simp |
| rw [he', one_smul] |
|
|
| lemma radialExpand_differentiableOn (c : ℝ) : |
| DifferentiableOn ℝ (radialExpand (E := E) c) (Metric.ball 0 1) := by |
| intro u hu |
| have hu' : ‖u‖ < 1 := by simpa using hu |
| have hpos : 0 < 1 - ⟪u,u⟫ := by |
| rw [real_inner_self_eq_norm_sq] |
| nlinarith [norm_nonneg u] |
| apply DifferentiableAt.differentiableWithinAt |
| unfold radialExpand |
| apply DifferentiableAt.smul |
| · simp only [div_eq_mul_inv] |
| apply DifferentiableAt.const_mul |
| apply DifferentiableAt.inv |
| · apply DifferentiableAt.sqrt |
| · exact (differentiableAt_const 1).sub |
| (DifferentiableAt.inner ℝ differentiableAt_id differentiableAt_id) |
| · exact ne_of_gt hpos |
| · exact ne_of_gt (Real.sqrt_pos.2 hpos) |
| · exact differentiableAt_id |
| |
| variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| |
| lemma radialShrink_preimage_null {c : ℝ} (hc : 0 < c) {s : Set E} |
| (hs : volume s = 0) : volume (radialShrink c ⁻¹' s) = 0 := by |
| have hn : volume (s ∩ Metric.ball (0:E) 1) = 0 := |
| measure_mono_null Set.inter_subset_left hs |
| have hi := addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero volume |
| ((radialExpand_differentiableOn (E:=E) c).mono Set.inter_subset_right) hn |
| apply measure_mono_null _ hi |
| intro v hv |
| refine ⟨radialShrink c v, ⟨hv, radialShrink_mem_ball hc v⟩, ?_⟩ |
| exact radialExpand_shrink hc v |
| end RadialCoordinates |
|
|
| section Projection |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| lemma orthogonal_normalized_preimage_null (K : Submodule ℝ E) [Nontrivial Kᗮ] |
| {s : Set K} (hs : MeasurableSet s) (hs0 : volume s = 0) : |
| volume {x : E | ‖x‖⁻¹ • K.orthogonalProjectionOnto x ∈ s} = 0 := by |
| let e : (Kᗮ × K) ≃ₗ[ℝ] E := |
| (LinearEquiv.prodComm ℝ Kᗮ K).trans (K.prodEquivOfIsCompl Kᗮ K.isCompl_orthogonal) |
| let t : Set (Kᗮ × K) := {x | radialShrink ‖x.1‖ x.2 ∈ s} |
| have ht : MeasurableSet t := by |
| apply hs.preimage |
| dsimp [radialShrink] |
| fun_prop |
| have ht0 : ((volume : Measure Kᗮ).prod (volume : Measure K)) t = 0 := by |
| rw [Measure.prod_apply ht] |
| rw [← lintegral_zero (μ := (volume : Measure Kᗮ))] |
| apply lintegral_congr_ae |
| have hk : ∀ᵐ k : Kᗮ ∂volume, k ≠ 0 := by |
| simpa only [Ne, ← Set.mem_singleton_iff] using |
| (measure_eq_zero_iff_ae_notMem.mp (measure_singleton (0:Kᗮ))) |
| filter_upwards [hk] with k hk |
| exact radialShrink_preimage_null (norm_pos_iff.mpr hk) hs0 |
| have hn : MeasurableSet {x : E | ‖x‖⁻¹ • K.orthogonalProjectionOnto x ∈ s} := by |
| apply hs.preimage |
| exact measurable_id.norm.inv.smul K.orthogonalProjectionOnto.continuous.measurable |
| have hnt : e ⁻¹' {x : E | ‖x‖⁻¹ • K.orthogonalProjectionOnto x ∈ s} = t := by |
| ext x |
| have hp : K.orthogonalProjectionOnto (e x) = x.2 := by |
| change K.orthogonalProjectionOnto ((x.2:E)+(x.1:E)) = x.2 |
| rw [map_add, Submodule.orthogonalProjectionOnto_mem_subspace_eq_self, |
| Submodule.orthogonalProjectionOnto_apply_of_mem_orthogonal x.1.property, add_zero] |
| have hnorm : ‖e x‖ = Real.sqrt (‖x.2‖^2 + ‖x.1‖^2) := by |
| have h := norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (x.2:E) (x.1:E) |
| (Submodule.inner_right_of_mem_orthogonal x.2.property x.1.property) |
| have he : ‖e x‖^2 = ‖x.2‖^2 + ‖x.1‖^2 := by |
| change ‖(x.2:E)+(x.1:E)‖^2 = _ |
| simpa only [pow_two, Submodule.norm_coe] using h |
| rw [← he, Real.sqrt_sq (norm_nonneg _)] |
| simp only [Set.mem_preimage, Set.mem_ofPred_eq, hp, hnorm, t, radialShrink] |
| let : ((((volume : Measure Kᗮ).prod (volume : Measure K)).map e)).IsAddHaarMeasure := |
| Measure.MapLinearEquiv.isAddHaarMeasure _ e |
| have heac : (volume : Measure E) ≪ |
| (((volume : Measure Kᗮ).prod (volume : Measure K)).map e) := |
| Measure.absolutelyContinuous_isAddHaarMeasure _ _ |
| apply heac |
| rw [Measure.map_apply (f := e) e.toContinuousLinearEquiv.continuous.measurable hn, hnt, ht0] |
| |
| lemma sphere_projection_ac (K : Submodule ℝ E) [Nontrivial E] [Nontrivial Kᗮ] : |
| ((sphereLaw (E:=E)).map (fun x => K.orthogonalProjectionOnto x.val)) ≪ |
| (volume : Measure K) := by |
| apply Measure.AbsolutelyContinuous.mk |
| intro s hs hs0 |
| have hm : Measurable (fun x : Metric.sphere (0:E) 1 => K.orthogonalProjectionOnto x.val) := |
| K.orthogonalProjectionOnto.continuous.measurable.comp measurable_subtype_coe |
| rw [Measure.map_apply hm hs, sphereLaw_apply (hm hs)] |
| have hn := orthogonal_normalized_preimage_null K hs hs0 |
| have hcone : volume (Set.Ioo (0:ℝ) 1 • |
| (Subtype.val '' ((fun x : Metric.sphere (0:E) 1 => K.orthogonalProjectionOnto x.val) ⁻¹' s))) = 0 := by |
| apply measure_mono_null _ hn |
| rintro _ ⟨r, hr, _, ⟨x, hx, rfl⟩, rfl⟩ |
| change ‖r • x.val‖⁻¹ • K.orthogonalProjectionOnto (r • x.val) ∈ s |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos hr.1, |
| mem_sphere_zero_iff_norm.mp x.property, mul_one, map_smul, smul_smul, |
| inv_mul_cancel₀ hr.1.ne', one_smul] |
| exact hx |
| rw [unitBallLaw, Measure.smul_apply, smul_eq_mul, |
| Measure.restrict_apply' Metric.isClosed_closedBall.measurableSet] |
| rw [measure_mono_null Set.inter_subset_left hcone, mul_zero] |
|
|
| lemma sphere_rows_ac [Nontrivial E] {m : ℕ} (A : Fin m → E) |
| (hA : LinearIndependent ℝ A) (hm : m < Module.finrank ℝ E) : |
| ((sphereLaw (E:=E)).map |
| (fun x : Metric.sphere (0:E) 1 => fun j => ⟪x.val,A j⟫)) ≪ |
| (volume : Measure (Fin m → ℝ)) := by |
| let K : Submodule ℝ E := Submodule.span ℝ (Set.range A) |
| have hd : Module.finrank ℝ K = m := by |
| simpa only [K, Fintype.card_fin] using finrank_span_eq_card hA |
| have hK : K ≠ ⊤ := by |
| intro he |
| rw [he, finrank_top] at hd |
| omega |
| have : Nontrivial Kᗮ := Submodule.nontrivial_iff_ne_bot.mpr |
| (fun h => hK (Submodule.orthogonal_eq_bot_iff.mp h)) |
| let L : K →ₗ[ℝ] (Fin m → ℝ) := |
| LinearMap.pi (fun j => (innerSL ℝ (A j)).toLinearMap.comp K.subtype) |
| have hLi : Function.Injective L := by |
| apply (injective_iff_map_eq_zero L).mpr |
| intro x hx |
| have hle : K ≤ (innerSL ℝ (x:E)).toLinearMap.ker := by |
| apply Submodule.span_le.mpr |
| rintro _ ⟨j,rfl⟩ |
| change ⟪(x:E),A j⟫ = 0 |
| have hj := congrFun hx j |
| change ⟪A j,(x:E)⟫ = 0 at hj |
| rwa [real_inner_comm] |
| have hz : ⟪(x:E),(x:E)⟫ = 0 := hle x.property |
| exact Subtype.ext (inner_self_eq_zero.mp hz) |
| have hdim : Module.finrank ℝ K = Module.finrank ℝ (Fin m → ℝ) := by |
| simpa using hd |
| let e : K ≃ₗ[ℝ] (Fin m → ℝ) := |
| LinearEquiv.ofBijective L ⟨hLi, |
| (LinearMap.injective_iff_surjective_of_finrank_eq_finrank hdim).mp hLi⟩ |
| have he : (fun x : Metric.sphere (0:E) 1 => fun j => ⟪x.val,A j⟫) = |
| e ∘ (fun x : Metric.sphere (0:E) 1 => K.orthogonalProjectionOnto x.val) := by |
| funext x j |
| change ⟪x.val,A j⟫ = ⟪A j,(K.orthogonalProjectionOnto x.val:E)⟫ |
| have hp := K.inner_orthogonalProjectionOnto_eq_of_mem_left |
| ⟨A j, Submodule.subset_span ⟨j,rfl⟩⟩ x.val |
| change ⟪A j,(K.orthogonalProjectionOnto x.val:E)⟫ = ⟪A j,x.val⟫ at hp |
| rw [hp, real_inner_comm] |
| rw [he, ← Measure.map_map (f := fun x : Metric.sphere (0:E) 1 => |
| K.orthogonalProjectionOnto x.val) (g := e) |
| e.toContinuousLinearEquiv.continuous.measurable (by fun_prop)] |
| apply ((sphere_projection_ac K).map |
| (show Measurable e from e.toContinuousLinearEquiv.continuous.measurable)).trans |
| let : ((volume : Measure K).map e).IsAddHaarMeasure := |
| Measure.MapLinearEquiv.isAddHaarMeasure _ e |
| exact Measure.absolutelyContinuous_isAddHaarMeasure _ _ |
| end Projection |
|
|
| theorem spherical_projection_ac {d m : ℕ} (hd : 2 ≤ d) (hm : m < d) |
| (A : Fin m → Space d) (hA : LinearIndependent ℝ A) |
| (μ : Measure (Space d)) (hμ : μ ≪ UniformSphere d) : |
| μ.map (fun s => fun j => ⟪s,A j⟫) ≪ (volume : Measure (Fin m → ℝ)) := by |
| have : NeZero d := ⟨by omega⟩ |
| apply (hμ.map (by fun_prop)).trans |
| have h := sphere_rows_ac A hA (by simpa using hm) |
| simpa only [UniformSphere, Kernel.UniformSphere, sphereLaw, Measure.map_map (by fun_prop : |
| Measurable (fun s : Space d => fun j => ⟪s,A j⟫)) measurable_subtype_coe, |
| Function.comp_def] using h |
| end MemoryPrecision |
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|
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| end |
|
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| end OAI |
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|
