| import OAI.Probability.GaussianRegression.AffineTubes |
|
|
| namespace OAI |
|
|
| /-! |
| Normalized spherical surface measure, cap bounds, and rotational symmetry. |
| -/ |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open InnerProductSpace |
|
|
| noncomputable section |
|
|
| universe u |
|
|
| namespace NoiselessRegression |
|
|
| section SphereBounds |
| open scoped Pointwise |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| def unitBallLaw : Measure E := |
| ((volume : Measure E) (Metric.ball 0 1))⁻¹ • volume.restrict (Metric.closedBall 0 1) |
|
|
| lemma unitBallLaw_closedBall (z : E) {r : ℝ} (hr : 0 < r) : |
| unitBallLaw (Metric.closedBall z r) ≤ ENNReal.ofReal (r ^ Module.finrank ℝ E) := by |
| have h0 : (volume : Measure E) (Metric.ball 0 1) ≠ 0 := |
| (Metric.measure_ball_pos volume 0 zero_lt_one).ne' |
| have ht : (volume : Measure E) (Metric.ball 0 1) ≠ ∞ := measure_ball_lt_top.ne |
| calc |
| _ ≤ ((volume : Measure E) (Metric.ball 0 1))⁻¹ * volume (Metric.closedBall z r) := by |
| exact mul_le_mul' le_rfl (Measure.restrict_apply_le _ _) |
| _ = _ := by |
| rw [Measure.addHaar_closedBall _ _ hr.le, mul_comm _ (_ * _), mul_assoc, |
| ENNReal.mul_inv_cancel h0 ht, mul_one] |
|
|
| lemma unitBallLaw_local : HasLocalMassBound (unitBallLaw : Measure E) |
| (Module.finrank ℝ E : ℝ) := by |
| intro z _ r hr |
| simpa only [Real.rpow_natCast] using unitBallLaw_closedBall z hr |
|
|
| lemma unitBallLaw_support : (unitBallLaw : Measure E).support ⊆ Metric.closedBall 0 1 := by |
| apply Measure.support_subset_of_isClosed Metric.isClosed_closedBall |
| rw [mem_ae_iff, unitBallLaw, Measure.smul_apply, smul_eq_mul, |
| Measure.restrict_apply Metric.isClosed_closedBall.measurableSet.compl] |
| simp |
|
|
| instance unitBallLaw_probability : IsProbabilityMeasure (unitBallLaw : Measure E) := by |
| constructor |
| rw [unitBallLaw, Measure.smul_apply, smul_eq_mul, Measure.restrict_apply_univ, |
| Measure.addHaar_closedBall _ _ (by norm_num : (0 : ℝ) ≤ 1)] |
| simp only [one_pow, ENNReal.ofReal_one, one_mul] |
| exact ENNReal.inv_mul_cancel (Metric.measure_ball_pos volume 0 zero_lt_one).ne' |
| measure_ball_lt_top.ne |
|
|
| instance sphereLaw_probability [Nontrivial E] : IsProbabilityMeasure (sphereLaw (E := E)) := by |
| constructor |
| rw [sphereLaw, Measure.smul_apply, smul_eq_mul] |
| apply ENNReal.inv_mul_cancel _ (measure_ne_top _ _) |
| rw [Measure.toSphere_apply_univ] |
| exact mul_ne_zero (by exact_mod_cast (Module.finrank_pos (R := ℝ) (M := E)).ne') |
| (Metric.measure_ball_pos volume 0 zero_lt_one).ne' |
|
|
| lemma sphereLaw_apply [Nontrivial E] {s : Set (Metric.sphere (0 : E) 1)} |
| (hs : MeasurableSet s) : |
| sphereLaw s = unitBallLaw (Set.Ioo (0 : ℝ) 1 • (Subtype.val '' s)) := by |
| have hd0 : (Module.finrank ℝ E : ℝ≥0∞) ≠ 0 := by |
| exact_mod_cast (Module.finrank_pos (R := ℝ) (M := E)).ne' |
| have hdt : (Module.finrank ℝ E : ℝ≥0∞) ≠ ∞ := ENNReal.natCast_ne_top _ |
| have hsub : Set.Ioo (0 : ℝ) 1 • (Subtype.val '' s) ⊆ Metric.closedBall (0 : E) 1 := by |
| rintro _ ⟨r, hr, _, ⟨x, hx, rfl⟩, rfl⟩ |
| rw [Metric.mem_closedBall, dist_zero_right, norm_smul, |
| (mem_sphere_zero_iff_norm.mp x.property), mul_one, Real.norm_eq_abs, |
| abs_of_pos hr.1] |
| exact hr.2.le |
| rw [sphereLaw, Measure.smul_apply, smul_eq_mul, Measure.toSphere_apply_univ, |
| Measure.toSphere_apply' _ hs, ENNReal.mul_inv (Or.inl hd0) (Or.inl hdt), unitBallLaw, Measure.smul_apply, |
| smul_eq_mul, Measure.restrict_apply' Metric.isClosed_closedBall.measurableSet, |
| Set.inter_eq_left.mpr hsub] |
| calc |
| _ = ((volume : Measure E) (Metric.ball 0 1))⁻¹ * |
| (((Module.finrank ℝ E : ℝ≥0∞))⁻¹ * (Module.finrank ℝ E)) * |
| volume (Set.Ioo (0 : ℝ) 1 • (Subtype.val '' s)) := by ring |
| _ = _ := by rw [ENNReal.inv_mul_cancel hd0 hdt, mul_one] |
|
|
| lemma sphereLaw_cap_raw [Nontrivial E] (z : E) {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) |
| (hd : 1 ≤ Module.finrank ℝ E) : |
| sphereLaw {s : Metric.sphere (0 : E) 1 | dist s.val z ≤ r} ≤ |
| ENNReal.ofReal ((3 : ℝ) * 4 ^ Module.finrank ℝ E) * |
| ENNReal.ofReal (r ^ (Module.finrank ℝ E - 1)) := by |
| classical |
| let L := (Submodule.span ℝ ({z} : Set E)).toAffineSubspace |
| let : Nonempty L := ⟨⟨0, (Submodule.span ℝ ({z} : Set E)).zero_mem⟩⟩ |
| have hl : Module.finrank ℝ L.direction ≤ 1 := by |
| dsimp only [L] |
| rw [Submodule.toAffineSubspace_direction] |
| simpa using (finrank_span_le_card (R := ℝ) ({z} : Set E)) |
| have hm : (Module.finrank ℝ L.direction : ℝ) + (Module.finrank ℝ E - 1 : ℕ) ≤ |
| (Module.finrank ℝ E : ℝ) := by |
| exact_mod_cast (by omega : Module.finrank ℝ L.direction + (Module.finrank ℝ E - 1) ≤ Module.finrank ℝ E) |
| have hsub : Set.Ioo (0 : ℝ) 1 • |
| (Subtype.val '' {s : Metric.sphere (0 : E) 1 | dist s.val z ≤ r}) ⊆ |
| {t | Metric.infDist t L ≤ r} := by |
| rintro _ ⟨c, hc, _, ⟨s, hs, rfl⟩, rfl⟩ |
| have hz : c • z ∈ L := Submodule.smul_mem _ c (Submodule.subset_span (by simp)) |
| exact (Metric.infDist_le_dist_of_mem hz).trans (by |
| rw [dist_smul₀, Real.norm_eq_abs, abs_of_pos hc.1] |
| exact (mul_le_mul_of_nonneg_left hs hc.1.le).trans (mul_le_of_le_one_left hr.le hc.2.le)) |
| rw [sphereLaw_apply (measurableSet_le (by fun_prop) measurable_const)] |
| apply (measure_mono hsub).trans |
| apply (unitBallLaw_local.affine_tube_pow unitBallLaw_support L _ hm hr hr1).trans |
| simp only [Real.rpow_natCast] |
| apply mul_le_mul' _ le_rfl |
| apply ENNReal.ofReal_le_ofReal |
| exact mul_le_mul_of_nonneg_right (by simpa using pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 3) hl) |
| (by positivity) |
|
|
| end SphereBounds |
|
|
| open scoped Pointwise |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| instance ambientSphereLaw_probability [Nontrivial E] : IsProbabilityMeasure (ambientSphereLaw (E := E)) := |
| inferInstanceAs (IsProbabilityMeasure ((sphereLaw (E := E)).map Subtype.val)) |
|
|
| lemma ambientSphereLaw_support : (ambientSphereLaw (E := E)).support ⊆ Metric.sphere 0 1 := by |
| apply Measure.support_subset_of_isClosed Metric.isClosed_sphere |
| rw [mem_ae_iff, ambientSphereLaw, Measure.map_apply (by fun_prop) |
| Metric.isClosed_sphere.measurableSet.compl] |
| have h : (Subtype.val : Metric.sphere (0 : E) 1 → E) ⁻¹' (Metric.sphere 0 1)ᶜ = ∅ := by |
| ext x |
| simp |
| rw [h, measure_empty] |
|
|
| lemma ambientSphereLaw_closedBall [Nontrivial E] (z : E) {r : ℝ} (hr : 0 < r) |
| (k : ℕ) (hk : k ≤ Module.finrank ℝ E - 1) : |
| ambientSphereLaw (Metric.closedBall z r) ≤ |
| ENNReal.ofReal ((3 : ℝ) * 4 ^ Module.finrank ℝ E) * ENNReal.ofReal (r^k) := by |
| by_cases hr1 : r ≤ 1 |
| · rw [ambientSphereLaw, Measure.map_apply (by fun_prop) Metric.isClosed_closedBall.measurableSet] |
| apply (sphereLaw_cap_raw z hr hr1 (Module.finrank_pos (R := ℝ) (M := E))).trans |
| apply mul_le_mul' le_rfl |
| exact ENNReal.ofReal_le_ofReal (pow_le_pow_of_le_one hr.le hr1 hk) |
| · have hr1 : 1 ≤ r := (le_of_not_ge hr1) |
| calc |
| _ ≤ 1 := prob_le_one |
| _ ≤ _ := by |
| apply one_le_mul |
| · rw [← ENNReal.ofReal_one] |
| apply ENNReal.ofReal_le_ofReal |
| have : (1 : ℝ) ≤ 4 ^ Module.finrank ℝ E := one_le_pow₀ (by norm_num) |
| linarith |
| · rw [← ENNReal.ofReal_one] |
| exact ENNReal.ofReal_le_ofReal (one_le_pow₀ hr1) |
|
|
| lemma terminal_cap_mass [Nontrivial E] (y z : E) {ε r : ℝ} (hε : 0 < ε) (hr : 0 < r) |
| (a : ℕ) (ha : 2*a ≤ Module.finrank ℝ E - 1) : |
| ((ambientSphereLaw (E := E)).restrict (Metric.closedBall y ε)) (Metric.closedBall z r) ≤ |
| ENNReal.ofReal ((3 : ℝ) * 4 ^ Module.finrank ℝ E) * |
| ENNReal.ofReal (ε^a) * ENNReal.ofReal (r^a) := by |
| rw [Measure.restrict_apply Metric.isClosed_closedBall.measurableSet] |
| by_cases h : r ≤ ε |
| · apply (measure_mono Set.inter_subset_left).trans |
| apply (ambientSphereLaw_closedBall z hr (2*a) ha).trans |
| rw [mul_assoc, ← ENNReal.ofReal_mul (pow_nonneg hε.le a)] |
| apply mul_le_mul' le_rfl |
| apply ENNReal.ofReal_le_ofReal |
| rw [mul_comm 2 a, pow_mul, pow_two] |
| exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hr.le h a) (by positivity) |
| · apply (measure_mono Set.inter_subset_right).trans |
| apply (ambientSphereLaw_closedBall y hε (2*a) ha).trans |
| rw [mul_assoc, ← ENNReal.ofReal_mul (pow_nonneg hε.le a)] |
| apply mul_le_mul' le_rfl |
| apply ENNReal.ofReal_le_ofReal |
| rw [mul_comm 2 a, pow_mul, pow_two] |
| exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hε.le (le_of_not_ge h) a) (by positivity) |
|
|
| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma chord_le_arccos (s t : Metric.sphere (0 : E) 1) : |
| dist s.val t.val ≤ Real.arccos ⟪s.val,t.val⟫ := by |
| have hs : ‖s.val‖ = 1 := mem_sphere_zero_iff_norm.mp s.property |
| have ht : ‖t.val‖ = 1 := mem_sphere_zero_iff_norm.mp t.property |
| have habs : |⟪s.val,t.val⟫| ≤ 1 := by |
| simpa [hs, ht] using abs_real_inner_le_norm s.val t.val |
| have hcos := Real.one_sub_sq_div_two_le_cos (x := Real.arccos ⟪s.val,t.val⟫) |
| rw [Real.cos_arccos (abs_le.mp habs).1 (abs_le.mp habs).2] at hcos |
| have hd : (dist s.val t.val)^2 = 2 - 2*⟪s.val,t.val⟫ := by |
| rw [dist_eq_norm, norm_sub_sq_real, hs, ht] |
| ring |
| nlinarith [dist_nonneg (x := s.val) (y := t.val), Real.arccos_nonneg ⟪s.val,t.val⟫] |
|
|
| lemma unitBallLaw_isometry_apply (e : E ≃ₗᵢ[ℝ] E) (s : Set E) : |
| unitBallLaw (e ⁻¹' s) = unitBallLaw s := by |
| have hb : e ⁻¹' Metric.closedBall (0 : E) 1 = Metric.closedBall 0 1 := by |
| ext x |
| simp [Metric.mem_closedBall, dist_zero_right] |
| have hset : e ⁻¹' s ∩ Metric.closedBall (0 : E) 1 = |
| e ⁻¹' (s ∩ Metric.closedBall 0 1) := by rw [Set.preimage_inter, hb] |
| rw [unitBallLaw, Measure.smul_apply, Measure.smul_apply, |
| Measure.restrict_apply' Metric.isClosed_closedBall.measurableSet, |
| Measure.restrict_apply' Metric.isClosed_closedBall.measurableSet, hset] |
| congr 1 |
| calc |
| _ = (Measure.map e volume) (s ∩ Metric.closedBall 0 1) := |
| (e.toMeasurableEquiv.measurableEmbedding.map_apply _ _).symm |
| _ = _ := by rw [e.measurePreserving.map_eq] |
|
|
| lemma ambientSphereLaw_apply [Nontrivial E] {s : Set E} (hs : MeasurableSet s) : |
| ambientSphereLaw s = unitBallLaw (Set.Ioo (0 : ℝ) 1 • (Metric.sphere 0 1 ∩ s)) := by |
| rw [ambientSphereLaw, Measure.map_apply (by fun_prop) hs, |
| sphereLaw_apply (measurable_subtype_coe hs)] |
| congr 2 |
| ext x |
| simp only [Set.mem_image, Set.mem_preimage, Set.mem_inter_iff] |
| constructor |
| · rintro ⟨y, hy, rfl⟩ |
| exact ⟨y.property, hy⟩ |
| · rintro ⟨hx, hs⟩ |
| exact ⟨⟨x,hx⟩,hs,rfl⟩ |
|
|
| lemma ambientSphereLaw_isometry [Nontrivial E] (e : E ≃ₗᵢ[ℝ] E) : |
| MeasurePreserving e (ambientSphereLaw (E := E)) (ambientSphereLaw (E := E)) := by |
| refine ⟨e.continuous.measurable, Measure.ext fun s hs => ?_⟩ |
| rw [Measure.map_apply e.continuous.measurable hs, |
| ambientSphereLaw_apply (e.continuous.measurable hs), ambientSphereLaw_apply hs] |
| have hset : Set.Ioo (0 : ℝ) 1 • (Metric.sphere 0 1 ∩ e ⁻¹' s) = |
| e ⁻¹' (Set.Ioo (0 : ℝ) 1 • (Metric.sphere 0 1 ∩ s)) := by |
| ext x |
| constructor |
| · rintro ⟨r, hr, y, ⟨hy, hys⟩, he⟩ |
| refine ⟨r,hr,e y,⟨?_,hys⟩,?_⟩ |
| · simpa only [mem_sphere_zero_iff_norm, e.norm_map] using hy |
| · simpa only [map_smul] using congrArg e he |
| · rintro ⟨r,hr,y,⟨hy,hys⟩,he⟩ |
| refine ⟨r,hr,e.symm y,⟨?_,?_⟩,?_⟩ |
| · simpa only [mem_sphere_zero_iff_norm, e.symm.norm_map] using hy |
| · simpa using hys |
| · simpa only [map_smul, e.symm_apply_apply] using congrArg e.symm he |
| rw [hset, unitBallLaw_isometry_apply] |
|
|
| lemma ambientSphereLaw_ae_norm [Nontrivial E] : |
| ∀ᵐ x : E ∂ambientSphereLaw, ‖x‖ = 1 := by |
| filter_upwards [Measure.support_mem_ae (μ := ambientSphereLaw (E := E))] with x hx |
| exact mem_sphere_zero_iff_norm.mp (ambientSphereLaw_support hx) |
|
|
| lemma sphere_inner_sq_integrable [Nontrivial E] (v : E) : |
| Integrable (fun x => ⟪v,x⟫^2) (ambientSphereLaw (E := E)) := by |
| apply (integrable_const (‖v‖^2)).mono' (by fun_prop) |
| filter_upwards [ambientSphereLaw_ae_norm (E := E)] with x hx |
| have h := abs_real_inner_le_norm v x |
| rw [hx, mul_one] at h |
| simpa only [Real.norm_eq_abs, abs_pow, sq_abs] using |
| (sq_le_sq₀ (abs_nonneg _) (norm_nonneg _) |>.2 h) |
|
|
| lemma sphere_inner_sq_eq [Nontrivial E] {v w : E} (h : ‖v‖ = ‖w‖) : |
| (∫ x, ⟪v,x⟫^2 ∂ambientSphereLaw) = ∫ x, ⟪w,x⟫^2 ∂ambientSphereLaw := by |
| let e := (ℝ ∙ (v-w))ᗮ.reflection |
| have he : e v = w := Submodule.reflection_sub h |
| have hi := (ambientSphereLaw_isometry e).integral_comp |
| e.toMeasurableEquiv.measurableEmbedding (fun x => ⟪w,x⟫^2) |
| rw [← he] at hi |
| simp only [e.inner_map_map] at hi |
| simpa only [he] using hi |
|
|
| lemma sphere_inner_sq_unit [Nontrivial E] {v : E} (hv : ‖v‖ = 1) : |
| (∫ x, ⟪v,x⟫^2 ∂ambientSphereLaw) = 1 / (Module.finrank ℝ E : ℝ) := by |
| let b := stdOrthonormalBasis ℝ E |
| have hsum : ∀ x : E, (∑ i, ⟪b i,x⟫^2) = ‖x‖^2 := by |
| intro x |
| simpa only [real_inner_comm x, pow_two, real_inner_self_eq_norm_sq] using |
| b.sum_inner_mul_inner x x |
| have h : (Module.finrank ℝ E : ℝ) * (∫ x, ⟪v,x⟫^2 ∂ambientSphereLaw) = 1 := by |
| calc |
| _ = ∑ i : Fin (Module.finrank ℝ E), ∫ x, ⟪b i,x⟫^2 ∂ambientSphereLaw := by |
| have heq : ∀ i, (∫ x, ⟪b i,x⟫^2 ∂ambientSphereLaw) = |
| ∫ x, ⟪v,x⟫^2 ∂ambientSphereLaw := fun i => |
| sphere_inner_sq_eq (by rw [b.norm_eq_one, hv]) |
| simp only [heq, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
| _ = ∫ x, ∑ i, ⟪b i,x⟫^2 ∂ambientSphereLaw := |
| (integral_finsetSum _ (fun i _ => sphere_inner_sq_integrable (b i))).symm |
| _ = ∫ _ : E, (1 : ℝ) ∂ambientSphereLaw := by |
| apply integral_congr_ae |
| filter_upwards [ambientSphereLaw_ae_norm (E := E)] with x hx |
| rw [hsum, hx, one_pow] |
| _ = 1 := by simp |
| have hd : (Module.finrank ℝ E : ℝ) ≠ 0 := by |
| exact_mod_cast (Module.finrank_pos (R := ℝ) (M := E)).ne' |
| exact (eq_div_iff hd).2 (by simpa only [mul_comm] using h) |
|
|
| lemma sphere_projection_sq_integrable [Nontrivial E] (U : Submodule ℝ E) : |
| Integrable (fun x => ‖U.orthogonalProjectionOnto x‖^2) (ambientSphereLaw (E := E)) := by |
| let b := stdOrthonormalBasis ℝ U |
| have hsum : ∀ x : E, (∑ i, ⟪(b i : E),x⟫^2) = ‖U.orthogonalProjectionOnto x‖^2 := by |
| intro x |
| have h := b.sum_inner_mul_inner (U.orthogonalProjectionOnto x) (U.orthogonalProjectionOnto x) |
| rw [real_inner_self_eq_norm_sq] at h |
| have hi : ∀ i, ⟪U.orthogonalProjectionOnto x, b i⟫ = ⟪b i, U.orthogonalProjectionOnto x⟫ := |
| fun i => real_inner_comm _ _ |
| simpa only [hi, U.inner_orthogonalProjectionOnto_eq_of_mem_left, pow_two] using h |
| simp_rw [← hsum] |
| exact integrable_finsetSum _ (fun i _ => sphere_inner_sq_integrable (b i : E)) |
|
|
| lemma sphere_projection_sq_integral [Nontrivial E] (U : Submodule ℝ E) : |
| (∫ x, ‖U.orthogonalProjectionOnto x‖^2 ∂ambientSphereLaw) = |
| (Module.finrank ℝ U : ℝ) / (Module.finrank ℝ E : ℝ) := by |
| let b := stdOrthonormalBasis ℝ U |
| have hsum : ∀ x : E, (∑ i, ⟪(b i : E),x⟫^2) = ‖U.orthogonalProjectionOnto x‖^2 := by |
| intro x |
| have h := b.sum_inner_mul_inner (U.orthogonalProjectionOnto x) (U.orthogonalProjectionOnto x) |
| rw [real_inner_self_eq_norm_sq] at h |
| have hi : ∀ i, ⟪U.orthogonalProjectionOnto x, b i⟫ = ⟪b i, U.orthogonalProjectionOnto x⟫ := |
| fun i => real_inner_comm _ _ |
| simpa only [hi, U.inner_orthogonalProjectionOnto_eq_of_mem_left, pow_two] using h |
| simp_rw [← hsum] |
| rw [integral_finsetSum _ (fun i _ => sphere_inner_sq_integrable (b i : E))] |
| have hi : ∀ i, (∫ x, ⟪(b i : E), x⟫^2 ∂ambientSphereLaw) = |
| 1 / (Module.finrank ℝ E : ℝ) := fun i => |
| sphere_inner_sq_unit (by exact b.norm_eq_one i) |
| simp only [hi, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, mul_one_div] |
|
|
| def sphereIsometry (e : E ≃ₗᵢ[ℝ] E) : Metric.sphere (0 : E) 1 ≃ₜ Metric.sphere (0 : E) 1 := |
| e.toHomeomorph.subtype (fun x => by |
| simp only [mem_sphere_zero_iff_norm] |
| change ‖x‖ = 1 ↔ ‖e x‖ = 1 |
| rw [e.norm_map]) |
|
|
| lemma sphereIsometry_preserving [Nontrivial E] (e : E ≃ₗᵢ[ℝ] E) : |
| MeasurePreserving (sphereIsometry e) (sphereLaw (E := E)) (sphereLaw (E := E)) := by |
| refine ⟨(sphereIsometry e).continuous.measurable, ?_⟩ |
| apply (MeasurableEmbedding.subtype_coe Metric.isClosed_sphere.measurableSet).map_injective |
| rw [Measure.map_map (by fun_prop) (by fun_prop)] |
| change (sphereLaw (E := E)).map (e ∘ Subtype.val) = ambientSphereLaw |
| rw [← Measure.map_map e.continuous.measurable (by fun_prop)] |
| exact (ambientSphereLaw_isometry e).map_eq |
|
|
| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma reflection_dist (U : Submodule ℝ E) (x : E) : |
| dist x (U.reflection x) = 2 * ‖Uᗮ.starProjection x‖ := by |
| rw [dist_eq_norm, Submodule.reflection_apply] |
| have heq : x - (2 • U.starProjection x - x) = (2 : ℝ) • Uᗮ.starProjection x := by |
| rw [Submodule.starProjection_orthogonal] |
| simp only [sub_apply, ContinuousLinearMap.id_apply, two_smul ℝ] |
| abel |
| rw [heq, norm_smul] |
| norm_num |
|
|
| lemma sphere_projection_large [Nontrivial E] (U : Submodule ℝ E) |
| (hU : 4 * Module.finrank ℝ U ≤ Module.finrank ℝ E) : |
| sphereLaw {x : Metric.sphere (0 : E) 1 | (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto x.val‖^2} ≤ |
| ENNReal.ofReal (5/16 : ℝ) := by |
| have hi := ofReal_integral_eq_lintegral_ofReal (sphere_projection_sq_integrable U) |
| (Filter.Eventually.of_forall fun x => sq_nonneg ‖U.orthogonalProjectionOnto x‖) |
| rw [sphere_projection_sq_integral] at hi |
| have hmark := meas_ge_le_lintegral_div |
| (μ := ambientSphereLaw (E := E)) (f := fun x => ENNReal.ofReal (‖U.orthogonalProjectionOnto x‖^2)) |
| (by fun_prop) (by norm_num : ENNReal.ofReal (4/5 : ℝ) ≠ 0) (ENNReal.ofReal_ne_top : ENNReal.ofReal (4/5 : ℝ) ≠ ∞) |
| rw [← hi] at hmark |
| have hset : {x : E | ENNReal.ofReal (4/5 : ℝ) ≤ ENNReal.ofReal (‖U.orthogonalProjectionOnto x‖^2)} = |
| {x : E | (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto x‖^2} := by |
| ext x |
| change ENNReal.ofReal (4/5 : ℝ) ≤ _ ↔ _ |
| exact ENNReal.ofReal_le_ofReal_iff (sq_nonneg _) |
| rw [hset, ambientSphereLaw, |
| (MeasurableEmbedding.subtype_coe Metric.isClosed_sphere.measurableSet).map_apply] at hmark |
| apply hmark.trans |
| have hd : (0 : ℝ) < Module.finrank ℝ E := by |
| exact_mod_cast Module.finrank_pos (R := ℝ) (M := E) |
| have hdiv : (Module.finrank ℝ U : ℝ) / (Module.finrank ℝ E : ℝ) ≤ 1/4 := by |
| rw [div_le_iff₀ hd] |
| have : (4 : ℝ) * Module.finrank ℝ U ≤ Module.finrank ℝ E := by exact_mod_cast hU |
| linarith |
| calc |
| _ ≤ ENNReal.ofReal (1/4 : ℝ) / ENNReal.ofReal (4/5 : ℝ) := by gcongr |
| _ = ENNReal.ofReal (5/16 : ℝ) := by |
| rw [← ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 4/5)] |
| norm_num |
|
|
| lemma sphere_reflection_success_bound [Nontrivial E] (U : Submodule ℝ E) |
| (hU : 4 * Module.finrank ℝ U ≤ Module.finrank ℝ E) |
| (S : Set (Metric.sphere (0 : E) 1)) (hS : NullMeasurableSet S sphereLaw) |
| (hsep : ∀ s ∈ S, sphereIsometry U.reflection s ∈ S → |
| dist s.val (U.reflection s.val) ≤ 1/5) : |
| sphereLaw S ≤ ENNReal.ofReal (21/32 : ℝ) := by |
| let f := sphereIsometry U.reflection |
| have hp := sphereIsometry_preserving U.reflection |
| have hi : S ∩ f ⁻¹' S ⊆ |
| {x : Metric.sphere (0 : E) 1 | (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto x.val‖^2} := by |
| intro s hs |
| have hd := hsep s hs.1 hs.2 |
| rw [reflection_dist] at hd |
| have hn := U.norm_sq_eq_add_norm_sq_starProjection s.val |
| rw [mem_sphere_zero_iff_norm.mp s.property] at hn |
| change (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto s.val‖^2 |
| have he : ‖U.starProjection s.val‖ = ‖U.orthogonalProjectionOnto s.val‖ := rfl |
| rw [he] at hn |
| nlinarith [norm_nonneg (Uᗮ.starProjection s.val)] |
| have hinter := (measure_mono hi).trans (sphere_projection_large U hU) |
| have heq := measure_union_add_inter₀ S (hS.preimage hp.quasiMeasurePreserving) |
| rw [hp.measure_preimage hS] at heq |
| calc |
| sphereLaw S = ENNReal.ofReal (1/2 : ℝ) * (sphereLaw S + sphereLaw S) := by |
| rw [← two_mul, ← mul_assoc] |
| have hc : ENNReal.ofReal (1/2 : ℝ) * 2 = 1 := by |
| rw [← ENNReal.ofReal_ofNat, ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1/2)] |
| norm_num |
| rw [hc, one_mul] |
| _ = ENNReal.ofReal (1/2 : ℝ) * |
| (sphereLaw (S ∪ f ⁻¹' S) + sphereLaw (S ∩ f ⁻¹' S)) := by rw [heq] |
| _ ≤ ENNReal.ofReal (1/2 : ℝ) * (1 + ENNReal.ofReal (5/16 : ℝ)) := by |
| exact mul_le_mul' le_rfl (add_le_add prob_le_one hinter) |
| _ = ENNReal.ofReal (21/32 : ℝ) := by |
| rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1) |
| (by norm_num : (0 : ℝ) ≤ 5/16), |
| ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1/2)] |
| norm_num |
|
|
| end NoiselessRegression |
|
|
| end |
|
|
| end OAI |
|
|
