| import OAI.Probability.InvariantIsing.Haar.PublishedInputs |
| import OAI.Probability.IsingPerceptron.EnrichedFieldCap |
| import OAI.Probability.IsingPerceptron.NormalizedRestrictionPi |
|
|
| namespace OAI |
|
|
| /-! Finite Gaussian cutoff partition functions in the Euclidean norm. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped NNReal InnerProductSpace |
|
|
| namespace InvariantIsing |
|
|
| / |
| weights and deterministic base energy. In the application the finite state |
| space is a positive-mass restriction of the spin/cascade-leaf prior. -/ |
| def finiteGaussianLogPartition {X : Type*} [MeasurableSpace X] |
| (ν : Measure X) (H : X → ℝ) {d : ℕ} |
| (A : X → EuclideanSpace ℝ (Fin d)) (g : EuclideanSpace ℝ (Fin d)) : ℝ := |
| Real.log (∫ x, Real.exp (H x + ⟪A x, g⟫_ℝ) ∂ν) |
|
|
| theorem finiteGaussianLogPartition_lipschitz {X : Type*} [Fintype X] |
| [MeasurableSpace X] [MeasurableSingletonClass X] |
| (ν : Measure X) [IsProbabilityMeasure ν] (H : X → ℝ) {d : ℕ} |
| (A : X → EuclideanSpace ℝ (Fin d)) (L : ℝ≥0) (hA : ∀ x, ‖A x‖ ≤ L) : |
| LipschitzWith L (finiteGaussianLogPartition ν H A) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro g h |
| have hb (x : X) : |(H x + ⟪A x, g⟫_ℝ) - (H x + ⟪A x, h⟫_ℝ)| ≤ |
| (L : ℝ) * ‖g - h‖ := by |
| rw [add_sub_add_left_eq_sub, ← inner_sub_right] |
| exact (abs_real_inner_le_norm _ _).trans |
| (mul_le_mul_of_nonneg_right (hA x) (norm_nonneg _)) |
| have hcomp := logMean_abs_sub_le ν (by norm_num : (0 : ℝ) < 1) |
| (show Integrable (fun x => Real.exp (1 * (H x + ⟪A x, g⟫_ℝ))) ν from Integrable.of_finite) |
| (show Integrable (fun x => Real.exp (1 * (H x + ⟪A x, h⟫_ℝ))) ν from Integrable.of_finite) hb |
| simpa only [Real.dist_eq, dist_eq_norm, Real.norm_eq_abs, logMean, one_mul, div_one, |
| finiteGaussianLogPartition] using hcomp |
|
|
| / |
| log partition variance by the coefficient squared norm, |
| uniformly in the cutoff and base energy. -/ |
| theorem finiteGaussianLogPartition_variance_le (hpub : GaussianLipschitzVarianceInput) |
| {X : Type*} [Fintype X] [MeasurableSpace X] [MeasurableSingletonClass X] |
| (ν : Measure X) [IsProbabilityMeasure ν] (H : X → ℝ) {d : ℕ} |
| (A : X → EuclideanSpace ℝ (Fin d)) (B : ℝ≥0) |
| (hA : ∀ x, ‖A x‖ ^ 2 ≤ B) : |
| variance (finiteGaussianLogPartition ν H A) |
| (stdGaussian (EuclideanSpace ℝ (Fin d))) ≤ B := by |
| have hs : ((NNReal.sqrt B : ℝ≥0) : ℝ) ^ 2 = B := by exact_mod_cast NNReal.sq_sqrt B |
| have hnorm (x : X) : ‖A x‖ ≤ ((NNReal.sqrt B : ℝ≥0) : ℝ) := by |
| have hn := NNReal.coe_nonneg (NNReal.sqrt B) |
| nlinarith [hA x, norm_nonneg (A x)] |
| exact (hpub d (NNReal.sqrt B) _ |
| (finiteGaussianLogPartition_lipschitz ν H A (NNReal.sqrt B) hnorm)).trans_eq hs |
|
|
| / |
| finite prefix, whose Gaussian law is the standard Euclidean Gaussian. -/ |
| theorem finite_cylinder_cgf_variance_le (hpub : GaussianLipschitzVarianceInput) |
| {X : Type*} [Fintype X] [MeasurableSpace X] [MeasurableSingletonClass X] |
| (ν : Measure X) [IsProbabilityMeasure ν] (A : X → ℕ →₀ ℝ) |
| (B : ℝ) (hB : 0 ≤ B) (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ) : |
| variance (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g) ν t) |
| gaussianCoordinates ≤ t ^ 2 * B := by |
| classical |
| obtain ⟨n, hn⟩ := finite_family_support_bound A |
| have hs (x : X) : ∀ j ∈ (A x).support, j < n + 1 := |
| fun j hj => hn x j (Finsupp.mem_support_iff.mp hj) |
| let V : X → EuclideanSpace ℝ (Fin (n + 1)) := |
| fun x => WithLp.toLp 2 (fun j : Fin (n + 1) => t * A x j) |
| let B' : ℝ≥0 := ⟨t ^ 2 * B, mul_nonneg (sq_nonneg t) hB⟩ |
| have hv (x : X) : ‖V x‖ ^ 2 ≤ B' := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp only [V, mul_pow, ← Finset.mul_sum, |
| ← cylinder_variance_prefix (A x) (hs x)] |
| exact mul_le_mul_of_nonneg_left (hA x) (sq_nonneg t) |
| let F := finiteGaussianLogPartition ν (fun _ => 0) V |
| have hF : Measurable F := by |
| have hsqrt : (((NNReal.sqrt B' : ℝ≥0) : ℝ)) ^ 2 = B' := |
| by exact_mod_cast NNReal.sq_sqrt B' |
| have hnorm (x : X) : ‖V x‖ ≤ ((NNReal.sqrt B' : ℝ≥0) : ℝ) := by |
| have hn' := NNReal.coe_nonneg (NNReal.sqrt B') |
| nlinarith [hv x, norm_nonneg (V x)] |
| exact (finiteGaussianLogPartition_lipschitz ν (fun _ => 0) V (NNReal.sqrt B') hnorm).continuous.measurable |
| have hto : MeasurePreserving |
| (WithLp.toLp 2 : (Fin (n + 1) → ℝ) → EuclideanSpace ℝ (Fin (n + 1))) |
| (Measure.pi (fun _ : Fin (n + 1) => gaussianReal 0 1)) |
| (stdGaussian (EuclideanSpace ℝ (Fin (n + 1)))) := |
| ⟨by fun_prop, map_pi_eq_stdGaussian⟩ |
| have hmp := hto.comp (gaussian_prefix_measurePreserving (n + 1)) |
| have he : (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g) ν t) = |
| F ∘ ((WithLp.toLp 2) ∘ (fun g : ℕ → ℝ => fun j : Fin (n + 1) => g j)) := by |
| funext g |
| have hi (x : X) : ⟪V x, WithLp.toLp 2 (fun j : Fin (n + 1) => g j)⟫_ℝ = |
| t * cylinderField (A x) g := by |
| rw [cylinderField_eq_prefix (A x) (hs x)] |
| simp only [V, PiLp.inner_apply, RCLike.inner_apply', conj_trivial] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| simp only [Function.comp_apply, F, finiteGaussianLogPartition, cgf, mgf, zero_add, hi] |
| rw [he] |
| change variance (fun g => F ((WithLp.toLp 2 ∘ |
| (fun g : ℕ → ℝ => fun j : Fin (n + 1) => g j)) g)) gaussianCoordinates ≤ _ |
| rw [hmp.variance_fun_comp hF.aemeasurable] |
| exact finiteGaussianLogPartition_variance_le hpub ν (fun _ => 0) V B' hv |
| |
| end InvariantIsing |
| |
| end |
| |
| end OAI |
| |
