| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.MeasurableTests |
|
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| namespace OAI |
|
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| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
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| noncomputable section |
| universe u v |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision |
| open NoiselessRegression |
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| def realNormalize {d : ℕ} (ν : Measure (Space d)) (β H : ℝ) (z : Space d) (r : ℝ) : |
| Measure (Space d) := |
| (ENNReal.ofReal (H * r^β))⁻¹ • |
| (ν.restrict (Metric.closedBall z r)).map (fun x => r⁻¹ • (x-z)) |
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| lemma realNormalize_local {d : ℕ} {ν : Measure (Space d)} {β H : ℝ} |
| (hν : AmbientBallBound ν H β) (hH : 0 < H) (z : Space d) {r : ℝ} (hr : 0 < r) : |
| AmbientBallBound (realNormalize ν β H z r) 1 β := by |
| have hc0 : ENNReal.ofReal (H*r^β) ≠ 0 := by positivity |
| intro t ρ hρ |
| rw [realNormalize, Measure.smul_apply, smul_eq_mul, |
| Measure.map_apply (by fun_prop) Metric.isClosed_closedBall.measurableSet] |
| have hsub : (fun x : Space d => r⁻¹ • (x-z)) ⁻¹' Metric.closedBall t ρ ⊆ |
| Metric.closedBall (z + r • t) (r*ρ) := by |
| intro x hx |
| have he : x-(z+r•t) = r • (r⁻¹ • (x-z)-t) := by |
| rw [smul_sub, smul_smul, mul_inv_cancel₀ hr.ne', one_smul] |
| abel |
| rw [Metric.mem_closedBall, dist_eq_norm, he, norm_smul, Real.norm_eq_abs, abs_of_pos hr] |
| exact mul_le_mul_of_nonneg_left |
| (by simpa only [Set.mem_preimage, Metric.mem_closedBall, dist_eq_norm] using hx) hr.le |
| calc |
| _ ≤ (ENNReal.ofReal (H*r^β))⁻¹ * ν (Metric.closedBall (z+r•t) (r*ρ)) := |
| mul_le_mul' le_rfl ((Measure.restrict_le_self _).trans (measure_mono hsub)) |
| _ ≤ (ENNReal.ofReal (H*r^β))⁻¹ * ENNReal.ofReal (H*(r*ρ)^β) := |
| mul_le_mul' le_rfl (hν _ _ (mul_pos hr hρ)) |
| _ = ENNReal.ofReal (1*ρ^β) := by |
| rw [Real.mul_rpow hr.le hρ.le, ← mul_assoc, |
| ENNReal.ofReal_mul (by positivity : 0 ≤ H*r^β), ← mul_assoc, |
| ENNReal.inv_mul_cancel hc0 ENNReal.ofReal_ne_top, one_mul, one_mul] |
|
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| lemma realNormalize_support {d : ℕ} (ν : Measure (Space d)) (β H : ℝ) |
| (z : Space d) {r : ℝ} (hr : 0 < r) : |
| (realNormalize ν β H z r).support ⊆ Metric.closedBall 0 1 := by |
| apply Measure.support_subset_of_isClosed Metric.isClosed_closedBall |
| rw [mem_ae_iff, realNormalize, Measure.smul_apply, Measure.map_apply (by fun_prop) |
| Metric.isClosed_closedBall.measurableSet.compl, |
| Measure.restrict_apply ((by fun_prop : Measurable (fun x : Space d => r⁻¹ • (x-z))) |
| Metric.isClosed_closedBall.measurableSet.compl)] |
| have he : (fun x : Space d => r⁻¹ • (x-z)) ⁻¹' (Metric.closedBall 0 1)ᶜ ∩ |
| Metric.closedBall z r = ∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro x ⟨hx,hb⟩ |
| apply hx |
| rw [Metric.mem_closedBall, dist_zero_right, norm_smul, Real.norm_eq_abs, |
| abs_of_pos (inv_pos.mpr hr)] |
| exact (inv_mul_le_iff₀ hr).2 (by simpa [dist_eq_norm] using hb) |
| rw [he, measure_empty, smul_zero] |
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|
| lemma realNormalize_finite {d : ℕ} (ν : Measure (Space d)) [IsFiniteMeasure ν] |
| {β H : ℝ} (hH : 0 < H) (z : Space d) {r : ℝ} (hr : 0 < r) : |
| IsFiniteMeasure (realNormalize ν β H z r) := by |
| constructor |
| rw [realNormalize, Measure.smul_apply, smul_eq_mul] |
| apply ENNReal.mul_lt_top |
| · exact ENNReal.inv_lt_top.mpr (by positivity) |
| · exact measure_lt_top _ _ |
|
|
| lemma realNormalize_reconstruct {d : ℕ} (ν : Measure (Space d)) {β H : ℝ} |
| (hH : 0 < H) (z : Space d) {r : ℝ} (hr : 0 < r) : |
| ENNReal.ofReal (H*r^β) • (realNormalize ν β H z r).map (fun t => z+r•t) = |
| ν.restrict (Metric.closedBall z r) := by |
| have hc0 : ENNReal.ofReal (H*r^β) ≠ 0 := by positivity |
| rw [realNormalize, Measure.map_smul _ (by fun_prop), smul_smul, |
| ENNReal.mul_inv_cancel hc0 ENNReal.ofReal_ne_top, one_smul, |
| Measure.map_map (by fun_prop) (by fun_prop)] |
| have he : (fun t : Space d => z+r•t) ∘ (fun x => r⁻¹ • (x-z)) = id := by |
| ext x |
| simp [smul_smul, hr.ne'] |
| rw [he, Measure.map_id] |
|
|
| lemma normalized_scale_le {d m p : ℕ} (hp : 2 ≤ p) {β K : ℝ} (hK : 1 ≤ K) |
| (hgap : (m:ℝ)+(p:ℝ)-2 < β) |
| (hJ : ∀ j : ℕ, j ≤ p-2 → J β m j ≤ K^d) : |
| ProjectionNormScale m p 1 β 1 ≤ K^d := by |
| have hp0 : (0:ℝ) < p := by exact_mod_cast (by omega : 0 < p) |
| have hprod0 : 0 ≤ ∏ j ∈ Finset.range (p-1), J β m j := by |
| apply Finset.prod_nonneg |
| intro j hj |
| apply (J_pos ?_).le |
| have hj' : (j:ℝ)+2 ≤ p := by exact_mod_cast (show j+2 ≤ p by have := Finset.mem_range.mp hj; omega) |
| linarith |
| have hprod : (∏ j ∈ Finset.range (p-1), J β m j) ≤ (K^d)^(p-1) := by |
| calc |
| _ ≤ ∏ _j ∈ Finset.range (p-1), K^d := by |
| apply Finset.prod_le_prod₀ |
| · intro j hj |
| apply (J_pos ?_).le |
| have hj' : (j:ℝ)+2 ≤ p := by exact_mod_cast (show j+2 ≤ p by have := Finset.mem_range.mp hj; omega) |
| linarith |
| · intro j hj |
| exact hJ j (by have := Finset.mem_range.mp hj; omega) |
| _ = _ := by simp |
| have hpeak : (2*Real.pi) ^ (-(m:ℝ)*((p:ℝ)-1)/(2*(p:ℝ))) ≤ 1 := by |
| apply Real.rpow_le_one_of_one_le_of_nonpos |
| · linarith [Real.pi_gt_three] |
| · have : (1:ℝ) ≤ p := by exact_mod_cast (by omega : 1 ≤ p) |
| apply div_nonpos_of_nonpos_of_nonneg |
| · exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (Nat.cast_nonneg m)) (by linarith) |
| · positivity |
| unfold ProjectionNormScale |
| simp only [mul_one, Real.one_rpow] |
| calc |
| _ ≤ 1 * (∏ j ∈ Finset.range (p-1), J β m j)^(1/(p:ℝ)) := |
| mul_le_mul_of_nonneg_right hpeak (Real.rpow_nonneg hprod0 _) |
| _ ≤ ((K^d)^(p-1))^(1/(p:ℝ)) := by rw [one_mul]; exact Real.rpow_le_rpow hprod0 hprod (by positivity) |
| _ = (K^d)^(((p-1:ℕ):ℝ)/(p:ℝ)) := by |
| rw [← Real.rpow_natCast_mul (by positivity : 0 ≤ K^d)] |
| congr 1 |
| ring |
| _ ≤ (K^d)^(1:ℝ) := by |
| apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ hK) |
| rw [div_le_one hp0] |
| exact_mod_cast Nat.sub_le p 1 |
| _ = _ := Real.rpow_one _ |
|
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| lemma normalized_weighted_selection {d m p N : ℕ} (ν : Fin N → Measure (Space d)) |
| [∀ w, IsFiniteMeasure (ν w)] (hp : 2 ≤ p) (hm : 1 ≤ m) (hmd : m ≤ d) |
| {β K : ℝ} (hK : 1 ≤ K) (hgap : (m:ℝ)+(p:ℝ)-2 < β) |
| (hJ : ∀ j : ℕ, j ≤ p-2 → J β m j ≤ K^d) |
| (hball : ∀ w, AmbientBallBound (ν w) 1 β) |
| (hsupport : ∀ w, (ν w).support ⊆ Metric.closedBall 0 1) |
| (g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞) |
| (hg : ∀ w, Measurable (g w)) (hgsum : ∀ x, ∑ w, g w x ≤ 1) : |
| (∑ w, ∫⁻ x, g w x ∂projectionLaw (ν w) m) ≤ |
| ENNReal.ofReal (K*192)^d * (N:ℝ≥0∞)^(1/(p:ℝ)) := by |
| classical |
| obtain ⟨D,hD,hDs,hDV⟩ := exists_projection_support (E := Space d) m (by simpa using hmd) |
| have hg1 (w) (x) : g w x ≤ 1 := |
| (Finset.single_le_sum (fun _ _ => zero_le) (Finset.mem_univ w)).trans (hgsum x) |
| have hpoint (w) : (∫⁻ x, g w x ∂projectionLaw (ν w) m) ≤ |
| ENNReal.ofReal (K^d) * (∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ)) := by |
| obtain ⟨F,hF,hF0,hden,_,hnorm⟩ := exact_finite_density d m p hp hm β 1 1 hgap |
| (by norm_num) (by norm_num) 0 (ν w) (hsupport w) (hball w) |
| have hFD : ∀ᵐ x ∂matrixLabelReference m, x ∉ D → F x = 0 := by |
| have hz : projectionLaw (ν w) m Dᶜ = 0 := by |
| rw [← compl_mem_ae_iff, compl_compl] |
| unfold projectionLaw |
| apply (ae_map_iff (by fun_prop) hD).2 |
| filter_upwards [Measure.quasiMeasurePreserving_snd.ae (ν w).support_mem_ae] with x hx |
| exact hDs x.1 x.2 (by simpa using hsupport w hx) |
| change (matrixLabelReference m).withDensity _ = projectionLaw (ν w) m at hden |
| rw [← hden, withDensity_apply _ hD.compl] at hz |
| have ha := (lintegral_eq_zero_iff' hF.ennreal_ofReal.aemeasurable).mp hz |
| change ∀ᵐ x ∂(matrixLabelReference m).restrict Dᶜ, ENNReal.ofReal (F x) = 0 at ha |
| rw [ae_restrict_iff' hD.compl] at ha |
| filter_upwards [ha] with x hx hnot |
| exact le_antisymm (ENNReal.ofReal_eq_zero.mp (hx hnot)) (hF0 x) |
| have hn : eLpNorm F (p:ℝ≥0∞) (matrixLabelReference m) ≤ ENNReal.ofReal (K^d) := |
| hnorm.trans (ENNReal.ofReal_le_ofReal (normalized_scale_le hp hK hgap hJ)) |
| have ht := density_test_bound (matrixLabelReference m) F hF hF0 p hp |
| (ENNReal.ofReal (K^d)) hn D hD hFD (g w) (hg w).aemeasurable (hg1 w) |
| change (matrixLabelReference m).withDensity _ = projectionLaw (ν w) m at hden |
| rwa [hden] at ht |
| have hq0 : 0 ≤ 1-1/(p:ℝ) := by |
| have hp1 : (1:ℝ) ≤ p := by exact_mod_cast (by omega : 1 ≤ p) |
| linarith [one_div_le_one_div_of_le (by norm_num : (0:ℝ)<1) hp1] |
| have hsum : (∑ w, ∫⁻ x in D, g w x ∂matrixLabelReference m) ≤ |
| matrixLabelReference (E := Space d) m D := by |
| rw [← lintegral_finsetSum _ (fun w _ => hg w)] |
| simpa using lintegral_mono (μ := (matrixLabelReference (E := Space d) m).restrict D) hgsum |
| have hv : (matrixLabelReference (E := Space d) m D)^(1-1/(p:ℝ)) ≤ (192:ℝ≥0∞)^d := by |
| calc |
| _ ≤ ((192:ℝ≥0∞)^d)^(1-1/(p:ℝ)) := ENNReal.rpow_le_rpow (by simpa using hDV) hq0 |
| _ ≤ ((192:ℝ≥0∞)^d)^(1:ℝ) := |
| ENNReal.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) (sub_le_self _ (by positivity)) |
| _ = _ := ENNReal.rpow_one _ |
| calc |
| _ ≤ ∑ w, ENNReal.ofReal (K^d) * (∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ)) := |
| Finset.sum_le_sum (fun w _ => hpoint w) |
| _ = ENNReal.ofReal (K^d) * ∑ w, (∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ)) := by rw [Finset.mul_sum] |
| _ ≤ ENNReal.ofReal (K^d) * ((N:ℝ≥0∞)^(1/(p:ℝ)) * |
| (∑ w, ∫⁻ x in D, g w x ∂matrixLabelReference m)^(1-1/(p:ℝ))) := by |
| apply mul_le_mul' le_rfl |
| simpa using finite_rpow_holder (fun w => ∫⁻ x in D, g w x ∂matrixLabelReference m) |
| (by exact_mod_cast (by omega : 1 < p)) |
| _ ≤ ENNReal.ofReal (K^d) * ((N:ℝ≥0∞)^(1/(p:ℝ)) * (192:ℝ≥0∞)^d) := |
| mul_le_mul' le_rfl (mul_le_mul' le_rfl ((ENNReal.rpow_le_rpow hsum hq0).trans hv)) |
| _ = _ := by |
| rw [ENNReal.ofReal_pow (by linarith : 0 ≤ K), mul_left_comm, ← mul_pow, |
| ENNReal.ofReal_mul (by linarith : 0 ≤ K)] |
| norm_num |
| ring |
| lemma localized_weighted_selection {d m p N : ℕ} (ν : Fin N → Measure (Space d)) |
| [∀ w, IsFiniteMeasure (ν w)] (hp : 2 ≤ p) (hm : 1 ≤ m) (hmd : m ≤ d) |
| {β K H : ℝ} (hK : 1 ≤ K) (hH : 0 < H) (hgap : (m:ℝ)+(p:ℝ)-2 < β) |
| (hJ : ∀ j : ℕ, j ≤ p-2 → J β m j ≤ K^d) |
| (hball : ∀ w, AmbientBallBound (ν w) H β) |
| (g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞) |
| (hg : ∀ w, Measurable (g w)) (hgsum : ∀ x, ∑ w, g w x ≤ 1) |
| (z : Space d) {r : ℝ} (hr : 0 < r) : |
| (∑ w, ∫⁻ x, g w x ∂projectionLaw ((ν w).restrict (Metric.closedBall z r)) m) ≤ |
| ENNReal.ofReal ((K*192)^d * (N:ℝ)^(1/(p:ℝ)) * H * r^β) := by |
| let τ w := realNormalize (ν w) β H z r |
| let (w : Fin N) : IsFiniteMeasure (τ w) := realNormalize_finite (ν w) hH z hr |
| let F : ((Fin m → Space d) × (Fin m → ℝ)) → ((Fin m → Space d) × (Fin m → ℝ)) := |
| fun x => (x.1, fun i => ⟪z,x.1 i⟫ + r*x.2 i) |
| have hF : Measurable F := by fun_prop |
| have hb := normalized_weighted_selection τ hp hm hmd hK hgap hJ |
| (fun w => realNormalize_local (hball w) hH z hr) |
| (fun w => realNormalize_support (ν w) β H z hr) |
| (fun w => g w ∘ F) (fun w => (hg w).comp hF) (fun x => hgsum (F x)) |
| have he (w) : (∫⁻ x, g w x ∂projectionLaw ((ν w).restrict (Metric.closedBall z r)) m) = |
| ENNReal.ofReal (H*r^β) * ∫⁻ x, g w (F x) ∂projectionLaw (τ w) m := by |
| rw [← realNormalize_reconstruct (ν w) (β := β) hH z hr, projectionLaw_smul, |
| projectionLaw_affine, lintegral_smul_measure, smul_eq_mul, lintegral_map (hg w) hF] |
| simp_rw [he] |
| rw [← Finset.mul_sum] |
| refine (mul_le_mul' le_rfl hb).trans_eq ?_ |
| rw [← ENNReal.ofReal_natCast N, ENNReal.ofReal_rpow_of_nonneg (Nat.cast_nonneg _) (by positivity), |
| ← ENNReal.ofReal_pow (by positivity : 0 ≤ K*192), |
| ← ENNReal.ofReal_mul (by positivity : 0 ≤ (K*192)^d), |
| ← ENNReal.ofReal_mul (by positivity : 0 ≤ H*r^β)] |
| congr 1 |
| ring |
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| end MemoryPrecision |
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| end |
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| end OAI |
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