import OAI.Probability.GaussianReplacement.FiberModel namespace OAI open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology noncomputable section namespace CurrentProjection open InnerProductSpace open InformationTheory section GaussianDirection variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [finiteDimensionalRealE : FiniteDimensional ℝ E] [measurableSpaceE : MeasurableSpace E] [borelSpaceE : BorelSpace E] def directionComplement (u : E) : E →L[ℝ] E := ContinuousLinearMap.id ℝ E - (innerSL ℝ u).smulRight u omit finiteDimensionalRealE measurableSpaceE borelSpaceE in lemma directionComplement_apply (u g : E) : directionComplement u g = g - ⟪u,g⟫ • u := rfl omit finiteDimensionalRealE measurableSpaceE borelSpaceE in lemma directionComplement_inner (u g y : E) : ⟪y,directionComplement u g⟫ = ⟪y-⟪u,y⟫ • u,g⟫ := by simp only [directionComplement_apply,inner_sub_right,inner_smul_right,inner_sub_left, inner_smul_left,conj_trivial,real_inner_comm u y] ring lemma gaussian_direction_indep {u : E} (hu : ‖u‖ = 1) : IndepFun (fun g : E => ⟪u,g⟫) (directionComplement u) (stdGaussian E) := by have hG := (IsGaussian.hasGaussianLaw_id (μ:=stdGaussian E)).map_fun ((innerSL ℝ u).prod (directionComplement u)) change HasGaussianLaw (fun g : E => (⟪u,g⟫,directionComplement u g)) (stdGaussian E) at hG apply hG.indepFun_of_covariance_inner intro x y have he : (fun g : E => ⟪x,⟪u,g⟫⟫) = fun g => ⟪x • u,g⟫ := by funext g simp only [Real.inner_apply,inner_smul_left,conj_trivial] simp_rw [he,directionComplement_inner] rw [←covarianceBilin_apply_eq_cov IsGaussian.memLp_two_id,covarianceBilin_stdGaussian] change ⟪x • u,y-⟪u,y⟫ • u⟫ = 0 simp only [inner_smul_left,conj_trivial,inner_sub_right,inner_smul_right, real_inner_self_eq_norm_sq,hu,one_pow,mul_one] ring lemma gaussian_direction_real {u : E} (hu : ‖u‖ = 1) : (stdGaussian E).map (fun g => ⟪u,g⟫) = gaussianReal 0 1 := by have hG := (IsGaussian.hasGaussianLaw_id (μ:=stdGaussian E)).map_fun (innerSL ℝ u) change HasGaussianLaw (fun g : E => ⟪u,g⟫) (stdGaussian E) at hG rw [hG.map_eq_gaussianReal] have hi : (∫ g : E, ⟪u,g⟫ ∂stdGaussian E) = 0 := integral_strongDual_stdGaussian (innerSL ℝ u) have hv : Var[fun g : E => ⟪u,g⟫; stdGaussian E] = 1 := by rw [←covariance_self (by fun_prop),←covarianceBilin_apply_eq_cov IsGaussian.memLp_two_id, covarianceBilin_stdGaussian,innerSL_apply_apply,real_inner_self_eq_norm_sq,hu,one_pow] simp only [hi,hv,Real.toNNReal_one] lemma gaussian_direction_split {u : E} (hu : ‖u‖ = 1) : (stdGaussian E).map (fun g => (⟪u,g⟫,directionComplement u g)) = (gaussianReal 0 1).prod ((stdGaussian E).map (directionComplement u)) := by rw [(gaussian_direction_indep hu).map_prod_eq_prod_map_map (by fun_prop) (by fun_prop), gaussian_direction_real hu] end GaussianDirection section GaussianDirectionRows variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma gaussian_direction_rows_split {u : E} (hu : ‖u‖ = 1) (k : ℕ) : (Measure.pi (fun _ : Fin k => stdGaussian E)).map (fun G => ((fun i => ⟪u,G i⟫),(fun i => directionComplement u (G i)))) = (Measure.pi (fun _ : Fin k => gaussianReal 0 1)).prod (Measure.pi (fun _ : Fin k => (stdGaussian E).map (directionComplement u))) := by let e := MeasurableEquiv.arrowProdEquivProdArrow ℝ E (Fin k) have he : (fun G : Fin k→E => ((fun i => ⟪u,G i⟫),(fun i => directionComplement u (G i)))) = e ∘ (fun G => fun i => (⟪u,G i⟫,directionComplement u (G i))) := rfl rw [he,←Measure.map_map e.measurable (by fun_prop), Measure.pi_map_pi (f := fun (_ : Fin k) => fun g : E => (⟪u,g⟫,directionComplement u g)) (μ := fun _ => stdGaussian E) (fun i => by fun_prop)] simp_rw [gaussian_direction_split hu] exact (measurePreserving_arrowProdEquivProdArrow ℝ E (Fin k) (fun _ => gaussianReal 0 1) (fun _ => (stdGaussian E).map (directionComplement u))).map_eq lemma gaussian_direction_rows_recover {u : E} (hu : ‖u‖ = 1) (k : ℕ) : ((Measure.pi (fun _ : Fin k => gaussianReal 0 1)).prod (Measure.pi (fun _ : Fin k => (stdGaussian E).map (directionComplement u)))).map (fun z => fun i => z.1 i • u+z.2 i) = Measure.pi (fun _ : Fin k => stdGaussian E) := by rw [←gaussian_direction_rows_split hu k,Measure.map_map (by fun_prop) (by fun_prop)] have he : (fun z : (Fin k→ℝ)×(Fin k→E) => fun i => z.1 i • u+z.2 i) ∘ (fun G : Fin k→E => ((fun i => ⟪u,G i⟫),(fun i => directionComplement u (G i)))) = id := by funext G i simp [directionComplement_apply] rw [he,Measure.map_id] end GaussianDirectionRows section FiberAverage variable {X : Type*} [PseudoMetricSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] (μ : Measure X) lemma tendsto_normalized_closedBall {f : X → ℝ≥0∞} {x : X} (hf : ContinuousAt f x) (hvol : ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≠ 0 ∧ μ (Metric.closedBall x r) ≠ ∞) : Tendsto (fun r : ℝ => (μ (Metric.closedBall x r))⁻¹ * ∫⁻ y in Metric.closedBall x r, f y ∂μ) (𝓝[>] 0) (𝓝 (f x)) := by apply tendsto_order.2 constructor · intro a ha obtain ⟨b,hab,hb⟩ := exists_between ha have hnb : {y | b < f y} ∈ 𝓝 x := hf (Ioi_mem_nhds hb) have he : ∀ᶠ r in 𝓝[>] (0:ℝ), Metric.closedBall x r ⊆ _ := (eventually_closedBall_subset hnb).filter_mono inf_le_left filter_upwards [he, self_mem_nhdsWithin] with r hr hr0 have hv := hvol r hr0 apply hab.trans_le calc b = (μ (Metric.closedBall x r))⁻¹ * (b * μ (Metric.closedBall x r)) := by rw [mul_left_comm, ENNReal.inv_mul_cancel hv.1 hv.2, mul_one] _ ≤ _ := mul_le_mul_right (by rw [←setLIntegral_const] apply setLIntegral_mono' Metric.isClosed_closedBall.measurableSet intro y hy exact (hr hy).le) _ · intro a ha obtain ⟨b,hb,hba⟩ := exists_between ha have hnb : {y | f y < b} ∈ 𝓝 x := hf (Iio_mem_nhds hb) have he : ∀ᶠ r in 𝓝[>] (0:ℝ), Metric.closedBall x r ⊆ _ := (eventually_closedBall_subset hnb).filter_mono inf_le_left filter_upwards [he, self_mem_nhdsWithin] with r hr hr0 have hv := hvol r hr0 apply lt_of_le_of_lt _ hba calc _ ≤ (μ (Metric.closedBall x r))⁻¹ * (b * μ (Metric.closedBall x r)) := by apply mul_le_mul_right rw [←setLIntegral_const] apply setLIntegral_mono' Metric.isClosed_closedBall.measurableSet intro y hy exact (hr hy).le _ = b := by rw [mul_left_comm, ENNReal.inv_mul_cancel hv.1 hv.2,mul_one] end FiberAverage lemma ennreal_double_mul_le_sq_add (a b : ℝ≥0∞) : 2*(a*b) ≤ a^2+b^2 := by have h := ENNReal.young_inequality a b Real.HolderConjugate.two_two norm_num only [ENNReal.rpow_two,ENNReal.ofReal_ofNat] at h calc _ ≤ 2*(a^2/2+b^2/2) := mul_le_mul_right h 2 _ = _ := by rw [mul_add,div_eq_mul_inv,div_eq_mul_inv] rw [mul_left_comm 2, mul_left_comm 2] rw [ENNReal.mul_inv_cancel (by norm_num) (by norm_num)] simp lemma cube_energy_contraction {k : ℕ} (p K : (Fin k → ℝ) → ℝ≥0∞) (hp : Measurable p) (hK : Measurable K) (hKn : ∫⁻ z, K z = 1) : (∫⁻ x, ∫⁻ y, p x*p y*K (y-x)) ≤ ∫⁻ x, p x^2 := by have hA : (∫⁻ x, ∫⁻ y, p x^2*K (y-x)) = ∫⁻ x, p x^2 := by congr 1 funext x rw [lintegral_const_mul _ (show Measurable (fun y => K (y-x)) by fun_prop), lintegral_sub_right_eq_self,hKn,mul_one] have hB : (∫⁻ x, ∫⁻ y, p y^2*K (y-x)) = ∫⁻ y, p y^2 := by rw [lintegral_lintegral_swap (by fun_prop)] congr 1 funext y rw [lintegral_const_mul _ (show Measurable (fun x => K (y-x)) by fun_prop), lintegral_sub_left_eq_self,hKn,mul_one] have he : 2*(∫⁻ x, ∫⁻ y, p x*p y*K (y-x)) ≤ 2*(∫⁻ x, p x^2) := by calc _ = ∫⁻ x, ∫⁻ y, 2*(p x*p y)*K (y-x) := by simp_rw [mul_assoc,lintegral_const_mul' (2:ℝ≥0∞) _ (by norm_num)] _ ≤ ∫⁻ x, ∫⁻ y, (p x^2+p y^2)*K (y-x) := by apply lintegral_mono intro x apply lintegral_mono intro y exact mul_le_mul_left (ennreal_double_mul_le_sq_add _ _) _ _ = (∫⁻ x, ∫⁻ y, p x^2*K (y-x)) + (∫⁻ x, ∫⁻ y, p y^2*K (y-x)) := by simp_rw [add_mul] rw [show (fun x => ∫⁻ y, p x^2*K (y-x)+p y^2*K (y-x)) = (fun x => (∫⁻ y, p x^2*K (y-x))+(∫⁻ y, p y^2*K (y-x))) by funext x; exact lintegral_add_left (by fun_prop) _] exact lintegral_add_left (by fun_prop) _ _ = _ := by rw [hA,hB,two_mul] exact (ENNReal.mul_le_mul_iff_right (by norm_num : (2:ℝ≥0∞) ≠ 0) (by norm_num)).mp he lemma continuous_lintegral_bounded {A X : Type*} [TopologicalSpace A] [FirstCountableTopology A] [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] (f : A → X → ℝ≥0∞) (hf : ∀ x, Continuous (fun a => f a x)) (hm : ∀ a, Measurable (f a)) (M : ℝ≥0∞) (hM : M ≠ ∞) (hb : ∀ a x, f a x ≤ M) : Continuous (fun a => ∫⁻ x, f a x ∂μ) := by apply continuous_iff_continuousAt.2 intro a exact tendsto_lintegral_filter_of_dominated_convergence (fun _ => M) (Eventually.of_forall hm) (Eventually.of_forall (fun a => ae_of_all _ (hb a))) (by simp only [lintegral_const]; exact ENNReal.mul_ne_top hM (measure_ne_top _ _)) (ae_of_all _ (fun x => (hf x).continuousAt)) lemma tendsto_scaled_cube_average {k : ℕ} (F : (Fin k→ℝ) → ℝ≥0∞) (hF : ContinuousAt F 0) {r : ℝ} (hr : 0 < r) : Tendsto (fun δ : ℝ => ENNReal.ofReal ((2*δ)⁻¹ ^ k) * ∫⁻ z in Metric.closedBall (0 : Fin k→ℝ) (δ/r), F z) (𝓝[>] 0) (𝓝 (ENNReal.ofReal (r⁻¹ ^ k) * F 0)) := by have hv : ∀ a : ℝ, 0 < a → (volume : Measure (Fin k→ℝ)) (Metric.closedBall 0 a) ≠ 0 ∧ (volume : Measure (Fin k→ℝ)) (Metric.closedBall 0 a) ≠ ∞ := by intro a ha rw [Real.volume_pi_closedBall 0 ha.le] constructor · exact (ENNReal.ofReal_pos.mpr (by positivity)).ne' · exact ENNReal.ofReal_ne_top have hz : Tendsto (fun δ : ℝ => δ/r) (𝓝[>] (0:ℝ)) (𝓝 (0/r)) := ((tendsto_id : Tendsto (fun δ : ℝ => δ) (𝓝 (0:ℝ)) (𝓝 (0:ℝ))).mono_left (show 𝓝[>] (0:ℝ) ≤ 𝓝 (0:ℝ) from inf_le_left)).div_const r rw [zero_div] at hz have ht := (tendsto_normalized_closedBall volume hF hv).comp (show Tendsto (fun δ : ℝ => δ/r) (𝓝[>] 0) (𝓝[>] 0) from tendsto_nhdsWithin_iff.2 ⟨hz, by filter_upwards [self_mem_nhdsWithin] with δ hδ; exact div_pos hδ hr⟩) have ht' := ENNReal.Tendsto.const_mul ht (Or.inr (ENNReal.ofReal_ne_top : ENNReal.ofReal (r⁻¹ ^ k) ≠ ∞)) apply ht'.congr' filter_upwards [self_mem_nhdsWithin] with δ hδ change 0 < δ at hδ dsimp only [Function.comp_def] rw [Real.volume_pi_closedBall 0 (div_nonneg hδ.le hr.le)] simp only [Fintype.card_fin] rw [←mul_assoc] congr 1 rw [←ENNReal.ofReal_inv_of_pos (by positivity : (0:ℝ) < (2*(δ/r))^k), ←ENNReal.ofReal_mul (by positivity)] congr 1 simp only [inv_pow] rw [←mul_inv_rev,←mul_pow] congr 2 field_simp section GaussianSlab variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [finiteDimensionalRealE : FiniteDimensional ℝ E] [measurableSpaceE : MeasurableSpace E] [borelSpaceE : BorelSpace E] def gaussianReconstruct (u : E) {k : ℕ} (zG : (Fin k→ℝ)×(Fin k→E)) : Fin k→E := fun i => zG.1 i • u + directionComplement u (zG.2 i) lemma gaussian_reconstruct_law {u : E} (hu : ‖u‖ = 1) (k : ℕ) : ((Measure.pi (fun _ : Fin k => gaussianReal 0 1)).prod (Measure.pi (fun _ : Fin k => stdGaussian E))).map (gaussianReconstruct u) = Measure.pi (fun _ : Fin k => stdGaussian E) := by have hm : Measurable (fun G : Fin k→E => fun i => directionComplement u (G i)) := by fun_prop have he := Measure.pi_map_pi (μ := fun _ : Fin k => stdGaussian E) (f := fun _ => directionComplement u) (fun _ => (directionComplement u).measurable.aemeasurable) calc _ = (((Measure.pi (fun _ : Fin k => gaussianReal 0 1)).prod ((Measure.pi (fun _ : Fin k => stdGaussian E)).map (fun G => fun i => directionComplement u (G i)))).map (fun z => fun i => z.1 i • u+z.2 i)) := by rw [show Measure.pi (fun _ : Fin k => gaussianReal 0 1) = (Measure.pi (fun _ : Fin k => gaussianReal 0 1)).map id by rw [Measure.map_id], Measure.map_prod_map _ _ measurable_id hm, Measure.map_map (by fun_prop) (by fun_prop)] rw [Measure.map_id] rfl _ = _ := by rw [he]; exact gaussian_direction_rows_recover hu k omit finiteDimensionalRealE measurableSpaceE borelSpaceE in lemma gaussian_reconstruct_pairing {u : E} (hu : ‖u‖ = 1) {k : ℕ} (zG : (Fin k→ℝ)×(Fin k→E)) (i : Fin k) : ⟪u,gaussianReconstruct u zG i⟫ = zG.1 i := by simp [gaussianReconstruct,directionComplement_apply,inner_add_right, inner_sub_right,inner_smul_right,hu] lemma continuous_gaussian_slice {u : E} {k : ℕ} (H : BoundedContinuousFunction (Fin k→E) ℝ≥0) : Continuous (fun z : Fin k→ℝ => ∫⁻ G : Fin k→E, (H (gaussianReconstruct u (z,G)) : ℝ≥0∞) ∂Measure.pi (fun _ => stdGaussian E)) := by apply continuous_lintegral_bounded _ (fun G => by apply ENNReal.continuous_coe.comp exact H.continuous.comp (by unfold gaussianReconstruct; fun_prop)) (fun z => by apply ENNReal.continuous_coe.measurable.comp exact H.continuous.measurable.comp (by unfold gaussianReconstruct; fun_prop)) ((nndist H 0 : ℝ≥0) : ℝ≥0∞) ENNReal.coe_ne_top intro z G apply ENNReal.coe_le_coe.mpr simpa using (BoundedContinuousFunction.nndist_coe_le_nndist (f:=H) (g:=0) (gaussianReconstruct u (z,G))) end GaussianSlab def fiberEnergyBound (d : ℕ) : ℝ≥0∞ := momentConstant 9 d (3*4^d) (3*4^d) lemma fiberEnergyBound_ne_top (d : ℕ) : fiberEnergyBound d ≠ ∞ := ENNReal.ofReal_ne_top lemma uniformSphere_ae_ne {d : ℕ} (hd : 2 ≤ d) (s : Sphere d) : ∀ᵐ t ∂uniformSphere d, t ≠ s := by have hz := (uniformSphere_growth (by omega : 1 ≤ d) (le_refl (d-1))).sphereLift.affine_span_null (by positivity) (⊥ : Submodule ℝ (Vector d)) (s:Vector d) (by simp; omega) (sphereLift_ae_norm_sub_le (uniformSphere d) s) simp only [Submodule.mem_bot,sub_eq_zero] at hz rw [sphereLift,Measure.map_apply continuous_subtype_val.measurable (show MeasurableSet {u : Vector d | u = (s:Vector d)} from isClosed_eq continuous_id continuous_const |>.measurableSet)] at hz simpa only [ae_iff,not_not,Set.preimage_ofPred_eq,Subtype.val_inj] using hz lemma spherical_inverse_bound {d k : ℕ} (hk : 1 ≤ k) (hkd : k < d-1) (s : Sphere d) : (∫⁻ t : Sphere d, ENNReal.ofReal (‖(t:Vector d)-(s:Vector d)‖⁻¹ ^ k) ∂uniformSphere d) ≤ fiberEnergyBound d := by have hB : AmbientGrowth (sphereLift (uniformSphere d)) (0+k) (3*4^d) := by simpa only [zero_add] using (uniformSphere_growth (by omega : 1 ≤ d) hkd.le).sphereLift have hh := hB.mixed_inverse_uniform ((uniformSphere_growth (by omega : 1 ≤ d) (le_refl (d-1))).sphereLift) (by positivity) (by positivity) (m:=0) (by simpa using hkd) (Nat.sub_le d 1) (⊥ : Submodule ℝ (Vector d)) (s:Vector d) (by simpa using hk) (sphereLift_ae_norm_sub_le (uniformSphere d) s) rw [sphereLift,lintegral_map (measurable_mixedInverse _ _ _ _) continuous_subtype_val.measurable] at hh simpa only [mixedInverse,pow_zero,mul_one,fiberEnergyBound] using hh lemma spherical_cube_expectation {d k : ℕ} (hk : 1 ≤ k) (hkd : k < d-1) (s : Sphere d) {δ : ℝ} (hδ : 0 < δ) : (∫⁻ C, cubeAverage (uniformSphere d) C (label C (s:Vector d)) δ ∂gaussianRows k d) ≤ fiberEnergyBound d := by have he : ∀ C : Rows k d, cubeAverage (uniformSphere d) C (label C (s:Vector d)) δ = ∫⁻ t : Sphere d, rowKernel (cubeKernel δ) (s:Vector d) (t:Vector d) C ∂uniformSphere d := by intro C rw [cubeAverage_eq_kernel _ _ hδ.le,sphereLift, lintegral_map (measurable_rowKernel_left _ (measurable_cubeKernel δ) _ _ C) continuous_subtype_val.measurable] simp_rw [he] have hK := measurable_cubeKernel δ rw [lintegral_lintegral_swap (by dsimp [rowKernel]; fun_prop)] apply le_trans _ (spherical_inverse_bound hk hkd s) apply lintegral_mono_ae filter_upwards [uniformSphere_ae_ne (by omega) s] with t ht exact gaussian_fresh_kernel_bound (cubeKernel δ) (measurable_cubeKernel δ) (lintegral_cubeKernel hδ) k (s:Vector d) (t:Vector d) (Subtype.coe_injective.ne ht) lemma spherical_extended_energy {d k : ℕ} (hk : 1 ≤ k) (hkd : k < d-1) : (∫⁻ z : Rows k d×Sphere d, extendedCubeDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d)) ∂(gaussianRows k d).prod (uniformSphere d)) ≤ fiberEnergyBound d := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos (by omega) change (∫⁻ z, liminf (fun j : ℕ => cubeAverage (uniformSphere d) z.1 (label z.1 (z.2:Vector d)) (1/(j:ℝ))) atTop ∂(gaussianRows k d).prod (uniformSphere d)) ≤ _ apply (lintegral_liminf_le (fun j => measurable_own_cube (uniformSphere d) _)).trans apply liminf_le_of_frequently_le' apply Eventually.frequently filter_upwards [eventually_ge_atTop 1] with j hj rw [lintegral_prod _ (measurable_own_cube (k:=k) (uniformSphere d) _).aemeasurable, lintegral_lintegral_swap (measurable_own_cube (k:=k) (uniformSphere d) _).aemeasurable] calc _ ≤ ∫⁻ s : Sphere d, fiberEnergyBound d ∂uniformSphere d := by apply lintegral_mono intro s exact spherical_cube_expectation hk hkd s (by positivity) _ = _ := by simp lemma spherical_projection_ac {d k : ℕ} (hk : 1 ≤ k) (hkd : k < d-1) : ∀ᵐ C ∂gaussianRows k d, labelLaw (uniformSphere d) C ≪ volume := by have hf := ae_lt_top (measurable_own_extended (k:=k) (uniformSphere d)) (ne_of_lt (lt_of_le_of_lt (spherical_extended_energy hk hkd) (lt_top_iff_ne_top.mpr (fiberEnergyBound_ne_top d)))) have hff := Measure.ae_ae_of_ae_prod hf filter_upwards [hff] with C hC exact labelLaw_ac_of_finite_extended (uniformSphere d) C hC lemma projection_convolution_contraction {k : ℕ} (μ : Measure (Fin k→ℝ)) (p K : (Fin k→ℝ) → ℝ≥0∞) (hp : Measurable p) (hK : Measurable K) (hμ : volume.withDensity p = μ) (hKn : ∫⁻ z, K z = 1) : (∫⁻ x, ∫⁻ y, K (y-x) ∂μ ∂μ) ≤ ∫⁻ x, p x ∂μ := by rw [←hμ] have hi (x : Fin k→ℝ) : (∫⁻ y, K (y-x) ∂volume.withDensity p) = ∫⁻ y, p y*K (y-x) := by exact lintegral_withDensity_eq_lintegral_mul _ hp (by fun_prop) simp_rw [hi] rw [lintegral_withDensity_eq_lintegral_mul _ hp (by fun_prop), lintegral_withDensity_eq_lintegral_mul _ hp hp] simp only [Pi.mul_apply] have he : (∫⁻ x, p x*(∫⁻ y, p y*K (y-x))) = ∫⁻ x, ∫⁻ y, p x*p y*K (y-x) := by congr 1 funext x rw [←lintegral_const_mul _ (show Measurable (fun y => p y*K (y-x)) by fun_prop)] simp only [mul_assoc] rw [he] simpa only [pow_two] using cube_energy_contraction p K hp hK hKn lemma cubeVectorKernel_normalized {k : ℕ} {δ : ℝ} (hδ : 0 < δ) : (∫⁻ z : Fin k→ℝ, ∏ i, cubeKernel δ (z i)) = 1 := by rw [volume_pi,lintegral_fin_prod (volume : Measure ℝ) k (fun _ => cubeKernel δ) (fun _ => (measurable_cubeKernel δ))] simp [lintegral_cubeKernel hδ] lemma cubeAverage_labelKernel {d k : ℕ} (ρ : Measure (Sphere d)) (C : Rows k d) {δ : ℝ} (hδ : 0 < δ) (s : Sphere d) : cubeAverage ρ C (label C (s:Vector d)) δ = ∫⁻ y, ∏ i, cubeKernel δ (y i-label C (s:Vector d) i) ∂labelLaw ρ C := by rw [cubeAverage_eq_kernel _ _ hδ.le,sphereLift,labelLaw, lintegral_map (measurable_rowKernel_left _ (measurable_cubeKernel δ) _ _ C) continuous_subtype_val.measurable, lintegral_map (by have := measurable_cubeKernel δ; fun_prop) (by unfold label; fun_prop)] congr 1 funext t simp only [rowKernel,label,inner_sub_left] lemma spherical_cube_energy_eq {d k : ℕ} (ρ : Measure (Sphere d)) [IsFiniteMeasure ρ] (C : Rows k d) {δ : ℝ} (hδ : 0 < δ) : (∫⁻ s : Sphere d, cubeAverage ρ C (label C (s:Vector d)) δ ∂ρ) = ∫⁻ x, ∫⁻ y, (∏ i, cubeKernel δ ((y-x) i)) ∂labelLaw ρ C ∂labelLaw ρ C := by have hK : Measurable (fun z : Fin k→ℝ => ∏ i, cubeKernel δ (z i)) := by have := measurable_cubeKernel δ fun_prop have hf : Measurable (fun s : Sphere d => label C (s:Vector d)) := by unfold label; fun_prop have hm : Measurable (fun x : Fin k→ℝ => ∫⁻ y, (∏ i, cubeKernel δ ((y-x) i)) ∂labelLaw ρ C) := Measurable.lintegral_prod_right (hK.comp (measurable_snd.sub measurable_fst)) calc _ = ∫⁻ s : Sphere d, ∫⁻ y, (∏ i, cubeKernel δ ((y-label C (s:Vector d)) i)) ∂labelLaw ρ C ∂ρ := by apply lintegral_congr intro s exact cubeAverage_labelKernel ρ C hδ s _ = _ := (lintegral_map hm hf).symm lemma lintegral_labelLaw {d k : ℕ} (ρ : Measure (Sphere d)) (C : Rows k d) (f : (Fin k→ℝ) → ℝ≥0∞) (hf : Measurable f) : (∫⁻ y, f y ∂labelLaw ρ C) = ∫⁻ s : Sphere d, f (label C (s:Vector d)) ∂ρ := lintegral_map hf (by unfold label; fun_prop) lemma spherical_cube_energy_le {d k : ℕ} (ρ : Measure (Sphere d)) [IsFiniteMeasure ρ] (C : Rows k d) (hC : labelLaw ρ C ≪ volume) {δ : ℝ} (hδ : 0 < δ) : (∫⁻ s : Sphere d, cubeAverage ρ C (label C (s:Vector d)) δ ∂ρ) ≤ ∫⁻ s : Sphere d, exactDensity ρ C (label C (s:Vector d)) ∂ρ := by have hpair : Measurable (fun y : Fin k→ℝ => (C,y)) := measurable_const.prodMk measurable_id have hp := (measurable_exactDensity (k:=k) ρ).comp hpair change Measurable (fun y : Fin k→ℝ => exactDensity ρ C y) at hp have hK : Measurable (fun z : Fin k→ℝ => ∏ i, cubeKernel δ (z i)) := by have := measurable_cubeKernel δ fun_prop rw [spherical_cube_energy_eq ρ C hδ] calc _ ≤ ∫⁻ y, exactDensity ρ C y ∂labelLaw ρ C := projection_convolution_contraction (labelLaw ρ C) (fun y => exactDensity ρ C y) (fun z => ∏ i, cubeKernel δ (z i)) hp hK (labelLaw_withDensity_exact ρ C hC) (cubeVectorKernel_normalized (k:=k) hδ) _ = _ := lintegral_labelLaw ρ C _ hp lemma spherical_quadratic_limit {d k : ℕ} (ρ : Measure (Sphere d)) [IsFiniteMeasure ρ] (H : BoundedContinuousFunction (Rows k d) ℝ≥0) (hAC : ∀ᵐ C ∂gaussianRows k d, labelLaw ρ C ≪ volume) : Tendsto (fun j : ℕ => ∫⁻ z : Rows k d×Sphere d, (H z.1 : ℝ≥0∞)*cubeAverage ρ z.1 (label z.1 (z.2:Vector d)) (1/(j:ℝ)) ∂(gaussianRows k d).prod ρ) atTop (𝓝 (∫⁻ z : Rows k d×Sphere d, (H z.1 : ℝ≥0∞)*exactDensity ρ z.1 (label z.1 (z.2:Vector d)) ∂(gaussianRows k d).prod ρ)) := by have hH : Measurable (fun C => (H C : ℝ≥0∞)) := ENNReal.continuous_coe.measurable.comp H.continuous.measurable have hm (j : ℕ) : Measurable (fun z : Rows k d×Sphere d => (H z.1 : ℝ≥0∞)*cubeAverage ρ z.1 (label z.1 (z.2:Vector d)) (1/(j:ℝ))) := (hH.comp measurable_fst).mul (measurable_own_cube ρ _) have hlim : ∀ᵐ z : Rows k d×Sphere d ∂(gaussianRows k d).prod ρ, Tendsto (fun j : ℕ => cubeAverage ρ z.1 (label z.1 (z.2:Vector d)) (1/(j:ℝ))) atTop (𝓝 (exactDensity ρ z.1 (label z.1 (z.2:Vector d)))) := by apply (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun (fun j : ℕ => measurable_own_cube (k:=k) ρ (1/(j:ℝ))) (measurable_own_exact ρ))).mpr filter_upwards [hAC] with C hC exact ae_of_ae_map (by unfold label; fun_prop) (hC.ae_le (volume_ae_cube_limit ρ C)) have hlow : (∫⁻ z : Rows k d×Sphere d, (H z.1 : ℝ≥0∞)*exactDensity ρ z.1 (label z.1 (z.2:Vector d)) ∂(gaussianRows k d).prod ρ) ≤ liminf (fun j : ℕ => ∫⁻ z : Rows k d×Sphere d, (H z.1 : ℝ≥0∞)*cubeAverage ρ z.1 (label z.1 (z.2:Vector d)) (1/(j:ℝ)) ∂(gaussianRows k d).prod ρ) atTop := by calc _ = ∫⁻ z : Rows k d×Sphere d, liminf (fun j : ℕ => (H z.1 : ℝ≥0∞)*cubeAverage ρ z.1 (label z.1 (z.2:Vector d)) (1/(j:ℝ))) atTop ∂(gaussianRows k d).prod ρ := by apply lintegral_congr_ae filter_upwards [hlim] with z hz exact (ENNReal.Tendsto.const_mul hz (Or.inr ENNReal.coe_ne_top)).liminf_eq.symm _ ≤ _ := lintegral_liminf_le hm apply tendsto_of_le_liminf_of_limsup_le hlow apply limsup_le_of_le (by isBoundedDefault) filter_upwards [eventually_ge_atTop 1] with j hj have hmExact : Measurable (fun z : Rows k d×Sphere d => (H z.1 : ℝ≥0∞)*exactDensity ρ z.1 (label z.1 (z.2:Vector d))) := (hH.comp measurable_fst).mul (measurable_own_exact ρ) rw [lintegral_prod _ (hm j).aemeasurable,lintegral_prod _ hmExact.aemeasurable] apply lintegral_mono_ae filter_upwards [hAC] with C hC rw [lintegral_const_mul' _ _ ENNReal.coe_ne_top,lintegral_const_mul' _ _ ENNReal.coe_ne_top] exact mul_le_mul_right (spherical_cube_energy_le ρ C hC (by positivity)) _ end CurrentProjection end end OAI