| import Mathlib |
| import OAI.Probability.Perceptron.Cavity.CavityCompensator |
| import OAI.Probability.Perceptron.Cavity.CavityFiniteEvaluation |
| import OAI.Probability.Perceptron.Cavity.CavityTrueShellLower |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def cavityOldIncrement (n L M : ℕ) (f : Jet3) (a : Fin M→Fin (n+1)→ℝ) |
| (W : NormalizedSpin (n+1)→ℝ) (y : Fin M→Spin L) (p : CavityShellSpin n L) : ℝ := |
| cavityShellPatternEnergy n L M f.f (cavityJoinedPatterns (n+1) L M a y) p- |
| normalizedPatternEnergy (n+1) M f.f a p.1+W p.1 |
|
|
| lemma cavityOldIncrement_measurable (n L M : ℕ) (f : Jet3) (a : Fin M→Fin (n+1)→ℝ) |
| (W : NormalizedSpin (n+1)→ℝ) (hW : Measurable W) : |
| Measurable (fun p : CavityShellSpin n L×(Fin M→Spin L)=>cavityOldIncrement n L M f a W p.2 p.1) := by |
| have hj : Measurable (fun p : CavityShellSpin n L×(Fin M→Spin L)=> |
| cavityJoinedPatterns (n+1) L M a p.2) := by |
| unfold cavityJoinedPatterns |
| apply Measurable.of_eval |
| intro i |
| apply Measurable.of_eval |
| intro j |
| refine Fin.addCases (fun j=>?_) (fun j=>?_) j |
| · simpa only [Fin.addCases_left] using (measurable_const : Measurable (fun _ : CavityShellSpin n L×(Fin M→Spin L)=>a i j)) |
| · simp only [Fin.addCases_right] |
| fun_prop |
| have he := (normalizedPatternEnergy_continuous (n+1+L) M f.f).measurable.comp |
| (hj.prodMk ((cavityShellSpinEmbedding_measurable n L).comp measurable_fst)) |
| have hb : Measurable (fun p : CavityShellSpin n L×(Fin M→Spin L)=> |
| normalizedPatternEnergy (n+1) M f.f a p.1.1) := (normalizedPatternEnergy_continuous (n+1) M f.f).measurable.comp |
| (f:=fun p : CavityShellSpin n L×(Fin M→Spin L)=>(a,p.1.1)) |
| (measurable_const.prodMk measurable_fst.fst) |
| exact (he.sub hb).add (hW.comp measurable_fst.fst) |
|
|
| lemma cavityOldIncrement_bound (n L M : ℕ) (f : Jet3) (a : Fin M→Fin (n+1)→ℝ) |
| (W : NormalizedSpin (n+1)→ℝ) {E : ℝ} (hW : ∀ x,|W x|≤E) |
| (y : Fin M→Spin L) (p : CavityShellSpin n L) : |
| |cavityOldIncrement n L M f a W y p|≤2*M*‖f.f‖+E := by |
| apply (abs_add_le _ _).trans |
| apply (add_le_add ((abs_sub _ _).trans (add_le_add |
| (cavityShellPatternEnergy_bound n L M f.f _ p) |
| (normalizedPatternEnergy_bound (n+1) M f.f a p.1))) (hW p.1)).trans_eq |
| ring |
|
|
| theorem cavity_conditional_old_increment_lower (n k M : ℕ) (f : Jet3) |
| (h1 : HasCompactSupport (f.d1 : ℝ→ℝ)) (h2 : HasCompactSupport (f.d2 : ℝ→ℝ)) |
| (h3 : HasCompactSupport (f.d3 : ℝ→ℝ)) (hn : 2*((k+1)+1)≤n+1) |
| (hp : (cavitySphereLaw n (k+1) : Measure (Spin (k+1))) (cavityShell (k+1))≠0) |
| (μ : Measure (NormalizedSpin (n+1))) [IsProbabilityMeasure μ] |
| (a : Fin M→Fin (n+1)→ℝ) (W : NormalizedSpin (n+1)→ℝ) |
| (hW : Measurable W) {E : ℝ} (hE : 0≤E) (hWE : ∀ x,|W x|≤E) : |
| let K : ℝ:=M/(n+1:ℕ)*‖f.dilationMark h1‖ |
| let A : NormalizedSpin (n+1)→Fin M→ℝ:=fun x i=>f.d1 (∑ j,a i j*x.val j) |
| (∫ y : Fin M→Spin (k+1),Real.log (∫ p,Real.exp (cavityOldIncrement n (k+1) M f a W y p) |
| ∂μ.prod (cavityShellProbability n (k+1))) ∂Measure.pi (fun _=>stdGaussian (Spin (k+1))))≥ |
| (∫ y : Fin M→Spin (k+1),Real.log (∫ x,Real.exp (W x+(k+1:ℕ)*bulkC (n+1) M f a x/2)* |
| sphericalExp k (cavityVectorField M (k+1) (A x) (Real.sqrt (n+1:ℕ))⁻¹ y) |
| (Real.sqrt (k+1:ℕ)) ∂μ) ∂Measure.pi (fun _=>stdGaussian (Spin (k+1))))-K/2- |
| Real.exp (2*(E+((k+1:ℕ)+1)*K/2)+(Real.sqrt (n+1:ℕ))⁻¹^2*((k+1:ℕ)+1)^2*M*‖f.d1‖^2)* |
| Real.sqrt ((1/(2*(n+1:ℕ)))^2*(((k+1:ℕ)+1)^2)^2*squareGaussianVariance*M*‖f.d2‖^2)- |
| (∫ y : Fin M→Spin (k+1),∑ i,cavityRemainderMajorant |
| (cavityRemainderCoefficients f h1 h2 h3 (k+1) ((k+1)+1)) (n+1:ℕ) ‖y i‖ |
| ∂Measure.pi (fun _=>stdGaussian (Spin (k+1)))) := by |
| dsimp only |
| have := cavityShellProbability_probability n (k+1) hp |
| let A : NormalizedSpin (n+1)→Fin M→ℝ:=fun x i=>f.d1 (∑ j,a i j*x.val j) |
| let B : NormalizedSpin (n+1)→Fin M→ℝ:=fun x i=>f.d2 (∑ j,a i j*x.val j) |
| let C : NormalizedSpin (n+1)→ℝ:=fun x=>bulkC (n+1) M f a x |
| let K : ℝ:=M/(n+1:ℕ)*‖f.dilationMark h1‖ |
| have hK : 0≤K := by dsimp [K]; positivity |
| have hA : Measurable A := by have hm:=f.d1.continuous.measurable; unfold A; fun_prop |
| have hB : Measurable B := by have hm:=f.d2.continuous.measurable; unfold B; fun_prop |
| have hC : Measurable C := (bulkC_measurable n M f).comp |
| (f:=fun x : NormalizedSpin (n+1)=>((a,0),x)) (measurable_const.prodMk measurable_id) |
| have hAB (x) (i) : |A x i|≤‖f.d1‖ := f.d1.norm_coe_le_norm _ |
| have hBB (x) (i) : |B x i|≤‖f.d2‖ := f.d2.norm_coe_le_norm _ |
| have hCK (x) : |C x|≤K := bulkC_bound n M f h1 (a,0) x |
| let U : CavityShellSpin n (k+1)→ℝ:=fun p=>W p.1+‖p.2.val‖^2*C p.1/2 |
| let V : (Fin M→Spin (k+1))→CavityShellSpin n (k+1)→ℝ:=fun y p=>U p+ |
| cavityLinearField M (k+1) (A p.1) (Real.sqrt (n+1:ℕ))⁻¹ p.2.val y+ |
| cavityQuadraticField M (k+1) (B p.1) (1/(2*(n+1:ℕ))) p.2.val y |
| have hU : Measurable U := by |
| have hw : Measurable (fun p : CavityShellSpin n (k+1)=>W p.1) := hW.comp measurable_fst |
| have hc : Measurable (fun p : CavityShellSpin n (k+1)=>C p.1) := hC.comp measurable_fst |
| unfold U |
| fun_prop |
| have hUB (p) : |U p|≤E+((k+1:ℕ)+1)*K/2 := |
| cavity_shell_compensator_bound k _ _ _ _ hK (hCK p.1) (hWE p.1) p.2 |
| have hVi : Integrable (fun y=>Real.log (∫ p,Real.exp (V y p) |
| ∂μ.prod (cavityShellProbability n (k+1)))) (Measure.pi (fun _=>stdGaussian (Spin (k+1)))) := |
| cavity_quadratic_log_integrable (μ.prod (cavityShellProbability n (k+1))) M (k+1) |
| (fun p=>A p.1) (fun p=>B p.1) (fun p=>p.2.val) U |
| (hA.comp measurable_fst) (hB.comp measurable_fst) measurable_snd.subtype_val hU _ _ |
| (norm_nonneg _) (norm_nonneg _) (by positivity) (by positivity) |
| (fun p=>hAB p.1) (fun p=>hBB p.1) (fun p=>cavity_shell_norm_bound (k+1) p.2) hUB |
| have hFi := cavity_log_integrable_of_bound (μ.prod (cavityShellProbability n (k+1))) |
| (Measure.pi (fun _ : Fin M=>stdGaussian (Spin (k+1)))) |
| (fun p=>cavityOldIncrement n (k+1) M f a W p.2 p.1) |
| (cavityOldIncrement_measurable n (k+1) M f a W hW) |
| (fun _=>2*M*‖f.f‖+E) (integrable_const _) (fun _=>by positivity) |
| (fun p y=>cavityOldIncrement_bound n (k+1) M f a W hWE y p) |
| let R : (Fin M→Spin (k+1))→ℝ:=fun y=>∑ i,cavityRemainderMajorant |
| (cavityRemainderCoefficients f h1 h2 h3 (k+1) ((k+1)+1)) (n+1:ℕ) ‖y i‖ |
| have hRi : Integrable R (Measure.pi (fun _=>stdGaussian (Spin (k+1)))) := by |
| apply integrable_finsetSum |
| intro i _ |
| exact ((measurePreserving_eval (fun _ : Fin M=>stdGaussian (Spin (k+1))) i).integrable_comp |
| (cavityRemainderMajorant_integrable _ _ _).aestronglyMeasurable).mpr |
| (cavityRemainderMajorant_integrable _ _ _) |
| have hpoint (y : Fin M→Spin (k+1)) : Real.log (∫ p,Real.exp (V y p) |
| ∂μ.prod (cavityShellProbability n (k+1)))-R y≤ |
| Real.log (∫ p,Real.exp (cavityOldIncrement n (k+1) M f a W y p) |
| ∂μ.prod (cavityShellProbability n (k+1))) := by |
| have hv : Measurable (V y) := by |
| dsimp [V] |
| unfold cavityLinearField cavityQuadraticField |
| fun_prop |
| have herr (p : CavityShellSpin n (k+1)) : |
| |cavityOldIncrement n (k+1) M f a W y p-V y p|≤R y := by |
| have hh:=cavity_true_shell_expansion n (k+1) M f h1 h2 h3 hn a y p |
| convert hh using 1 <;> try dsimp [cavityOldIncrement,V,U,A,B,C,cavityShellTaylorModel] |
| · congr 1 |
| ring |
| · simp only [R,Nat.cast_add,Nat.cast_one] |
| have hh:=cavity_log_uniform_change (μ.prod (cavityShellProbability n (k+1))) |
| ((cavityOldIncrement_measurable n (k+1) M f a W hW).comp |
| (measurable_id.prodMk measurable_const)) hv (A:=2*M*‖f.f‖+E) (by positivity) |
| (cavityOldIncrement_bound n (k+1) M f a W hWE y) herr |
| simp only [Function.comp_def,id_eq] at hh |
| linarith [(abs_le.mp hh).1] |
| have hi := integral_mono (hVi.sub hRi) hFi hpoint |
| simp only [Pi.sub_apply] at hi |
| rw [integral_sub hVi hRi] at hi |
| have hs := cavity_expected_shell_lower μ n k M hp A B W C hA hB hW hC |
| (Real.sqrt (n+1:ℕ))⁻¹ (1/(2*(n+1:ℕ))) (norm_nonneg _) (norm_nonneg _) hE hK hAB hBB hWE hCK |
| exact (sub_le_sub_right hs _).trans hi |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
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|
