openAI-math-dataset / lean /OAI /Probability /Perceptron /Interpolation /VectorReplicaDirectionBound.lean
| import Mathlib | |
| import OAI.Probability.Perceptron.Variational.IntegratedConvex | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped Topology BigOperators | |
| namespace SphericalPerceptronFreeEnergy | |
| lemma euclideanGaussian_replica_direction_bound {I : Type} [Fintype I] {S : Type*} [MeasurableSpace S] | |
| (μ : Measure S) [IsProbabilityMeasure μ] (r : ℕ) | |
| {H : S→ℝ} {V : S→EuclideanSpace ℝ I} {W : (Fin r→S)→EuclideanSpace ℝ I} | |
| (hH : Measurable H) (hV : Measurable V) (hW : Measurable W) | |
| {A C D K : ℝ} (hC : 0≤C) (hD : 0≤D) (hK : 0≤K) | |
| (hA : ∀ x, |H x|≤A) (hVC : ∀ x, ‖V x‖≤C) (hWD : ∀ x, ‖W x‖≤D) | |
| (hWK : ∀ x y, |inner ℝ (W x) (V y)|≤K) : | |
| |∫ g, gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r | |
| (fun x => inner ℝ (W x) g) ∂stdGaussian (EuclideanSpace ℝ I)|≤2*r*K := by | |
| let ν := Measure.pi (fun _ : Fin r => μ) | |
| let HH := replicaPotential H r | |
| let VV := fun x : Fin r→S => ∑ l, V (x l) | |
| have hHH : Measurable HH := replicaPotential_measurable hH r | |
| have hVV : Measurable VV := Finset.measurable_sum _ fun l _ => hV.comp (measurable_pi_apply l) | |
| have hHHA (x : Fin r→S) : |HH x|≤r*A := by | |
| apply (Finset.abs_sum_le_sum_abs _ _).trans | |
| calc | |
| _ ≤ ∑ _ : Fin r, A := Finset.sum_le_sum fun l _ => hA (x l) | |
| _ = _ := by simp | |
| have hVVC (x : Fin r→S) : ‖VV x‖^2≤(r*C)^2 := by | |
| have hb : ‖VV x‖≤r*C := by | |
| apply (norm_sum_le _ _).trans | |
| calc | |
| _ ≤ ∑ _ : Fin r, C := Finset.sum_le_sum fun l _ => hVC (x l) | |
| _ = _ := by simp | |
| exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (Nat.cast_nonneg r) hC)).2 hb | |
| have hWWD (x : Fin r→S) : ‖W x‖^2≤D^2 := | |
| (sq_le_sq₀ (norm_nonneg _) hD).2 (hWD x) | |
| have hc (x y : Fin r→S) : |inner ℝ (W x) (VV y)|≤r*K := by | |
| dsimp only [VV] | |
| rw [inner_sum] | |
| apply (Finset.abs_sum_le_sum_abs _ _).trans | |
| calc | |
| _ ≤ ∑ _ : Fin r, K := Finset.sum_le_sum fun l _ => hWK x (y l) | |
| _ = _ := by simp | |
| have he := euclideanGaussian_tiltMean_ibp ν hHH hVV hW hHHA hVVC hWWD | |
| have hham (g : EuclideanSpace ℝ I) : (fun x => HH x+inner ℝ (VV x) g)= | |
| replicaPotential (fun x => H x+inner ℝ (V x) g) r := by | |
| funext x | |
| simp only [HH,VV,replicaPotential,sum_inner,Finset.sum_add_distrib] | |
| have hl (g : EuclideanSpace ℝ I) : | |
| gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r (fun x => inner ℝ (W x) g)= | |
| tiltMean ν (fun x => HH x+inner ℝ (VV x) g) (fun x => inner ℝ (W x) g) 1 := by | |
| rw [hham] | |
| rfl | |
| simp_rw [hl] | |
| rw [he] | |
| apply abs_integral_le_integral_abs.trans | |
| calc | |
| _ ≤ ∫ _ : EuclideanSpace ℝ I, (2*r*K:ℝ) ∂stdGaussian (EuclideanSpace ℝ I) := by | |
| apply integral_mono_of_nonneg (ae_of_all _ fun _ => abs_nonneg _) (integrable_const _) | |
| apply ae_of_all | |
| intro g | |
| have hm : Measurable (fun x => HH x+inner ℝ (VV x) g) := hHH.add (hVV.inner measurable_const) | |
| have hm1 : Measurable (fun x => inner ℝ (W x) (VV x)) := hW.inner hVV | |
| have hm2 : Measurable (fun x : Fin 2→(Fin r→S) => inner ℝ (W (x 1)) (VV (x 0))) := | |
| (hW.comp (measurable_pi_apply 1)).inner (hVV.comp (measurable_pi_apply 0)) | |
| have hb1 := tiltMean_bound_general ν hm hm1 (t:=1) (by positivity : 0≤(r:ℝ)*K) (fun x => hc x x) | |
| have hb2 : |gibbsReplicaMean ν (fun x => HH x+inner ℝ (VV x) g) 2 | |
| (fun x => inner ℝ (W (x 1)) (VV (x 0)))|≤(r:ℝ)*K := | |
| tiltMean_bound_general (Measure.pi (fun _ : Fin 2 => ν)) | |
| (replicaPotential_measurable hm 2) hm2 (by positivity) | |
| (fun x => hc (x 1) (x 0)) | |
| exact (abs_sub _ _).trans (by linarith) | |
| _ = _ := by simp | |
| end SphericalPerceptronFreeEnergy | |
| end | |
| end OAI | |