| import Mathlib |
| import OAI.Probability.Perceptron.Cascade.PoissonCountMoments |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma summable_nat_mul_pow_div_factorial (r : ℝ) : |
| Summable (fun n : ℕ => (n:ℝ)*r^n/(n.factorial:ℝ)) := by |
| apply (summable_nat_add_iff 1).mp |
| have hs := (Real.summable_pow_div_factorial r).mul_left r |
| apply hs.congr |
| intro n |
| simp only [Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one,pow_succ] |
| field_simp |
|
|
| lemma summable_linear_pow_div_factorial (A C r : ℝ) : |
| Summable (fun n : ℕ => (A+C*(n:ℝ))*r^n/(n.factorial:ℝ)) := by |
| have h := ((Real.summable_pow_div_factorial r).mul_left A).add |
| ((summable_nat_mul_pow_div_factorial r).mul_left C) |
| apply h.congr |
| intro n |
| ring |
|
|
| lemma bounded_increment_linear_bound {F : ℕ→ℝ} {C : ℝ} (hC : 0≤C) |
| (hF : ∀ n, |F (n+1)-F n|≤C) (n : ℕ) : |F n|≤|F 0|+C*(n:ℝ) := by |
| have h := nat_bounded_increment_difference hC hF n 0 |
| simp only [Nat.cast_zero,sub_zero,Nat.abs_cast] at h |
| calc |F n| ≤ |F n-F 0|+|F 0| := by simpa using abs_sub_le (F n) (F 0) 0 |
| _ ≤ |F 0|+C*(n:ℝ) := by linarith |
|
|
| lemma bounded_increment_factorial_summable {F : ℕ→ℝ} {C : ℝ} (hC : 0≤C) |
| (hF : ∀ n, |F (n+1)-F n|≤C) (t : ℝ) : |
| Summable (fun n => t^n/(n.factorial:ℝ)*F n) := by |
| refine Summable.of_norm_bounded (summable_linear_pow_div_factorial |F 0| C |t|) fun n => ?_ |
| simp only [Real.norm_eq_abs,abs_mul,abs_div,abs_pow,Nat.abs_cast] |
| simpa only [div_eq_mul_inv,mul_comm,mul_left_comm,mul_assoc] using |
| mul_le_mul_of_nonneg_left (bounded_increment_linear_bound hC hF n) |
| (show 0≤|t|^n/(n.factorial:ℝ) by positivity) |
|
|
| lemma hasDerivAt_factorial_series {F : ℕ→ℝ} {C : ℝ} (hC : 0≤C) |
| (hF : ∀ n, |F (n+1)-F n|≤C) (t : ℝ) : |
| HasDerivAt (fun s => ∑' n, s^n/(n.factorial:ℝ)*F n) |
| (∑' n, t^n/(n.factorial:ℝ)*F (n+1)) t := by |
| let R := |t|+1 |
| let g := fun n : ℕ => fun s : ℝ => s^(n+1)/((n+1).factorial:ℝ)*F (n+1) |
| let g' := fun n : ℕ => fun s : ℝ => s^n/(n.factorial:ℝ)*F (n+1) |
| let b := fun n : ℕ => (|F 0|+C+C*(n:ℝ))*R^n/(n.factorial:ℝ) |
| have hb : Summable b := summable_linear_pow_div_factorial (|F 0|+C) C R |
| have hg : ∀ n s, s∈Ioo (-R) R → HasDerivAt (g n) (g' n s) s := by |
| intro n s _ |
| unfold g g' |
| apply HasDerivAt.congr_deriv ((((hasDerivAt_id s).pow (n+1)).div_const ((n+1).factorial:ℝ)).mul_const (F (n+1))) |
| simp only [Nat.add_sub_cancel,Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one,mul_one] |
| field_simp |
| simp [id_eq,mul_comm] |
| have hbound : ∀ n s, s∈Ioo (-R) R → ‖g' n s‖≤b n := by |
| intro n s hs |
| have ha : |s|≤R := (abs_le.mpr ⟨hs.1.le,hs.2.le⟩) |
| have hf := bounded_increment_linear_bound hC hF (n+1) |
| simp only [Nat.cast_add,Nat.cast_one] at hf |
| simp only [g',b,Real.norm_eq_abs,abs_mul,abs_div,abs_pow,Nat.abs_cast] |
| calc _ ≤ (R^n/(n.factorial:ℝ))*(|F 0|+C+C*(n:ℝ)) := by |
| apply mul_le_mul |
| · exact div_le_div_of_nonneg_right (pow_le_pow_left₀ (abs_nonneg _) ha _) (by positivity) |
| · linarith |
| · positivity |
| · positivity |
| _ = _ := by ring |
| have hz : (0:ℝ)∈Ioo (-R) R := by dsimp [R]; constructor <;> linarith [abs_nonneg t] |
| have ht : t∈Ioo (-R) R := by dsimp [R]; constructor <;> linarith [le_abs_self t,neg_abs_le t] |
| have hzero : Summable (fun n => g n 0) := by simp [g] |
| have hd := hasDerivAt_tsum_of_isPreconnected hb isOpen_Ioo (convex_Ioo (-R) R).isPreconnected |
| hg hbound hz hzero ht |
| have he : ∀ s : ℝ, (F 0+(∑' n, g n s))=(∑' n, s^n/(n.factorial:ℝ)*F n) := by |
| intro s |
| rw [tsum_eq_zero_add' ((summable_nat_add_iff 1).mpr (bounded_increment_factorial_summable hC hF s))] |
| simp [g] |
| have hh := hd.const_add (F 0) |
| simp_rw [he] at hh |
| exact hh |
|
|
| def poissonRealMean (F : ℕ→ℝ) (t : ℝ) : ℝ := |
| Real.exp (-t)*(∑' n, t^n/(n.factorial:ℝ)*F n) |
|
|
| lemma poissonRealMean_eq_integral {F : ℕ→ℝ} {C : ℝ} (hC : 0≤C) |
| (hF : ∀ n, |F (n+1)-F n|≤C) (t : ℝ≥0) : |
| poissonRealMean F t = ∫ n, F n ∂poissonMeasure t := by |
| rw [integral_countable ((poisson_bounded_increment_memLp t hC hF).integrable (by norm_num))] |
| simp only [poissonMeasure_real_singleton,smul_eq_mul,poissonRealMean] |
| rw [← tsum_mul_left] |
| congr 1 |
| ext n |
| ring |
|
|
| lemma hasDerivAt_poissonRealMean {F : ℕ→ℝ} {C : ℝ} (hC : 0≤C) |
| (hF : ∀ n, |F (n+1)-F n|≤C) (t : ℝ) : |
| HasDerivAt (poissonRealMean F) |
| (poissonRealMean (fun n => F (n+1)-F n) t) t := by |
| have hs := bounded_increment_factorial_summable hC hF t |
| have hs' := bounded_increment_factorial_summable (F := fun n => F (n+1)) hC (fun n => hF (n+1)) t |
| have hd := (((hasDerivAt_id t).neg).exp).mul (hasDerivAt_factorial_series hC hF t) |
| apply hd.congr_deriv |
| simp only [poissonRealMean,mul_sub,hs'.tsum_sub hs,Pi.neg_apply,id_eq] |
| ring |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
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| end OAI |
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