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)