| import OAI.Geometry.Convex.GeneralMahler.Angle |
|
|
| namespace OAI |
| /-! Estimates for one-dimensional layers under changing coordinates. -/ |
| noncomputable section |
| open Set Filter MeasureTheory MeasureTheory.Measure Real Metric Matrix |
| open scoped ENNReal NNReal Topology MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace |
| namespace GeneralMahler |
| open Layers ProjField |
| variable {m : ℕ} |
|
|
| lemma exp_est_dir (m : ℕ) [NeZero m] : |
| ∃ D ≥ 0, ∀ f : Rn m → ℝ, (∀ x y, ‖f x - f y‖ ≤ 2 * ‖x-y‖) → |
| ∀ v : Rn m, ‖v‖ ≤ 2 → ∀ j : ℝ, 1 ≤ j → |
| (∫ x, ⟪x,v⟫*f x ∂normal m) ≤ |
| D*(j*(∫ x, ‖f x‖ ∂normal m) + Real.exp (-(j^2)/4)) := by |
| classical |
| obtain ⟨C,hC,h₁⟩ := normal_tail m 2 |
| have hJ := j_nonneg m |
| let D := 2+2*J m+4*(1+J m)*C |
| use D |
| refine ⟨by unfold D; positivity, fun f hf v hv j hj => ?_⟩ |
| have hk : LipschitzWith 2 f := LipschitzWith.of_dist_le_mul fun x y => by |
| simpa only [dist_eq_norm,NNReal.coe_ofNat] using hf x y |
| let μ := normal m |
| have hi : Integrable f μ := (PolyBound.lipschitz hk).gaussian_integrable |
| hk.continuous.aestronglyMeasurable |
| let F := ∫ x, ‖f x‖ ∂μ |
| have hh : 0 ≤ F := integral_nonneg fun x => norm_nonneg _ |
| let E := fun x : Rn m => (1+‖x‖)^2 |
| let S : Set (Rn m) := {x| j ≤ ‖x‖} |
| have hm : MeasurableSet S := (isClosed_le continuous_const continuous_norm).measurableSet |
| let b := 4*(1+J m) |
| let g := fun x : Rn m => (2*j)*‖f x‖ + (2*F)*‖x‖ + b*S.indicator E x |
| have hl (x) : ⟪x,v⟫*f x ≤ g x := by |
| have h₀ : ⟪x,v⟫*f x ≤ (‖x‖*2)*‖f x‖ := calc |
| _ ≤ |⟪x,v⟫*f x| := le_abs_self _ |
| _ ≤ _ := by |
| rw [abs_mul, Real.norm_eq_abs (f x)] |
| apply mul_le_mul_of_nonneg_right _ (abs_nonneg (f x)) |
| exact (abs_real_inner_le_norm x v).trans (mul_le_mul_of_nonneg_left hv (norm_nonneg x)) |
| apply h₀.trans |
| by_cases hx:x∈S |
| · change _ ≤ _+_+b*_ |
| rw [indicator_of_mem hx] |
| have he := norm_le_mean_error hi hf (by norm_num) x |
| change ‖f x‖ ≤ F+_ at he |
| have hv : ‖x‖+J m ≤ (1+J m)*(1+‖x‖) := by nlinarith [norm_nonneg x] |
| have ht : (‖x‖*2)*‖f x‖ ≤ (‖x‖*2)*(F+2*((1+J m)*(1+‖x‖))) := |
| mul_le_mul_of_nonneg_left (by linarith) (by positivity) |
| unfold b E |
| nlinarith [norm_nonneg x,mul_nonneg hJ (norm_nonneg x),norm_nonneg (f x)] |
| change _ ≤ _+_+b*_ |
| rw [indicator_of_notMem hx] |
| change ¬j≤‖x‖ at hx |
| nlinarith [norm_nonneg x,norm_nonneg (f x)] |
| have hA : Integrable (fun x=> (2*j)*‖f x‖) μ := hi.norm.const_mul _ |
| have hB : Integrable (fun x:Rn m=> (2*F)*‖x‖) μ := (i_norm _).const_mul _ |
| have hE : Integrable (fun x:Rn m=> b*S.indicator E x) μ := |
| ((integrable_envelope 2).indicator hm).const_mul _ |
| have hg : Integrable (fun x:Rn m=>⟪x,v⟫*f x) μ := by |
| have he : PolyBound (fun x:Rn m=>⟪x,v⟫) := by |
| simp_rw [real_inner_comm v] |
| exact PolyBound.clm (f := (show Rn m→L[ℝ] ℝ from innerSL ℝ v)) |
| exact (he.mul (PolyBound.lipschitz hk)).gaussian_integrable |
| (((continuous_id.inner continuous_const).mul hk.continuous).aestronglyMeasurable) |
| apply (integral_mono hg ((hA.add hB).add hE) hl).trans |
| change ∫ x, _+_+b*S.indicator E x ∂μ ≤ _ |
| rw [integral_add (show Integrable (fun x=>2*j*‖f x‖+(2*F)*‖x‖) μ from hA.add hB) hE, integral_add hA hB] |
| simp_rw [integral_const_mul]; rw [integral_indicator hm] |
| change 2*j*F+2*F*J m+b*_ ≤ D*(j*F+_) |
| have hx := h₁ j (by linarith) |
| change ∫ x in S, E x ∂μ ≤ _ at hx |
| have he := mul_le_mul_of_nonneg_left hx (show 0 ≤ b by unfold b; positivity) |
| unfold D |
| set e := Real.exp (-(j^2)/4) |
| have hp : 0 ≤ e := Real.exp_nonneg _ |
| unfold b at * |
| have hu : F ≤ j*F := by nlinarith |
| calc |
| _ ≤ 2*j*F + 2*(j*F)*J m + 4*(1+J m)*(C*e) := add_le_add (by nlinarith) he |
| _ ≤ _ := by |
| rw [← sub_nonneg] |
| ring_nf |
| positivity |
|
|
| namespace Param |
|
|
| variable {R:ℝ} (x:Param m R) (hr:0<R) (q₀:ProjField m) (v:Rn m) (hv:‖v‖=1) |
| lemma normS : ‖x.S‖ ≤ 2 ∧ ‖(equivPD x.S_pos).symm.toContinuousLinearMap‖ ≤ 2 := |
| inv_norm_box 2 (by norm_num) (A := x.S) (by simpa using x.S_box) |
| (equivPD x.S_pos) (equivPD_spec _) |
| include hv in lemma vp_two : ‖(x.instanceQ hr q₀).vp v‖ ≤ 2 := by |
| apply ((equivPD x.S_pos).symm.toContinuousLinearMap.le_opNorm v).trans |
| rw [hv,mul_one]; exact x.normS.2 |
|
|
| lemma Z_two (w:Rn m) : ‖(x.instanceQ hr q₀).Z w‖ ≤ 2*‖w‖ := by |
| rw [(x.instanceQ hr q₀).Z_def] |
| apply ((op (x.instanceQ hr q₀).S).le_opNorm w).trans |
| rw [op_norm]; apply mul_le_mul_of_nonneg_right x.normS.1 (norm_nonneg _) |
|
|
| include hv in lemma coordLipX (z:ℝ) (a b:Rn m) : |
| ‖⟪v,(x.instanceQ hr q₀).XT z a⟫-⟪v,(x.instanceQ hr q₀).XT z b⟫‖ ≤ 2*‖a-b‖ := by |
| rw [← inner_sub_right] |
| apply (abs_real_inner_le_norm ..).trans |
| rw [hv,one_mul] |
| apply le_trans _ (x.Z_two hr q₀ (a-b)) |
| have he : affineN (x.instanceQ hr q₀).root ((x.instanceQ hr q₀).shift z) a-affineN (x.instanceQ hr q₀).root ((x.instanceQ hr q₀).shift z) b=(x.instanceQ hr q₀).Z (a-b) := by |
| unfold affineN Z; rw [_root_.map_sub]; abel |
| rw [← he] |
| simpa only [XT,sample,dist_eq_norm, NNReal.coe_one,one_mul] using (coneProj_lip (x.instanceQ hr q₀).C).dist_le_mul |
| (affineN (x.instanceQ hr q₀).root ((x.instanceQ hr q₀).shift z) a) (affineN (x.instanceQ hr q₀).root ((x.instanceQ hr q₀).shift z) b) |
|
|
| include hv in lemma coordLipY (z:ℝ) (a b:Rn m) : |
| ‖-⟪v,(x.instanceQ hr q₀).YT z a⟫- -⟪v,(x.instanceQ hr q₀).YT z b⟫‖ ≤ 2*‖a-b‖ := by |
| rw [sub_neg_eq_add, ← sub_eq_neg_add, ← inner_sub_right] |
| apply (abs_real_inner_le_norm ..).trans |
| rw [hv,one_mul, norm_sub_rev ((x.instanceQ hr q₀).YT z b)] |
| apply le_trans _ (x.Z_two hr q₀ (a-b)) |
| let f := fun t => -((x.instanceQ hr q₀).Z t + (x.instanceQ hr q₀).shift z) |
| have he : f a-f b= -((x.instanceQ hr q₀).Z (a-b)) := by unfold f Z; rw [_root_.map_sub]; abel |
| have h := (coneProj_lip (x.instanceQ hr q₀).D).dist_le_mul (f a) (f b) |
| rw [dist_eq_norm,dist_eq_norm,NNReal.coe_one, one_mul,he,norm_neg] at h |
| exact h |
|
|
| include hv in lemma small_X {K} (hk:q₀.Bound K) (he : op x.B v=R⁻¹ • v) (z:ℝ) : |
| (∫ t, ‖⟪v,(x.instanceQ hr q₀).XT z t⟫‖ ∂normal m) ≤ K*m/R*a z := by |
| let e := x.transE hr |
| have hb (w : Rn m) (hw : w∈(x.instanceQ hr q₀).C) : ‖⟪v,w⟫‖ ≤ R⁻¹*K*⟪(x.instanceQ hr q₀).V,w⟫ := by |
| obtain ⟨w,rfl⟩ := e.surjective w |
| change e.toLinearEquiv w ∈ pushCone e.toLinearEquiv q₀.C at hw |
| have hi := hk.inD w (by simpa using hw) |
| have HH : ⟪(x.instanceQ hr q₀).V,e w⟫=⟪q₀.V,w⟫ := by |
| rw [real_inner_comm, real_inner_comm w]; apply pair_contra e.toLinearEquiv |
| rw [HH,show e w= _ from x.trans_apply hr w, |
| ← op_iff_hermitian.mp x.in_box.1.1,he,real_inner_smul_left,norm_mul, |
| Real.norm_eq_abs, abs_of_nonneg (inv_pos.mpr hr).le,mul_assoc] |
| apply mul_le_mul_of_nonneg_left _ (inv_pos.mpr hr).le |
| apply (abs_real_inner_le_norm v w).trans |
| rwa [hv,one_mul] |
| let l : Rn m→L[ℝ]ℝ := innerSL ℝ (x.instanceQ hr q₀).V |
| have hi := (x.instanceQ hr q₀).X_int z |
| calc |
| _ ≤ ∫ t,R⁻¹*K*l ((x.instanceQ hr q₀).XT z t) ∂normal m := integral_mono |
| (ContinuousLinearMap.integrable_comp (show Rn m→L[ℝ]ℝ from innerSL ℝ v) hi).norm |
| ((l.integrable_comp hi).const_mul _) (fun t=> hb _ (proj_mem ..)) |
| _ = _ := by |
| rw [integral_const_mul,l.integral_comp_comm hi,(x.instanceQ hr q₀).XT_mean] |
| change _*⟪(x.instanceQ hr q₀).V,a z • (x.instanceQ hr q₀).U⟫=_ |
| rw [real_inner_smul_right,real_inner_comm,(x.instanceQ hr q₀).ip] |
| ring |
|
|
| include hv in lemma large_Y [NeZero m] {K} (hk:q₀.Bound K) (he : op x.B v=R • v) (z:ℝ) : |
| (∫ t, ‖-⟪v,(x.instanceQ hr q₀).YT z t⟫‖ ∂normal m) ≤ K*m/R*(x.instanceQ hr q₀).Bt z := by |
| let e := x.transE hr |
| let b := contra e.toLinearEquiv |
| have hb (w : Rn m) (hw : w∈(x.instanceQ hr q₀).D) : ‖-⟪v,w⟫‖ ≤ R⁻¹*K*⟪(x.instanceQ hr q₀).U,w⟫ := by |
| obtain ⟨w,rfl⟩ := b.surjective w |
| change b w ∈ posDual (pushCone e.toLinearEquiv q₀.C) at hw |
| rw [dual_pushCone] at hw |
| have hi := hk.inC w (by simpa only [b,mem_pushCone,LinearEquiv.symm_apply_apply] using hw) |
| have HH : ⟪(x.instanceQ hr q₀).U,b w⟫=⟪q₀.U,w⟫ := pair_contra e.toLinearEquiv .. |
| have hv' : e v= R • v := (x.trans_apply hr _).trans he |
| have HH' : ⟪v,b w⟫= R⁻¹*⟪v,w⟫ := by |
| have h := pair_contra e.toLinearEquiv v w |
| change ⟪e v,b w⟫=_ at h |
| rw [hv',real_inner_smul_left] at h |
| rw [← h]; field_simp |
| rw [norm_neg,HH,HH',norm_mul,Real.norm_eq_abs,abs_of_nonneg (inv_pos.mpr hr).le,mul_assoc] |
| apply mul_le_mul_of_nonneg_left _ (inv_pos.mpr hr).le |
| apply (abs_real_inner_le_norm v w).trans |
| rwa [hv,one_mul] |
| let l : Rn m→L[ℝ]ℝ := innerSL ℝ (x.instanceQ hr q₀).U |
| have hi := (x.instanceQ hr q₀).Y_int z |
| calc |
| _ ≤ ∫ t,R⁻¹*K*l ((x.instanceQ hr q₀).YT z t) ∂normal m := integral_mono |
| (ContinuousLinearMap.integrable_comp (show Rn m→L[ℝ]ℝ from innerSL ℝ v) hi).neg.norm |
| ((l.integrable_comp hi).const_mul _) (fun t=> hb _ (proj_mem ..)) |
| _ = _ := by |
| rw [integral_const_mul,l.integral_comp_comm hi] |
| dsimp [l,Bt]; have H:= m_pos (m:=m); field_simp |
| end Param |
|
|
| lemma angle_est [NeZero m] (q₀:ProjField m) : |
| ∃ D ≥ 0, ∀ R, ∀ hr:0<R, ∀ x : Param m R, ∀ v:Rn m, ‖v‖=1 → |
| let q := x.instanceQ hr q₀ |
| ∀ (z:ℝ) (j:ℝ), 1 ≤ j → |
| (op x.B v= R⁻¹ • v → q.dimH v z ≤ D*(j/R*a z+ Real.exp (-(j^2)/4))) ∧ |
| (op x.B v= R • v → 1-q.dimH v z ≤ D*(j/R*q.Bt z+ Real.exp (-(j^2)/4))) := by |
| obtain ⟨K,hK⟩ := q₀.Field_isBound |
| obtain ⟨C,hC,hc⟩ := exp_est_dir m |
| have hv := hK.positive |
| use C*(1+K*m) |
| refine ⟨by positivity, fun R hr x v hv z j hj => ?_⟩ |
| let q := x.instanceQ hr q₀ |
| have HH (F g : ℝ) (hf : F ≤ K*m/R*g) (hg : 0 ≤ g) : |
| C*(j*F+Real.exp (-(j^2)/4)) ≤ C*(1+K*m)*(j/R*g+Real.exp (-(j^2)/4)) := by |
| rw [mul_assoc C] |
| apply mul_le_mul_of_nonneg_left _ hC |
| apply le_trans (add_le_add (mul_le_mul_of_nonneg_left hf (by linarith)) le_rfl) |
| rw [← sub_nonneg]; ring_nf; positivity |
| constructor |
| · intro he |
| change q.dimH _ _ ≤ _ |
| rw [q.dimH_X] |
| exact (hc _ (x.coordLipX hr q₀ v hv z) (q.vp v) (x.vp_two hr q₀ v hv) j hj).trans |
| (HH _ _ (x.small_X hr q₀ v hv hK he z) (a_pos z).le) |
| intro he |
| change 1-q.dimH v z≤_ |
| rw [q.dimH_Y v hv] |
| exact (hc _ (x.coordLipY hr q₀ v hv z) (q.vp v) (x.vp_two hr q₀ v hv) j hj).trans |
| (HH _ _ (x.large_Y hr q₀ v hv hK he z) (q.Bt_nonneg z)) |
| end GeneralMahler |
|
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| end |
|
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| end OAI |
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