| import OAI.Probability.ProjectionMoments.Caps |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| namespace ProjectionMoments |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| lemma cap_mass_inv_bound [Nontrivial E] {c : E} (hc : ‖c‖ = 1) |
| {r : ℝ} (hr : 0 < r) (hr2 : r ≤ 2) : |
| (NoiselessRegression.ambientSphereLaw (Metric.closedBall c r))⁻¹ ≤ |
| ENNReal.ofReal ((r^(Module.finrank ℝ E-1))⁻¹ * (16:ℝ)^Module.finrank ℝ E) := by |
| let p := NoiselessRegression.ambientSphereLaw (Metric.closedBall c r) |
| let a := ENNReal.ofReal (r^(Module.finrank ℝ E-1)) |
| have ha0 : a ≠ 0 := (ENNReal.ofReal_pos.mpr (pow_pos hr _)).ne' |
| have hat : a ≠ ∞ := ENNReal.ofReal_ne_top |
| have h := mul_le_mul' (le_rfl : a⁻¹ ≤ a⁻¹) (sphere_cap_lower hc hr hr2) |
| change a⁻¹*a ≤ a⁻¹ * (ENNReal.ofReal ((16:ℝ)^Module.finrank ℝ E) * p) at h |
| rw [ENNReal.inv_mul_cancel ha0 hat, ← mul_assoc] at h |
| have hh : p⁻¹ ≤ a⁻¹ * ENNReal.ofReal ((16:ℝ)^Module.finrank ℝ E) := by |
| apply (ENNReal.mul_le_mul_iff_right (cap_mass_pos hc hr hr2) (measure_ne_top _ _)).mp |
| change p*p⁻¹ ≤ p*(a⁻¹ * ENNReal.ofReal ((16:ℝ)^Module.finrank ℝ E)) |
| rw [ENNReal.mul_inv_cancel (cap_mass_pos hc hr hr2) (measure_ne_top _ _)] |
| simpa only [mul_comm] using h |
| simpa only [a, ENNReal.ofReal_mul (inv_nonneg.mpr (pow_nonneg hr.le _)), |
| ENNReal.ofReal_inv_of_pos (pow_pos hr _)] using hh |
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| lemma cap_ratio_algebra (d n : ℕ) {r ρ α : ℝ} (hr : 0 < r) (hρ : 0 < ρ) |
| (hα : 2*α = n) : |
| (r^n)⁻¹ * (16:ℝ)^d * (3*4^d*ρ^n) * (r/ρ)^α = |
| 3*64^d*(ρ/r)^α := by |
| have hre : r^n = (r^α)^2 := by |
| rw [← Real.rpow_natCast, ← Real.rpow_natCast _ 2, ← Real.rpow_mul hr.le] |
| congr 1 |
| norm_num only [Nat.cast_ofNat] |
| linarith |
| have hρe : ρ^n = (ρ^α)^2 := by |
| rw [← Real.rpow_natCast, ← Real.rpow_natCast _ 2, ← Real.rpow_mul hρ.le] |
| congr 1 |
| norm_num only [Nat.cast_ofNat] |
| linarith |
| rw [hre,hρe,Real.div_rpow hr.le hρ.le, Real.div_rpow hρ.le hr.le] |
| have hc : (16:ℝ)^d * 4^d = 64^d := by rw [← mul_pow]; norm_num |
| field_simp [(Real.rpow_pos_of_pos hr α).ne', (Real.rpow_pos_of_pos hρ α).ne'] |
| nlinarith [hc] |
| |
| lemma cap_cost_ratio [Nontrivial E] {c c' : E} (hc : ‖c‖ = 1) |
| {r ρ α : ℝ} (hr : 0 < r) (hr2 : r ≤ 2) (hρ : 0 < ρ) |
| (hα : 2*α = (Module.finrank ℝ E-1:ℕ)) : |
| (NoiselessRegression.ambientSphereLaw (Metric.closedBall c r))⁻¹ * |
| NoiselessRegression.ambientSphereLaw (Metric.closedBall c' ρ) * ENNReal.ofReal ((r/ρ)^α) ≤ |
| ENNReal.ofReal (3*64^Module.finrank ℝ E*(ρ/r)^α) := by |
| have hu := NoiselessRegression.ambientSphereLaw_closedBall c' hρ (Module.finrank ℝ E-1) le_rfl |
| have h := mul_le_mul' (mul_le_mul' (cap_mass_inv_bound hc hr hr2) hu) |
| (le_rfl : ENNReal.ofReal ((r/ρ)^α) ≤ _) |
| rw [← ENNReal.ofReal_mul (by positivity)] at hu |
| apply le_trans (mul_le_mul' (mul_le_mul' (cap_mass_inv_bound hc hr hr2) hu) le_rfl) |
| rw [← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_mul (by positivity), |
| cap_ratio_algebra _ _ hr hρ hα] |
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| lemma cap_small_scale_constant (d : ℕ) (hd : 1 ≤ d) {α ρ r t : ℝ} |
| (hα0 : 0 ≤ α) (hαd : α ≤ d) (hr : 0 < r) (ht : 0 < t) |
| (hρ : 0 ≤ ρ) (hρt : ρ ≤ 8*r*t) : |
| 3*64^d*(ρ/r)^α ≤ (2048:ℝ)^d*t^α := by |
| have hratio : ρ/r ≤ 8*t := (div_le_iff₀ hr).mpr (by nlinarith) |
| have h8 : (8:ℝ)^α ≤ (8:ℝ)^(d:ℝ) := Real.rpow_le_rpow_of_exponent_le (by norm_num) hαd |
| have h3 : (3:ℝ) ≤ 3^d := by simpa using pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 3) hd |
| calc |
| _ ≤ 3*64^d*(8*t)^α := mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow (div_nonneg hρ hr.le) hratio hα0) (by positivity) |
| _ = 3*64^d*8^α*t^α := by rw [Real.mul_rpow (by norm_num) ht.le]; ring |
| _ ≤ 3^d*64^d*8^d*t^α := by rw [Real.rpow_natCast] at h8; gcongr |
| _ = (1536:ℝ)^d*t^α := by rw [← mul_pow, ← mul_pow]; norm_num |
| _ ≤ _ := by gcongr; norm_num |
| |
| lemma weighted_cap_set_le [Nontrivial E] {c : E} (hc : ‖c‖ = 1) |
| {r b : ℝ} (hr : 0 < r) (hr2 : r ≤ 2) (h : E → ℝ≥0∞) |
| (hb : ∫⁻ x, h x ∂capMeasure c r ≤ ENNReal.ofReal b) : |
| (NoiselessRegression.ambientSphereLaw.withDensity h) (Metric.closedBall c r) ≤ |
| NoiselessRegression.ambientSphereLaw (Metric.closedBall c r) * ENNReal.ofReal b := by |
| rw [capMeasure, lintegral_smul_measure, smul_eq_mul] at hb |
| rw [withDensity_apply _ Metric.isClosed_closedBall.measurableSet] |
| exact (ENNReal.inv_mul_le_iff (cap_mass_pos hc hr hr2) (measure_ne_top _ _)).mp hb |
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| def capTestMeasure (h : E → ℝ≥0∞) (c : E) (r : ℝ) : Measure E := |
| ((capMeasure c r).withDensity h).map (rescale c r) |
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| lemma capTestMeasure_mass {h : E → ℝ≥0∞} {c : E} {r α : ℝ} (hr : 0 < r) |
| (D : CapDictionary c r α) |
| (hdual : ∀ i, ∫⁻ x, h x ∂capMeasure (D.center i) (D.radius i) ≤ ENNReal.ofReal ((r/D.radius i)^α)) : |
| capTestMeasure h c r Set.univ ≤ 1 := by |
| have hp := hdual D.parent |
| rw [D.parent_center,D.parent_radius,div_self hr.ne',Real.one_rpow,ENNReal.ofReal_one] at hp |
| simpa only [capTestMeasure,Measure.map_apply (measurable_rescale c r) MeasurableSet.univ, |
| Set.preimage_univ, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] using hp |
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| lemma capTestMeasure_finite {h : E → ℝ≥0∞} {c : E} {r α : ℝ} (hr : 0 < r) |
| (D : CapDictionary c r α) |
| (hdual : ∀ i, ∫⁻ x, h x ∂capMeasure (D.center i) (D.radius i) ≤ ENNReal.ofReal ((r/D.radius i)^α)) : |
| IsFiniteMeasure (capTestMeasure h c r) := |
| ⟨(capTestMeasure_mass hr D hdual).trans_lt (by simp)⟩ |
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| lemma capTestMeasure_support (h : E → ℝ≥0∞) (c : E) {r : ℝ} (hr : 0 < r) : |
| (capTestMeasure h c r).support ⊆ Metric.closedBall 0 1 := by |
| apply Measure.support_subset_of_isClosed Metric.isClosed_closedBall |
| apply (ae_map_iff (measurable_rescale c r).aemeasurable Metric.isClosed_closedBall.measurableSet).mpr |
| filter_upwards [((capMeasure c r).withDensity h).support_mem_ae] with x hx |
| have hx' := capMeasure_support c r ((withDensity_absolutelyContinuous (capMeasure c r) h).support_mono hx) |
| change x ∈ rescale c r ⁻¹' Metric.closedBall 0 1 |
| rw [rescale_preimage_closedBall c 0 hr, dilate, smul_zero, zero_add, mul_one] |
| exact hx' |
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| lemma capTestMeasure_ball (h : E → ℝ≥0∞) (c : E) {r : ℝ} (hr : 0 < r) (b : E) (t : ℝ) : |
| capTestMeasure h c r (Metric.closedBall b t) = |
| (NoiselessRegression.ambientSphereLaw (Metric.closedBall c r))⁻¹ * |
| (NoiselessRegression.ambientSphereLaw.withDensity h) |
| (Metric.closedBall (dilate c r b) (r*t) ∩ Metric.closedBall c r) := by |
| rw [capTestMeasure, Measure.map_apply (measurable_rescale c r) Metric.isClosed_closedBall.measurableSet, |
| rescale_preimage_closedBall c b hr, capMeasure, withDensity_smul_measure, |
| Measure.smul_apply, smul_eq_mul, ← restrict_withDensity Metric.isClosed_closedBall.measurableSet, |
| Measure.restrict_apply Metric.isClosed_closedBall.measurableSet] |
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| lemma capTest_allBall [Nontrivial E] {h : E → ℝ≥0∞} {c : E} (hc : ‖c‖ = 1) |
| {r α : ℝ} (hr : 0 < r) (hr2 : r ≤ 2) (hα0 : 0 ≤ α) |
| (hα : 2*α = (Module.finrank ℝ E-1:ℕ)) (D : CapDictionary c r α) |
| (hdual : ∀ i, ∫⁻ x, h x ∂capMeasure (D.center i) (D.radius i) ≤ ENNReal.ofReal ((r/D.radius i)^α)) : |
| AllBallMass (capTestMeasure h c r) ((2048:ℝ)^Module.finrank ℝ E) α := by |
| have hd : 1 ≤ Module.finrank ℝ E := Module.finrank_pos (R := ℝ) (M := E) |
| have hαd : α ≤ Module.finrank ℝ E := by |
| have hn : ((Module.finrank ℝ E-1:ℕ):ℝ) ≤ Module.finrank ℝ E := by exact_mod_cast Nat.sub_le (Module.finrank ℝ E) 1 |
| linarith |
| intro b t ht |
| by_cases ht4 : t ≤ 1/4 |
| · let ξ := NoiselessRegression.ambientSphereLaw.withDensity h |
| let T := (Metric.closedBall (dilate c r b) (r*t) ∩ Metric.closedBall c r) ∩ Metric.sphere 0 1 |
| have he : ξ (Metric.closedBall (dilate c r b) (r*t) ∩ Metric.closedBall c r) = ξ T := by |
| apply measure_congr |
| filter_upwards [(withDensity_absolutelyContinuous NoiselessRegression.ambientSphereLaw h).ae_le |
| (NoiselessRegression.ambientSphereLaw_ae_norm (E := E))] with x hx |
| have hs : x ∈ Metric.sphere (0:E) 1 := by simpa using hx |
| apply propext |
| simp only [T, Set.mem_inter_iff, hs, and_true] |
| rw [capTestMeasure_ball h c hr, he] |
| by_cases hT : T.Nonempty |
| · obtain ⟨s,hs⟩ := hT |
| have hsn : ‖s‖ = 1 := by simpa using hs.2 |
| obtain ⟨i,hi1,hi2,hsi⟩ := D.cover s hsn t ht ht4 |
| have hsub : T ⊆ Metric.closedBall (D.center i) (D.radius i) := by |
| intro x hx |
| change dist x (D.center i) ≤ D.radius i |
| have hxB : dist x (dilate c r b) ≤ r*t := hx.1.1 |
| have hsB : dist s (dilate c r b) ≤ r*t := hs.1.1 |
| calc |
| _ ≤ dist x s + dist s (D.center i) := dist_triangle _ _ _ |
| _ ≤ (dist x (dilate c r b) + dist (dilate c r b) s) + D.radius i/2 := |
| add_le_add (dist_triangle _ _ _) hsi |
| _ ≤ D.radius i := by rw [dist_comm (dilate c r b) s]; linarith |
| apply (mul_le_mul' le_rfl (measure_mono hsub)).trans |
| have htest := weighted_cap_set_le (D.center_norm i) (D.radius_pos i) |
| ((D.radius_le i).trans hr2) h (hdual i) |
| apply (mul_le_mul' le_rfl htest).trans |
| rw [← mul_assoc] |
| exact (cap_cost_ratio hc hr hr2 (D.radius_pos i) hα).trans |
| (ENNReal.ofReal_le_ofReal (cap_small_scale_constant _ hd hα0 hαd hr ht (D.radius_pos i).le hi2.le)) |
| · rw [Set.not_nonempty_iff_eq_empty.mp hT, measure_empty, mul_zero] |
| exact bot_le |
| · apply (measure_mono (Set.subset_univ _)).trans ((capTestMeasure_mass hr D hdual).trans ?_) |
| have h4t : 1 ≤ 4*t := by linarith |
| have hpow : 1 ≤ (4*t)^α := Real.one_le_rpow h4t hα0 |
| have h4 : (4:ℝ)^α ≤ (2048:ℝ)^Module.finrank ℝ E := by |
| calc |
| _ ≤ (4:ℝ)^(Module.finrank ℝ E:ℝ) := Real.rpow_le_rpow_of_exponent_le (by norm_num) hαd |
| _ = (4:ℝ)^Module.finrank ℝ E := Real.rpow_natCast _ _ |
| _ ≤ _ := by gcongr; norm_num |
| rw [Real.mul_rpow (by norm_num) ht.le] at hpow |
| rw [← ENNReal.ofReal_one] |
| exact ENNReal.ofReal_le_ofReal (hpow.trans (mul_le_mul_of_nonneg_right h4 (Real.rpow_nonneg ht.le α))) |
| |
| lemma rescaled_projection_ac {ν : Measure E} {k : ℕ} (A : Fin k → E) (c : E) {r : ℝ} (hr : 0 < r) |
| (hν : ν.map (coordinateMap A) ≪ (volume : Measure (Fin k → ℝ))) : |
| (ν.map (rescale c r)).map (coordinateMap A) ≪ (volume : Measure (Fin k → ℝ)) := by |
| let T : (Fin k → ℝ) → (Fin k → ℝ) := fun y => r⁻¹ • (y-coordinateMap A c) |
| have hT : Measurable T := by fun_prop |
| have hv : (volume : Measure (Fin k → ℝ)).map T ≪ volume := by |
| have he : T = (fun y : Fin k → ℝ => r⁻¹ • y) ∘ (fun y => y + -coordinateMap A c) := by |
| funext y; simp [T,sub_eq_add_neg] |
| rw [he, ← Measure.map_map (by fun_prop) (by fun_prop), |
| Measure.IsAddRightInvariant.map_add_right_eq_self, Measure.map_addHaar_smul volume (inv_ne_zero hr.ne')] |
| exact Measure.smul_absolutelyContinuous |
| have hh := (hν.map hT).trans hv |
| rw [Measure.map_map hT (by fun_prop)] at hh |
| rw [Measure.map_map (by fun_prop) (measurable_rescale c r)] |
| convert hh using 1 |
| congr 1 |
| funext s |
| ext i |
| simp [T, rescale, coordinateMap_apply, real_inner_smul_left, inner_sub_left] |
|
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| lemma capTest_frame_ac [Nontrivial E] (h : E → ℝ≥0∞) (c : E) {r : ℝ} (hr : 0 < r) |
| {k : ℕ} (hk : k+2 ≤ Module.finrank ℝ E) (A : Fin k → E) (hA : Orthonormal ℝ A) : |
| (capTestMeasure h c r).map (rowMap A) ≪ (volume : Measure (Label k)) := by |
| apply (rowMap_ac_iff _).mpr |
| apply rescaled_projection_ac A c hr |
| have hh := (((withDensity_absolutelyContinuous (capMeasure c r) h).trans (capMeasure_ac c r)).map |
| (show Measurable (coordinateMap A) by fun_prop)).trans (sphere_frame_ac hk hA) |
| exact hh |
|
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| end ProjectionMoments |
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| end |
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| end OAI |
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