openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianMeanInterpolation.lean
| import OAI.Probability.InvariantIsing.Gaussian.BaseGaussianVariance | |
| namespace OAI | |
| /-! Finite Gaussian mean interpolation with arbitrary base energy and weights. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped BigOperators | |
| namespace InvariantIsing | |
| variable {X : Type*} [Fintype X] | |
| def affineGaussianCoefficients {d : ℕ} (C A : X → Fin d → ℝ) (s : ℝ) : X → Fin d → ℝ := | |
| fun x i => C x i + s * A x i | |
| omit [Fintype X] in | |
| lemma linearGaussian_affine {d : ℕ} (C A : X → Fin d → ℝ) (s : ℝ) | |
| (g : Fin d → ℝ) (x : X) : | |
| linearGaussian (affineGaussianCoefficients C A s) g x = | |
| linearGaussian C g x + s * linearGaussian A g x := by | |
| simp only [linearGaussian, affineGaussianCoefficients, add_mul, Finset.sum_add_distrib, | |
| Finset.mul_sum, mul_assoc] | |
| def affineGaussianMean {d : ℕ} (w H : X → ℝ) (C A : X → Fin d → ℝ) (s : ℝ) : ℝ := | |
| ∫ g, Real.log (finitePartition w (fun x => H x + | |
| linearGaussian (affineGaussianCoefficients C A s) g x)) | |
| ∂Measure.pi (fun _ : Fin d => gaussianReal 0 1) | |
| def affineGaussianSlope {d : ℕ} (w H : X → ℝ) (C A : X → Fin d → ℝ) (s : ℝ) : ℝ := | |
| ∫ g, ∑ x, gaussianGibbs w H (affineGaussianCoefficients C A s) g x * linearGaussian A g x | |
| ∂Measure.pi (fun _ : Fin d => gaussianReal 0 1) | |
| lemma affineGaussianMean_hasDerivAt {d : ℕ} {w : X → ℝ} (hw : GibbsReference w) | |
| (H : X → ℝ) (C A : X → Fin d → ℝ) (t : ℝ) : | |
| HasDerivAt (affineGaussianMean w H C A) (affineGaussianSlope w H C A t) t := by | |
| let μ := Measure.pi (fun _ : Fin d => gaussianReal 0 1) | |
| have hm (s : ℝ) : AEStronglyMeasurable (fun g => Real.log (finitePartition w | |
| (fun x => H x + linearGaussian (affineGaussianCoefficients C A s) g x))) μ := by | |
| simpa only [one_mul] using | |
| (continuous_log_finitePartition hw H (affineGaussianCoefficients C A s) 1).aestronglyMeasurable | |
| have hi : Integrable (fun g => Real.log (finitePartition w | |
| (fun x => H x + linearGaussian (affineGaussianCoefficients C A t) g x))) μ := by | |
| simpa only [one_mul] using integrable_log_finitePartition hw H (affineGaussianCoefficients C A t) 1 | |
| apply (hasDerivAt_integral_of_dominated_loc_of_deriv_le (μ := μ) (s := Set.univ) | |
| (F' := fun s g => ∑ x, gaussianGibbs w H (affineGaussianCoefficients C A s) g x * | |
| linearGaussian A g x) (bound := fun g => ∑ x, |linearGaussian A g x|) | |
| (Filter.univ_mem) (Filter.Eventually.of_forall hm) hi ?_ ?_ ?_ ?_).2 | |
| · exact (continuous_finsetSum _ (fun x _ => | |
| (continuous_gaussianGibbs hw H _ x).mul (by unfold linearGaussian; fun_prop))).aestronglyMeasurable | |
| · apply ae_of_all | |
| intro g s _ | |
| rw [Real.norm_eq_abs] | |
| exact finiteGibbs_average_bound hw _ _ (fun x => | |
| Finset.single_le_sum (fun y _ => abs_nonneg (linearGaussian A g y)) (Finset.mem_univ x)) | |
| · exact integrable_finsetSum _ (fun x _ => (integrable_linearGaussian A x).abs) | |
| · apply ae_of_all | |
| intro g s _ | |
| apply hasDerivAt_log_finitePartition hw | |
| intro x | |
| simp_rw [linearGaussian_affine] | |
| simpa only [one_mul, id_eq] using (((hasDerivAt_id s).mul_const (linearGaussian A g x)).const_add | |
| (linearGaussian C g x)).const_add (H x) | |
| lemma expectedGaussianEnergy_abs_le {d : ℕ} {w : X → ℝ} (hw : GibbsReference w) | |
| (H : X → ℝ) (A C : X → Fin (d + 1) → ℝ) {K : ℝ} | |
| (hK : ∀ x y, |gaussianCross A C x y| ≤ K) : | |
| |∫ g, ∑ x, gaussianGibbs w H C g x * linearGaussian A g x | |
| ∂Measure.pi (fun _ : Fin (d + 1) => gaussianReal 0 1)| ≤ 2 * K := by | |
| have hb (g : Fin (d + 1) → ℝ) (x : X) : | |
| |gaussianCross A C x x - ∑ y, gaussianGibbs w H C g y * gaussianCross A C x y| ≤ 2 * K := by | |
| have hh := finiteGibbs_average_bound hw (fun y => H y + linearGaussian C g y) | |
| (fun y => gaussianCross A C x y) (hK x) | |
| exact (abs_sub _ _).trans (by dsimp only [gaussianGibbs] at *; linarith [hK x x]) | |
| have hc (x : X) : Integrable (fun g => gaussianGibbs w H C g x * | |
| (gaussianCross A C x x - ∑ y, gaussianGibbs w H C g y * gaussianCross A C x y)) | |
| (Measure.pi (fun _ : Fin (d + 1) => gaussianReal 0 1)) := by | |
| apply Integrable.of_bound | |
| ((continuous_gaussianGibbs hw H C x).mul | |
| (continuous_const.sub (continuous_finsetSum _ (fun y _ => | |
| (continuous_gaussianGibbs hw H C y).mul continuous_const)))).aestronglyMeasurable (2 * K) | |
| apply ae_of_all | |
| intro g | |
| change ‖gaussianGibbs w H C g x * | |
| (gaussianCross A C x x - ∑ y, gaussianGibbs w H C g y * gaussianCross A C x y)‖ ≤ _ | |
| rw [Real.norm_eq_abs, abs_mul, | |
| abs_of_nonneg (show 0 ≤ gaussianGibbs w H C g x from finiteGibbs_nonneg hw _ x)] | |
| exact (mul_le_mul_of_nonneg_right (finiteGibbs_le_one hw _ x) (abs_nonneg _)).trans | |
| (by simpa only [one_mul] using hb g x) | |
| rw [integral_finsetSum _ (fun x _ => by | |
| simpa only [mul_comm] using integrable_linearGaussian_mul_gibbs hw H A C x x)] | |
| simp_rw [mul_comm (gaussianGibbs w H C _ _) (linearGaussian A _ _), | |
| gaussian_gibbs_insertion hw H A C] | |
| rw [← integral_finsetSum _ (fun x _ => hc x)] | |
| simpa only [Real.norm_eq_abs, probReal_univ, mul_one] using | |
| (norm_integral_le_of_norm_le_const (μ := Measure.pi (fun _ : Fin (d + 1) => gaussianReal 0 1)) | |
| (f := fun g => ∑ x, gaussianGibbs w H C g x * | |
| (gaussianCross A C x x - ∑ y, gaussianGibbs w H C g y * gaussianCross A C x y)) | |
| (C := 2 * K) (ae_of_all _ (fun g => by | |
| rw [Real.norm_eq_abs] | |
| exact finiteGibbs_average_bound hw (fun x => H x + linearGaussian C g x) _ (hb g)))) | |
| /-- A dimension-independent mean comparison. The cross-covariance of the | |
| increment with the interpolated field, rather than the Gaussian coordinate | |
| count, determines the bound. -/ | |
| theorem affineGaussianMean_abs_sub_le {d : ℕ} {w : X → ℝ} (hw : GibbsReference w) | |
| (H : X → ℝ) (C A : X → Fin (d + 1) → ℝ) {K : ℝ} | |
| (hK : ∀ s ∈ Set.Icc (0 : ℝ) 1, ∀ x y, | |
| |gaussianCross A (affineGaussianCoefficients C A s) x y| ≤ K) : | |
| |affineGaussianMean w H C A 1 - affineGaussianMean w H C A 0| ≤ 2 * K := by | |
| have hd (s : ℝ) := affineGaussianMean_hasDerivAt hw H C A s | |
| have hb (s : ℝ) (hs : s ∈ Set.Icc (0 : ℝ) 1) : |affineGaussianSlope w H C A s| ≤ 2 * K := | |
| expectedGaussianEnergy_abs_le hw H A (affineGaussianCoefficients C A s) (hK s hs) | |
| simpa only [Real.norm_eq_abs] using | |
| norm_image_sub_le_of_norm_deriv_le_segment_01' | |
| (fun s _ => (hd s).hasDerivWithinAt) (fun s hs => by simpa only [Real.norm_eq_abs] using hb s (Set.Ico_subset_Icc_self hs)) | |
| end InvariantIsing | |
| end | |
| end OAI | |