| import Mathlib |
| import OAI.Probability.Perceptron.Variational.GridGap |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def heightIndex {d : ℕ} (h : Fin (d+1) → ℝ) (t : ℝ) : Fin (d+2) := |
| ⟨(Finset.univ.filter (fun r => h r < t)).card,by |
| have H := Finset.card_le_card (Finset.filter_subset (fun r => h r < t) Finset.univ) |
| simp only [Finset.card_univ,Fintype.card_fin] at H |
| omega⟩ |
|
|
| lemma heightIndex_monotone {d : ℕ} (h : Fin (d+1) → ℝ) : Monotone (heightIndex h) := by |
| intro s t hst |
| change (Finset.univ.filter (fun r => h r < s)).card ≤ (Finset.univ.filter (fun r => h r < t)).card |
| apply Finset.card_le_card |
| intro r hr |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hr ⊢ |
| exact hr.trans_le hst |
|
|
| lemma heightIndex_le_iff {d : ℕ} (h : Fin (d+1) → ℝ) (hh : Monotone h) |
| (t : ℝ) (i : Fin (d+1)) : heightIndex h t ≤ i.castSucc ↔ t ≤ h i := by |
| constructor |
| · intro hi |
| by_contra ht |
| have hsub : Finset.Iic i ⊆ Finset.univ.filter (fun r => h r < t) := by |
| intro r hr |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and] |
| exact (hh (Finset.mem_Iic.mp hr)).trans_lt (lt_of_not_ge ht) |
| have H := Finset.card_le_card hsub |
| rw [Fin.card_Iic] at H |
| change (Finset.univ.filter (fun r => h r < t)).card ≤ i.val at hi |
| omega |
| · intro hi |
| have hsub : Finset.univ.filter (fun r => h r < t) ⊆ Finset.Iio i := by |
| intro r hr |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hr |
| apply Finset.mem_Iio.mpr |
| by_contra hir |
| exact (not_lt_of_ge (hi.trans (hh (le_of_not_gt hir)))) hr |
| have H := Finset.card_le_card hsub |
| rw [Fin.card_Iio] at H |
| exact H |
|
|
| lemma lt_heightIndex_iff {d : ℕ} (h : Fin (d+1) → ℝ) (hh : Monotone h) |
| (t : ℝ) (i : Fin (d+1)) : i.castSucc < heightIndex h t ↔ h i < t := by |
| simpa only [not_le] using not_congr (heightIndex_le_iff h hh t i) |
|
|
| def heightVariances {K : ℕ} (z : Fin (K+1) → ℝ) (j : Fin K) : ℝ := |
| 2*(z j.succ-z j.castSucc) |
|
|
| lemma heightVariances_nonneg {K : ℕ} (z : Fin (K+1) → ℝ) (hz : Monotone z) |
| (j : Fin K) : 0 ≤ heightVariances z j := by |
| exact mul_nonneg (by norm_num) (sub_nonneg.mpr (hz (Fin.castSucc_le_succ j))) |
|
|
| lemma heightVariances_sum {K : ℕ} (z : Fin (K+1) → ℝ) : |
| (∑ j, heightVariances z j)=2*(z (Fin.last K)-z 0) := by |
| simp only [heightVariances,← Finset.mul_sum,finite_sum_successive_differences] |
|
|
| lemma heightVariances_prefix {K : ℕ} (z : Fin (K+1) → ℝ) (hz : Monotone z) |
| (r : ℝ) (hr : r ∈ Set.range z) : |
| (∑ j, if z j.succ ≤ r then heightVariances z j else 0)=2*(r-z 0) := by |
| obtain ⟨a,rfl⟩ := hr |
| have he (j : Fin K) : (if z j.succ ≤ z a then heightVariances z j else 0)= |
| 2*(min (z a) (z j.succ)-min (z a) (z j.castSucc)) := by |
| by_cases hja : j.val < a.val |
| · have hs : j.succ ≤ a := by simp only [Fin.le_def,Fin.val_succ]; omega |
| have hc : j.castSucc ≤ a := (Fin.castSucc_le_succ j).trans hs |
| simp [ite_eq_left (hz hs),min_eq_right (hz hs),min_eq_right (hz hc),heightVariances] |
| · have hc : a ≤ j.castSucc := by simp only [Fin.le_def,Fin.val_castSucc]; omega |
| have hs : a ≤ j.succ := hc.trans (Fin.castSucc_le_succ j) |
| rw [min_eq_left (hz hc),min_eq_left (hz hs),sub_self,mul_zero] |
| split_ifs with ht |
| · have heq : z j.succ=z j.castSucc := le_antisymm (ht.trans (hz hc)) (hz (Fin.castSucc_le_succ j)) |
| simp [heightVariances,heq] |
| · rfl |
| simp only [he,← Finset.mul_sum] |
| rw [finite_sum_successive_differences K (fun i => min (z a) (z i))] |
| rw [min_eq_left (hz (Fin.le_last a)),min_eq_right (hz (Fin.zero_le a))] |
|
|
| lemma heightIndex_grid_prefix {d K : ℕ} (h : Fin (d+1) → ℝ) (hh : Monotone h) |
| (z : Fin (K+1) → ℝ) (hz : Monotone z) (hz0 : z 0=0) |
| (hm : ∀ i, h i ∈ Set.range z) (i : Fin (d+1)) : |
| varianceLevels (fun j => gridVariance |
| (List.ofFn fun r => (heightIndex h (z r.succ),heightVariances z r)) j.castSucc) i=h i := by |
| rw [varianceLevels_gridVariance] |
| simp only [List.map_ofFn,List.sum_ofFn,Function.comp_apply,heightIndex_le_iff h hh] |
| rw [heightVariances_prefix z hz (h i) (hm i),hz0,sub_zero] |
| ring |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
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|
