| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.AnalyticBase |
| import OAI.Probability.LogConcave.JetEstimates.Squared |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory |
| open scoped Classical BigOperators NNReal |
| open TensorEnergy |
|
|
| lemma finset_sum_le_sum_of_eq {α : Type} (s t : Finset α) (f g : α → ℝ) |
| (he : s=t) (h : ∀a∈s,f a ≤ g a) : (∑a∈s,f a) ≤ ∑a∈t,g a := by |
| subst t |
| exact Finset.sum_le_sum h |
|
|
| lemma adjointCommutator_squared {S : Type} [Fintype S] {d j : ℕ} |
| {H : Point d → ℝ} (hH : ContDiff ℝ (⊤:ℕ∞) H) |
| (F : (S ⊕ Unit → Fin d) → Point d → ℝ) (hF : ∀c,ContDiff ℝ (⊤:ℕ∞) (F c)) |
| (y : Point d) (M : ℕ → ℝ) |
| (hM : ∀e,0<e → e ≤ j → AllSplitBound (spatialTensor (scoreField H) (List.finRange e) y) (M e)) : |
| squared (adjointCommutator H F (List.finRange j) y) ≤ |
| 2^j*∑s∈(Finset.univ : Finset (Fin j)).powerset.erase ∅, |
| (M s.card)^2*spatialSquared F (j-s.card) y := by |
| let : DecidableEq (Fin j) := Classical.decEq _ |
| have heq : (List.finRange j).toFinset=(Finset.univ : Finset (Fin j)) := by ext k; simp |
| unfold adjointCommutator |
| rw [heq] |
| apply (squared_sum _ _).trans |
| have hc : (((Finset.univ : Finset (Fin j)).powerset.erase ∅).card : ℝ) ≤ 2^j := by |
| exact_mod_cast (Finset.card_erase_le.trans_eq (by simp : |
| (Finset.univ : Finset (Fin j)).powerset.card=2^j)) |
| apply (mul_le_mul_of_nonneg_right hc (Finset.sum_nonneg (fun s _ => squared_nonneg _))).trans |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply finset_sum_le_sum_of_eq |
| · ext s; simp |
| intro s hs |
| have hs0 : s.Nonempty := Finset.nonempty_iff_ne_empty.mpr (Finset.mem_erase.mp hs).1 |
| let : Nonempty {k // k∈s} := ⟨⟨hs0.choose,hs0.choose_spec⟩⟩ |
| have hh := scoreField_arbitrary_allSplit hH (selectList (fun k => k∈s) (List.finRange j)) |
| (selectList_nodup _ (List.nodup_finRange _)) (selectList_full _ (by simp)) y (M s.card) |
| (by |
| have hc : Fintype.card {k : Fin j // k∈s}=s.card := Fintype.card_of_subtype s (fun _ => Iff.rfl) |
| rw [hc] |
| exact hM s.card hs0.card_pos ((Finset.card_le_univ s).trans_eq (Fintype.card_fin j))) |
| rw [squared_reindex] |
| have hb := spatialContractTerm_squared (fun k => k∈s) (scoreField H) F hF _ (List.nodup_finRange _) (by simp) y hh |
| have hc : Fintype.card {k : Fin j // k∉s}=j-s.card := by |
| rw [Fintype.card_subtype_compl] |
| have hs' : Fintype.card {k : Fin j // k∈s}=s.card := Fintype.card_of_subtype s (fun _ => Iff.rfl) |
| rw [hs',Fintype.card_fin] |
| simp only [Fintype.card_eq_nat_card] at hb hc |
| rw [hc] at hb |
| exact hb |
|
|
| lemma tensorAdjoint_polySmooth {S : Type} {d : ℕ} {H : Point d → ℝ} |
| (hH : PolySmooth H) (F : (S ⊕ Unit → Fin d) → Point d → ℝ) (hF : ∀c,PolySmooth (F c)) |
| (a : S → Fin d) : PolySmooth |
| (tensorAdjoint H (EuclideanSpace.basisFun (Fin d) ℝ) (fun z => F (Sum.elim a (fun _ => z)))) := |
| hH.tensorAdjoint (fun _ => hF _) _ |
|
|
| lemma spatialEnergy_adjoint {d j : ℕ} {S : Type} [Fintype S] |
| {H : Point d → ℝ} (hH : PolySmooth H) (ht : HasGaussianLowerTail H) |
| {K : ℝ≥0} (hL : LipschitzWith K (gradient H)) |
| (F : (S ⊕ Unit → Fin d) → Point d → ℝ) (hF : ∀c,PolySmooth (F c)) |
| (M : ℕ → ℝ) |
| (hM : ∀e,0<e → e ≤ j → ∀y, AllSplitBound (spatialTensor (scoreField H) (List.finRange e) y) (M e)) : |
| spatialEnergy (fun a => tensorAdjoint H (EuclideanSpace.basisFun (Fin d) ℝ) |
| (fun z => F (Sum.elim a (fun _ => z)))) j (gibbs H) ≤ |
| 2*(spatialEnergy F (j+1) (gibbs H)+(K:ℝ)*spatialEnergy F j (gibbs H)+ |
| 2^j*∑s∈(Finset.univ : Finset (Fin j)).powerset.erase ∅, |
| (M s.card)^2*spatialEnergy F (j-s.card) (gibbs H)) := by |
| have hlead : Integrable (fun y => squared (spatialAdjointLeading H F (List.finRange j) y)) (gibbs H) := |
| integrable_finsetSum _ (fun c _ => |
| (hH.tensorAdjoint (fun z => spatialInside_polySmooth F hF _ c z) _).sq.integrable hH.smooth.continuous ht) |
| have hf k := (spatialSquared_polySmooth F hF k).integrable hH.smooth.continuous ht |
| have hsum := (integrable_finsetSum ((Finset.univ : Finset (Fin j)).powerset.erase ∅) |
| (fun s _ => (hf (j-s.card)).const_mul ((M s.card)^2))).const_mul ((2:ℝ)^j) |
| have hbase := (spatialSquared_polySmooth _ (tensorAdjoint_polySmooth hH F hF) j).integrable hH.smooth.continuous ht |
| have hp (y : Point d) : spatialSquared (fun a => tensorAdjoint H (EuclideanSpace.basisFun (Fin d) ℝ) |
| (fun z => F (Sum.elim a (fun _ => z)))) j y ≤ |
| 2*(squared (spatialAdjointLeading H F (List.finRange j) y)+ |
| 2^j * ∑s∈(Finset.univ : Finset (Fin j)).powerset.erase ∅, |
| (M s.card)^2*spatialSquared F (j-s.card) y) := by |
| unfold spatialSquared |
| rw [spatialTensor_adjoint_sum hH F hF _ (List.nodup_finRange _) y] |
| apply (squared_add _ _).trans |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| apply add_le_add le_rfl |
| exact adjointCommutator_squared hH.smooth F (fun c => (hF c).smooth) y M (fun e he hj => hM e he hj y) |
| have hh := integral_mono hbase ((hlead.add hsum).const_mul 2) hp |
| simp only [Pi.add_apply] at hh |
| rw [integral_const_mul,integral_add hlead hsum,integral_const_mul, |
| integral_finsetSum _ (fun s _ => (hf (j-s.card)).const_mul ((M s.card)^2))] at hh |
| simp only [integral_const_mul] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_left (add_le_add |
| (spatialAdjointLeading_energy hH ht hL F hF) le_rfl) (by norm_num)) |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open TensorEnergy |
| open scoped Classical |
|
|
| lemma spatialTensor_finRange_of_arrayBound {d n : ℕ} {S : Type} [Fintype S] [DecidableEq S] |
| {g : Point d → (S → Fin d) → ℝ} (hg : ContDiff ℝ (⊤:ℕ∞) g) |
| (y : Point d) {B : ℝ} |
| (hb : AllSplitBound (arrayTensor (iteratedFDeriv ℝ n g y)) B) : |
| AllSplitBound (spatialTensor (fun c z => g z c) (List.finRange n) y) B := by |
| rw [spatialTensor_eq_arrayTensor hg y] |
| exact hb.reindex (Equiv.sumComm S (Fin n)) |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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