import OAI.Probability.GaussianReplacement.HybridModel namespace OAI open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology noncomputable section namespace CurrentProjection open InnerProductSpace open InformationTheory section KernelGeometry variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [finiteDimensionalRealE : FiniteDimensional ℝ E] [measurableSpaceE : MeasurableSpace E] [borelSpaceE : BorelSpace E] def rowSpan {r : ℕ} (A : Fin r → E) : Submodule ℝ E := Submodule.span ℝ (Set.range A) def rowProjectionL {r : ℕ} (A : Fin r → E) : E →L[ℝ] E := ∑ i, (‖gramSchmidt ℝ A i‖ ^ 2)⁻¹ • InnerProductSpace.rankOne ℝ (gramSchmidt ℝ A i) (gramSchmidt ℝ A i) omit measurableSpaceE borelSpaceE in lemma rowProjectionL_eq {r : ℕ} (A : Fin r → E) : rowProjectionL A = (rowSpan A).starProjection := by ext v have hgs := gramSchmidt_eq_sub_starProjection (Fin.snoc A v) (Fin.last r) rw [image_Iio_last] at hgs simp only [Fin.snoc_castSucc, Fin.snoc_last] at hgs have he := gramSchmidt_def ℝ (Fin.snoc A v) (Fin.last r) rw [Fin.Iio_last_eq_map, Finset.sum_map] at he simp only [Fin.castSuccEmb_apply, Fin.snoc_last, gramSchmidt_castSucc, Fin.snoc_castSucc, Submodule.starProjection_singleton] at he have hz : (rowSpan A).starProjection v = ∑ i, (⟪gramSchmidt ℝ A i, v⟫ / ‖gramSchmidt ℝ A i‖ ^ 2) • gramSchmidt ℝ A i := by dsimp only [rowSpan] have := hgs.symm.trans he exact (sub_right_inj).mp this rw [hz] simp only [rowProjectionL, sum_apply, smul_apply, InnerProductSpace.rankOne_apply, smul_smul, div_eq_mul_inv, mul_comm] lemma measurable_rowProjectionL (r : ℕ) : Measurable (rowProjectionL (E:=E) (r:=r)) := by apply Finset.measurable_sum intro i _ have hg := measurable_gramSchmidt (E:=E) i have hr : Measurable (fun A : Fin r → E => InnerProductSpace.rankOne ℝ (gramSchmidt ℝ A i) (gramSchmidt ℝ A i)) := by exact (show Continuous (fun x : E => InnerProductSpace.rankOne ℝ x x) from (ContinuousLinearMap.smulRightL ℝ E E).continuous₂.comp₂ (innerSL ℝ).continuous continuous_id).measurable.comp hg exact (hg.norm.pow_const 2 |>.inv).smul hr omit finiteDimensionalRealE measurableSpaceE borelSpaceE in lemma rowSpan_isometry {r : ℕ} (A : Fin r → E) (f : E ≃ₗᵢ[ℝ] E) : rowSpan (fun i => f (A i)) = (rowSpan A).map f.toLinearEquiv.toLinearMap := by rw [rowSpan, rowSpan, Submodule.map_span, ← Set.range_comp] rfl omit measurableSpaceE borelSpaceE in lemma rowProjectionL_isometry {r : ℕ} (A : Fin r → E) (f : E ≃ₗᵢ[ℝ] E) (v : E) : rowProjectionL (fun i => f (A i)) (f v) = f (rowProjectionL A v) := by rw [rowProjectionL_eq, rowProjectionL_eq, rowSpan_isometry] exact (f.toLinearIsometry.map_starProjection' (rowSpan A) v).symm omit measurableSpaceE borelSpaceE in lemma rowProjectionL_contraction {r : ℕ} (A : Fin r → E) (v : E) : ‖rowProjectionL A v‖ ≤ ‖v‖ := by rw [rowProjectionL_eq] exact (rowSpan A).norm_starProjection_apply_le v omit measurableSpaceE borelSpaceE in lemma rowProjectionL_perpendicular {r : ℕ} (A : Fin r → E) (v : E) : v - rowProjectionL A v ∈ (rowSpan A)ᗮ := by rw [rowProjectionL_eq] exact (rowSpan A).sub_starProjection_mem_orthogonal v omit measurableSpaceE borelSpaceE in lemma rowProjectionL_infDist {r : ℕ} (A : Fin r → E) (v : E) : ‖rowProjectionL A v‖ = Metric.infDist v ((rowSpan A)ᗮ : Set E) := by rw [rowProjectionL_eq] have he : v - (rowSpan A)ᗮ.starProjection v = (rowSpan A).starProjection v := by rw [Submodule.starProjection_orthogonal_val] abel rw [← he, Submodule.starProjection_minimal, Metric.infDist_eq_iInf] simp only [dist_eq_norm] rfl lemma gaussianRowIsometry_preserving (r : ℕ) (f : E ≃ₗᵢ[ℝ] E) : MeasurePreserving (fun A : Fin r → E => fun i => f (A i)) (Measure.pi fun _ => stdGaussian E) (Measure.pi fun _ => stdGaussian E) := measurePreserving_pi _ _ (fun _ => ⟨f.continuous.measurable, stdGaussian_map f⟩) lemma rowProjection_smallBall_eq {r : ℕ} (v w : E) (h : ‖v‖ = ‖w‖) (t : ℝ) : (Measure.pi fun _ : Fin r => stdGaussian E) {A | ‖rowProjectionL A v‖ ≤ t} = (Measure.pi fun _ : Fin r => stdGaussian E) {A | ‖rowProjectionL A w‖ ≤ t} := by let f := Submodule.reflection (ℝ ∙ (v-w))ᗮ have hf : f v = w := Submodule.reflection_sub h have hm : MeasurableSet {A : Fin r → E | ‖rowProjectionL A w‖ ≤ t} := measurableSet_le ((measurable_rowProjectionL r).apply_continuousLinearMap w).norm measurable_const rw [← (gaussianRowIsometry_preserving r f).measure_preimage hm.nullMeasurableSet] congr 1 ext A simp only [Set.mem_preimage, Set.mem_ofPred_eq, ← hf, rowProjectionL_isometry, f.norm_map] end KernelGeometry section GaussianRank variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma stdGaussian_submodule_null (V : Submodule ℝ E) (hV : Module.finrank ℝ V < Module.finrank ℝ E) : stdGaussian E (V:Set E) = 0 := by have hne : Vᗮ ≠ ⊥ := by intro h have hn := V.finrank_add_finrank_orthogonal rw [h, finrank_bot, add_zero] at hn omega obtain ⟨w, hw, hw0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hne let f : StrongDual ℝ E := innerSL ℝ w have hf : ‖f‖ ≠ 0 := by simpa [f] using norm_ne_zero_iff.mpr hw0 have hp : (‖f‖^2).toNNReal ≠ 0 := by exact ne_of_gt (Real.toNNReal_pos.mpr (sq_pos_of_ne_zero hf)) have hmap : (stdGaussian E).map f = gaussianReal 0 (‖f‖^2).toNNReal := by rw [IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian, variance_dual_stdGaussian] have hn : stdGaussian E {x | f x = 0} = 0 := by have hs : (gaussianReal 0 (‖f‖^2).toNNReal) {0} = 0 := by let := nullSingletonClass_gaussianReal (μ:=0) hp exact measure_singleton 0 rw [← hmap, Measure.map_apply f.measurable (measurableSet_singleton 0)] at hs exact hs apply le_antisymm _ bot_le calc stdGaussian E (V:Set E) ≤ stdGaussian E {x | f x = 0} := by apply measure_mono intro x hx change ⟪w,x⟫ = 0 simpa only [real_inner_comm] using hw x hx _ = 0 := hn lemma gaussianRows_ae_gramSchmidt_ne_zero (k : ℕ) (hk : k ≤ Module.finrank ℝ E) : ∀ᵐ v : Fin k → E ∂Measure.pi (fun _ => stdGaussian E), ∀ i, gramSchmidt ℝ v i ≠ 0 := by induction k with | zero => exact Filter.Eventually.of_forall fun _ i => Fin.elim0 i | succ k ih => have hm : MeasurableSet {v : Fin (k+1) → E | ∀ i, gramSchmidt ℝ v i ≠ 0} := by simpa only [Set.compl_ofPred, Set.iInter_ofPred] using (MeasurableSet.iInter fun i : Fin (k+1) => (measurableSet_eq_fun (measurable_gramSchmidt (E:=E) i) (measurable_const (a:=(0:E)))).compl) rw [ae_pi_snoc (stdGaussian E) hm] filter_upwards [ih (by omega)] with v hv let V := rowSpan v have hd : Module.finrank ℝ V ≤ k := by change Module.finrank ℝ (Submodule.span ℝ (Set.range v)) ≤ k exact (finrank_range_le_card (R:=ℝ) v).trans_eq (Fintype.card_fin k) have hn : stdGaussian E (V:Set E) = 0 := stdGaussian_submodule_null V (by omega) have ha : ∀ᵐ x ∂stdGaussian E, x ∉ V := by rw [ae_iff] simp only [not_not] change stdGaussian E (V:Set E) = 0 exact hn filter_upwards [ha] with x hx intro i induction i using Fin.lastCases with | last => apply norm_ne_zero_iff.mp rw [norm_gramSchmidt_eq_infDist, image_Iio_last] simp only [Fin.snoc_castSucc, Fin.snoc_last] exact ne_of_gt ((V.closed_of_finiteDimensional.notMem_iff_infDist_pos ⟨0, V.zero_mem⟩).mp hx) | cast j => simpa only [gramSchmidt_castSucc, Fin.snoc_castSucc] using hv j lemma gaussianRows_ae_rowSpan_finrank (k : ℕ) (hk : k ≤ Module.finrank ℝ E) : ∀ᵐ A : Fin k → E ∂Measure.pi (fun _ => stdGaussian E), Module.finrank ℝ (rowSpan A) = k := by filter_upwards [gaussianRows_ae_gramSchmidt_ne_zero k hk] with A hA have hi : LinearIndependent ℝ A := by apply Matrix.linearIndependent_of_det_gram_ne_zero rw [det_gram_eq_prod_gramSchmidt] exact Finset.prod_ne_zero_iff.mpr fun i _ => pow_ne_zero _ (norm_ne_zero_iff.mpr (hA i)) change Module.finrank ℝ (Submodule.span ℝ (Set.range A)) = k exact (finrank_span_eq_card hi).trans (Fintype.card_fin k) end GaussianRank section SphereTubes open scoped Pointwise variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] lemma sphereLaw_submodule_tube (V : Submodule ℝ E) (k : ℕ) (hk : Module.finrank ℝ V + k ≤ Module.finrank ℝ E) {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) : sphereLaw {v : Metric.sphere (0:E) 1 | Metric.infDist (v:E) (V:Set E) ≤ t} ≤ ENNReal.ofReal ((12:ℝ)^Module.finrank ℝ E * t^k) := by let L := V.toAffineSubspace let : Nonempty L := ⟨⟨0,V.zero_mem⟩⟩ have hd : (Module.finrank ℝ L.direction : ℝ) + k ≤ Module.finrank ℝ E := by dsimp only [L] rw [Submodule.toAffineSubspace_direction] exact_mod_cast hk have hsub : Set.Ioo (0:ℝ) 1 • (Subtype.val '' {v : Metric.sphere (0:E) 1 | Metric.infDist (v:E) (V:Set E) ≤ t}) ⊆ {v:E | Metric.infDist v L ≤ t} := by rintro _ ⟨c,hc,_,⟨v,hv,rfl⟩,rfl⟩ change Metric.infDist (c • (v:E)) (V:Set E) ≤ t rw [infDist_submodule_smul, abs_of_pos hc.1] exact (mul_le_mul_of_nonneg_left hv hc.1.le).trans (mul_le_of_le_one_left ht.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 k hd ht ht1).trans simp only [Real.rpow_natCast] rw [← ENNReal.ofReal_mul (by positivity)] apply ENNReal.ofReal_le_ofReal apply mul_le_mul_of_nonneg_right _ (pow_nonneg ht.le _) calc (3:ℝ)^Module.finrank ℝ L.direction * 4^Module.finrank ℝ E ≤ 3^Module.finrank ℝ E * 4^Module.finrank ℝ E := by exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num) (by exact_mod_cast (show (Module.finrank ℝ L.direction:ℝ) ≤ Module.finrank ℝ E by linarith))) (by positivity) _ = _ := by rw [← mul_pow]; norm_num lemma sphereLaw_rowProjection {r : ℕ} (A : Fin r → E) (hr : Module.finrank ℝ (rowSpan A) = r) {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) : sphereLaw {v : Metric.sphere (0:E) 1 | ‖rowProjectionL A (v:E)‖ ≤ t} ≤ ENNReal.ofReal ((12:ℝ)^Module.finrank ℝ E * t^r) := by simp only [rowProjectionL_infDist] apply sphereLaw_submodule_tube (rowSpan A)ᗮ r _ ht ht1 have h := (rowSpan A).finrank_add_finrank_orthogonal omega end SphereTubes section KernelSmallBall variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [nontrivialE : Nontrivial E] omit nontrivialE in lemma measurable_rowProjection_pair (r : ℕ) : Measurable (fun z : (Fin r → E) × E => rowProjectionL z.1 z.2) := by have he : Continuous (fun z : (E →L[ℝ] E) × E => z.1 z.2) := continuous_fst.clm_apply continuous_snd exact he.measurable.comp ((measurable_rowProjectionL r).comp measurable_fst |>.prodMk measurable_snd) lemma gaussian_rowProjection_smallBall {r : ℕ} (hr : r ≤ Module.finrank ℝ E) (v : E) (hv : ‖v‖ = 1) {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) : (Measure.pi fun _ : Fin r => stdGaussian E) {A | ‖rowProjectionL A v‖ ≤ t} ≤ ENNReal.ofReal ((12:ℝ)^Module.finrank ℝ E * t^r) := by let G : Measure (Fin r → E) := Measure.pi fun _ => stdGaussian E let T : Set ((Fin r → E) × Metric.sphere (0:E) 1) := {z | ‖rowProjectionL z.1 (z.2:E)‖ ≤ t} have hT : MeasurableSet T := measurableSet_le ((measurable_rowProjection_pair r).comp (measurable_fst.prodMk (measurable_subtype_coe.comp measurable_snd))).norm measurable_const have hconst : ∀ w : Metric.sphere (0:E) 1, G ((fun A => (A,w)) ⁻¹' T) = G {A | ‖rowProjectionL A v‖ ≤ t} := by intro w exact rowProjection_smallBall_eq (w:E) v ((mem_sphere_zero_iff_norm.mp w.property).trans hv.symm) t have heq : G.prod sphereLaw T = G {A | ‖rowProjectionL A v‖ ≤ t} := by rw [Measure.prod_apply_symm hT] simp only [hconst, lintegral_const, measure_univ, mul_one] rw [← heq, Measure.prod_apply hT] calc _ ≤ ∫⁻ _ : Fin r → E, ENNReal.ofReal ((12:ℝ)^Module.finrank ℝ E * t^r) ∂G := by apply lintegral_mono_ae filter_upwards [gaussianRows_ae_rowSpan_finrank r hr] with A hA exact sphereLaw_rowProjection A hA ht ht1 _ = _ := by simp end KernelSmallBall section ProjectionWitness variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma exists_unit_sphere_net {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) : ∃ s : Finset E, (∀ v ∈ s, ‖v‖ = 1) ∧ (s.card:ℝ) ≤ (3/t)^Module.finrank ℝ E ∧ ∀ v:E, ‖v‖=1 → ∃ w ∈ s, dist v w ≤ t := by classical have hbound : ∀ s : Finset E, (∀ v ∈ s, ‖v‖ = 1) → (∀ v ∈ s, ∀ w ∈ s, v ≠ w → t < dist v w) → (s.card:ℝ) ≤ (3/t)^Module.finrank ℝ E := by intro s hs hsep refine (card_le_volume_ratio s ht zero_le_one (fun v hv => (hs v hv).le) (fun v hv w hw hne => (hsep v hv w hw hne).le)).trans ?_ apply pow_le_pow_left₀ (by positivity) apply (div_le_div_iff₀ (by positivity : 0 < t/2) ht).mpr nlinarith obtain ⟨s,hs,hsep,hnet⟩ := exists_net_of_finite_packing_bound {v:E | ‖v‖=1} ht.le ⌈(3/t)^Module.finrank ℝ E⌉₊ (by intro s hs hsep exact_mod_cast (hbound s hs hsep).trans (Nat.le_ceil _)) exact ⟨s,hs,hbound s hs hsep,hnet⟩ omit [MeasurableSpace E] [BorelSpace E] in lemma exists_unit_projection_le_distance (P R : Submodule ℝ E) (hP : P ≠ ⊥) (u:E) (hu : ‖u‖=1) (huR : u ∈ Rᗮ) : ∃ v : P, ‖v‖=1 ∧ ‖R.starProjection (v:E)‖ ≤ Metric.infDist u (P:Set E) := by by_cases hp : P.starProjection u = 0 · obtain ⟨w,hw,hw0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hP let v : P := ⟨‖w‖⁻¹ • w,P.smul_mem _ hw⟩ have hv : ‖v‖=1 := by change ‖‖w‖⁻¹ • w‖=1 rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (inv_nonneg.mpr (norm_nonneg w)), inv_mul_cancel₀ (norm_ne_zero_iff.mpr hw0)] refine ⟨v,hv,?_⟩ rw [infDist_eq_norm_sub_projection, hp, sub_zero, hu] exact (R.norm_starProjection_apply_le (v:E)).trans_eq hv · let p := P.starProjection u let v : P := ⟨‖p‖⁻¹ • p,P.smul_mem _ (P.starProjection_apply_mem u)⟩ have hp0 : 0 < ‖p‖ := norm_pos_iff.mpr hp have hv : ‖v‖=1 := by change ‖‖p‖⁻¹ • p‖=1 rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hp0), inv_mul_cancel₀ hp0.ne'] have hip : ⟪u,p⟫ = ‖p‖^2 := by rw [← real_inner_self_eq_norm_sq] exact (P.inner_starProjection_left_eq_right u p).trans (by rw [P.starProjection_eq_self_iff.mpr (P.starProjection_apply_mem u)]) |>.symm have hiv : ⟪u,(v:E)⟫ = ‖p‖ := by change ⟪u, ‖p‖⁻¹ • p⟫ = ‖p‖ rw [real_inner_smul_right, hip] field_simp have hic : ⟪u,Rᗮ.starProjection (v:E)⟫ = ‖p‖ := by rw [← Rᗮ.inner_starProjection_left_eq_right, (Rᗮ.starProjection_eq_self_iff.mpr huR)] exact hiv have hcs : ‖p‖ ≤ ‖Rᗮ.starProjection (v:E)‖ := by have h := real_inner_le_norm u (Rᗮ.starProjection (v:E)) rwa [hic,hu,one_mul] at h have ht := Submodule.norm_sq_eq_add_norm_sq_starProjection (v:E) R have hpyt := Submodule.norm_sq_eq_add_norm_sq_starProjection u P have hdist : Metric.infDist u (P:Set E) = ‖Pᗮ.starProjection u‖ := by rw [infDist_eq_norm_sub_projection, Submodule.starProjection_orthogonal_val] refine ⟨v,hv,?_⟩ rw [hdist] change ‖(v:E)‖=1 at hv rw [hv] at ht rw [hu] at hpyt have hs := sq_le_sq₀ (norm_nonneg p) (norm_nonneg (Rᗮ.starProjection (v:E))) |>.mpr hcs dsimp only [p] at hs nlinarith [norm_nonneg (Pᗮ.starProjection u), norm_nonneg (R.starProjection (v:E))] end ProjectionWitness section KernelSelectedSmallBall variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] lemma kernel_selected_smallBall_raw {r : ℕ} (hr : r ≤ Module.finrank ℝ E) (P : Submodule ℝ E) (hP : P ≠ ⊥) (Q : Measure ((Fin r → E) × E)) (hQ : Q.fst = Measure.pi (fun _ => stdGaussian E)) (hsupport : ∀ᵐ z ∂Q, ‖z.2‖=1 ∧ z.2 ∈ (rowSpan z.1)ᗮ) {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1/2) : Q {z | Metric.infDist z.2 (P:Set E) ≤ t} ≤ ENNReal.ofReal ((3/t)^Module.finrank ℝ P * ((12:ℝ)^Module.finrank ℝ E * (2*t)^r)) := by classical obtain ⟨s,hs,hcard,hnet⟩ := exists_unit_sphere_net (E:=P) ht (by linarith) let T : P → Set (Fin r → E) := fun w => {A | ‖rowProjectionL A (w:E)‖ ≤ 2*t} have hTm : ∀ w, MeasurableSet (T w) := fun w => measurableSet_le ((measurable_rowProjectionL r).apply_continuousLinearMap (w:E)).norm measurable_const have hUnion : MeasurableSet (⋃ w ∈ s, T w) := s.finite_toSet.measurableSet_biUnion (fun w _ => hTm w) have hsub : {z | Metric.infDist z.2 (P:Set E) ≤ t} ≤ᵐ[Q] Prod.fst ⁻¹' (⋃ w ∈ s, T w) := by filter_upwards [hsupport] with z hz intro hzt obtain ⟨v,hv,hvp⟩ := exists_unit_projection_le_distance P (rowSpan z.1) hP z.2 hz.1 hz.2 obtain ⟨w,hw,hvw⟩ := hnet v hv refine Set.mem_preimage.mpr (Set.mem_iUnion.mpr ⟨w,Set.mem_iUnion.mpr ⟨hw,?_⟩⟩) change ‖rowProjectionL z.1 (w:E)‖ ≤ 2*t have hdist : ‖(w:E)-(v:E)‖ ≤ t := by rw [dist_eq_norm] at hvw change ‖(w-v:P)‖ ≤ t simpa only [norm_sub_rev] using hvw have hproj : ‖rowProjectionL z.1 (v:E)‖ ≤ t := by rw [rowProjectionL_eq] exact hvp.trans hzt calc ‖rowProjectionL z.1 (w:E)‖ ≤ ‖rowProjectionL z.1 (v:E)‖ + ‖rowProjectionL z.1 ((w:E)-(v:E))‖ := by rw [map_sub] exact norm_le_insert' _ _ _ ≤ t+t := add_le_add hproj ((rowProjectionL_contraction z.1 _).trans hdist) _ = 2*t := by ring calc Q {z | Metric.infDist z.2 (P:Set E) ≤ t} ≤ Q (Prod.fst ⁻¹' (⋃ w ∈ s, T w)) := measure_mono_ae hsub _ = (Measure.pi fun _:Fin r => stdGaussian E) (⋃ w ∈ s, T w) := by rw [←Measure.fst_apply hUnion,hQ] _ ≤ ∑ w ∈ s, (Measure.pi fun _:Fin r => stdGaussian E) (T w) := measure_biUnion_finset_le _ _ _ ≤ ∑ _w ∈ s, ENNReal.ofReal ((12:ℝ)^Module.finrank ℝ E * (2*t)^r) := by exact Finset.sum_le_sum fun w hw => gaussian_rowProjection_smallBall hr (w:E) (hs w hw) (by positivity) (by linarith) _ = ENNReal.ofReal ((s.card:ℝ)*((12:ℝ)^Module.finrank ℝ E*(2*t)^r)) := by simp [ENNReal.ofReal_mul, nsmul_eq_mul] _ ≤ _ := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hcard (by positivity)) end KernelSelectedSmallBall def kernelSmallBallConstant : ℝ := 12^17 lemma kernelSmallBallConstant_ge_two : 2 ≤ kernelSmallBallConstant := by norm_num [kernelSmallBallConstant] lemma kernelSmallBall_coefficient {d r p : ℕ} (hd : d ≤ 8*r) (hp : 2*p ≤ r) {t : ℝ} (ht : 0 < t) : (3/t)^p * ((12:ℝ)^d * (2*t)^r) ≤ (kernelSmallBallConstant*t)^(r-p) := by have hpr : p ≤ r := by omega have hr : r ≤ 2*(r-p) := by omega have hpp : p ≤ r-p := by omega have hdd : d ≤ 16*(r-p) := by omega have hcoef : (3:ℝ)^p * 12^d * 2^r ≤ kernelSmallBallConstant^(r-p) := by calc (3:ℝ)^p*12^d*2^r ≤ 3^(r-p)*12^(16*(r-p))*2^(2*(r-p)) := by apply mul_le_mul (mul_le_mul (pow_le_pow_right₀ (by norm_num) hpp) (pow_le_pow_right₀ (by norm_num) hdd) (by positivity) (by positivity)) (pow_le_pow_right₀ (by norm_num) hr) (by positivity) (by positivity) _ = _ := by rw [pow_mul, pow_mul, ←mul_pow, ←mul_pow] congr 1 norm_num [kernelSmallBallConstant] have htPow : t^r = t^(r-p)*t^p := by rw [←pow_add, Nat.sub_add_cancel hpr] calc (3/t)^p * ((12:ℝ)^d*(2*t)^r) = (3^p*12^d*2^r)*t^(r-p) := by rw [div_pow, mul_pow, htPow] field_simp _ ≤ kernelSmallBallConstant^(r-p)*t^(r-p) := mul_le_mul_of_nonneg_right hcoef (by positivity) _ = _ := (mul_pow _ _ _).symm section KernelInverseOneStep variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] lemma kernel_selected_scaled_smallBall {r : ℕ} (hr : r ≤ Module.finrank ℝ E) (hd : Module.finrank ℝ E ≤ 8*r) (P : Submodule ℝ E) (hp : 2*Module.finrank ℝ P ≤ r) (hmargin : 2 ≤ r-Module.finrank ℝ P) (Q : Measure ((Fin r → E) × E)) (hQ : Q.fst = Measure.pi (fun _ => stdGaussian E)) (hsupport : ∀ᵐ z ∂Q, ‖z.2‖=1 ∧ z.2 ∈ (rowSpan z.1)ᗮ) {s:ℝ} (hs : 0 C*f x) 1 (measurable_const.mul hf) (fun x => mul_nonneg hC.le (hf0 x)) 1 (by intro s hs hs1 simpa only [one_mul] using htube s hs hs1) norm_num at h ⊢ exact h have heq (x:X) : (f x)⁻¹ = C*(C*f x)⁻¹ := by rw [mul_inv, ←mul_assoc, mul_inv_cancel₀ hC.ne',one_mul] constructor · calc (∫⁻ x, ENNReal.ofReal ((f x)⁻¹) ∂Q) = ENNReal.ofReal C * ∫⁻ x, ENNReal.ofReal ((C*f x)⁻¹) ∂Q := by simp_rw [heq, ENNReal.ofReal_mul hC.le] exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top _ ≤ ENNReal.ofReal C * 5 := mul_le_mul' le_rfl hm _ = _ := by rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤5)]; norm_num; ring · have hn : Q {x | f x = 0} = 0 := by have hb (n:ℕ) : Q {x | f x=0} ≤ ENNReal.ofReal (((1/2:ℝ)^n)^2) := by calc Q {x | f x=0} ≤ Q {x | C*f x ≤ (1/2:ℝ)^n} := by apply measure_mono intro x hx simp only [Set.mem_ofPred_eq] at hx ⊢ rw [hx,mul_zero] positivity _ ≤ _ := htube _ (by positivity) (pow_le_one₀ (by norm_num) (by norm_num)) have hl : Tendsto (fun n:ℕ => ENNReal.ofReal (((1/2:ℝ)^n)^2)) atTop (𝓝 0) := by convert ENNReal.tendsto_ofReal ((tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)<1)).pow 2) using 1 simp exact le_antisymm (ge_of_tendsto' hl hb) bot_le have ha : ∀ᵐ x ∂Q, f x ≠ 0 := by rw [ae_iff] simpa only [not_not] using hn filter_upwards [ha] with x hx exact lt_of_le_of_ne (hf0 x) hx.symm end GenericInverseStep section KernelInverseStepFinal variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] lemma kernel_selected_inverse_step {r : ℕ} (hr : r ≤ Module.finrank ℝ E) (hd : Module.finrank ℝ E ≤ 8*r) (P : Submodule ℝ E) (hp : 2*Module.finrank ℝ P ≤ r) (hmargin : 2 ≤ r-Module.finrank ℝ P) (Q : Measure ((Fin r → E) × E)) [IsProbabilityMeasure Q] (hQ : Q.fst = Measure.pi (fun _ => stdGaussian E)) (hsupport : ∀ᵐ z ∂Q, ‖z.2‖=1 ∧ z.2 ∈ (rowSpan z.1)ᗮ) : (∫⁻ z, ENNReal.ofReal ((Metric.infDist z.2 (P:Set E))⁻¹) ∂Q) ≤ ENNReal.ofReal (5*kernelSmallBallConstant) ∧ ∀ᵐ z ∂Q, 0 < Metric.infDist z.2 (P:Set E) := by apply inverse_moment_of_scaled_smallBall Q (fun z => Metric.infDist z.2 (P:Set E)) ((Metric.continuous_infDist_pt _).measurable.comp measurable_snd) (fun _ => Metric.infDist_nonneg) (by have := kernelSmallBallConstant_ge_two; linarith) exact fun s hs hs1 => kernel_selected_scaled_smallBall hr hd P hp hmargin Q hQ hsupport hs hs1 end KernelInverseStepFinal end CurrentProjection end end OAI