import Mathlib import OAI.Probability.GaussianRegression.Main namespace OAI open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise noncomputable section universe u v open MeasureTheory Set Filter EuclideanGeometry open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise namespace MemoryPrecision abbrev Space (d : ℕ) := EuclideanSpace ℝ (Fin d) def J (β : ℝ) (m j : ℕ) : ℝ := let two : ℝ := @OfNat.ofNat ℝ 2 (@instOfNatAtLeastTwo ℝ 2 Real.commRing.toNatCast (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0))) let three : ℝ := @OfNat.ofNat ℝ 3 (@instOfNatAtLeastTwo ℝ 3 Real.commRing.toNatCast (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 1))) 1 + three^j * two^β * (m : ℝ) / (β - (j : ℝ) - (m : ℝ)) def AmbientBallBound {d : ℕ} (μ : Measure (Space d)) (H β : ℝ) : Prop := ∀ x : Space d, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H * r^β) def InverseGram {d p : ℕ} (m : ℕ) (v : Space d × (Fin (p-1) → Space d)) : ℝ≥0∞ := let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2 (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0))) letI : AddCommGroup (Space d) := WithLp.instAddCommGroup exponent (Fin d → ℝ) letI : Inner ℝ (Space d) := @InnerProductSpace.toInner ℝ (Space d) _ (PiLp.seminormedAddCommGroup exponent (fun _ : Fin d => ℝ)) (PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ)) ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ (fun i => v.2 i - v.1)).det)⁻¹ ^ m) lemma sharp_inverse_unit {X : Type*} [MeasurableSpace X] (μ : Measure X) (f : X → ℝ) (hf : Measurable f) (hf0 : ∀ x, 0 ≤ f x) {m a C : ℝ} (hm : 0 < m) (ha : m < a) (hC : 0 ≤ C) (htube : ∀ t : ℝ, 0 < t → t ≤ 1 → μ {x | f x ≤ t} ≤ ENNReal.ofReal (C * t^a)) : (∫⁻ x, ENNReal.ofReal (f x ^ (-m)) ∂μ) ≤ μ univ + ENNReal.ofReal (C * m / (a-m)) := by have htail (t : ℝ) (ht : 1 < t) : μ {x | t < f x ^ (-m)} ≤ ENNReal.ofReal (C * t ^ (-a/m)) := by have ht0 : 0 < t := by linarith have hr0 : 0 < t ^ ((-m)⁻¹) := Real.rpow_pos_of_pos ht0 _ have hr1 : t ^ ((-m)⁻¹) ≤ 1 := by calc _ ≤ t ^ (0 : ℝ) := Real.rpow_le_rpow_of_exponent_le ht.le (inv_nonpos.mpr (by linarith)) _ = 1 := Real.rpow_zero _ have hsub : {x | t < f x ^ (-m)} ⊆ {x | f x ≤ t ^ ((-m)⁻¹)} := by intro x hx have hfx : 0 < f x := by by_contra h have heq : f x = 0 := le_antisymm (le_of_not_gt h) (hf0 x) simp only [mem_ofPred_eq, heq, Real.zero_rpow (neg_ne_zero.mpr hm.ne')] at hx linarith exact ((Real.lt_rpow_inv_iff_of_neg hfx ht0 (by linarith)).2 hx).le calc _ ≤ μ {x | f x ≤ t ^ ((-m)⁻¹)} := measure_mono hsub _ ≤ ENNReal.ofReal (C * (t ^ ((-m)⁻¹)) ^ a) := htube _ hr0 hr1 _ = _ := by rw [← Real.rpow_mul ht0.le] congr 2 ring_nf have hb : -a/m < -1 := by rw [div_lt_iff₀ hm] linarith have hint : IntegrableOn (fun t : ℝ => C * t ^ (-a/m)) (Ioi 1) := (integrableOn_Ioi_rpow_of_lt hb (by norm_num)).const_mul C have hnn : 0 ≤ᵐ[volume.restrict (Ioi (1:ℝ))] (fun t : ℝ => C * t ^ (-a/m)) := by filter_upwards [ae_restrict_mem (μ := (volume : Measure ℝ)) measurableSet_Ioi] with t ht exact mul_nonneg hC (Real.rpow_nonneg (by have ht' : (1:ℝ) < t := ht; linarith : 0 ≤ t) _) have htailint : (∫⁻ t in Ioi (1:ℝ), ENNReal.ofReal (C * t ^ (-a/m))) = ENNReal.ofReal (C * m / (a-m)) := by rw [← ofReal_integral_eq_lintegral_ofReal hint hnn, integral_const_mul, integral_Ioi_rpow_of_lt hb (by norm_num), Real.one_rpow] congr 1 field_simp [hm.ne', show a - m ≠ 0 by linarith, show -a + m ≠ 0 by linarith] ring have hsplit : Ioi (0:ℝ) = Ioc 0 1 ∪ Ioi 1 := by ext t simp only [mem_Ioi, mem_union, mem_Ioc] constructor · intro ht by_cases h : t ≤ 1 · exact Or.inl ⟨ht,h⟩ · exact Or.inr (lt_of_not_ge h) · rintro (⟨ht,_⟩ | ht) <;> linarith rw [lintegral_eq_lintegral_meas_lt μ (Filter.Eventually.of_forall fun x => Real.rpow_nonneg (hf0 x) _) (hf.pow_const _).aemeasurable, hsplit] apply (lintegral_union_le _ _ _).trans apply add_le_add · calc _ ≤ ∫⁻ _t in Ioc (0:ℝ) 1, μ univ := lintegral_mono fun t => measure_mono (subset_univ _) _ = μ univ := by simp [Real.volume_Ioc] · calc _ ≤ ∫⁻ t in Ioi (1:ℝ), ENNReal.ofReal (C * t ^ (-a/m)) := by apply lintegral_mono_ae filter_upwards [ae_restrict_mem (μ := (volume : Measure ℝ)) measurableSet_Ioi] with t ht exact htail t ht _ = _ := htailint lemma ae_pos_of_power_tube {X : Type*} [MeasurableSpace X] (μ : Measure X) (f : X → ℝ) {a C : ℝ} (ha : 0 < a) (htube : ∀ t : ℝ, 0 < t → t ≤ 1 → μ {x | f x ≤ t} ≤ ENNReal.ofReal (C * t^a)) : ∀ᵐ x ∂μ, 0 < f x := by have hb (n : ℕ) : μ {x | f x ≤ 0} ≤ ENNReal.ofReal (C * ((1/2 : ℝ)^n)^a) := by calc μ {x | f x ≤ 0} ≤ μ {x | f x ≤ (1/2 : ℝ)^n} := measure_mono fun x hx => (show f x ≤ 0 from hx).trans (by positivity) _ ≤ _ := htube _ (by positivity) (pow_le_one₀ (by norm_num) (by norm_num)) have hlim : Tendsto (fun n : ℕ => ENNReal.ofReal (C * ((1/2 : ℝ)^n)^a)) atTop (𝓝 (0:ℝ≥0∞)) := by have h := (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (1/2:ℝ) < 1)).rpow_const_nhds_zero ha simpa using ENNReal.tendsto_ofReal (tendsto_const_nhds.mul h) have hz := le_antisymm (ge_of_tendsto' hlim hb) (bot_le : (0:ℝ≥0∞) ≤ μ {x | f x ≤ 0}) simpa only [ae_iff, not_lt] using hz lemma sharp_inverse_scaled {X : Type*} [MeasurableSpace X] (μ : Measure X) (f : X → ℝ) (hf : Measurable f) (hf0 : ∀ x, 0 ≤ f x) {m a B C R : ℝ} (hm : 0 < m) (ha : m < a) (hB : 0 ≤ B) (hC : 0 ≤ C) (hR : 0 < R) (hmass : μ univ ≤ ENNReal.ofReal B) (htube : ∀ t : ℝ, 0 < t → t ≤ 1 → μ {x | f x ≤ R*t} ≤ ENNReal.ofReal (C * t^a)) : (∀ᵐ x ∂μ, 0 < f x) ∧ (∫⁻ x, (ENNReal.ofReal (f x)) ^ (-m) ∂μ) ≤ ENNReal.ofReal ((B + C*m/(a-m)) * R^(-m)) := by have hg0 (x : X) : 0 ≤ f x / R := div_nonneg (hf0 x) hR.le have hg : Measurable (fun x => f x / R) := hf.div_const R have hgtube (t : ℝ) (ht : 0 < t) (ht1 : t ≤ 1) : μ {x | f x / R ≤ t} ≤ ENNReal.ofReal (C*t^a) := by have heq : {x | f x / R ≤ t} = {x | f x ≤ R*t} := by ext x simp only [mem_ofPred_eq, div_le_iff₀ hR, mul_comm] rw [heq] exact htube t ht ht1 have hgp := ae_pos_of_power_tube μ (fun x => f x / R) (hm.trans ha) hgtube have hfp : ∀ᵐ x ∂μ, 0 < f x := by filter_upwards [hgp] with x hx exact (div_pos_iff_of_pos_right hR).mp hx refine ⟨hfp, ?_⟩ have hi := sharp_inverse_unit μ (fun x => f x/R) hg hg0 hm ha hC hgtube have hcpos : 0 < R^(-m) := Real.rpow_pos_of_pos hR _ calc _ = ∫⁻ x, ENNReal.ofReal (R^(-m)) * ENNReal.ofReal ((f x/R)^(-m)) ∂μ := by apply lintegral_congr_ae filter_upwards [hfp] with x hx rw [ENNReal.ofReal_rpow_of_pos hx, ← ENNReal.ofReal_mul hcpos.le, Real.div_rpow hx.le hR.le] congr 1 exact (mul_div_cancel₀ _ hcpos.ne').symm _ = ENNReal.ofReal (R^(-m)) * (∫⁻ x, ENNReal.ofReal ((f x/R)^(-m)) ∂μ) := lintegral_const_mul _ (hg.pow_const _).ennreal_ofReal _ ≤ ENNReal.ofReal (R^(-m)) * (ENNReal.ofReal B + ENNReal.ofReal (C*m/(a-m))) := mul_le_mul' le_rfl (hi.trans (add_le_add hmass le_rfl)) _ = _ := by rw [← ENNReal.ofReal_add hB (by positivity), ← ENNReal.ofReal_mul hcpos.le] congr 1 ring section Geometry variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma affine_tube_cover_radius (L : AffineSubspace ℝ E) [Nonempty L] (z : E) {R t : ℝ} (hR : 0 < R) (ht : 0 < t) (htR : t ≤ R) : ∃ s : Finset L.direction, (s.card : ℝ) ≤ (3*R/t)^Module.finrank ℝ L.direction ∧ ∀ x : E, dist x z ≤ R → Metric.infDist x L ≤ t → ∃ y ∈ s, dist x (R • (y:E) + (orthogonalProjection L z : E)) ≤ 2*t := by classical have hr : 0 < t/R := div_pos ht hR have hr1 : t/R ≤ 1 := (div_le_one hR).2 htR obtain ⟨s, _, hcard, hnet⟩ := NoiselessRegression.exists_unit_ball_net (E := L.direction) hr hr1 refine ⟨s, ?_, ?_⟩ · convert hcard using 1 congr 1 field_simp · intro x hx hdist let v : L.direction := L.direction.orthogonalProjectionOnto (x-z) have hv : ‖v‖ ≤ R := by apply (L.direction.norm_orthogonalProjectionOnto_apply_le (x-z)).trans simpa only [dist_eq_norm] using hx have hvnorm : ‖R⁻¹ • v‖ ≤ 1 := by rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hR)] calc _ ≤ R⁻¹ * R := mul_le_mul_of_nonneg_left hv (inv_nonneg.mpr hR.le) _ = 1 := inv_mul_cancel₀ hR.ne' obtain ⟨y, hy, hclose⟩ := hnet (R⁻¹ • v) hvnorm refine ⟨y, hy, ?_⟩ have hscaled : dist (v:E) (R • (y:E)) ≤ t := by have heq : (v:E) = R • (R⁻¹ • (v:E)) := by simp [smul_smul, hR.ne'] rw [heq, dist_smul₀, Real.norm_eq_abs, abs_of_pos hR] calc _ ≤ R * (t/R) := mul_le_mul_of_nonneg_left hclose hR.le _ = t := mul_div_cancel₀ _ hR.ne' have hproj : (orthogonalProjection L x : E) = (v:E) + (orthogonalProjection L z : E) := by have h := (orthogonalProjection L).map_vadd z (x-z) simpa only [vadd_eq_add, sub_add_cancel, orthogonalProjection_contLinear, AffineSubspace.coe_vadd] using congrArg (fun q : L => (q:E)) h have hpd : dist (orthogonalProjection L x : E) (R • (y:E) + (orthogonalProjection L z : E)) ≤ t := by rw [hproj, dist_add_right] exact hscaled have hxp : dist x (orthogonalProjection L x : E) ≤ t := by rwa [dist_orthogonalProjection_eq_infDist] calc _ ≤ dist x (orthogonalProjection L x : E) + dist (orthogonalProjection L x : E) (R • (y:E) + (orthogonalProjection L z : E)) := dist_triangle _ _ _ _ ≤ t+t := add_le_add hxp hpd _ = 2*t := by ring end Geometry section AmbientMoments variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma ambient_tube_radius_factors (j : ℕ) (β : ℝ) {R t : ℝ} (ht : 0 < t) : (3*R/t)^j * (2*t)^β = (3:ℝ)^j * (2:ℝ)^β * R^j * t^(β-(j:ℝ)) := by rw [div_pow, mul_pow, Real.mul_rpow (by norm_num) ht.le, Real.rpow_sub_natCast ht.ne'] ring lemma ambient_tube {μ : Measure E} {H β R : ℝ} {z : E} (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) (L : AffineSubspace ℝ E) [Nonempty L] {t : ℝ} (ht : 0 < t) (htR : t ≤ R) : μ {x | Metric.infDist x L ≤ t} ≤ ENNReal.ofReal ((3:ℝ)^Module.finrank ℝ L.direction * (2:ℝ)^β * H * R^Module.finrank ℝ L.direction * t^(β-(Module.finrank ℝ L.direction:ℝ))) := by classical obtain ⟨s, hcard, hcover⟩ := affine_tube_cover_radius L z hR ht htR let c : L.direction → E := fun y => R • (y:E) + (orthogonalProjection L z : E) have hsub : {x | Metric.infDist x L ≤ t} ≤ᵐ[μ] ⋃ y ∈ s, Metric.closedBall (c y) (2*t) := by filter_upwards [μ.support_mem_ae] with x hx intro hxL have hxR : dist x z ≤ R := hsupport hx obtain ⟨y, hy, hxy⟩ := hcover x hxR hxL exact mem_iUnion₂.mpr ⟨y, hy, hxy⟩ calc _ ≤ μ (⋃ y ∈ s, Metric.closedBall (c y) (2*t)) := measure_mono_ae hsub _ ≤ ∑ y ∈ s, μ (Metric.closedBall (c y) (2*t)) := measure_biUnion_finset_le _ _ _ ≤ ∑ _y ∈ s, ENNReal.ofReal (H*(2*t)^β) := Finset.sum_le_sum fun y _ => hball (c y) _ (by positivity) _ = (s.card : ℝ≥0∞) * ENNReal.ofReal (H*(2*t)^β) := by simp _ ≤ ENNReal.ofReal ((3*R/t)^Module.finrank ℝ L.direction) * ENNReal.ofReal (H*(2*t)^β) := by gcongr exact_mod_cast ENNReal.ofReal_le_ofReal hcard _ = _ := by rw [← ENNReal.ofReal_mul (by positivity)] congr 1 calc _ = H * ((3*R/t)^Module.finrank ℝ L.direction * (2*t)^β) := by ring _ = _ := by rw [ambient_tube_radius_factors _ _ ht]; ring omit [BorelSpace E] in lemma ambient_mass_bound {μ : Measure E} {H β R : ℝ} {z : E} (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) : μ univ ≤ ENNReal.ofReal (H * R^β) := by calc _ ≤ μ (Metric.closedBall z R) := by apply measure_mono_ae filter_upwards [μ.support_mem_ae] with x hx intro _ exact hsupport hx _ ≤ _ := hball z R hR lemma ambient_tube_scaled {μ : Measure E} {H β R : ℝ} {z : E} (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) (L : AffineSubspace ℝ E) [Nonempty L] {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) : μ {x | Metric.infDist x L ≤ R*t} ≤ ENNReal.ofReal (((3:ℝ)^Module.finrank ℝ L.direction * (2:ℝ)^β * H * R^β) * t^(β-(Module.finrank ℝ L.direction:ℝ))) := by refine (ambient_tube hR hsupport hball L (mul_pos hR ht) (by nlinarith)).trans_eq ?_ congr 1 rw [Real.mul_rpow hR.le ht.le] have heq : R^Module.finrank ℝ L.direction * R^(β-(Module.finrank ℝ L.direction:ℝ)) = R^β := by rw [← Real.rpow_natCast, ← Real.rpow_add hR] congr 1 ring calc _ = ((3:ℝ)^Module.finrank ℝ L.direction * (2:ℝ)^β * H) * (R^Module.finrank ℝ L.direction * R^(β-(Module.finrank ℝ L.direction:ℝ))) * t^(β-(Module.finrank ℝ L.direction:ℝ)) := by ring _ = _ := by rw [heq] lemma ambient_inverse_moment {μ : Measure E} {H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) (L : AffineSubspace ℝ E) [Nonempty L] {m : ℕ} (hm : 0 < m) (hgap : (Module.finrank ℝ L.direction:ℝ) + (m:ℝ) < β) : (∀ᵐ x ∂μ, 0 < Metric.infDist x L) ∧ (∫⁻ x, (ENNReal.ofReal (Metric.infDist x L)) ^ (-(m:ℝ)) ∂μ) ≤ ENNReal.ofReal (J β m (Module.finrank ℝ L.direction) * H * R^(β-(m:ℝ))) := by have hm' : (0:ℝ) < m := by exact_mod_cast hm have hi := sharp_inverse_scaled μ (fun x => Metric.infDist x L) (Metric.continuous_infDist_pt _).measurable (fun _ => Metric.infDist_nonneg) hm' (show (m:ℝ) < β-(Module.finrank ℝ L.direction:ℝ) by linarith) (show 0 ≤ H*R^β by positivity) (show 0 ≤ (3:ℝ)^Module.finrank ℝ L.direction * (2:ℝ)^β * H * R^β by positivity) hR (ambient_mass_bound hR hsupport hball) (fun t ht ht1 => ambient_tube_scaled hR hsupport hball L ht ht1) refine ⟨hi.1, hi.2.trans_eq ?_⟩ congr 1 have heq : H*R^β + ((3:ℝ)^Module.finrank ℝ L.direction * (2:ℝ)^β * H * R^β) * (m:ℝ)/(β-(Module.finrank ℝ L.direction:ℝ)-(m:ℝ)) = J β m (Module.finrank ℝ L.direction) * H * R^β := by unfold J ring rw [heq, mul_assoc, ← Real.rpow_add hR] congr 2 end AmbientMoments lemma J_pos {β : ℝ} {m j : ℕ} (hgap : (j:ℝ)+(m:ℝ)<β) : 0 < J β m j := by unfold J have hd : 0 < β-(j:ℝ)-(m:ℝ) := by linarith positivity lemma J_mono {β : ℝ} {m j k : ℕ} (hjk : j ≤ k) (hgap : (k:ℝ)+(m:ℝ)<β) : J β m j ≤ J β m k := by have hjk' : (j:ℝ) ≤ k := by exact_mod_cast hjk unfold J apply add_le_add le_rfl apply div_le_div₀ · positivity · gcongr norm_num · linarith · linarith section ReplicaExact variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] open NoiselessRegression InnerProductSpace lemma ambient_inverse_moment_le {μ : Measure E} {H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) (L : AffineSubspace ℝ E) [Nonempty L] {m k : ℕ} (hm : 0 < m) (hdim : Module.finrank ℝ L.direction ≤ k) (hgap : (k:ℝ)+(m:ℝ)<β) : (∫⁻ x, ENNReal.ofReal ((Metric.infDist x L)⁻¹ ^ m) ∂μ) ≤ ENNReal.ofReal (J β m k * H * R^(β-(m:ℝ))) := by have hdim' : (Module.finrank ℝ L.direction:ℝ) ≤ k := by exact_mod_cast hdim obtain ⟨ha, hi⟩ := ambient_inverse_moment hH hR hsupport hball L hm (by linarith) calc _ = ∫⁻ x, (ENNReal.ofReal (Metric.infDist x L)) ^ (-(m:ℝ)) ∂μ := by apply lintegral_congr_ae filter_upwards [ha] with x hx rw [ENNReal.ofReal_rpow_of_pos hx, Real.rpow_neg hx.le, Real.rpow_natCast, inv_pow] _ ≤ ENNReal.ofReal (J β m (Module.finrank ℝ L.direction) * H * R^(β-(m:ℝ))) := hi _ ≤ _ := by apply ENNReal.ofReal_le_ofReal gcongr exact J_mono hdim hgap lemma replica_integral_exact {μ : Measure E} [IsFiniteMeasure μ] {H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) {m : ℕ} (hm : 0 < m) (p : E) (n : ℕ) (hgap : (n:ℝ)+(m:ℝ)-1<β) : (∫⁻ v : Fin n → E, replicaWeight m p v ∂Measure.pi (fun _ => μ)) ≤ ∏ j : Fin n, ENNReal.ofReal (J β m j * H * R^(β-(m:ℝ))) := by induction n with | zero => simp [replicaWeight] | succ n ih => have hngap : (n:ℝ)+(m:ℝ)<β := by push_cast at hgap; linarith have hinner (v : Fin n → E) : (∫⁻ x, replicaWeight m p (Fin.snoc v x) ∂μ) ≤ replicaWeight m p v * ENNReal.ofReal (J β m n * H * R^(β-(m:ℝ))) := by simp_rw [replicaWeight_snoc] rw [lintegral_const_mul _ (by fun_prop)] apply mul_le_mul' le_rfl apply ambient_inverse_moment_le hH hR hsupport hball _ hm _ hngap rw [AffineSubspace.direction_mk'] simpa only [Fintype.card_fin, Set.finrank] using finrank_range_le_card (R := ℝ) (fun i : Fin n => v i - p) calc _ = ∫⁻ v : Fin n → E, ∫⁻ x, replicaWeight m p (Fin.snoc v x) ∂μ ∂Measure.pi (fun _ => μ) := lintegral_pi_snoc μ _ (measurable_replicaWeight m p (n+1)) _ ≤ ∫⁻ v : Fin n → E, replicaWeight m p v * ENNReal.ofReal (J β m n * H * R^(β-(m:ℝ))) ∂Measure.pi (fun _ => μ) := lintegral_mono hinner _ = (∫⁻ v : Fin n → E, replicaWeight m p v ∂Measure.pi (fun _ => μ)) * ENNReal.ofReal (J β m n * H * R^(β-(m:ℝ))) := lintegral_mul_const _ (measurable_replicaWeight m p n) _ ≤ (∏ j : Fin n, ENNReal.ofReal (J β m j * H * R^(β-(m:ℝ)))) * ENNReal.ofReal (J β m n * H * R^(β-(m:ℝ))) := mul_le_mul' (ih (by linarith)) le_rfl _ = _ := by rw [Fin.prod_univ_castSucc]; rfl lemma ae_replica_nondegenerate_exact {μ : Measure E} [IsFiniteMeasure μ] {H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) {m : ℕ} (hm : 0 < m) (p : E) (n : ℕ) (hgap : (n:ℝ)+(m:ℝ)-1<β) : ∀ᵐ v : Fin n → E ∂Measure.pi (fun _ => μ), ∀ i, gramSchmidt ℝ (fun j => v j-p) i ≠ 0 := by induction n with | zero => exact Filter.Eventually.of_forall fun _ i => Fin.elim0 i | succ n ih => rw [ae_pi_snoc μ (measurableSet_gramSchmidt_ne_zero p (n+1))] have hngap : (n:ℝ)+(m:ℝ)<β := by push_cast at hgap; linarith filter_upwards [ih (by linarith)] with v hv let L := AffineSubspace.mk' p (Submodule.span ℝ (Set.range (fun i : Fin n => v i-p))) have hdim : (Module.finrank ℝ L.direction:ℝ) ≤ n := by dsimp [L] rw [AffineSubspace.direction_mk'] have hh := finrank_range_le_card (R := ℝ) (fun i : Fin n => v i-p) simp only [Fintype.card_fin, Set.finrank] at hh exact_mod_cast hh have ha := (ambient_inverse_moment hH hR hsupport hball L hm (by linarith)).1 filter_upwards [ha] with x hx intro i induction i using Fin.lastCases with | last => apply norm_ne_zero_iff.mp rw [norm_gramSchmidt_sub_eq_affine_infDist, image_Iio_last] simp only [Fin.snoc_castSucc, Fin.snoc_last] exact hx.ne' | cast j => simpa only [gramSchmidt_castSucc, Fin.snoc_castSucc] using hv j lemma measurable_joint_replicaWeight (m n : ℕ) : Measurable (fun v : E × (Fin n → E) => replicaWeight m v.1 v.2) := by have hs : Measurable (fun v : E × (Fin n → E) => fun j => v.2 j-v.1) := by fun_prop exact Finset.measurable_prod _ fun i _ => (((measurable_gramSchmidt i).comp hs).norm.inv.pow_const m).ennreal_ofReal lemma ae_joint_replica_independent {μ : Measure E} [IsFiniteMeasure μ] {H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) {m : ℕ} (hm : 0 < m) (n : ℕ) (hgap : (n:ℝ)+(m:ℝ)-1<β) : ∀ᵐ v : E × (Fin n → E) ∂μ.prod (Measure.pi (fun _ => μ)), LinearIndependent ℝ (fun j => v.2 j-v.1) := by have hmeas : MeasurableSet {v : E × (Fin n → E) | ∀ i, gramSchmidt ℝ (fun j => v.2 j-v.1) i ≠ 0} := by rw [Set.ofPred_forall] apply MeasurableSet.iInter intro i have h := (measurable_gramSchmidt i).comp (show Measurable (fun v : E × (Fin n → E) => fun j => v.2 j-v.1) by fun_prop) exact (measurableSet_eq_fun h measurable_const).compl have ha : ∀ᵐ v : E × (Fin n → E) ∂μ.prod (Measure.pi (fun _ => μ)), ∀ i, gramSchmidt ℝ (fun j => v.2 j-v.1) i ≠ 0 := by rw [Measure.ae_prod_iff_ae_ae hmeas] exact Filter.Eventually.of_forall fun p => ae_replica_nondegenerate_exact hH hR hsupport hball hm p n hgap filter_upwards [ha] with v hv 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 (hv i)) lemma joint_replica_integral_exact {μ : Measure E} [IsFiniteMeasure μ] {H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R) (hsupport : μ.support ⊆ Metric.closedBall z R) (hball : ∀ x : E, ∀ r : ℝ, 0 < r → μ (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) {m : ℕ} (hm : 0 < m) (n : ℕ) (hgap : (n:ℝ)+(m:ℝ)-1<β) : (∫⁻ v : E × (Fin n → E), replicaWeight m v.1 v.2 ∂μ.prod (Measure.pi (fun _ => μ))) ≤ ENNReal.ofReal (H^(n+1) * R^(β*(n+1:ℕ)-(m:ℝ)*(n:ℝ)) * ∏ j : Fin n, J β m j) := by have hJ (j : Fin n) : 0 < J β m j := by apply J_pos have hj : (j:ℝ)+1 ≤ n := by exact_mod_cast Nat.succ_le_of_lt j.isLt linarith have hfac (j : Fin n) : 0 ≤ J β m j * H * R^(β-(m:ℝ)) := by exact le_of_lt (mul_pos (mul_pos (hJ j) hH) (Real.rpow_pos_of_pos hR _)) calc _ = ∫⁻ p, ∫⁻ v : Fin n → E, replicaWeight m p v ∂Measure.pi (fun _ => μ) ∂μ := lintegral_prod _ (measurable_joint_replicaWeight m n).aemeasurable _ ≤ ∫⁻ _p, ∏ j : Fin n, ENNReal.ofReal (J β m j * H * R^(β-(m:ℝ))) ∂μ := lintegral_mono fun p => replica_integral_exact hH hR hsupport hball hm p n hgap _ = (∏ j : Fin n, ENNReal.ofReal (J β m j * H * R^(β-(m:ℝ)))) * μ univ := lintegral_const _ _ ≤ (∏ j : Fin n, ENNReal.ofReal (J β m j * H * R^(β-(m:ℝ)))) * ENNReal.ofReal (H*R^β) := mul_le_mul' le_rfl (ambient_mass_bound hR hsupport hball) _ = _ := by rw [← ENNReal.ofReal_prod_of_nonneg (fun j _ => hfac j), ← ENNReal.ofReal_mul (Finset.prod_nonneg fun j _ => hfac j)] congr 1 simp only [Finset.prod_mul_distrib, Finset.prod_const, Finset.card_univ, Fintype.card_fin] rw [← Real.rpow_mul_natCast hR.le] have heq : R^((β-(m:ℝ))*(n:ℝ)) * R^β = R^(β*(n+1:ℕ)-(m:ℝ)*(n:ℝ)) := by rw [← Real.rpow_add hR] congr 1 push_cast ring calc _ = (H^n*H) * (R^((β-(m:ℝ))*(n:ℝ)) * R^β) * ∏ j : Fin n, J β m j := by ring _ = _ := by rw [← pow_succ, heq] end ReplicaExact lemma specialized_J_bound (d m p : ℕ) (hd : 16 ≤ d) (hp : p=d/8) (_ : 1 ≤ m) (hmp : m ≤ p) (β : ℝ) (hβlo : ((d:ℝ)-1)/2 ≤ β) (hβhi : β ≤ (d:ℝ)/2) : ∀ j : ℕ, j ≤ p-2 → ((d:ℝ)-1)/2 - 2*(p:ℝ) + 2 ≤ β-(j:ℝ)-(m:ℝ) ∧ (d:ℝ)/4 + 3/2 ≤ ((d:ℝ)-1)/2 - 2*(p:ℝ) + 2 ∧ J β m j ≤ (7:ℝ)^d := by have hp2 : 2 ≤ p := by omega have hp8 : p*8 ≤ d := by rw [hp]; exact Nat.div_mul_le_self d 8 have hp8' : (p:ℝ)*8 ≤ d := by exact_mod_cast hp8 have hd' : (16:ℝ) ≤ d := by exact_mod_cast hd have hm' : (m:ℝ) ≤ p := by exact_mod_cast hmp intro j hj have hj' : (j:ℝ)+2 ≤ p := by exact_mod_cast (show j+2 ≤ p by omega) have hmargin : ((d:ℝ)-1)/2-2*(p:ℝ)+2 ≤ β-(j:ℝ)-(m:ℝ) := by linarith have hdim : (d:ℝ)/4+3/2 ≤ ((d:ℝ)-1)/2-2*(p:ℝ)+2 := by linarith refine ⟨hmargin, hdim, ?_⟩ have hgap : 0 < β-(j:ℝ)-(m:ℝ) := by linarith have hquot : (m:ℝ)/(β-(j:ℝ)-(m:ℝ)) ≤ 1 := by rw [div_le_one hgap] linarith have h3 : (3:ℝ)^j ≤ (3:ℝ)^d := by apply pow_le_pow_right₀ (by norm_num) omega have h2 : (2:ℝ)^β ≤ (2:ℝ)^d := by rw [← Real.rpow_natCast] apply Real.rpow_le_rpow_of_exponent_le (by norm_num) linarith calc J β m j = 1 + (3:ℝ)^j * (2:ℝ)^β * ((m:ℝ)/(β-(j:ℝ)-(m:ℝ))) := by unfold J; ring _ ≤ 1 + (3:ℝ)^j * (2:ℝ)^β * 1 := by gcongr _ ≤ 1 + (3:ℝ)^d * (2:ℝ)^d := by rw [mul_one]; gcongr _ = 1 + (6:ℝ)^d := by rw [← mul_pow]; norm_num _ ≤ (7:ℝ)^d := by simpa only [one_pow, show (1:ℝ)+6=7 by norm_num] using pow_add_pow_le (show (0:ℝ) ≤ 1 by norm_num) (show (0:ℝ) ≤ 6 by norm_num) (show d ≠ 0 by omega) lemma specialized_gram_product_bound (d m p : ℕ) (hd : 16 ≤ d) (hp : p=d/8) (hm : 1 ≤ m) (hmp : m ≤ p) (β : ℝ) (hβlo : ((d:ℝ)-1)/2 ≤ β) (hβhi : β ≤ (d:ℝ)/2) (H R : ℝ) (hH : 0 < H) (hR : 0 < R) : H^p * R^(β*(p:ℝ)-(m:ℝ)*(p-1:ℕ)) * (∏ j : Fin (p-1), J β m j) ≤ (7:ℝ)^(d*(p-1)) * H^p * R^(β*(p:ℝ)-(m:ℝ)*(p-1:ℕ)) := by have hp2 : 2 ≤ p := by omega have hs := specialized_J_bound d m p hd hp hm hmp β hβlo hβhi have hprod : (∏ j : Fin (p-1), J β m j) ≤ (7:ℝ)^(d*(p-1)) := by calc _ ≤ ∏ _j : Fin (p-1), (7:ℝ)^d := by apply Finset.prod_le_prod₀ · intro j _ apply (J_pos ?_).le have hh := hs j (by have := j.isLt; omega) have hd' : (16:ℝ) ≤ d := by exact_mod_cast hd linarith [hh.1, hh.2.1] · intro j _ exact (hs j (by have := j.isLt; omega)).2.2 _ = _ := by simp [← pow_mul] calc _ ≤ H^p * R^(β*(p:ℝ)-(m:ℝ)*(p-1:ℕ)) * (7:ℝ)^(d*(p-1)) := by gcongr _ = _ := by ring def AmbientGramStatement : Prop := let two : ℝ := @OfNat.ofNat ℝ 2 (@instOfNatAtLeastTwo ℝ 2 Real.commRing.toNatCast (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0))) let three : ℝ := @OfNat.ofNat ℝ 3 (@instOfNatAtLeastTwo ℝ 3 Real.commRing.toNatCast (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 1))) let four : ℝ := @OfNat.ofNat ℝ 4 (@instOfNatAtLeastTwo ℝ 4 Real.commRing.toNatCast (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 2))) let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2 (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0))) (∀ (d m p : ℕ), letI : MeasurableSpace (Space d) := WithLp.measurableSpace exponent (Fin d → ℝ) letI : PseudoMetricSpace (Space d) := PiLp.instPseudoMetricSpace (p := exponent) (β := fun _ : Fin d => ℝ) letI : AddCommGroup (Space d) := WithLp.instAddCommGroup exponent (Fin d → ℝ) letI : Module ℝ (Space d) := WithLp.instModule (p := exponent) (K := ℝ) (V := Fin d → ℝ) 1 ≤ d → 1 ≤ m → 2 ≤ p → ∀ β R H : ℝ, (m : ℝ) + (p : ℝ) - two < β → 0 < R → 0 < H → ∀ (μ : Measure (Space d)), IsFiniteMeasure μ → ∀ z : Space d, μ.support ⊆ Metric.closedBall z R → AmbientBallBound μ H β → (∀ (j : ℕ), j ≤ p-2 → ∀ L : AffineSubspace ℝ (Space d), Nonempty L → Module.finrank ℝ L.direction = j → (∀ t : ℝ, 0 < t → t ≤ R → μ {s | Metric.infDist s L ≤ t} ≤ ENNReal.ofReal (three^j * two^β * H * R^j * t^(β-(j:ℝ)))) ∧ (∫⁻ s, (ENNReal.ofReal (Metric.infDist s L)) ^ (-(m:ℝ)) ∂μ) ≤ ENNReal.ofReal (J β m j * H * R^(β-(m:ℝ)))) ∧ (∀ᵐ v : Space d × (Fin (p-1) → Space d) ∂μ.prod (Measure.pi (fun _ : Fin (p-1) => μ)), LinearIndependent ℝ (fun i => v.2 i - v.1)) ∧ (∫⁻ v : Space d × (Fin (p-1) → Space d), InverseGram m v ∂μ.prod (Measure.pi (fun _ : Fin (p-1) => μ))) ≤ ENNReal.ofReal (H^p * R^(β*(p:ℝ)-(m:ℝ)*(p-1:ℕ)) * ∏ j : Fin (p-1), J β m j)) ∧ (∃ C : ℝ, 1 ≤ C ∧ ∀ (d m p : ℕ), 16 ≤ d → p = d/8 → 1 ≤ m → m ≤ p → ∀ β : ℝ, ((d:ℝ)-1)/two ≤ β → β ≤ (d:ℝ)/two → (∀ j : ℕ, j ≤ p-2 → ((d:ℝ)-1)/two - two*(p:ℝ) + two ≤ β-(j:ℝ)-(m:ℝ) ∧ (d:ℝ)/four + three/two ≤ ((d:ℝ)-1)/two - two*(p:ℝ) + two ∧ J β m j ≤ C^d) ∧ (∀ H R : ℝ, 0 < H → 0 < R → H^p * R^(β*(p:ℝ)-(m:ℝ)*(p-1:ℕ)) * (∏ j : Fin (p-1), J β m j) ≤ C^(d*(p-1)) * H^p * R^(β*(p:ℝ)-(m:ℝ)*(p-1:ℕ)))) theorem ambient_gram : AmbientGramStatement := by constructor · intro d m p _hd hm hp β R H hβ hR hH μ hfin z hsupport hball let : IsFiniteMeasure μ := hfin have hm0 : 0 < m := by omega have hp1 : 1 ≤ p := by omega have hgap : ((p-1:ℕ):ℝ)+(m:ℝ)-1<β := by rw [Nat.cast_sub hp1] push_cast linarith refine ⟨?_, ?_, ?_⟩ · intro j hj L hnonempty hdim let : Nonempty L := hnonempty have hj' : (j:ℝ)+2 ≤ p := by exact_mod_cast (show j+2 ≤ p by omega) constructor · intro t ht htR simpa only [hdim] using ambient_tube hR hsupport hball L ht htR · have hi := (ambient_inverse_moment hH hR hsupport hball L hm0 (by rw [hdim]; linarith)).2 simpa only [hdim] using hi · exact ae_joint_replica_independent hH hR hsupport hball hm0 (p-1) hgap · have hi := joint_replica_integral_exact hH hR hsupport hball hm0 (p-1) hgap simpa only [NoiselessRegression.replicaWeight_eq_det, InverseGram, Nat.sub_add_cancel hp1] using hi · refine ⟨7, by norm_num, ?_⟩ intro d m p hd hp hm hmp β hβlo hβhi exact ⟨specialized_J_bound d m p hd hp hm hmp β hβlo hβhi, specialized_gram_product_bound d m p hd hp hm hmp β hβlo hβhi⟩ end MemoryPrecision end end OAI