| import OAI.Probability.ProjectionMoments.BallMoments |
|
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| 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 |
|
|
