| import OAI.Geometry.ProjectionBodies.VariationalMinimizers |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped NNReal RealInnerProductSpace Gradient |
| namespace PettyProjection |
| open Spherical (Sphere sigma mean objective objectiveIntegrand) |
| namespace RelativeGauge |
| variable {n : ℕ} [NeZero n] |
|
|
| /-- Actual Euler equation for the normalized variational optimizer. The test |
| is only continuous, even and degree-one homogeneous, not assumed convex. -/ |
| theorem normalized_representation {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))) (hgn : supportFunctional μ g.body=1) |
| (hmin : ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 → |
| objective b g.toSeminorm≤objective b q.toSeminorm) |
| (φ : Space n → ℝ) (hφ : Continuous φ) (heven : ∀ x,φ (-x)=φ x) |
| (hh : ∀ a : ℝ,0<a → ∀ x,φ (a • x)=a*φ x) : |
| (∫ x,φ x ∂μ)*(if b then 1 else objective false g.toSeminorm)= |
| mean (fun u : Sphere n => (if b then 1 else g u)*φ (∇ (fun x => g x) (u:Space n))) := by |
| let s : C(Sphere n,ℝ) := ⟨fun u => support g.body u,(support_continuous g.compact).comp continuous_subtype_val⟩ |
| let φs : C(Sphere n,ℝ) := ⟨fun u => φ u,hφ.comp continuous_subtype_val⟩ |
| have hs : ∀ u,0<s u := support_pos_of_nhds g.compact g.top_nhds |
| have hw : wulff s=g.body := wulff_support g.compact g.convex g.nonempty |
| have hge : gauge (wulff s)=(fun x => g x) := by ext x; rw [hw,g.top_gauge] |
| let I := ∫ x,φ x ∂μ |
| let F : ℝ → ℝ := fun t => mean (fun u : Sphere n => objectiveIntegrand b (gauge (wulff ⇑(s+t • φs)) (u:Space n))) |
| let D := mean (fun u : Sphere n => (if b then 1 else g u)*φ (∇ (fun x => g x) (u:Space n))) |
| let H : ℝ → ℝ := fun t => if b then Real.log (1+t*I)+F t else (1+t*I)*F t |
| have hF0 : F 0=objective b g.toSeminorm := by |
| simp only [F,zero_smul,add_zero,hge,objective] |
| have hFder : HasDerivAt F (-D) 0 := by |
| have h := wulff_objective_derivative b s hs φ hφ hh |
| rw [hge] at h |
| exact h |
| have hlin : HasDerivAt (fun t : ℝ => 1+t*I) I 0 := by |
| simpa only [one_mul,id_eq] using (((hasDerivAt_id (0:ℝ)).mul_const I).const_add 1) |
| have hH0 : H 0=objective b g.toSeminorm := by |
| dsimp only [H]; rw [zero_mul,add_zero,Real.log_one,zero_add,one_mul]; split <;> exact hF0 |
| have hHder : HasDerivAt H (I*(if b then 1 else objective false g.toSeminorm)-D) 0 := by |
| cases b |
| · have hd := hlin.mul hFder |
| simpa only [Pi.mul_def,H,Bool.false_eq_true,ite_false,zero_mul,add_zero,one_mul,hF0,mul_neg,one_mul,sub_eq_add_neg] using hd |
| · have hd := (hlin.log (by norm_num)).add hFder |
| simpa only [Pi.add_def,H,ite_true,zero_mul,add_zero,div_one,mul_one,sub_eq_add_neg] using hd |
| have hHmin : IsLocalMin H 0 := by |
| have hpos : ∀ᶠ t in 𝓝 (0:ℝ),0<1+t*I := by |
| have hc : Continuous (fun t : ℝ => 1+t*I) := by fun_prop |
| have hh := hc.continuousAt.eventually (lt_mem_nhds (by norm_num : (0:ℝ)<1+0*I)) |
| exact hh |
| filter_upwards [positive_constraints_nhds s φs hs,hpos] with t ht htp |
| rw [hH0] |
| have hsym : ∀ u : Sphere n,(s+t • φs) ⟨-(u:Space n),by simp⟩=(s+t • φs) u := by |
| intro u |
| change support g.body (-(u:Space n))+t*φ (-(u:Space n))=support g.body u+t*φ u |
| rw [support_even g.compact g.nonempty (fun _ h => g.neg_mem h),heven] |
| let f := ofFullBody (wulff_compact (s+t • φs) ht) (wulff_convex _) (wulff_balanced _ hsym) (wulff_nhds (s+t • φs) ht) |
| have hfb : f.body=wulff ⇑(s+t • φs) := body_ofFullBody _ _ _ _ |
| have hbound : supportFunctional μ f.body≤1+t*I := by |
| rw [hfb] |
| have hb := functional_wulff_bound hμ g.compact g.nonempty φ hφ hh t ht |
| rw [hgn] at hb |
| exact hb |
| have hlo := minimizer_scaled_lower hμ (show (⊤ : Submodule ℝ (Space n))≠⊥ from top_ne_bot) |
| (fun x _ hx => hq x hx) b hmin f htp hbound |
| have hscale := Spherical.objective_scale b (g := f.toSeminorm) |
| (f := (f.scale (1+t*I) htp).toSeminorm) (f.ne_zero top_ne_bot) htp (fun _ => rfl) |
| rw [hscale] at hlo |
| have hfobj : objective b f.toSeminorm=F t := by |
| unfold objective |
| congr 1 |
| funext u |
| rw [← f.top_gauge,hfb] |
| rw [hfobj] at hlo |
| exact hlo |
| have hz := hHmin.hasDerivAt_eq_zero hHder |
| exact sub_eq_zero.mp hz |
|
|
| end RelativeGauge |
| end PettyProjection |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped NNReal RealInnerProductSpace Pointwise Gradient |
| namespace PettyProjection |
| open Spherical (Sphere norm_coe mean sigma) |
|
|
| lemma normalizedVolume_pos {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) |
| (hi : (interior K).Nonempty) : 0 < normalizedVolume K := |
| div_pos (volumeRoot_pos hK hi) (Real.rpow_pos_of_pos (kappa_pos n) _) |
|
|
| lemma normalizedVolume_pow {n : ℕ} [NeZero n] (K : Set (Space n)) : |
| normalizedVolume K^n=volume.real K/kappa n := by |
| rw [normalizedVolume_formula] |
| exact Real.rpow_inv_natCast_pow (div_nonneg measureReal_nonneg (kappa_pos n).le) (NeZero.ne n) |
|
|
| lemma normalizedVolume_smul {n : ℕ} [NeZero n] {a : ℝ} (ha : 0 ≤ a) (K : Set (Space n)) : |
| normalizedVolume (a • K)=a*normalizedVolume K := by |
| rw [normalizedVolume,volumeRoot_smul (NeZero.pos n) ha,normalizedVolume] |
| ring |
|
|
| namespace RelativeGauge |
| variable {n : ℕ} [NeZero n] |
|
|
| omit [NeZero n] in |
| lemma top_pos (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (u : Sphere n) : 0 < g u := |
| g.pos (Submodule.mem_top) (by intro h;simpa [h] using norm_coe u) |
|
|
| omit [NeZero n] in |
| lemma top_interior (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : (interior g.body).Nonempty := |
| ⟨0,mem_interior_iff_mem_nhds.mpr g.top_nhds⟩ |
|
|
| def volumeNormalize (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : |
| RelativeGauge (⊤ : Submodule ℝ (Space n)) := |
| g.scale (normalizedVolume g.body) (normalizedVolume_pos g.compact g.top_interior) |
|
|
| lemma volumeNormalize_body_volume (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : |
| volume.real g.volumeNormalize.body=kappa n := by |
| have ha := normalizedVolume_pos g.compact g.top_interior |
| rw [volumeNormalize,g.body_scale,volumeReal_smul_nonneg (inv_nonneg.mpr ha.le), |
| finrank_euclideanSpace_fin,inv_pow,normalizedVolume_pow] |
| field_simp [(volumeReal_pos_of_body g.compact g.top_interior).ne',(kappa_pos n).ne'] |
|
|
| lemma volumeNormalize_moment (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : |
| mean (fun u : Sphere n => (g.volumeNormalize u)⁻¹^n)=1 := by |
| have hh := gauge_volume g.volumeNormalize.convex g.volumeNormalize.top_nhds |
| (fun u => by rw [g.volumeNormalize.top_gauge];exact g.volumeNormalize.top_pos u) |
| simp_rw [g.volumeNormalize.top_gauge] at hh |
| rw [g.volumeNormalize_body_volume] at hh |
| exact (mul_left_cancel₀ (kappa_pos n).ne' (by simpa only [mul_one] using hh.symm)) |
| |
| omit [NeZero n] in |
| lemma full_body_of_sphere_one (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) |
| (hg : ∀ u : Sphere n,g u=1) : g.body=unitBall n := by |
| have he (x : Space n) : g x=‖x‖ := by |
| by_cases hx : x=0 |
| · simp [hx] |
| let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx]⟩ |
| have h := hg u |
| change g.toSeminorm (‖x‖⁻¹ • x)=1 at h |
| rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))] at h |
| have hn := norm_ne_zero_iff.mpr hx |
| field_simp at h |
| exact h |
| ext x |
| rw [g.top_body] |
| change g x≤1 ↔ x∈closedBall 0 1 |
| rw [he,mem_closedBall,dist_zero_right] |
| |
| def fullOfRange {A : Space n →L[ℝ] Space n} (hA : LinearMap.range A.toLinearMap=⊤) |
| (g : RelativeGauge (LinearMap.range A.toLinearMap)) : RelativeGauge (⊤ : Submodule ℝ (Space n)) where |
| toSeminorm := g.toSeminorm |
| projection x := by rw [Submodule.starProjection_top];rfl |
| lower := by simpa only [hA] using g.lower |
| |
| omit [NeZero n] in |
| lemma fullOfRange_body {A : Space n →L[ℝ] Space n} (hA : LinearMap.range A.toLinearMap=⊤) |
| (g : RelativeGauge (LinearMap.range A.toLinearMap)) : (fullOfRange hA g).body=g.body := by |
| simp only [body,fullOfRange,hA] |
| |
| end RelativeGauge |
| end PettyProjection |
| end |
| |
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped RealInnerProductSpace Pointwise Gradient |
| namespace PettyProjection |
| open Spherical (Sphere mean sigma gaugeField gaugeMoment) |
| |
| def regime (n : ℕ) : Bool := decide (n=4) |
| lemma exponent_pos (n : ℕ) : 0 < Scalar.exponent n := by unfold Scalar.exponent;split_ifs <;> norm_num |
| |
| namespace RelativeGauge |
| variable {n : ℕ} [NeZero n] {μ : Measure (Space n)} |
| |
| omit [NeZero n] in |
| lemma scale_field_ae (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) {a : ℝ} (ha : 0 < a) |
| {k : ℕ} (hk : 0 < k) : |
| gaugeField (g.scale a ha).toSeminorm k =ᵐ[sigma n] (fun u => a^k • gaugeField g.toSeminorm k u) := by |
| filter_upwards [Spherical.seminorm_ae_differentiable_sphere g.toSeminorm] with u hu |
| have hgrad : ∇ (fun x => g.scale a ha x) (u:Space n)=a • ∇ (fun x => g x) (u:Space n) := by |
| apply (InnerProductSpace.toDual ℝ (Space n)).injective |
| rw [toDual_gradient,map_smul,toDual_gradient] |
| exact fderiv_const_mul hu a |
| change (a*g u)^(k-1) • ∇ (fun x => g.scale a ha x) (u:Space n)=_ |
| rw [hgrad,mul_pow,smul_smul] |
| change (a^(k-1)*g u^(k-1)*a) • _=a^k • (g u^(k-1) • _) |
| rw [smul_smul] |
| congr 1 |
| have hp : a^k=a^(k-1)*a := by |
| conv_lhs => rw [show k=(k-1)+1 by omega,pow_succ] |
| rw [hp] |
| ring |
| |
| omit [NeZero n] in |
| lemma scale_moment (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) {a : ℝ} (ha : 0 < a) (k : ℕ) : |
| gaugeMoment (g.scale a ha).toSeminorm k=a^k*gaugeMoment g.toSeminorm k := by |
| change mean (fun u : Sphere n => (a*g u)^k)=_ |
| simp only [mul_pow,gaugeMoment,mean,integral_const_mul] |
| |
| lemma minimizer_field_representation |
| (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) |
| (φ : Space n → ℝ) (hφ : Continuous φ) (heven : ∀ x,φ (-x)=φ x) |
| (hh : ∀ a : ℝ,0 < a → ∀ x,φ (a • x)=a*φ x) : |
| (∫ x,φ x ∂μ)*gaugeMoment g.toSeminorm (Scalar.exponent n-1)= |
| mean (fun u : Sphere n => φ (gaugeField g.toSeminorm (Scalar.exponent n) u)) := by |
| have hv := normalized_representation hμ hq (regime n) g hgn hmin φ hφ heven hh |
| have he : mean (fun u : Sphere n => φ (gaugeField g.toSeminorm (Scalar.exponent n) u))= |
| mean (fun u : Sphere n => (if regime n then 1 else g u)*φ (∇ (fun x => g x) (u:Space n))) := by |
| congr 1 |
| funext u |
| by_cases hn : n=4 |
| · simp [gaugeField,Scalar.exponent,regime,hn] |
| · simp only [gaugeField,Scalar.exponent,ite_eq_right hn,Nat.reduceSub,pow_one, |
| regime,decide_eq_false hn,Bool.false_eq_true,ite_false] |
| exact hh _ (g.top_pos u) _ |
| rw [he] |
| convert hv using 1 |
| congr 1 |
| by_cases hn : n=4 |
| · simp [gaugeMoment,Scalar.exponent,regime,hn,Spherical.mean_const] |
| · simp [gaugeMoment,Scalar.exponent,regime,hn,Spherical.objective,Spherical.objectiveIntegrand] |
| |
| lemma scaled_field_representation |
| (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) |
| (φ : Space n → ℝ) (hφ : Continuous φ) (heven : ∀ x,φ (-x)=φ x) |
| (hh : ∀ r : ℝ,0 < r → ∀ x,φ (r • x)=r*φ x) : |
| (∫ x,φ x ∂μ)*gaugeMoment (g.scale a ha).toSeminorm (Scalar.exponent n-1)= |
| supportFunctional μ (g.scale a ha).body* |
| mean (fun u : Sphere n => φ (gaugeField (g.scale a ha).toSeminorm (Scalar.exponent n) u)) := by |
| have he : mean (fun u : Sphere n => φ (gaugeField (g.scale a ha).toSeminorm (Scalar.exponent n) u))= |
| a^(Scalar.exponent n)*mean (fun u : Sphere n => φ (gaugeField g.toSeminorm (Scalar.exponent n) u)) := by |
| simp only [mean] |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [g.scale_field_ae ha (exponent_pos n)] with u hu |
| rw [hu,hh _ (pow_pos ha _)] |
| rw [g.scale_moment ha,g.functional_scale,hgn,mul_one,he] |
| have hr := minimizer_field_representation hμ hq g hgn hmin φ hφ heven hh |
| rw [← hr] |
| have hk : a^(Scalar.exponent n)=a^(Scalar.exponent n-1)*a := by |
| rw [← pow_succ];congr 1;have := exponent_pos n;omega |
| rw [hk] |
| field_simp |
| |
| end RelativeGauge |
| end PettyProjection |
| end |
| |
| end OAI |
| |
