import OAI.Probability.ProjectionMoments.BallMoments namespace OAI noncomputable section open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace open scoped ENNReal NNReal RealInnerProductSpace Topology namespace ProjectionMoments section variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma sphere_packing_annulus (s : Finset E) {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) (hs : ∀ x ∈ s, ‖x‖ = 1) (hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r ≤ dist x y) : (s.card : ℝ) * (r/2)^Module.finrank ℝ E ≤ (1+r/2)^Module.finrank ℝ E - (1-r/2)^Module.finrank ℝ E := by classical let δ := r/2 let U : Set E := ⋃ c ∈ s, Metric.ball c δ let V : Set E := Metric.ball 0 (1-δ) have hδ : 0 < δ := by dsimp [δ]; positivity have hinner : 0 < 1-δ := by dsimp [δ]; linarith have houter : 0 < 1+δ := by positivity have hdis : Set.Pairwise (s : Set E) (fun c d => Disjoint (Metric.ball c δ) (Metric.ball d δ)) := by intro c hc d hd hcd apply Metric.ball_disjoint_ball simpa [δ] using hsep c hc d hd hcd have hUV : Disjoint U V := by apply Set.disjoint_left.mpr intro x hx hxV obtain ⟨c,hc,hxc⟩ := Set.mem_iUnion₂.mp hx have hxc' : dist x c < δ := hxc have hx0 : ‖x‖ < 1-δ := by simpa [V, Metric.mem_ball, dist_zero_right] using hxV have hc1 := hs c hc have htri := dist_triangle c x (0 : E) rw [dist_zero_right, dist_zero_right, dist_comm c x, hc1] at htri linarith have hsub : U ∪ V ⊆ Metric.ball (0 : E) (1+δ) := by rintro x (hx | hx) · obtain ⟨c,hc,hxc⟩ := Set.mem_iUnion₂.mp hx apply Metric.mem_ball.mpr have htri := dist_triangle x c (0 : E) rw [dist_zero_right c, hs c hc] at htri exact htri.trans_lt (by have hh : dist x c < δ := hxc; linarith) · exact (Metric.ball_subset_ball (by linarith : 1-δ ≤ 1+δ)) hx have hU : MeasurableSet U := MeasurableSet.biUnion s.finite_toSet.countable (fun c _ => Metric.isOpen_ball.measurableSet) have hvol : ((s.card : ℝ≥0∞)*ENNReal.ofReal (δ^Module.finrank ℝ E) + ENNReal.ofReal ((1-δ)^Module.finrank ℝ E)) * volume (Metric.ball (0 : E) 1) ≤ ENNReal.ofReal ((1+δ)^Module.finrank ℝ E)*volume (Metric.ball (0 : E) 1) := by calc _ = volume (U ∪ V) := by rw [measure_union hUV Metric.isOpen_ball.measurableSet, show volume U = (s.card : ℝ≥0∞)*ENNReal.ofReal (δ^Module.finrank ℝ E)* volume (Metric.ball (0 : E) 1) by rw [measure_biUnion_finset hdis (fun _ _ => Metric.isOpen_ball.measurableSet)] simp only [Measure.addHaar_ball_of_pos volume _ hδ, Finset.sum_const, nsmul_eq_mul, mul_assoc], Measure.addHaar_ball_of_pos volume 0 hinner, add_mul] _ ≤ volume (Metric.ball (0 : E) (1+δ)) := measure_mono hsub _ = _ := Measure.addHaar_ball_of_pos volume 0 houter have hv0 : volume (Metric.ball (0 : E) 1) ≠ 0 := (Metric.measure_ball_pos volume 0 zero_lt_one).ne' have hvt : volume (Metric.ball (0 : E) 1) ≠ ∞ := measure_ball_lt_top.ne have hh := (ENNReal.mul_le_mul_iff_left hv0 hvt).mp hvol have hreal := ENNReal.toReal_le_of_le_ofReal (pow_nonneg houter.le _) hh simp only [ENNReal.toReal_add (by finiteness : (s.card : ℝ≥0∞)*ENNReal.ofReal (δ^Module.finrank ℝ E) ≠ ∞) ENNReal.ofReal_ne_top, ENNReal.toReal_mul, ENNReal.toReal_natCast, ENNReal.toReal_ofReal (pow_nonneg hδ.le _), ENNReal.toReal_ofReal (pow_nonneg hinner.le _)] at hreal dsimp [δ] at hreal linarith lemma sphere_packing_bound (s : Finset E) {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) (hd : 1 ≤ Module.finrank ℝ E) (hs : ∀ x ∈ s, ‖x‖ = 1) (hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r ≤ dist x y) : (s.card : ℝ)*r^(Module.finrank ℝ E-1) ≤ (8 : ℝ)^Module.finrank ℝ E := by let d := Module.finrank ℝ E have hi : 0 ≤ 1-r/2 := by linarith have ho : 0 ≤ 1+r/2 := by positivity have hio : 1-r/2 ≤ 1+r/2 := by linarith have h := sphere_packing_annulus s hr hr1 hs hsep have hdiff : (1+r/2)^d - (1-r/2)^d ≤ r*d*(1+r/2)^(d-1) := by have hh := abs_pow_sub_pow_le (a := (1+r/2)) (b := (1-r/2)) (n := d) rw [abs_of_nonneg (sub_nonneg.mpr (pow_le_pow_left₀ hi hio d)), abs_of_nonneg ho, abs_of_nonneg hi, max_eq_left hio, show (1+r/2)-(1-r/2) = r by ring, abs_of_pos hr] at hh exact hh have hpow : (1+r/2)^(d-1) ≤ (2 : ℝ)^d := by exact (pow_le_pow_left₀ ho (by linarith : 1+r/2 ≤ 2) _).trans (pow_le_pow_right₀ (by norm_num) (Nat.sub_le d 1)) have hd2 : (d : ℝ) ≤ (2 : ℝ)^d := by have hn : ∀ n : ℕ, (n : ℝ) ≤ (2 : ℝ)^n := by intro n induction n with | zero => norm_num | succ n ih => have hp : (1 : ℝ) ≤ 2^n := one_le_pow₀ (by norm_num) rw [Nat.cast_succ, pow_succ] linarith exact hn d have hupper : r*d*(1+r/2)^(d-1) ≤ r*(4 : ℝ)^d := by calc _ ≤ r*2^d*2^d := mul_le_mul (mul_le_mul_of_nonneg_left hd2 hr.le) hpow (by positivity) (by positivity) _ = _ := by rw [mul_assoc, ← mul_pow]; norm_num have hall := h.trans (hdiff.trans hupper) have htwo : 0 < (2 : ℝ)^d := by positivity rw [div_pow, ← mul_div_assoc, div_le_iff₀ htwo] at hall have hp : r^d = r^(d-1)*r := by rw [← pow_succ, Nat.sub_add_cancel hd] rw [hp, ← mul_assoc, mul_assoc r, ← mul_pow] at hall norm_num only at hall nlinarith lemma sphere_cap_mass_center [Nontrivial E] {c c' : E} (hc : ‖c‖ = 1) (hc' : ‖c'‖ = 1) (r : ℝ) : NoiselessRegression.ambientSphereLaw (Metric.closedBall c r) = NoiselessRegression.ambientSphereLaw (Metric.closedBall c' r) := by obtain ⟨O,hO⟩ := rotation_transitive (show Orthonormal ℝ (fun _ : Fin 1 => c) by simpa using hc) (show Orthonormal ℝ (fun _ : Fin 1 => c') by simpa using hc') let e : E ≃ₗᵢ[ℝ] E := Unitary.linearIsometryEquiv O have he : e c = c' := hO 0 have hh := (NoiselessRegression.ambientSphereLaw_isometry e).map_eq have hpre : e ⁻¹' Metric.closedBall c' r = Metric.closedBall c r := by ext x simp only [Set.mem_preimage, Metric.mem_closedBall, ← he, e.dist_map] have h := congrArg (fun μ : Measure E => μ (Metric.closedBall c' r)) hh rw [Measure.map_apply (by fun_prop) Metric.isClosed_closedBall.measurableSet, hpre] at h exact h lemma sphere_finite_net {r : ℝ} (hr : 0 < r) : ∃ s : Finset E, (∀ c ∈ s, ‖c‖ = 1) ∧ (∀ c ∈ s, ∀ c' ∈ s, c ≠ c' → r < dist c c') ∧ ∀ x : E, ‖x‖ = 1 → ∃ c ∈ s, dist x c ≤ r := by classical let N := ⌈((1+r/2)/(r/2))^Module.finrank ℝ E⌉₊ obtain ⟨s,hs,hsep,hnet⟩ := NoiselessRegression.exists_net_of_finite_packing_bound {x : E | ‖x‖ = 1} hr.le N (by intro s hs hsep have hh := NoiselessRegression.card_le_volume_ratio s hr (by norm_num : (0:ℝ) ≤ 1) (fun c hc => (hs c hc).le) (fun c hc c' hc' hne => (hsep c hc c' hc' hne).le) exact_mod_cast hh.trans (Nat.le_ceil _)) exact ⟨s,hs,hsep,hnet⟩ lemma sphere_cap_lower_one [Nontrivial E] {c : E} (hc : ‖c‖ = 1) {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : ENNReal.ofReal (r^(Module.finrank ℝ E-1)) ≤ ENNReal.ofReal ((8:ℝ)^Module.finrank ℝ E) * NoiselessRegression.ambientSphereLaw (Metric.closedBall c r) := by classical let μ : Measure E := NoiselessRegression.ambientSphereLaw obtain ⟨s,hs,hsep,hnet⟩ := sphere_finite_net (E := E) hr have hcard := sphere_packing_bound s hr hr1 (Module.finrank_pos (R := ℝ) (M := E)) hs (fun x hx y hy hxy => (hsep x hx y hy hxy).le) have hcover : μ (⋃ x ∈ s, Metric.closedBall x r) = 1 := by rw [← measure_univ (μ := μ)] apply measure_congr filter_upwards [NoiselessRegression.ambientSphereLaw_ae_norm (E := E)] with x hx apply propext change (x ∈ ⋃ y ∈ s, Metric.closedBall y r) ↔ x ∈ Set.univ simp only [Set.mem_univ, iff_true] obtain ⟨y,hy,hxy⟩ := hnet x hx exact Set.mem_iUnion₂.mpr ⟨y,hy,hxy⟩ have hone : (1 : ℝ≥0∞) ≤ (s.card : ℝ≥0∞)*μ (Metric.closedBall c r) := by calc _ = μ (⋃ x ∈ s, Metric.closedBall x r) := hcover.symm _ ≤ ∑ x ∈ s, μ (Metric.closedBall x r) := measure_biUnion_finset_le _ _ _ = _ := by have hm : ∀ x ∈ s, μ (Metric.closedBall x r) = μ (Metric.closedBall c r) := fun x hx => sphere_cap_mass_center (hs x hx) hc r simp_rw [Finset.sum_congr rfl hm] simp calc _ = ENNReal.ofReal (r^(Module.finrank ℝ E-1))*1 := (mul_one _).symm _ ≤ ENNReal.ofReal (r^(Module.finrank ℝ E-1))* ((s.card : ℝ≥0∞)*μ (Metric.closedBall c r)) := mul_le_mul' le_rfl hone _ = ENNReal.ofReal ((s.card:ℝ)*r^(Module.finrank ℝ E-1))* μ (Metric.closedBall c r) := by rw [ENNReal.ofReal_mul (by positivity), ENNReal.ofReal_natCast]; ring _ ≤ _ := mul_le_mul' (ENNReal.ofReal_le_ofReal hcard) le_rfl lemma sphere_cap_lower [Nontrivial E] {c : E} (hc : ‖c‖ = 1) {r : ℝ} (hr : 0 < r) (hr2 : r ≤ 2) : ENNReal.ofReal (r^(Module.finrank ℝ E-1)) ≤ ENNReal.ofReal ((16:ℝ)^Module.finrank ℝ E) * NoiselessRegression.ambientSphereLaw (Metric.closedBall c r) := by by_cases hr1 : r ≤ 1 · exact (sphere_cap_lower_one hc hr hr1).trans (mul_le_mul' (ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ (by norm_num : (0:ℝ) ≤ 8) (by norm_num : (8:ℝ) ≤ 16) _)) le_rfl) · have h1 := sphere_cap_lower_one hc (by norm_num : (0:ℝ) < 1) le_rfl simp only [one_pow, ENNReal.ofReal_one] at h1 have h2 : r^(Module.finrank ℝ E-1) ≤ (2:ℝ)^Module.finrank ℝ E := (pow_le_pow_left₀ hr.le hr2 _).trans (pow_le_pow_right₀ (by norm_num) (Nat.sub_le _ _)) calc _ ≤ ENNReal.ofReal ((2:ℝ)^Module.finrank ℝ E)*1 := by simpa using ENNReal.ofReal_le_ofReal h2 _ ≤ ENNReal.ofReal ((2:ℝ)^Module.finrank ℝ E)* (ENNReal.ofReal ((8:ℝ)^Module.finrank ℝ E)* NoiselessRegression.ambientSphereLaw (Metric.closedBall c 1)) := mul_le_mul' le_rfl h1 _ ≤ ENNReal.ofReal ((2:ℝ)^Module.finrank ℝ E)* (ENNReal.ofReal ((8:ℝ)^Module.finrank ℝ E)* NoiselessRegression.ambientSphereLaw (Metric.closedBall c r)) := by gcongr exact le_of_not_ge hr1 _ = _ := by rw [← mul_assoc, ← ENNReal.ofReal_mul (by positivity), ← mul_pow]; norm_num end universe u variable {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma cap_mass_pos [Nontrivial E] {c : E} (hc : ‖c‖ = 1) {r : ℝ} (hr : 0 < r) (hr2 : r ≤ 2) : NoiselessRegression.ambientSphereLaw (Metric.closedBall c r) ≠ 0 := by intro h have hh := sphere_cap_lower hc hr hr2 rw [h,mul_zero] at hh exact (ENNReal.ofReal_pos.mpr (pow_pos hr _)).not_ge hh lemma capMeasure_probability [Nontrivial E] {c : E} (hc : ‖c‖ = 1) {r : ℝ} (hr : 0 < r) (hr2 : r ≤ 2) : IsProbabilityMeasure (capMeasure c r) := by constructor simp only [capMeasure, Measure.smul_apply, Measure.restrict_apply_univ, smul_eq_mul] exact ENNReal.inv_mul_cancel (cap_mass_pos hc hr hr2) (measure_ne_top _ _) lemma capMeasure_ac (c : E) (r : ℝ) : capMeasure c r ≪ NoiselessRegression.ambientSphereLaw := Measure.smul_absolutelyContinuous.trans Measure.absolutelyContinuous_restrict lemma capMeasure_support (c : E) (r : ℝ) : (capMeasure c r).support ⊆ Metric.closedBall c r := by intro x hx have hh := (Measure.support_restrict_subset (Measure.smul_absolutelyContinuous.support_mono hx)).1 simpa only [Metric.isClosed_closedBall.closure_eq] using hh lemma reciprocal_scale {t : ℝ} (ht : 0 < t) (ht4 : t ≤ 1/4) : ∃ j : ℕ, 4*t ≤ 1/((j:ℝ)+1) ∧ 1/((j:ℝ)+1) < 8*t := by let n := ⌊1/(4*t)⌋₊ have hx : 1 ≤ 1/(4*t) := (le_div_iff₀ (by positivity)).mpr (by linarith) have hn : 1 ≤ n := (Nat.one_le_floor_iff _).mpr hx have hnle : (n:ℝ) ≤ 1/(4*t) := Nat.floor_le (by positivity) have hnlt : 1/(4*t) < (n:ℝ)+1 := Nat.lt_floor_add_one _ have he : ((n-1:ℕ):ℝ)+1 = n := by exact_mod_cast (Nat.sub_add_cancel hn) refine ⟨n-1, ?_, ?_⟩ · rw [he] rw [le_div_iff₀ (by exact_mod_cast (show 0 < n by omega))] rw [le_div_iff₀ (by positivity)] at hnle nlinarith · rw [he, div_lt_iff₀ (by exact_mod_cast (show 0 < n by omega))] rw [div_lt_iff₀ (by positivity)] at hnlt have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn nlinarith lemma exists_capDictionary {c : E} (hc : ‖c‖ = 1) {r α : ℝ} (hr : 0 < r) (hα : 1 ≤ α) : Nonempty (CapDictionary c r α) := by classical let ρ : ℕ → ℝ := fun j => r/((j:ℝ)+1) have hρ : ∀ j, 0 < ρ j := fun j => div_pos hr (by positivity) choose net hnetnorm hnetsep hnetcover using (fun j : ℕ => sphere_finite_net (E := E) (half_pos (hρ j))) let I := Option (Σ j : ℕ, {s : E // s ∈ net j}) let center : I → E := fun i => match i with | none => c | some p => p.2.val let radius : I → ℝ := fun i => match i with | none => r | some p => ρ p.1 have hw (j : ℕ) : (r/ρ j)^α = ((j:ℝ)+1)^α := by congr 1 dsimp [ρ] field_simp have hproper : ∀ R : ℝ, {i : I | (r/radius i)^α ≤ R}.Finite := by intro R have hlevel (j : ℕ) : (j:ℝ) ≤ (r/ρ j)^α := by rw [hw] calc _ ≤ (j:ℝ)+1 := by linarith _ = ((j:ℝ)+1)^(1:ℝ) := (Real.rpow_one _).symm _ ≤ _ := Real.rpow_le_rpow_of_exponent_le (by linarith [show (0:ℝ) ≤ j from Nat.cast_nonneg j]) hα have hsubset : {i : I | (r/radius i)^α ≤ R} ⊆ insert none (⋃ j ∈ Finset.range (⌈R⌉₊+1), Option.some '' (Sigma.mk j '' (Set.univ : Set {s : E // s ∈ net j}))) := by intro i hi cases i with | none => exact Set.mem_insert _ _ | some p => right have hjR : (p.1:ℝ) ≤ R := (hlevel p.1).trans hi have hj : p.1 ∈ Finset.range (⌈R⌉₊+1) := by apply Finset.mem_range.mpr have hh : p.1 ≤ ⌈R⌉₊ := by exact_mod_cast hjR.trans (Nat.le_ceil R) omega exact Set.mem_iUnion₂.mpr ⟨p.1,hj,⟨p,⟨p.2,Set.mem_univ _,rfl⟩,rfl⟩⟩ apply Set.Finite.subset _ hsubset apply Set.Finite.insert apply Set.Finite.biUnion (Finset.finite_toSet _) intro j hj exact ((Set.toFinite (Set.univ : Set {s : E // s ∈ net j})).image (Sigma.mk j)).image Option.some refine ⟨{ I := I countable := inferInstance center := center radius := radius center_norm := ?_ radius_pos := ?_ radius_le := ?_ parent := none parent_center := rfl parent_radius := rfl proper := hproper cover := ?_ }⟩ · rintro (_|⟨j,s,hs⟩) · exact hc · exact hnetnorm j s hs · rintro (_|⟨j,s⟩) · exact hr · exact hρ j · rintro (_|⟨j,s⟩) · exact le_rfl · change r/((j:ℝ)+1) ≤ r exact div_le_self hr.le (by linarith [show (0:ℝ) ≤ j from Nat.cast_nonneg j]) · intro s hs t ht ht4 obtain ⟨j,hj1,hj2⟩ := reciprocal_scale ht ht4 obtain ⟨s',hs',hss'⟩ := hnetcover j s hs refine ⟨some ⟨j,s',hs'⟩,?_,?_,hss'⟩ · change 4*r*t ≤ r/((j:ℝ)+1) have hh := mul_le_mul_of_nonneg_left hj1 hr.le rw [mul_one_div] at hh nlinarith only [hh] · change r/((j:ℝ)+1) < 8*r*t have hh := mul_lt_mul_of_pos_left hj2 hr rw [mul_one_div] at hh nlinarith only [hh] end ProjectionMoments end end OAI