| import OAI.Probability.ProjectionMoments.CapTests |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace Matrix |
| open MeasureTheory |
| open scoped ENNReal NNReal |
| 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] |
|
|
| @[fun_prop] lemma measurable_radialLabel {k : ℕ} : |
| Measurable (fun z : (Fin k → E) × Label k => radialLabel z.1 z.2) := by |
| have hH : Measurable (fun z : (Fin k → E) × Label k => polarRadial z.1) := (continuous_polarRadial (E := E) (k := k)).measurable.comp measurable_fst |
| apply (MeasurableEquiv.toLp 2 (Fin k → ℝ)).measurable.comp |
| apply Measurable.of_eval |
| intro i |
| change Measurable (fun z : (Fin k → E) × Label k => ∑ j, polarRadial z.1 i j * z.2.ofLp j) |
| apply Finset.measurable_sum |
| intro j hj |
| exact ((measurable_pi_apply j).comp ((measurable_pi_apply i).comp hH)).mul (by fun_prop) |
|
|
| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma rowMap_mixRows [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {k : ℕ} (M : Matrix (Fin k) (Fin k) ℝ) (X : Fin k → E) (s : E) : |
| rowMap (mixRows M X) s = WithLp.toLp 2 (M *ᵥ (rowMap X s).ofLp) := by |
| ext i |
| simp [rowMap, mixRows, Matrix.mulVec, dotProduct] |
|
|
| lemma polar_label_identity {k : ℕ} {X : Fin k → E} (hX : LinearIndependent ℝ X) |
| (u : Fin k → E) (c s : E) (r : ℝ) : |
| rowMap X (dilate c r s) = rowMap X c + r • radialLabel X (rowMap (polarRows u X) s) := by |
| have he : radialLabel X (rowMap (polarRows u X) s) = rowMap X s := by |
| rw [radialLabel, ← rowMap_mixRows] |
| simp only [polarRows, ite_eq_left hX, polar_reconstruction hX] |
| rw [he, dilate, map_add, map_smul] |
| exact add_comm _ _ |
|
|
| lemma normalizedDensity_unitBall (ν : Measure E) [IsFiniteMeasure ν] {k : ℕ} |
| {u : Fin k → E} (hu : Orthonormal ℝ u) (hs : ν.support ⊆ Metric.closedBall 0 1) |
| (X : Fin k → E) (hac : ν.map (rowMap (polarRows u X)) ≪ (volume : Measure (Label k))) : |
| (unitBallLaw k).withDensity (fun y => normalizedDensity ν k u (X,y)) = |
| ν.map (rowMap (polarRows u X)) := by |
| have he : (Metric.closedBall (0:Label k) 1).indicator |
| (fun y => normalizedDensity ν k u (X,y)) = fun y => normalizedDensity ν k u (X,y) := by |
| funext y |
| by_cases hy : y ∈ Metric.closedBall (0:Label k) 1 |
| · simp [hy] |
| · rw [Set.indicator_of_notMem hy] |
| have hh : 1 < ‖y‖ := by simpa using hy |
| exact (normalizedDensity_support ν hu hs X y hh).symm |
| rw [unitBallLaw, withDensity_smul_measure, ← withDensity_indicator Metric.isClosed_closedBall.measurableSet, he, |
| ← withDensity_smul_measure] |
| exact normalizedDensity_is_density ν X hac |
|
|
| @[fun_prop] lemma measurable_capReferenceLabel {k : ℕ} (c : E) (r : ℝ) : |
| Measurable (fun z : (Fin k → E) × Label k => capReferenceLabel c r z.1 z.2) := by |
| unfold capReferenceLabel |
| exact (show Measurable (fun z : (Fin k → E) × Label k => rowMap z.1 c) by |
| simp only [rowMap_apply]; fun_prop).add (measurable_radialLabel.const_smul r) |
|
|
| @[fun_prop] lemma measurable_capKernelValue {k : ℕ} {K : (Fin k → E) × Label k → ℝ≥0∞} |
| (hK : Measurable K) : Measurable (capKernelValue K) := by |
| apply Measurable.lintegral_prod_right |
| exact hK.comp (measurable_snd.prodMk (by simp only [rowMap_apply]; fun_prop)) |
|
|
| omit [BorelSpace E] in |
| lemma capReference_le_one [BorelSpace E] {k : ℕ} {K : (Fin k → E) × Label k → ℝ≥0∞} |
| (hK1 : ∀ z, K z ≤ 1) (c : E) (r : ℝ) : capReference K c r ≤ 1 := by |
| calc |
| _ ≤ ∫⁻ X, ∫⁻ z : Label k, (1:ℝ≥0∞) ∂unitBallLaw k ∂Measure.pi (fun _ : Fin k => stdGaussian E) := |
| lintegral_mono (fun X => lintegral_mono (fun z => hK1 _)) |
| _ = 1 := by simp |
|
|
| omit [BorelSpace E] in |
| lemma capKernelValue_le_one [BorelSpace E] {k : ℕ} {K : (Fin k → E) × Label k → ℝ≥0∞} |
| (hK1 : ∀ z, K z ≤ 1) (s : E) : capKernelValue K s ≤ 1 := by |
| calc |
| _ ≤ ∫⁻ X : Fin k → E, (1:ℝ≥0∞) ∂Measure.pi (fun _ => stdGaussian E) := |
| lintegral_mono (fun X => hK1 _) |
| _ = 1 := by simp |
|
|
| lemma cap_kernel_test_identity {k : ℕ} (hk : k ≤ Module.finrank ℝ E) |
| {K : (Fin k → E) × Label k → ℝ≥0∞} (hK : Measurable K) |
| (h : E → ℝ≥0∞) (hh : Measurable h) (c : E) {r : ℝ} (hr : 0 < r) |
| {u : Fin k → E} (hu : Orthonormal ℝ u) |
| [IsFiniteMeasure ((capMeasure c r).withDensity h)] |
| (hac : ∀ X, (capTestMeasure h c r).map (rowMap (polarRows u X)) ≪ (volume : Measure (Label k))) : |
| (∫⁻ s, h s * capKernelValue K s ∂capMeasure c r) = |
| ∫⁻ X, ∫⁻ z, normalizedDensity (capTestMeasure h c r) k u (X,z) * |
| K (X,capReferenceLabel c r X z) ∂unitBallLaw k ∂Measure.pi (fun _ => stdGaussian E) := by |
| let μ := (capMeasure c r).withDensity h |
| let ν := capTestMeasure h c r |
| have : IsFiniteMeasure ν := Measure.isFiniteMeasure_map _ _ |
| have hf (X : Fin k → E) : Measurable (fun s => K (X,rowMap X s)) := by fun_prop |
| have he (X : Fin k → E) (hX : LinearIndependent ℝ X) : |
| (∫⁻ s, K (X,rowMap X s) ∂μ) = |
| ∫⁻ z, normalizedDensity ν k u (X,z)*K (X,capReferenceLabel c r X z) ∂unitBallLaw k := by |
| have hid := lintegral_withDensity_eq_lintegral_mul (unitBallLaw k) |
| (show Measurable (fun z => normalizedDensity ν k u (X,z)) by fun_prop) |
| (show Measurable (fun z => K (X,capReferenceLabel c r X z)) by fun_prop) |
| simp only [Pi.mul_apply] at hid |
| rw [← hid, |
| normalizedDensity_unitBall ν hu (capTestMeasure_support h c hr) X (hac X), |
| lintegral_map (by fun_prop) (by fun_prop)] |
| change _ = ∫⁻ s, K (X,capReferenceLabel c r X (rowMap (polarRows u X) s)) ∂μ.map (rescale c r) |
| rw [lintegral_map (by fun_prop) (measurable_rescale c r)] |
| apply lintegral_congr |
| intro s |
| congr 1 |
| rw [capReferenceLabel, ← polar_label_identity hX u c (rescale c r s) r, |
| dilate_rescale c hr.ne'] |
| have hid := lintegral_withDensity_eq_lintegral_mul (capMeasure c r) hh (measurable_capKernelValue hK) |
| simp only [Pi.mul_apply] at hid |
| rw [← hid] |
| change (∫⁻ s, ∫⁻ X, K (X,rowMap X s) ∂Measure.pi (fun _ => stdGaussian E) ∂μ) = _ |
| rw [lintegral_lintegral_swap (show AEMeasurable (fun z : E × (Fin k → E) => K (z.2,rowMap z.2 z.1)) _ by |
| apply Measurable.aemeasurable; exact hK.comp (measurable_snd.prodMk (by simp only [rowMap_apply]; fun_prop)))] |
| exact lintegral_congr_ae (by filter_upwards [gaussian_rows_ae_linearIndependent k hk] with X hX; exact he X hX) |
| |
| end |
| |
| lemma holder_bounded_kernel {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| {g f : Ω → ℝ≥0∞} (hg : Measurable g) (hf : Measurable f) (hf1 : ∀ x, f x ≤ 1) |
| {p : ℝ} (hp : 1 < p) {D : ℝ≥0∞} (hD : (∫⁻ x, g x^p ∂μ) ≤ D) : |
| (∫⁻ x, g x*f x ∂μ) ≤ D^(1/p) * (∫⁻ x, f x ∂μ)^(1-1/p) := by |
| have hpq := Real.HolderConjugate.conjExponent hp |
| have hh := ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hg.aemeasurable hf.aemeasurable |
| have he : 1 / Real.conjExponent p = 1-1/p := by |
| simpa only [one_div] using hpq.one_sub_inv.symm |
| have hpow : ∀ x, f x ^ Real.conjExponent p ≤ f x := by |
| intro x |
| simpa only [ENNReal.rpow_one] using ENNReal.rpow_le_rpow_of_exponent_ge (hf1 x) hpq.symm.lt.le |
| simp only [Pi.mul_apply, he] at hh |
| exact hh.trans (mul_le_mul' (ENNReal.rpow_le_rpow hD (by positivity)) |
| (ENNReal.rpow_le_rpow (lintegral_mono hpow) (by rw [← he]; exact one_div_nonneg.mpr hpq.symm.pos.le))) |
|
|
| lemma cap_exp_root (C d : ℝ) {p : ℝ} (hp : 0 < p) : |
| (ENNReal.ofReal (Real.exp (C*d*p))) ^ (1/p) = ENNReal.ofReal (Real.exp (C*d)) := by |
| rw [ENNReal.ofReal_rpow_of_pos (Real.exp_pos _), ← Real.exp_mul] |
| congr 2 |
| field_simp |
|
|
| lemma cap_dual_estimate : |
| ∃ C : ℝ, 0 < C ∧ ∀ d : ℕ, 32 ≤ d → |
| let k := d/16 |
| let α : ℝ := ((d:ℝ)-1)/2 |
| ∀ (c : EuclideanSpace ℝ (Fin d)), ‖c‖ = 1 → |
| ∀ r : ℝ, 0 < r → r ≤ 2 → |
| ∀ D : CapDictionary c r α, |
| ∀ K : (Fin k → EuclideanSpace ℝ (Fin d)) × Label k → ℝ≥0∞, |
| Measurable K → (∀ z, K z ≤ 1) → |
| ∀ h : EuclideanSpace ℝ (Fin d) → ℝ≥0∞, Measurable h → |
| (∀ i, ∫⁻ x, h x ∂capMeasure (D.center i) (D.radius i) ≤ ENNReal.ofReal ((r/D.radius i)^α)) → |
| (∫⁻ s, h s*capKernelValue K s ∂capMeasure c r) ≤ |
| ENNReal.ofReal (Real.exp (C*d)) * (capReference K c r)^(1-1/(k:ℝ)) := by |
| obtain ⟨C,hC,hmoment⟩ := normalized_unitBall_moment 2048 (by norm_num) |
| refine ⟨C,hC,?_⟩ |
| intro d hd k α c hc r hr hr2 D K hK hK1 h hh hdual |
| let : NeZero d := ⟨by omega⟩ |
| have hk : 2 ≤ k := by dsimp [k]; omega |
| have hkd : k ≤ d := by dsimp [k]; omega |
| have hkd2 : k+2 ≤ d := by dsimp [k]; omega |
| have hdR : (32:ℝ) ≤ d := by exact_mod_cast hd |
| have hα : 2*α = (Module.finrank ℝ (EuclideanSpace ℝ (Fin d))-1:ℕ) := by |
| simp only [finrank_euclideanSpace_fin, Nat.cast_sub (by omega : 1 ≤ d), Nat.cast_one] |
| dsimp [α] |
| ring |
| let u : Fin k → EuclideanSpace ℝ (Fin d) := (EuclideanSpace.basisFun (Fin d) ℝ) ∘ Fin.castLE hkd |
| have hu : Orthonormal ℝ u := (EuclideanSpace.basisFun (Fin d) ℝ).orthonormal.comp _ (Fin.castLE_injective hkd) |
| let ν := capTestMeasure h c r |
| have hmass := capTestMeasure_mass hr D hdual |
| let : IsFiniteMeasure ν := capTestMeasure_finite hr D hdual |
| have hs := capTestMeasure_support h c hr |
| have hball := capTest_allBall hc hr hr2 (by dsimp [α]; linarith : 0 ≤ α) hα D hdual |
| simp only [finrank_euclideanSpace_fin] at hball |
| have hac : ∀ X, ν.map (rowMap (polarRows u X)) ≪ (volume : Measure (Label k)) := |
| fun X => capTest_frame_ac h c hr (by simpa using hkd2) _ (polarRows_orthonormal hu X) |
| have hm := (hmoment d hd ν u hmass hs hball hu (Filter.Eventually.of_forall hac)).2.2.2 |
| have hp : (1:ℝ) < k := by exact_mod_cast (show 1 < k by omega) |
| let γ := Measure.pi (fun _ : Fin k => stdGaussian (EuclideanSpace ℝ (Fin d))) |
| let F : (Fin k → EuclideanSpace ℝ (Fin d)) × Label k → ℝ≥0∞ := |
| fun z => K (z.1,capReferenceLabel c r z.1 z.2) |
| have hF : Measurable F := hK.comp (measurable_fst.prodMk (measurable_capReferenceLabel c r)) |
| have hnorm : (∫⁻ z, normalizedDensity ν k u z ^ (k:ℝ) ∂γ.prod (unitBallLaw k)) ≤ |
| ENNReal.ofReal (Real.exp (C*d*k)) := by |
| rw [lintegral_prod _ (by fun_prop)] |
| simpa only [ENNReal.rpow_natCast] using hm |
| have hb := holder_bounded_kernel (γ.prod (unitBallLaw k)) |
| (measurable_normalizedDensity ν k u) hF (fun z => hK1 _) hp hnorm |
| rw [cap_exp_root C d (by positivity : (0:ℝ)<k), lintegral_prod _ (by fun_prop), |
| lintegral_prod _ (by fun_prop)] at hb |
| have hparent : (∫⁻ s, h s ∂capMeasure c r) ≤ 1 := by |
| simpa only [D.parent_center,D.parent_radius,div_self hr.ne',Real.one_rpow,ENNReal.ofReal_one] using hdual D.parent |
| let : IsFiniteMeasure ((capMeasure c r).withDensity h) := |
| isFiniteMeasure_withDensity (ne_of_lt (hparent.trans_lt (by simp))) |
| rw [cap_kernel_test_identity (by simpa using hkd) hK h hh c hr hu hac] |
| exact hb |
| |
| end ProjectionMoments |
| |
| end |
| |
| end OAI |
| |
