| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.AmbientGram |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
|
|
| noncomputable section |
| universe u v |
|
|
|
|
| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision |
| open NoiselessRegression InnerProductSpace |
|
|
| def gaussianPeak : ℝ≥0∞ := ENNReal.ofReal ((Real.sqrt (2 * Real.pi))⁻¹) |
|
|
| lemma sharp_gaussianReal_le_volume : |
| gaussianReal 0 1 ≤ gaussianPeak • (volume : Measure ℝ) := by |
| rw [gaussianReal_of_var_ne_zero _ (by norm_num)] |
| calc |
| _ ≤ (volume : Measure ℝ).withDensity (fun _ => gaussianPeak) := by |
| apply withDensity_mono |
| filter_upwards [] with x |
| apply ENNReal.ofReal_le_ofReal |
| unfold gaussianPDFReal |
| simp only [NNReal.coe_one, mul_one, sub_zero] |
| exact mul_le_of_le_one_right (by positivity) |
| (Real.exp_le_one_iff.mpr (div_nonpos_of_nonpos_of_nonneg |
| (neg_nonpos.mpr (sq_nonneg x)) (by positivity))) |
| _ = _ := by rw [withDensity_const] |
|
|
| lemma sharp_pi_gaussianReal_le_volume (n : ℕ) : |
| Measure.pi (fun _ : Fin n => gaussianReal 0 1) ≤ |
| gaussianPeak ^ n • (volume : Measure (Fin n → ℝ)) := by |
| have h := measure_pi_mono (fun _ : Fin n => sharp_gaussianReal_le_volume) |
| rw [measure_pi_smul _ _ (show gaussianPeak ≠ ∞ from ENNReal.ofReal_ne_top)] at h |
| simpa only [Fintype.card_fin, volume_pi] using h |
|
|
| section Gaussian |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| local instance sharp_reference_sigmaFinite (m : ℕ) : |
| SigmaFinite (matrixLabelReference (E := E) m) := by |
| unfold matrixLabelReference |
| infer_instance |
|
|
| lemma sharp_gaussian_pairings_le {n : ℕ} (v : Fin n → E) (hv : LinearIndependent ℝ v) : |
| (stdGaussian E).map (fun x => fun i => ⟪v i, x⟫) ≤ |
| (gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) • |
| (volume : Measure (Fin n → ℝ)) := by |
| let u := gramSchmidtNormed ℝ v |
| let R : Matrix (Fin n) (Fin n) ℝ := |
| Matrix.diagonal (fun i => ‖gramSchmidt ℝ v i‖) * gsUpper n v |
| have hu : Orthonormal ℝ u := gramSchmidtNormed_orthonormal hv |
| have hn (i : Fin n) : ‖gramSchmidt ℝ v i‖ ≠ 0 := |
| norm_ne_zero_iff.mpr (gramSchmidt_ne_zero i hv) |
| have hexp (j : Fin n) : ∑ i, R i j • u i = v j := by |
| rw [← gsUpper_expansion n v j] |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp only [R, u, Matrix.diagonal_mul, gramSchmidtNormed, smul_smul] |
| congr 1 |
| simp [hn i, mul_comm] |
| have hdet : R.transpose.det = ∏ i, ‖gramSchmidt ℝ v i‖ := by |
| dsimp only [R] |
| rw [Matrix.det_transpose, Matrix.det_mul, Matrix.det_diagonal, gsUpper_det, mul_one] |
| have hd : R.transpose.det ≠ 0 := by |
| rw [hdet] |
| exact Finset.prod_ne_zero_iff.mpr (fun i _ => hn i) |
| have hpos : 0 ≤ R.transpose.det := by rw [hdet]; positivity |
| have hf : Measurable (Matrix.toLin' R.transpose) := |
| (Matrix.toLin' R.transpose).continuous_of_finiteDimensional.measurable |
| have heq : (fun x : E => fun j => ⟪v j, x⟫) = |
| (Matrix.toLin' R.transpose) ∘ (fun x => fun i => ⟪u i, x⟫) := by |
| funext x j |
| rw [← hexp j] |
| simp [Matrix.toLin'_apply, Matrix.mulVec, dotProduct, sum_inner, |
| inner_smul_left] |
| rw [heq, ← Measure.map_map hf (by fun_prop), orthonormal_gaussian_map hu] |
| calc |
| _ ≤ (gaussianPeak ^ n • (volume : Measure (Fin n → ℝ))).map |
| (Matrix.toLin' R.transpose) := |
| Measure.map_mono (sharp_pi_gaussianReal_le_volume n) hf |
| _ = _ := by |
| rw [Measure.map_smul _ hf.aemeasurable, Real.map_matrix_volume_pi_eq_smul_volume_pi hd, |
| abs_of_nonneg (inv_nonneg.mpr hpos), hdet, smul_smul] |
| |
| lemma sharp_gaussian_rows_pairings_le {n : ℕ} (v : Fin n → E) |
| (hv : LinearIndependent ℝ v) (m : ℕ) : |
| (Measure.pi (fun _ : Fin m => stdGaussian E)).map |
| (fun A => fun j i => ⟪v i, A j⟫) ≤ |
| ((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) ^ m) • |
| (volume : Measure (Fin m → Fin n → ℝ)) := by |
| rw [Measure.pi_map_pi (fun _ => (by fun_prop : Measurable |
| (fun x : E => fun i => ⟪v i, x⟫)).aemeasurable)] |
| have h := measure_pi_mono (fun _ : Fin m => sharp_gaussian_pairings_le v hv) |
| rw [measure_pi_smul _ _ (ENNReal.mul_ne_top (ENNReal.pow_ne_top |
| (show gaussianPeak ≠ ∞ from ENNReal.ofReal_ne_top)) ENNReal.ofReal_ne_top)] at h |
| simpa only [Fintype.card_fin, volume_pi] using h |
| lemma sharp_gaussian_rows_kernel_bound {n : ℕ} (v : Fin n → E) (hv : LinearIndependent ℝ v) |
| (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1) |
| (m : ℕ) (y : Fin m → ℝ) : |
| ∫⁻ A : Fin m → E, (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫) |
| ∂Measure.pi (fun _ => stdGaussian E) ≤ |
| ((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) ^ m) * ∏ j, φ (y j) := by |
| let H : (Fin m → Fin n → ℝ) → ℝ≥0∞ := |
| fun z => ∏ j, φ (y j) * ∏ i, φ (y j - z j i) |
| have hH : Measurable H := by dsimp [H]; fun_prop |
| have heq (A : Fin m → E) : |
| (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫) = |
| H (fun j i => ⟪v i, A j⟫) := by |
| dsimp [H] |
| rw [Finset.prod_mul_distrib, Finset.prod_comm] |
| simp_rw [heq] |
| rw [← lintegral_map hH (by fun_prop)] |
| calc |
| _ ≤ ∫⁻ z, H z ∂(((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) ^ m) • |
| (volume : Measure (Fin m → Fin n → ℝ))) := |
| lintegral_mono' (sharp_gaussian_rows_pairings_le v hv m) (fun _ => le_rfl) |
| _ = _ := by |
| rw [lintegral_smul_measure] |
| change _ * (∫⁻ z : Fin m → Fin n → ℝ, ∏ j, φ (y j) * ∏ i, φ (y j - z j i)) = _ |
| rw [lintegral_product_differences φ hφ hφ1] |
|
|
| lemma sharp_gaussian_replica_kernel_bound {n : ℕ} (p : E) (v : Fin n → E) |
| (hv : LinearIndependent ℝ (fun i => v i - p)) |
| (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1) (m : ℕ) : |
| ∫⁻ A : Fin m → E, ∫⁻ y : Fin m → ℝ, |
| (∏ j, φ (y j - ⟪p, A j⟫)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫) |
| ∂volume ∂Measure.pi (fun _ => stdGaussian E) ≤ |
| ((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun i => v i - p) i‖)⁻¹)) ^ m) := by |
| have ht (A : Fin m → E) : |
| (∫⁻ y : Fin m → ℝ, (∏ j, φ (y j - ⟪p, A j⟫)) * |
| ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫)) = |
| ∫⁻ y : Fin m → ℝ, (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i - p, A j⟫) := by |
| have he (y : Fin m → ℝ) (i : Fin n) (j : Fin m) : |
| y j - ⟪p, A j⟫ - ⟪v i - p, A j⟫ = y j - ⟪v i, A j⟫ := by |
| rw [inner_sub_left] |
| ring |
| simpa only [Pi.sub_apply, he] using |
| lintegral_sub_right_eq_self |
| (fun y : Fin m → ℝ => (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i - p, A j⟫)) |
| (fun j => ⟪p, A j⟫) |
| simp_rw [ht] |
| rw [lintegral_lintegral_swap (by fun_prop)] |
| calc |
| _ ≤ ∫⁻ y : Fin m → ℝ, |
| ((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun i => v i - p) i‖)⁻¹)) ^ m) * |
| ∏ j, φ (y j) := lintegral_mono fun y => |
| sharp_gaussian_rows_kernel_bound (fun i => v i - p) hv φ hφ hφ1 m y |
| _ = _ := by |
| rw [lintegral_const_mul _ (by fun_prop), volume_pi, |
| lintegral_fin_prod volume m (fun _ => φ) (fun _ => hφ)] |
| simp [hφ1] |
|
|
| lemma sharp_projectionSmoothing_moment {ν : Measure E} [IsFiniteMeasure ν] |
| {H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R) |
| (hsupport : ν.support ⊆ Metric.closedBall z R) |
| (hball : ∀ x : E, ∀ r : ℝ, 0 < r → |
| ν (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β)) |
| (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ x, φ x = 1) |
| {m : ℕ} (hm : 0 < m) (n : ℕ) (hgap : (n:ℝ)+(m:ℝ)-1 < β) : |
| (∫⁻ Ay : (Fin m → E) × (Fin m → ℝ), |
| (projectionSmoothing ν φ m Ay)^(n+1) ∂matrixLabelReference m) ≤ |
| gaussianPeak ^ (n*m) * ENNReal.ofReal |
| (H^(n+1) * R^(β*(n+1:ℕ)-(m:ℝ)*(n:ℝ)) * ∏ j : Fin n, J β m j) := by |
| simp_rw [projectionSmoothing_pow ν φ hφ m n] |
| rw [lintegral_lintegral_swap (by fun_prop)] |
| calc |
| _ ≤ ∫⁻ w : E × (Fin n → E), gaussianPeak ^ (n*m) * replicaWeight m w.1 w.2 |
| ∂ν.prod (Measure.pi (fun _ => ν)) := by |
| apply lintegral_mono_ae |
| filter_upwards [ae_joint_replica_independent hH hR hsupport hball hm n hgap] with w hw |
| rw [matrixLabelReference, lintegral_prod _ (by fun_prop)] |
| convert sharp_gaussian_replica_kernel_bound w.1 w.2 hw φ hφ hφ1 m using 1 |
| rw [mul_pow, ← pow_mul, replicaWeight_eq_det, sqrt_det_gram, |
| ENNReal.ofReal_pow (by positivity)] |
| _ = gaussianPeak ^ (n*m) * |
| (∫⁻ w : E × (Fin n → E), replicaWeight m w.1 w.2 |
| ∂ν.prod (Measure.pi (fun _ => ν))) := |
| lintegral_const_mul _ (measurable_joint_replicaWeight m n) |
| _ ≤ _ := mul_le_mul' le_rfl |
| (joint_replica_integral_exact hH hR hsupport hball hm n hgap) |
| |
| end Gaussian |
| end MemoryPrecision |
| |
| |
| end |
| |
| end OAI |
| |
