| import OAI.Geometry.ProjectionBodies.PositionMatrices |
|
|
| namespace OAI |
|
|
| universe uX |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped RealInnerProductSpace |
| namespace PettyProjection |
| open Spherical (Sphere mean sigma gaugeField gaugeMoment absolutePair) |
|
|
| /-- The genuine unpolarized cosine pairing of two (not necessarily even) measures. -/ |
| def measurePair {n : ℕ} (μ ν : Measure (Space n)) : ℝ := |
| ∫ x,∫ y,|⟪x,y⟫| ∂ν ∂μ |
|
|
| def Represents {n : ℕ} (μ : Measure (Space n)) (g : Seminorm ℝ (Space n)) (k : ℕ) (s : ℝ) : Prop := |
| ∀ (φ : Space n → ℝ),Continuous φ → (∀ x,φ (-x)=φ x) → |
| (∀ r : ℝ,0 < r → ∀ x,φ (r • x)=r*φ x) → |
| (∫ x,φ x ∂μ)=s*mean (fun u : Sphere n => φ (gaugeField g k u)) |
|
|
| lemma represents_pair {n : ℕ} [NeZero n] {μ ν : Measure (Space n)} |
| (hν : Integrable (fun x : Space n => ‖x‖) ν) |
| (f g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k) {s t : ℝ} |
| (hf : Represents μ f k s) (hg : Represents ν g k t) : |
| measurePair μ ν=s*t*absolutePair (gaugeField f k) (gaugeField g k) := by |
| have hcos : measurePair μ ν=∫ x,cosineSeminorm ν hν x ∂μ := by |
| unfold measurePair |
| apply integral_congr_ae |
| filter_upwards [] with x |
| change (∫ y,|⟪x,y⟫| ∂ν)=∫ y,|⟪y,x⟫| ∂ν |
| simp only [real_inner_comm x] |
| rw [hcos,hf _ (Spherical.seminorm_continuous n _) (fun x => map_neg_eq_map _ x) |
| (fun r hr x => by rw [map_smul_eq_mul,Real.norm_of_nonneg hr.le])] |
| have he (u : Sphere n) : cosineSeminorm ν hν (gaugeField f k u)= |
| t*mean (fun v : Sphere n => |⟪gaugeField f k u,gaugeField g k v⟫|) := by |
| have hh := hg (fun x => |⟪gaugeField f k u,x⟫|) (by fun_prop) |
| (fun x => by simp) (fun r hr x => by simp only [inner_smul_right,abs_mul,abs_of_pos hr]) |
| change (∫ x,|⟪x,gaugeField f k u⟫| ∂ν)=_ |
| simpa only [real_inner_comm (gaugeField f k u)] using hh |
| simp_rw [he] |
| rw [mean,integral_const_mul,mul_assoc] |
| congr 1 |
| congr 1 |
| exact (integral_prod _ (Spherical.gaugeField_absolute_integrable f g hk)).symm |
|
|
| namespace RelativeGauge |
| variable {n : ℕ} [NeZero n] {μ : Measure (Space n)} |
|
|
| lemma scaled_represents |
| (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x) |
| (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (hgn : supportFunctional μ g.body=1) |
| (hmin : ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 → |
| Spherical.objective (regime n) g.toSeminorm≤Spherical.objective (regime n) q.toSeminorm) |
| {a : ℝ} (ha : 0 < a) : |
| Represents μ (g.scale a ha).toSeminorm (Scalar.exponent n) |
| (supportFunctional μ (g.scale a ha).body/gaugeMoment (g.scale a ha).toSeminorm (Scalar.exponent n-1)) := by |
| intro φ hφ he hh |
| have ht : 0 < gaugeMoment (g.scale a ha).toSeminorm (Scalar.exponent n-1) := |
| Position.mean_power_pos _ ((g.scale a ha).ne_zero top_ne_bot) _ |
| have h := scaled_field_representation hμ hq g hgn hmin ha φ hφ he hh |
| rw [div_mul_eq_mul_div] |
| exact (eq_div_iff ht.ne').mpr h |
| |
| end RelativeGauge |
| end PettyProjection |
| end |
| |
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped RealInnerProductSpace |
| namespace PettyProjection.Position |
| open Spherical (Sphere norm_coe) |
| |
| lemma compact_inward {X : Type uX} [TopologicalSpace X] [CompactSpace X] |
| {a b : X → ℝ} (ha : Continuous a) (hb : Continuous b) |
| (ha0 : ∀ x,0≤a x) (hab : ∀ x,a x=0 → b x<0) : |
| ∃ ε : ℝ,0<ε ∧ ∀ x,0<a x-ε*b x := by |
| let K : Set X := {x | 0≤b x} |
| have hK : IsCompact K := (isClosed_le continuous_const hb).isCompact |
| by_cases hne : K.Nonempty |
| · obtain ⟨x,hx,hmin⟩ := hK.exists_isMinOn hne ha.continuousOn |
| have hax : 0<a x := lt_of_le_of_ne (ha0 x) (fun h => (hab x h.symm).not_ge hx) |
| obtain ⟨M,hM,hbound⟩ := (isCompact_univ.image hb).isBounded.exists_pos_norm_le |
| let ε := a x/(2*M) |
| have hε : 0<ε := div_pos hax (by positivity) |
| refine ⟨ε,hε,fun y => ?_⟩ |
| by_cases hy : y∈K |
| · have hmy : a x≤a y := hmin hy |
| have hbM : b y≤M := (le_abs_self _).trans (by simpa only [Real.norm_eq_abs] using hbound (b y) ⟨y,mem_univ _,rfl⟩) |
| have hh : ε*M=a x/2 := by dsimp [ε]; field_simp |
| have hle := mul_le_mul_of_nonneg_left hbM hε.le |
| rw [hh] at hle |
| linarith |
| · have hby : b y<0 := lt_of_not_ge hy |
| have hneg := mul_neg_of_pos_of_neg hε hby |
| linarith [ha0 y] |
| · refine ⟨1,zero_lt_one,fun y => ?_⟩ |
| have hby : b y<0 := by |
| by_contra h |
| exact hne ⟨y,le_of_not_gt h⟩ |
| linarith [ha0 y] |
| |
| lemma inward_perturbation {n : ℕ} [NeZero n] {A F : Mat n} (hA : A∈domain n) |
| (hF : Symmetric F) (htr : tr F=0) |
| (hker : ∀ e : Sphere n,op A e=0 → quad F e<0) : |
| ∃ ε : ℝ,0<ε ∧ A-ε • F∈domain n ∧ ∀ x : Space n,x≠0 → 0<quad (A-ε • F) x := by |
| have hca : Continuous (fun u : Sphere n => quad A u) := |
| by simp_rw [quad_eq]; fun_prop |
| have hcf : Continuous (fun u : Sphere n => quad F u) := |
| by simp_rw [quad_eq]; fun_prop |
| obtain ⟨ε,hε,hp⟩ := compact_inward hca hcf (fun u => hA.1.2 u) |
| (fun u hu => hker u (op_kernel_of_quad_zero hA.1 hu)) |
| have hq : ∀ x : Space n,x≠0 → 0<quad (A-ε • F) x := by |
| intro x hx |
| let u : Sphere n := ⟨‖x‖⁻¹ • x,by rw [mem_sphere_zero_iff_norm,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)]⟩ |
| have hh := hp u |
| rw [← quad_smul,← quad_sub] at hh |
| have he : quad (A-ε • F) u=‖x‖⁻¹^2*quad (A-ε • F) x := by |
| simp only [quad,u,map_smul,inner_smul_left,inner_smul_right,conj_trivial] |
| ring |
| rw [he] at hh |
| exact (mul_pos_iff.mp hh).resolve_right (fun h => (sq_nonneg ‖x‖⁻¹).not_gt h.1) |>.2 |
| refine ⟨ε,hε,⟨⟨?_,?_⟩,?_⟩,hq⟩ |
| · intro i j |
| change A (i,j)-ε*F (i,j)=A (j,i)-ε*F (j,i) |
| rw [hA.1.1 i j,hF i j] |
| · intro x |
| by_cases hx : x=0 |
| · simp [hx,quad] |
| · exact (hq x hx).le |
| · change (∑ i,(A (i,i)-ε*F (i,i)))=1 |
| rw [Finset.sum_sub_distrib,← Finset.mul_sum] |
| change tr A-ε*tr F=1 |
| rw [hA.2,htr,mul_zero,sub_zero] |
| |
| /-- The topological mechanism. The field is genuine and continuous; |
| its negativity on every null direction forces an interior zero. -/ |
| theorem traceOne_field_zero {n : ℕ} [NeZero n] (F : {A : Mat n // A∈domain n} → Mat n) |
| (hFc : Continuous F) (hFs : ∀ A,Symmetric (F A)) (hFt : ∀ A,tr (F A)=0) |
| (hFk : ∀ A (e : Sphere n),op A.val e=0 → quad (F A) e<0) : |
| ∃ A : {A : Mat n // A∈domain n},F A=0 ∧ Function.Injective (op A.val) := by |
| let p := FixedPoint.nearest (domain n) domain_nonempty domain_compact.isComplete domain_convex |
| have hp (x : Mat n) : p x∈domain n := FixedPoint.nearest_mem _ _ _ _ x |
| have hpc : Continuous p := (FixedPoint.nearest_lipschitz _ _ _ _).continuous |
| let f : Mat n → Mat n := fun x => p (x-F ⟨p x,hp x⟩) |
| have hfc : Continuous f := hpc.comp (continuous_id.sub (hFc.comp (hpc.subtype_mk hp))) |
| obtain ⟨A,hA,hfix⟩ := FixedPoint.fixedPoint_compact_convex domain_nonempty domain_compact domain_convex |
| hfc.continuousOn (fun x _ => hp _) |
| let a : {A : Mat n // A∈domain n} := ⟨A,hA⟩ |
| have hpa : p A=A := FixedPoint.nearest_eq_of_mem _ _ _ _ hA |
| have hfix' : p (A-F a)=A := by simpa only [f,hpa] using hfix |
| have hi := FixedPoint.nearest_inner (domain n) domain_nonempty domain_compact.isComplete domain_convex (A-F a) |
| change ∀ z∈domain n,⟪A-F a-p (A-F a),z-p (A-F a)⟫≤0 at hi |
| rw [hfix'] at hi |
| obtain ⟨ε,hε,hB,_⟩ := inward_perturbation hA (hFs a) (hFt a) (hFk a) |
| have hh := hi (A-ε • F a) hB |
| have hf0 : F a=0 := by |
| simp only [sub_sub_cancel_left,inner_neg_left,inner_neg_right,inner_smul_right,real_inner_self_eq_norm_sq] at hh |
| have hs : ‖F a‖^2≤0 := nonpos_of_mul_nonpos_left (by nlinarith only [hh]) hε |
| have hn : ‖F a‖=0 := by nlinarith [norm_nonneg (F a)] |
| exact norm_eq_zero.mp hn |
| refine ⟨a,hf0,?_⟩ |
| apply (op A).ker_eq_bot.mp |
| apply LinearMap.ker_eq_bot'.mpr |
| intro x hx |
| change op A x=0 at hx |
| by_contra hx0 |
| let e : Sphere n := ⟨‖x‖⁻¹ • x,by rw [mem_sphere_zero_iff_norm,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx0)]⟩ |
| have he : op A e=0 := by change op A (‖x‖⁻¹ • x)=0; rw [map_smul,hx,smul_zero] |
| have hh := hFk a e he |
| rw [hf0] at hh |
| simp only [quad_eq] at hh |
| simp at hh |
| end PettyProjection.Position |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped RealInnerProductSpace |
| namespace PettyProjection.Position |
| open Spherical (Sphere mean seminorm_continuous) |
| open RelativeGauge |
| variable {n : ℕ} [NeZero n] {μ : Measure (Space n)} |
|
|
| abbrev Domain (n : ℕ) := {A : Mat n |
| instance domain_compactSpace : CompactSpace (Domain n) := isCompact_iff_compactSpace.mp domain_compact |
|
|
| def domainMap (A : Domain n) : NonzeroMap n := ⟨op A.val,domain_range_ne_bot A.property⟩ |
| omit [NeZero n] in |
| lemma domainMap_continuous : Continuous (domainMap : Domain n → NonzeroMap n) := |
| (op_continuous.comp continuous_subtype_val).subtype_mk _ |
|
|
| lemma optimizer_kernel (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) (A : Domain n) |
| (e : Space n) (he : op A.val e=0) : optimizer hμ hq b (domainMap A) e=0 := by |
| let V := LinearMap.range (op A.val).toLinearMap |
| have hp : V.starProjection e=0 := by |
| apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (V.zero_mem) |
| rintro y ⟨z,rfl⟩ |
| simp only [sub_zero] |
| have hs := symmetric_op A.property.1.1 e z |
| change ⟪op A.val e,z⟫=⟪e,op A.val z⟫ at hs |
| rw [he,inner_zero_left] at hs |
| exact hs.symm |
| have hh := (optimizer hμ hq b (domainMap A)).projection e |
| change (optimizer hμ hq b (domainMap A)) (V.starProjection e)=_ at hh |
| rw [hp] at hh |
| exact hh.symm.trans (map_zero _) |
|
|
| lemma optimizer_joint_continuous (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) : |
| Continuous (fun p : Domain n × Sphere n => optimizer hμ hq b (domainMap p.1) p.2) := by |
| exact ((optimizerSphere_continuous hμ hq b).comp domainMap_continuous).fst'.eval continuous_snd |
| |
| def positionField (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) (k : ℕ) (A : Domain n) : Mat n := |
| momentField (optimizer hμ hq b (domainMap A)).toSeminorm k |
| |
| lemma positionField_continuous (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) (k : ℕ) : |
| Continuous (positionField hμ hq b k) := by |
| have hg := optimizer_joint_continuous hμ hq b |
| have hm : Continuous (fun A : Domain n => mean (fun u : Sphere n => optimizer hμ hq b (domainMap A) u^k)) := by |
| simpa only [Measure.restrict_univ,mean] using |
| (continuous_parametric_integral_of_continuous (μ := Spherical.sigma n) (hg.pow k) isCompact_univ) |
| have hij (i j : Fin n) : Continuous (fun A : Domain n => mean (fun u : Sphere n => |
| optimizer hμ hq b (domainMap A) u^k*(u:Space n) i*(u:Space n) j)) := by |
| have hc : Continuous (fun p : Domain n × Sphere n => |
| optimizer hμ hq b (domainMap p.1) p.2^k*(p.2:Space n) i*(p.2:Space n) j) := |
| (hg.pow k).mul (by fun_prop) |>.mul (by fun_prop) |
| simpa only [Measure.restrict_univ,mean] using |
| (continuous_parametric_integral_of_continuous (μ := Spherical.sigma n) hc isCompact_univ) |
| apply (PiLp.continuous_toLp 2 _).comp |
| apply continuous_pi |
| intro ij |
| exact ((hij ij.1 ij.2).div hm (fun A => |
| (mean_power_pos _ ((optimizer hμ hq b (domainMap A)).ne_zero (domainMap A).property) k).ne')).sub continuous_const |
|
|
| /-- A genuine nonsingular affine position for the variational optimizer. The |
| trace-one representative is used here; determinant normalization is a scalar |
| change and is performed in the global assembly. -/ |
| theorem exists_optimizer_position (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) {k : ℕ} (hk : 0<k) : |
| ∃ A : Domain n,Function.Injective (op A.val) ∧ |
| momentField (optimizer hμ hq b (domainMap A)).toSeminorm k=0 := by |
| obtain ⟨A,hA,hinj⟩ := traceOne_field_zero (positionField hμ hq b k) |
| (positionField_continuous hμ hq b k) |
| (fun A => field_symmetric _ k) |
| (fun A => field_trace _ ((optimizer hμ hq b (domainMap A)).ne_zero (domainMap A).property) k) |
| (fun A e he => field_kernel ((optimizer hμ hq b (domainMap A)).ne_zero (domainMap A).property) |
| hk e (optimizer_kernel hμ hq b A e he)) |
| exact ⟨A,hinj,hA⟩ |
| end PettyProjection.Position |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped RealInnerProductSpace Pointwise |
| namespace PettyProjection |
| variable {n : ℕ} |
|
|
| lemma volumeReal_linearEquiv (T : Space n ≃L[ℝ] Space n) (K : Set (Space n)) : |
| volume.real (T '' K)=|LinearMap.det T.toLinearMap| * volume.real K := by |
| unfold Measure.real |
| rw [Measure.addHaar_image_continuousLinearEquiv,ENNReal.toReal_mul,ENNReal.toReal_ofReal (abs_nonneg _)] |
|
|
| lemma linearEquiv_body (T : Space n ≃L[ℝ] Space n) {K : Set (Space n)} (hK : IsConvexBody K) : |
| IsConvexBody (T '' K) := by |
| refine ⟨hK.1.image T.continuous,hK.2.1.linear_image T.toLinearMap,?_⟩ |
| change (interior (T.toHomeomorph '' K)).Nonempty |
| rw [← T.toHomeomorph.image_interior] |
| exact hK.2.2.image T |
|
|
| lemma linearEquiv_det_ne_zero (T : Space n ≃L[ℝ] Space n) : LinearMap.det T.toLinearMap≠0 := |
| T.toLinearEquiv.isUnit_det'.ne_zero |
| |
| def volumeFactor (T : Space n ≃L[ℝ] Space n) : ℝ := |LinearMap.det T.toLinearMap|^((n:ℝ)⁻¹) |
| lemma volumeFactor_pos (T : Space n ≃L[ℝ] Space n) : 0 < volumeFactor T := |
| Real.rpow_pos_of_pos (abs_pos.mpr (linearEquiv_det_ne_zero T)) _ |
| |
| lemma normalizedVolume_linearEquiv (T : Space n ≃L[ℝ] Space n) (K : Set (Space n)) : |
| normalizedVolume (T '' K)=volumeFactor T*normalizedVolume K := by |
| rw [normalizedVolume,volumeRoot,volumeReal_linearEquiv, |
| Real.mul_rpow (abs_nonneg _) measureReal_nonneg] |
| unfold volumeFactor normalizedVolume volumeRoot |
| ring |
| |
| lemma volumeFactor_symm (T : Space n ≃L[ℝ] Space n) : volumeFactor T.symm=(volumeFactor T)⁻¹ := by |
| unfold volumeFactor |
| rw [show T.symm.toLinearMap=(T.toLinearEquiv.symm : Space n →ₗ[ℝ] Space n) from rfl, |
| LinearEquiv.det_coe_symm,abs_inv,Real.inv_rpow (abs_nonneg _)] |
| |
| lemma symmetric_equiv_symm (T : Space n ≃L[ℝ] Space n) (hT : T.toLinearMap.IsSymmetric) : |
| T.symm.toLinearMap.IsSymmetric := by |
| intro x y |
| change ⟪T.symm x,y⟫=⟪x,T.symm y⟫ |
| have hh := hT (T.symm x) (T.symm y) |
| change ⟪T (T.symm x),T.symm y⟫=⟪T.symm x,T (T.symm y)⟫ at hh |
| simpa only [T.apply_symm_apply] using hh.symm |
| |
| lemma functional_map_symmetric {μ : Measure (Space n)} (T : Space n ≃L[ℝ] Space n) |
| (hT : T.toLinearMap.IsSymmetric) {K : Set (Space n)} (hK : IsCompact K) (hne : K.Nonempty) : |
| supportFunctional (μ.map T) K=supportFunctional μ (T '' K) := by |
| change supportFunctional (μ.map T.toContinuousLinearMap) K=_ |
| rw [functional_map T.toContinuousLinearMap hK] |
| unfold supportFunctional |
| apply integral_congr_ae |
| filter_upwards [] with x |
| change support K (T.toContinuousLinearMap x)=support (T.toContinuousLinearMap '' K) x |
| rw [support_linear_image hK hne T.toContinuousLinearMap,hT.clm_adjoint_eq] |
| |
| lemma measurePair_cosine {μ ν : Measure (Space n)} |
| (hν : Integrable (fun x : Space n => ‖x‖) ν) : |
| measurePair μ ν=∫ x,cosineSeminorm ν hν x ∂μ := by |
| unfold measurePair |
| apply integral_congr_ae |
| filter_upwards [] with x |
| change (∫ y,|⟪x,y⟫| ∂ν)=∫ y,|⟪y,x⟫| ∂ν |
| simp only [real_inner_comm x] |
| |
| lemma measurePair_map_symmetric {μ ν : Measure (Space n)} |
| (hν : Integrable (fun x : Space n => ‖x‖) ν) |
| (T : Space n ≃L[ℝ] Space n) (hT : T.toLinearMap.IsSymmetric) : |
| measurePair (μ.map T) (ν.map T.symm)=measurePair μ ν := by |
| change measurePair (μ.map T.toContinuousLinearMap) (ν.map T.symm.toContinuousLinearMap)=_ |
| rw [measurePair_cosine (norm_integrable_map hν T.symm.toContinuousLinearMap), |
| integral_map T.toContinuousLinearMap.continuous.measurable.aemeasurable (Spherical.seminorm_continuous n _).aestronglyMeasurable, |
| measurePair_cosine hν] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| rw [cosine_map ν hν T.symm.toContinuousLinearMap,(symmetric_equiv_symm T hT).clm_adjoint_eq] |
| simp |
| |
| end PettyProjection |
| end |
| |
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped RealInnerProductSpace |
| namespace PettyProjection.Position |
| open Spherical |
| variable {n : ℕ} [NeZero n] |
| |
| lemma mean_inner_sq (x : Space n) : (n:ℝ)*mean (fun u : Sphere n => ⟪x,(u:Space n)⟫^2)=‖x‖^2 := by |
| by_cases hx : x=0 |
| · simp [hx,mean] |
| let e : Sphere n := ⟨‖x‖⁻¹ • x,by rw [mem_sphere_zero_iff_norm,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)]⟩ |
| have he := sphere_kernel_moment (0 : Seminorm ℝ (Space n)) 0 e (by simp) |
| simp only [Nat.cast_zero,add_zero,pow_zero,one_mul,mean_const] at he |
| have hh : mean (fun u : Sphere n => ⟪(e:Space n),(u:Space n)⟫^2)= |
| ‖x‖⁻¹^2*mean (fun u : Sphere n => ⟪x,(u:Space n)⟫^2) := by |
| simp only [e,inner_smul_left,conj_trivial,mul_pow,mean,integral_const_mul] |
| rw [hh] at he |
| have hn := norm_ne_zero_iff.mpr hx |
| field_simp at he |
| nlinarith [sq_pos_of_ne_zero hn] |
| |
| omit [NeZero n] in |
| lemma symmetric_eq_zero_of_quad {A : Mat n} (hA : Symmetric A) (hq : ∀ x,quad A x=0) : A=0 := by |
| have hd (i : Fin n) : A (i,i)=0 := by rw [← quad_basis,hq] |
| apply WithLp.ofLp_injective |
| funext ij |
| have hh := hq (EuclideanSpace.single ij.1 1+EuclideanSpace.single ij.2 1) |
| have he := quad_basis_add A ij.1 ij.2 1 1 |
| simp only [one_smul,one_pow,one_mul,hd,hA ij.2 ij.1] at he |
| change A ij=0 |
| linarith |
| |
| lemma constantMomentField_zero : momentField (0 : Seminorm ℝ (Space n)) 0=0 := by |
| apply symmetric_eq_zero_of_quad (field_symmetric _ _) (fun x => ?_) |
| rw [field_quad] |
| simp only [pow_zero,one_mul,mean_const,div_one] |
| have hm := mean_inner_sq x |
| have hn : (n:ℝ)≠0 := Nat.cast_ne_zero.mpr (NeZero.ne n) |
| rw [sub_eq_zero,eq_div_iff hn] |
| linarith |
| |
| lemma coordinateProduct_mean (i j : Fin n) : mean (coordinateProduct n i j)=(if i=j then 1 else 0)/(n:ℝ) := by |
| have hh := congrArg (fun A : Mat n => A (i,j)) (constantMomentField_zero (n := n)) |
| change mean (fun u : Sphere n => (0 : Seminorm ℝ (Space n)) u^0*(u:Space n) i*(u:Space n) j)/ |
| mean (fun u : Sphere n => (0 : Seminorm ℝ (Space n)) u^0)-(if i=j then 1 else 0)/(n:ℝ)=0 at hh |
| simp only [pow_zero,one_mul,mean_const,div_one] at hh |
| exact sub_eq_zero.mp hh |
| |
| /-- The matrix isotropy condition gives precisely the actual H₂ projection, |
| not a replacement cancellation condition. -/ |
| lemma field_zero_harmonic {n : ℕ} (hn : 2≤n) [NeZero n] (g : Seminorm ℝ (Space n)) |
| (hg : g≠0) (k : ℕ) (hF : momentField g k=0) : |
| (harmonicSpace n 2).starProjection (toH n |
| (restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k)))=0 := by |
| let f : C(Sphere n,ℝ) := restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k) |
| let c := mean (fun u : Sphere n => g u^k) |
| let v := toH n f-toH n (ContinuousMap.const (Sphere n) c) |
| have ho : v∈(lowSpace n)ᗮ := by |
| apply (Submodule.mem_orthogonal _ _).mpr |
| intro w hw |
| induction hw using Submodule.span_induction with |
| | mem w hw => |
| obtain ⟨⟨i,j⟩,rfl⟩ := hw |
| rw [real_inner_comm,show v=toH n (f-ContinuousMap.const (Sphere n) c) by simp [v],inner_toH] |
| change mean (fun u : Sphere n => (g u^k-c)*((u:Space n) i*(u:Space n) j))=0 |
| have hij := congrArg (fun A : Mat n => A (i,j)) hF |
| change mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) j)/c-(if i=j then 1 else 0)/(n:ℝ)=0 at hij |
| have hc : c≠0 := (mean_power_pos g hg k).ne' |
| have hv := (div_eq_iff hc).mp (sub_eq_zero.mp hij) |
| change _=0 |
| simp_rw [sub_mul] |
| have hi : Integrable (fun u : Sphere n => g u^k*((u:Space n) i*(u:Space n) j)) (sigma n) := |
| continuous_integrable (((seminorm_continuous n g).comp continuous_subtype_val).pow k |>.mul |
| (by fun_prop : Continuous (fun u : Sphere n => (u:Space n) i*(u:Space n) j))) |
| have hi' : Integrable (fun u : Sphere n => c*((u:Space n) i*(u:Space n) j)) (sigma n) := |
| continuous_integrable (by fun_prop) |
| rw [mean,integral_sub hi hi',integral_const_mul] |
| change mean (fun u : Sphere n => g u^k*((u:Space n) i*(u:Space n) j))-c*mean (coordinateProduct n i j)=0 |
| rw [coordinateProduct_mean] |
| have he : mean (fun u : Sphere n => g u^k*((u:Space n) i*(u:Space n) j))= |
| mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) j) := by congr 1;funext u;ring |
| rw [he,hv]; ring |
| | zero => simp |
| | add x y _ _ hx hy => simp only [inner_add_left,hx,hy,add_zero] |
| | smul a x _ hx => simp only [inner_smul_left,conj_trivial,hx,mul_zero] |
| apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (Submodule.zero_mem _) |
| intro w hw |
| simp only [sub_zero] |
| have hvol := (Submodule.mem_orthogonal _ _).mp ho w (harmonicSpace_two_le_lowSpace n hw) |
| have hconst : ⟪toH n (ContinuousMap.const (Sphere n) c),w⟫=0 := |
| harmonic_inner_ne hn (d := 0) (by decide) (constant_mem_harmonicSpace_zero n c) hw |
| change ⟪toH n f,w⟫=0 |
| dsimp only [v] at hvol |
| have hconst' : ⟪w,toH n (ContinuousMap.const (Sphere n) c)⟫=0 := |
| (real_inner_comm _ _).trans hconst |
| rw [inner_sub_right,hconst',sub_zero] at hvol |
| exact (real_inner_comm _ _).trans hvol |
| end PettyProjection.Position |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped RealInnerProductSpace |
| namespace PettyProjection |
| namespace RelativeGauge |
| variable {n : ℕ} [NeZero n] |
|
|
| def IsMinimizer (μ : Measure (Space n)) (b : Bool) (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : Prop := |
| supportFunctional μ g.body=1 ∧ |
| ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 → |
| Spherical.objective b g.toSeminorm≤Spherical.objective b q.toSeminorm |
|
|
| omit [NeZero n] in |
| lemma copy_minimizer {V : Submodule ℝ (Space n)} (hV : V=⊤) |
| {μ : Measure (Space n)} {b : Bool} (g : RelativeGauge V) |
| (hgn : supportFunctional μ g.body=1) |
| (hmin : ∀ q : RelativeGauge V,supportFunctional μ q.body=1 → |
| Spherical.objective b g.toSeminorm≤Spherical.objective b q.toSeminorm) : |
| ∃ f : RelativeGauge (⊤ : Submodule ℝ (Space n)),f.toSeminorm=g.toSeminorm ∧ IsMinimizer μ b f := by |
| subst V |
| exact ⟨g,rfl,hgn,hmin⟩ |
|
|
| /-- Actual full-dimensional position of the logarithmic/linear optimizer, |
| with no determinant normalization needed in the later weighted assembly. -/ |
| theorem exists_symmetric_position {μ : Measure (Space n)} |
| (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x) : |
| ∃ T : Space n ≃L[ℝ] Space n,T.toLinearMap.IsSymmetric ∧ |
| ∃ g : RelativeGauge (⊤ : Submodule ℝ (Space n)),IsMinimizer (μ.map T) (regime n) g ∧ |
| Position.momentField g.toSeminorm (Scalar.exponent n)=0 := by |
| obtain ⟨A,hinj,hF⟩ := Position.exists_optimizer_position hμ hq (regime n) (exponent_pos n) |
| have hsurj : Function.Surjective (Position.op A.val) := |
| LinearMap.surjective_of_injective (f := (Position.op A.val).toLinearMap) hinj |
| have hfull : LinearMap.range (Position.op A.val).toLinearMap=⊤ := LinearMap.range_eq_top.mpr hsurj |
| let T : Space n ≃L[ℝ] Space n := |
| (LinearEquiv.ofBijective (Position.op A.val).toLinearMap ⟨hinj,hsurj⟩).toContinuousLinearEquiv |
| obtain ⟨g,hg,hmin⟩ := copy_minimizer hfull (optimizer hμ hq (regime n) (Position.domainMap A)) |
| (optimizer_normalized hμ hq (regime n) (Position.domainMap A)) |
| (optimizer_minimal hμ hq (regime n) (Position.domainMap A)) |
| refine ⟨T,Position.symmetric_op A.property.1.1,g,hmin,?_⟩ |
| rw [hg] |
| exact hF |
|
|
| lemma exists_full_minimizer {μ : Measure (Space n)} |
| (hμ : Integrable (fun x : Space n => ‖x‖) μ) |
| (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x) (b : Bool) : |
| ∃ g : RelativeGauge (⊤ : Submodule ℝ (Space n)),IsMinimizer μ b g := |
| exists_minimizer hμ top_ne_bot (fun x _ hx => hq x hx) b |
|
|
| end RelativeGauge |
| end PettyProjection |
| end |
|
|
| end OAI |
|
|
