| import OAI.Probability.GaussianReplacement.Gram |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace CurrentProjection |
| open InnerProductSpace |
|
|
| section MixedNondegenerate |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| theorem AmbientGrowth.affine_span_null {μ : Measure E} {a : ℕ} {A : ℝ} |
| (hμ : AmbientGrowth μ a A) (hA : 0 ≤ A) (V : Submodule ℝ E) (s : E) |
| (hmargin : Module.finrank ℝ V + 1 ≤ a) (hsupp : ∀ᵐ u ∂μ, ‖u-s‖ ≤ 2) : |
| μ {u | u-s ∈ V} = 0 := by |
| let C : ℝ := 3 ^ Module.finrank ℝ V * 2 ^ a * A * 2 ^ a |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hb (j : ℕ) : μ {u | u-s ∈ V} ≤ ENNReal.ofReal (C * (1/2:ℝ)^j) := by |
| have hj : 0 < (1/2:ℝ)^j := by positivity |
| have hj1 : (1/2:ℝ)^j ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| calc |
| μ {u | u-s ∈ V} ≤ μ {u | ‖u-s‖ ≤ 2 ∧ Metric.infDist (u-s) (V:Set E) ≤ (1/2:ℝ)^j * 2} := by |
| apply measure_mono_ae |
| filter_upwards [hsupp] with u hu |
| intro hv |
| exact ⟨hu, by rw [Metric.infDist_zero_of_mem hv]; positivity⟩ |
| _ ≤ ENNReal.ofReal C * ENNReal.ofReal (((1/2:ℝ)^j)^1) := |
| hμ.radial_tube_pow hA V s 1 hmargin (by norm_num) hj hj1 |
| _ = _ := by rw [pow_one, ENNReal.ofReal_mul hC] |
| have hlim : Tendsto (fun j : ℕ => ENNReal.ofReal (C * (1/2:ℝ)^j)) atTop (𝓝 0) := by |
| convert ENNReal.tendsto_ofReal ((tendsto_const_nhds (x:=C)).mul |
| (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2) |
| (by norm_num : (1/2:ℝ) < 1))) using 1 |
| simp |
| exact le_antisymm (ge_of_tendsto' hlim hb) bot_le |
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| theorem AmbientGrowth.ae_gramSchmidt_ne_zero {μ : Measure E} [SigmaFinite μ] {a : ℕ} {A : ℝ} |
| (hμ : AmbientGrowth μ a A) (hA : 0 ≤ A) (s : E) |
| (hsupp : ∀ᵐ u ∂μ, ‖u-s‖ ≤ 2) (k : ℕ) (hk : k ≤ a) : |
| ∀ᵐ v : Fin k → E ∂Measure.pi (fun _ => μ), |
| ∀ i, gramSchmidt ℝ (fun j => v j - s) i ≠ 0 := by |
| induction k with |
| | zero => exact Filter.Eventually.of_forall fun _ i => Fin.elim0 i |
| | succ k ih => |
| rw [ae_pi_snoc μ (measurableSet_gramSchmidt_ne_zero s (k+1))] |
| filter_upwards [ih (by omega)] with v hv |
| let V := Submodule.span ℝ (Set.range (fun i : Fin k => v i - s)) |
| have hdim : Module.finrank ℝ V ≤ k := by |
| simpa [V, Set.finrank] using finrank_range_le_card (R:=ℝ) (fun i : Fin k => v i-s) |
| have hnull : μ {u | u-s ∈ V} = 0 := hμ.affine_span_null hA V s (by omega) hsupp |
| have ha : ∀ᵐ u ∂μ, u-s ∉ V := by |
| rw [ae_iff] |
| simpa only [not_not] using hnull |
| filter_upwards [ha] with x hx |
| intro i |
| induction i using Fin.lastCases with |
| | last => |
| apply norm_ne_zero_iff.mp |
| rw [norm_gramSchmidt_eq_infDist, image_Iio_last] |
| simp only [Fin.snoc_castSucc, Fin.snoc_last] |
| exact ne_of_gt ((V.closed_of_finiteDimensional.notMem_iff_infDist_pos |
| ⟨0, V.zero_mem⟩).mp hx) |
| | cast j => simpa only [gramSchmidt_castSucc, Fin.snoc_castSucc] using hv j |
|
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| theorem AmbientGrowth.ae_replica_linearIndependent {μ : Measure E} [SigmaFinite μ] |
| {a : ℕ} {A : ℝ} (hμ : AmbientGrowth μ a A) (hA : 0 ≤ A) (s : E) |
| (hsupp : ∀ᵐ u ∂μ, ‖u-s‖ ≤ 2) (k : ℕ) (hk : k ≤ a) : |
| ∀ᵐ v : Fin k → E ∂Measure.pi (fun _ => μ), |
| LinearIndependent ℝ (fun j => v j-s) := by |
| filter_upwards [hμ.ae_gramSchmidt_ne_zero hA s hsupp k hk] with v hv |
| apply Matrix.linearIndependent_of_det_gram_ne_zero |
| rw [det_gram_eq_prod_gramSchmidt] |
| exact Finset.prod_ne_zero_iff.mpr fun i _ => pow_ne_zero _ (norm_ne_zero_iff.mpr (hv i)) |
|
|
| end MixedNondegenerate |
|
|
| section MixedReplica |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [measurableSpaceE : MeasurableSpace E] [borelSpaceE : BorelSpace E] |
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| def mixedReplicaWeight (m r : ℕ) (s : E) {k : ℕ} (v : Fin k → E) : ℝ≥0∞ := |
| ∏ i, ENNReal.ofReal (‖v i-s‖⁻¹ ^ r * ‖gramSchmidt ℝ (fun j => v j-s) i‖⁻¹ ^ m) |
|
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| lemma measurable_mixedReplicaWeight (m r : ℕ) (s : E) (k : ℕ) : |
| Measurable (mixedReplicaWeight m r s (k:=k)) := by |
| have hs : Measurable (fun v : Fin k → E => fun j => v j-s) := by fun_prop |
| exact Finset.measurable_prod _ fun i _ => |
| (((measurable_pi_apply i).sub measurable_const).norm.inv.pow_const r |>.mul |
| (((measurable_gramSchmidt i).comp hs).norm.inv.pow_const m)).ennreal_ofReal |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma mixedReplicaWeight_snoc (m r : ℕ) (s : E) {k : ℕ} (v : Fin k → E) (x : E) : |
| mixedReplicaWeight m r s (Fin.snoc v x) = mixedReplicaWeight m r s v * |
| mixedInverse s (Submodule.span ℝ (Set.range (fun i => v i-s))) m r x := by |
| unfold mixedReplicaWeight |
| rw [Fin.prod_univ_castSucc] |
| simp only [gramSchmidt_castSucc, Fin.snoc_castSucc] |
| rw [norm_gramSchmidt_eq_infDist, image_Iio_last] |
| simp only [Fin.snoc_last, Fin.snoc_castSucc] |
| rfl |
|
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| theorem AmbientGrowth.mixed_inverse_uniform {μ : Measure E} {d n m r : ℕ} {B D : ℝ} |
| (hBμ : AmbientGrowth μ (m+r) B) (hDμ : AmbientGrowth μ n D) |
| (hB : 0 < B) (hD : 0 < D) (hn : m+r < n) (hnd : n ≤ d) |
| (V : Submodule ℝ E) (s : E) (hmargin : Module.finrank ℝ V + (m+1) ≤ m+r) |
| (hsupp : ∀ᵐ u ∂μ, ‖u-s‖ ≤ 2) : |
| ∫⁻ u, mixedInverse s V m r u ∂μ ≤ momentConstant 9 d B D := by |
| refine (hBμ.mixed_inverse_le hDμ hB hD (by omega) hn V s hmargin hsupp).trans |
| (ENNReal.ofReal_le_ofReal ?_) |
| gcongr |
| exact shellConstant_exp_bound d _ n m r (by omega) (by omega) hnd (by omega) (by omega) |
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| theorem AmbientGrowth.mixed_replica_integral {μ : Measure E} [SigmaFinite μ] |
| {d n m r : ℕ} {B D : ℝ} |
| (hBμ : AmbientGrowth μ (m+r) B) (hDμ : AmbientGrowth μ n D) |
| (hB : 0 < B) (hD : 0 < D) (hn : m+r < n) (hnd : n ≤ d) |
| (s : E) (hsupp : ∀ᵐ u ∂μ, ‖u-s‖ ≤ 2) (k : ℕ) (hk : k ≤ r) : |
| ∫⁻ v : Fin k → E, mixedReplicaWeight m r s v ∂Measure.pi (fun _ => μ) ≤ |
| (momentConstant 9 d B D)^k := by |
| induction k with |
| | zero => simp [mixedReplicaWeight] |
| | succ k ih => |
| have hinner (v : Fin k → E) : |
| ∫⁻ x, mixedReplicaWeight m r s (Fin.snoc v x) ∂μ ≤ |
| mixedReplicaWeight m r s v * momentConstant 9 d B D := by |
| simp_rw [mixedReplicaWeight_snoc] |
| rw [lintegral_const_mul _ (measurable_mixedInverse _ _ _ _)] |
| gcongr |
| apply hBμ.mixed_inverse_uniform hDμ hB hD hn hnd _ s _ hsupp |
| have hdim := finrank_range_le_card (R:=ℝ) (fun i : Fin k => v i-s) |
| simp only [Fintype.card_fin, Set.finrank] at hdim |
| omega |
| calc |
| _ = ∫⁻ v : Fin k → E, ∫⁻ x, mixedReplicaWeight m r s (Fin.snoc v x) ∂μ |
| ∂Measure.pi (fun _ => μ) := lintegral_pi_snoc μ _ (measurable_mixedReplicaWeight m r s (k+1)) |
| _ ≤ ∫⁻ v : Fin k → E, mixedReplicaWeight m r s v * momentConstant 9 d B D |
| ∂Measure.pi (fun _ => μ) := lintegral_mono hinner |
| _ = (∫⁻ v : Fin k → E, mixedReplicaWeight m r s v ∂Measure.pi (fun _ => μ)) * |
| momentConstant 9 d B D := lintegral_mul_const _ (measurable_mixedReplicaWeight m r s k) |
| _ ≤ (momentConstant 9 d B D)^k * momentConstant 9 d B D := |
| mul_le_mul' (ih (by omega)) le_rfl |
| _ = _ := (pow_succ _ _).symm |
|
|
| end MixedReplica |
|
|
| section FixedGaussianKernel |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| lemma gaussian_fixed_kernel_bound {k : ℕ} (v : Fin k → E) (hv : LinearIndependent ℝ v) |
| (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1) (m : ℕ) : |
| ∫⁻ X : Fin m → E, ∏ i, ∏ j, φ ⟪v i, X j⟫ |
| ∂Measure.pi (fun _ => stdGaussian E) ≤ |
| ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹ ^ m) := by |
| let H : (Fin m → Fin k → ℝ) → ℝ≥0∞ := fun z => ∏ j, ∏ i, φ (z j i) |
| have hH : Measurable H := by dsimp [H]; fun_prop |
| have hH1 : ∫⁻ z, H z = 1 := by |
| rw [volume_pi, lintegral_fin_prod (volume : Measure (Fin k → ℝ)) m |
| (fun _ z => ∏ i, φ (z i)) (by intro j; fun_prop)] |
| conv_lhs => arg 2; ext j; rw [volume_pi, lintegral_fin_prod (volume : Measure ℝ) k |
| (fun _ z => φ z) (fun _ => hφ)] |
| simp [hφ1] |
| have heq (X : Fin m → E) : (∏ i, ∏ j, φ ⟪v i, X j⟫) = H (fun j i => ⟪v i, X j⟫) := by |
| exact Finset.prod_comm |
| simp_rw [heq] |
| rw [← lintegral_map hH (by fun_prop)] |
| calc |
| _ ≤ ∫⁻ z, H z ∂(ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹ ^ m) • |
| (volume : Measure (Fin m → Fin k → ℝ))) := |
| lintegral_mono' (gaussian_rows_pairings_le v hv m) (fun _ => le_rfl) |
| _ = _ := by rw [lintegral_smul_measure, hH1]; simp |
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| def rowKernel (φ : ℝ → ℝ≥0∞) {m : ℕ} (s u : E) (X : Fin m → E) : ℝ≥0∞ := |
| ∏ j, φ ⟪u-s, X j⟫ |
|
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| lemma measurable_rowKernel (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (m : ℕ) (s : E) : |
| Measurable (fun z : E × (Fin m → E) => rowKernel φ s z.1 z.2) := by |
| dsimp [rowKernel] |
| fun_prop |
|
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| lemma measurable_rowKernel_right (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) |
| (m : ℕ) (s u : E) : Measurable (rowKernel φ s u (m:=m)) := by |
| change Measurable (fun X : Fin m → E => ∏ j, φ ⟪u-s, X j⟫) |
| fun_prop |
|
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| lemma measurable_rowKernel_left (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) |
| (m : ℕ) (s : E) (X : Fin m → E) : Measurable (fun u => rowKernel φ s u X) := by |
| change Measurable (fun u : E => ∏ j, φ ⟪u-s, X j⟫) |
| fun_prop |
|
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| lemma gaussian_fresh_kernel_bound (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) |
| (hφ1 : ∫⁻ z, φ z = 1) (r : ℕ) (s u : E) (hu : u ≠ s) : |
| ∫⁻ Z : Fin r → E, rowKernel φ s u Z ∂Measure.pi (fun _ => stdGaussian E) ≤ |
| ENNReal.ofReal (‖u-s‖⁻¹ ^ r) := by |
| have hv : LinearIndependent ℝ (fun _ : Fin 1 => u-s) := |
| linearIndependent_unique_iff.mpr (sub_ne_zero.mpr hu) |
| have hh := gaussian_fixed_kernel_bound (fun _ : Fin 1 => u-s) hv φ hφ hφ1 r |
| have he : gramSchmidt ℝ (fun _ : Fin 1 => u-s) 0 = u-s := gramSchmidt_bot ℝ _ |
| simpa only [Fin.prod_univ_one, he, rowKernel] using hh |
|
|
| end FixedGaussianKernel |
|
|
| section MixedKernelMoment |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [finiteDimensionalRealE : FiniteDimensional ℝ E] [measurableSpaceE : MeasurableSpace E] [borelSpaceE : BorelSpace E] |
|
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma rowKernel_append (φ : ℝ → ℝ≥0∞) {m r : ℕ} (s u : E) |
| (X : Fin m → E) (Z : Fin r → E) : |
| rowKernel φ s u (Fin.append X Z) = rowKernel φ s u X * rowKernel φ s u Z := by |
| simp only [rowKernel, Fin.prod_univ_add, Fin.append_left, Fin.append_right] |
|
|
| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma mixedReplicaWeight_factor (m r : ℕ) (s : E) {k : ℕ} (v : Fin k → E) : |
| mixedReplicaWeight m r s v = |
| (∏ i, ENNReal.ofReal (‖v i-s‖⁻¹ ^ r)) * |
| ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun j => v j-s) i‖)⁻¹ ^ m) := by |
| unfold mixedReplicaWeight |
| simp_rw [ENNReal.ofReal_mul (by positivity : 0 ≤ ‖v _-s‖⁻¹ ^ r)] |
| rw [Finset.prod_mul_distrib, ← Finset.prod_inv_distrib, ← Finset.prod_pow, |
| ENNReal.ofReal_prod_of_nonneg (fun i _ => by positivity)] |
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|
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| theorem AmbientGrowth.mixed_kernel_moment {μ : Measure E} [SigmaFinite μ] |
| {d n m r q : ℕ} {B D : ℝ} |
| (hBμ : AmbientGrowth μ (m+r) B) (hDμ : AmbientGrowth μ n D) |
| (hB : 0 < B) (hD : 0 < D) (hr : 1 ≤ r) (hn : m+r < n) (hnd : n ≤ d) |
| (hq : q ≤ r) (s : E) (hsupp : ∀ᵐ u ∂μ, ‖u-s‖ ≤ 2) |
| (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1) : |
| ∫⁻ X : Fin m → E, |
| (∫⁻ Z : Fin r → E, ∫⁻ u, rowKernel φ s u (Fin.append X Z) ∂μ |
| ∂Measure.pi (fun _ => stdGaussian E))^q |
| ∂Measure.pi (fun _ => stdGaussian E) ≤ (momentConstant 9 d B D)^q := by |
| have hnull : μ {u | u=s} = 0 := by |
| have hh := hBμ.affine_span_null hB.le (⊥ : Submodule ℝ E) s (by simpa using (by omega : 1 ≤ m+r)) hsupp |
| simpa only [Submodule.mem_bot, sub_eq_zero] using hh |
| have ha : ∀ᵐ u ∂μ, u ≠ s := by |
| rw [ae_iff] |
| simpa only [not_not] using hnull |
| let F : E → (Fin m → E) → ℝ≥0∞ := fun u X => |
| ENNReal.ofReal (‖u-s‖⁻¹ ^ r) * rowKernel φ s u X |
| have hF : Measurable (Function.uncurry F) := by dsimp [F, rowKernel, Function.uncurry]; fun_prop |
| have hFX (X : Fin m → E) : Measurable (fun u => F u X) := |
| hF.comp (measurable_id.prodMk measurable_const) |
| have hfresh (X : Fin m → E) : |
| ∫⁻ Z : Fin r → E, ∫⁻ u, rowKernel φ s u (Fin.append X Z) ∂μ |
| ∂Measure.pi (fun _ => stdGaussian E) ≤ ∫⁻ u, F u X ∂μ := by |
| simp_rw [rowKernel_append] |
| rw [lintegral_lintegral_swap (by simp only [rowKernel]; fun_prop)] |
| apply lintegral_mono_ae |
| filter_upwards [ha] with u hu |
| rw [lintegral_const_mul _ (measurable_rowKernel_right φ hφ r s u)] |
| simpa only [F, mul_comm] using mul_le_mul' (le_rfl : rowKernel φ s u X ≤ rowKernel φ s u X) |
| (gaussian_fresh_kernel_bound φ hφ hφ1 r s u hu) |
| have hpow (X : Fin m → E) : (∫⁻ u, F u X ∂μ)^q = |
| ∫⁻ v : Fin q → E, ∏ i, F (v i) X ∂Measure.pi (fun _ => μ) := |
| lintegral_pow_pi μ _ (hFX X) q |
| calc |
| _ ≤ ∫⁻ X : Fin m → E, (∫⁻ u, F u X ∂μ)^q |
| ∂Measure.pi (fun _ => stdGaussian E) := by |
| apply lintegral_mono |
| intro X |
| exact pow_le_pow_left' (hfresh X) q |
| _ = ∫⁻ v : Fin q → E, ∫⁻ X : Fin m → E, ∏ i, F (v i) X |
| ∂Measure.pi (fun _ => stdGaussian E) ∂Measure.pi (fun _ => μ) := by |
| simp_rw [hpow] |
| rw [lintegral_lintegral_swap (by dsimp [F, rowKernel]; fun_prop)] |
| _ ≤ ∫⁻ v : Fin q → E, mixedReplicaWeight m r s v ∂Measure.pi (fun _ => μ) := by |
| apply lintegral_mono_ae |
| filter_upwards [hBμ.ae_replica_linearIndependent hB.le s hsupp q (by omega)] with v hv |
| simp only [F, Finset.prod_mul_distrib] |
| rw [lintegral_const_mul _ (by simp only [rowKernel]; fun_prop), mixedReplicaWeight_factor] |
| exact mul_le_mul' le_rfl (gaussian_fixed_kernel_bound (fun i => v i-s) hv φ hφ hφ1 m) |
| _ ≤ _ := hBμ.mixed_replica_integral hDμ hB hD hn hnd s hsupp q hq |
|
|
| end MixedKernelMoment |
|
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| end CurrentProjection |
|
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| end |
|
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| end OAI |
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