| import Mathlib |
| import OAI.Probability.Perceptron.Variational.NormSquareBand |
| import OAI.Probability.Perceptron.Variational.EntropicHeat |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def logSphericalExp (n : ℕ) (R : ℝ) (x : Spin (n+1)) : ℝ := |
| Real.log (sphericalExp n x R) |
|
|
| lemma sphericalExp_le_mul (n : ℕ) (R : ℝ) (x y : Spin (n+1)) : |
| sphericalExp n x R ≤ Real.exp (|R| *‖x-y‖)*sphericalExp n y R := by |
| unfold sphericalExp |
| rw [← integral_const_mul] |
| apply integral_mono (unitSphere_continuous_integrable (by fun_prop)) |
| (unitSphere_continuous_integrable (by fun_prop)) |
| intro u |
| dsimp only |
| rw [← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have hu : ‖(u:Spin (n+1))‖=1 := by simp |
| have hh := abs_real_inner_le_norm (x-y) (u:Spin (n+1)) |
| rw [hu,mul_one,inner_sub_left] at hh |
| have hm := mul_le_mul_of_nonneg_left hh (abs_nonneg R) |
| rw [← abs_mul] at hm |
| have hl := (le_abs_self (R*(inner ℝ x (u:Spin (n+1))-inner ℝ y (u:Spin (n+1))))).trans hm |
| linarith |
|
|
| lemma logSphericalExp_lipschitz (n : ℕ) (R : ℝ) : |
| LipschitzWith ⟨|R|,abs_nonneg R⟩ (logSphericalExp n R) := by |
| have hle (x y : Spin (n+1)) : logSphericalExp n R x-logSphericalExp n R y ≤ |R| *‖x-y‖ := by |
| have hx : 0 < sphericalExp n x R := lt_of_lt_of_le zero_lt_one (one_le_sphericalExp n x R) |
| have hy : 0 < sphericalExp n y R := lt_of_lt_of_le zero_lt_one (one_le_sphericalExp n y R) |
| have hh := Real.log_le_log hx (sphericalExp_le_mul n R x y) |
| rw [Real.log_mul (Real.exp_pos _).ne' hy.ne',Real.log_exp] at hh |
| change Real.log (sphericalExp n x R)-Real.log (sphericalExp n y R) ≤ _ |
| linarith |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| change |logSphericalExp n R x-logSphericalExp n R y| ≤ |R| *‖x-y‖ |
| apply abs_le.mpr |
| constructor |
| · have hh := hle y x |
| rw [norm_sub_rev] at hh |
| linarith |
| · exact hle x y |
|
|
| lemma logSphericalExp_nonneg (n : ℕ) (R : ℝ) (x : Spin (n+1)) : |
| 0 ≤ logSphericalExp n R x := Real.log_nonneg (one_le_sphericalExp n x R) |
|
|
| lemma logSphericalExp_zero (n : ℕ) (R : ℝ) : logSphericalExp n R 0=0 := by |
| simp [logSphericalExp,sphericalExp] |
|
|
| lemma integral_exp_inner_stdGaussian {E : Type*} [NormedAddCommGroup E] |
| [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (v : E) (a : ℝ) : (∫ x, Real.exp (a*inner ℝ v x) ∂stdGaussian E)= |
| Real.exp (a^2*‖v‖^2/2) := by |
| have H := mgf_gaussianReal |
| ⟨(innerSL ℝ v).continuous.measurable.aemeasurable, |
| IsGaussian.map_eq_gaussianReal (μ := stdGaussian E) (innerSL ℝ v)⟩ a |
| simp only [integral_strongDual_stdGaussian,variance_dual_stdGaussian,innerSL_apply_norm,Real.coe_toNNReal (‖v‖^2) (sq_nonneg ‖v‖),zero_mul,zero_add] at H |
| change (∫ x, Real.exp (a*inner ℝ v x) ∂stdGaussian E)=Real.exp (‖v‖^2*a^2/2) at H |
| convert H using 1 |
| congr 1 |
| ring |
|
|
| lemma sphericalExp_gaussian_average (n : ℕ) (R σ : ℝ) (x : Spin (n+1)) : |
| (∫ y, sphericalExp n (x+σ • y) R ∂stdGaussian (Spin (n+1)))= |
| Real.exp (R^2*σ^2/2)*sphericalExp n x R := by |
| let μ := stdGaussian (Spin (n+1)) |
| let ν := unitSphereLaw (n+1) |
| have hI : Integrable (fun p : Spin (n+1)×Metric.sphere (0:Spin (n+1)) 1 => |
| Real.exp (R*inner ℝ (x+σ • p.1) (p.2:Spin (n+1)))) (μ.prod ν) := by |
| apply (((gaussian_integrable_exp_norm μ (|R| *|σ|)).const_mul (Real.exp (|R| *‖x‖))).comp_fst ν).mono' (by fun_prop) |
| exact ae_of_all _ fun p => by |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have hu : ‖(p.2:Spin (n+1))‖=1 := by simp |
| have hh := abs_real_inner_le_norm (x+σ • p.1) (p.2:Spin (n+1)) |
| rw [hu,mul_one] at hh |
| have ht := norm_add_le x (σ • p.1) |
| rw [norm_smul,Real.norm_eq_abs] at ht |
| have hb := mul_le_mul_of_nonneg_left (hh.trans ht) (abs_nonneg R) |
| rw [← abs_mul] at hb |
| have hl := (le_abs_self (R*inner ℝ (x+σ • p.1) (p.2:Spin (n+1)))).trans hb |
| nlinarith |
| have H := integral_integral_swap (f := fun (y : Spin (n+1)) (u : Metric.sphere (0:Spin (n+1)) 1) => |
| Real.exp (R*inner ℝ (x+σ • y) (u:Spin (n+1)))) hI |
| change (∫ y, ∫ u, Real.exp (R*inner ℝ (x+σ • y) (u:Spin (n+1))) ∂ν ∂μ)=_ at H |
| change (∫ y, ∫ u, Real.exp (R*inner ℝ (x+σ • y) (u:Spin (n+1))) ∂ν ∂μ)=_ |
| rw [H] |
| have he (u : Metric.sphere (0:Spin (n+1)) 1) : |
| (∫ y : Spin (n+1), Real.exp (R*inner ℝ (x+σ • y) (u:Spin (n+1))) ∂μ)= |
| Real.exp (R^2*σ^2/2)*Real.exp (R*inner ℝ x (u:Spin (n+1))) := by |
| simp only [inner_add_left,real_inner_smul_left,mul_add,Real.exp_add] |
| rw [integral_const_mul] |
| have hh := integral_exp_inner_stdGaussian (u:Spin (n+1)) (R*σ) |
| have hu : ‖(u:Spin (n+1))‖=1 := by simp |
| rw [hu] at hh |
| simp only [one_pow,mul_one] at hh |
| have hinner : (fun y : Spin (n+1) => Real.exp (R*(σ*inner ℝ y (u:Spin (n+1)))))= |
| (fun y => Real.exp ((R*σ)*inner ℝ (u:Spin (n+1)) y)) := by |
| funext y |
| rw [real_inner_comm y (u:Spin (n+1))] |
| congr 1 |
| ring |
| rw [hinner,hh,mul_pow] |
| ring |
| simp_rw [he] |
| rw [integral_const_mul] |
| rfl |
| |
| lemma gaussianEntropic_logSphericalExp_one (n : ℕ) (R σ : ℝ) (x : Spin (n+1)) : |
| gaussianEntropic (stdGaussian (Spin (n+1))) 1 σ (logSphericalExp n R) x= |
| logSphericalExp n R x+R^2*σ^2/2 := by |
| rw [gaussianEntropic_pos _ (logSphericalExp_lipschitz n R) (by norm_num)] |
| simp only [one_mul,div_one,logSphericalExp] |
| simp_rw [Real.exp_log (lt_of_lt_of_le zero_lt_one (one_le_sphericalExp n _ R))] |
| rw [sphericalExp_gaussian_average,Real.log_mul (Real.exp_pos _).ne' |
| (lt_of_lt_of_le zero_lt_one (one_le_sphericalExp n x R)).ne',Real.log_exp] |
| ring |
| |
| end SphericalPerceptronFreeEnergy |
| end |
| |
| end OAI |
| |
