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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import Mathlib
import OAI.Probability.LogConcave.Sampling.SpatialTensorArrayBound
import OAI.Probability.LogConcave.JetEstimates.MatrixArray
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.TensorEnergy
open scoped Classical BigOperators RealInnerProductSpace
def arrayRelabel {d : ℕ} {S T : Type} [Fintype S] [DecidableEq S]
[Fintype T] [DecidableEq T] (e : S ≃ T) :
((S → Fin d) → ℝ) →L[ℝ] ((T → Fin d) → ℝ) :=
ContinuousLinearMap.pi (fun c => ContinuousLinearMap.proj (c ∘ e))
lemma arrayTensor_relabel {d n : ℕ} {S T : Type} [Fintype S] [DecidableEq S]
[Fintype T] [DecidableEq T] (e : S ≃ T)
{f : Point d → (S → Fin d) → ℝ} (hf : ContDiff ℝ (⊤:ℕ∞) f) (y : Point d)
(c : T ⊕ Fin n → Fin d) :
arrayTensor (iteratedFDeriv ℝ n ((arrayRelabel e) ∘ f) y) c=
arrayTensor (iteratedFDeriv ℝ n f y) (c ∘ Equiv.sumCongr e (Equiv.refl (Fin n))) := by
rw [(arrayRelabel e).iteratedFDeriv_comp_left hf.contDiffAt
(by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))]
rfl
lemma AllSplitBound.array_relabel {d n : ℕ} {S T : Type} [Fintype S] [DecidableEq S]
[Fintype T] [DecidableEq T] (e : S ≃ T)
{f : Point d → (S → Fin d) → ℝ} (hf : ContDiff ℝ (⊤:ℕ∞) f) (y : Point d) {B : ℝ}
(h : AllSplitBound (arrayTensor (iteratedFDeriv ℝ n f y)) B) :
AllSplitBound (arrayTensor (iteratedFDeriv ℝ n ((arrayRelabel e) ∘ f) y)) B := by
have hh := h.reindex (Equiv.sumCongr e (Equiv.refl (Fin n)))
convert hh using 1
ext c
exact arrayTensor_relabel e hf y c
lemma matrixArray_comp {d : ℕ} (A B : Point d →L[ℝ] Point d) :
matrixArray d (A.comp B)=singleContract (matrixArray d A)
(arrayRelabel (Equiv.sumComm Unit Unit) (matrixArray d B)) := by
funext c
let v := B (EuclideanSpace.basisFun (Fin d) ℝ (c (Sum.inr ())))
have he := (EuclideanSpace.basisFun (Fin d) ℝ).sum_repr v
have hh := congrArg (fun z => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ (c (Sum.inl ()))) (A z)) he
simpa only [map_sum,map_smul,inner_sum,real_inner_smul_right,
OrthonormalBasis.repr_apply_apply,real_inner_comm,singleContract,
matrixArray_apply,arrayRelabel,ContinuousLinearMap.pi_apply,
ContinuousLinearMap.proj_apply,Function.comp_apply,Equiv.sumComm_apply,
Sum.swap_inl,Sum.swap_inr,Sum.elim_inl,Sum.elim_inr,mul_comm,v,ContinuousLinearMap.comp_apply] using hh.symm
lemma matrixComposition_split {d n : ℕ} {f g : Point d → Point d →L[ℝ] Point d}
(hf : ContDiff ℝ (⊤:ℕ∞) f) (hg : ContDiff ℝ (⊤:ℕ∞) g)
(A B : ℕ → ℝ) (hA : ∀k,0≤A k) (hB : ∀k,0≤B k) (y : Point d)
(ha : ∀k≤n,AllSplitBound (arrayTensor (iteratedFDeriv ℝ k ((matrixArray d) ∘ f) y)) (A k))
(hb : ∀k≤n,AllSplitBound (arrayTensor (iteratedFDeriv ℝ k ((matrixArray d) ∘ g) y)) (B k)) :
AllSplitBound (arrayTensor (iteratedFDeriv ℝ n (fun z => matrixArray d ((f z).comp (g z))) y))
(∑s∈(Finset.univ : Finset (Fin n)).powerset,A s.card*B (n-s.card)) := by
have hfg := (matrixArray d).contDiff.comp hf
have hgg := (matrixArray d).contDiff.comp hg
have hh := LogConcaveSampling.arrayBound_spatialContract hfg
((arrayRelabel (Equiv.sumComm Unit Unit)).contDiff.comp hgg) A B hA hB y ha
(fun k hk => AllSplitBound.array_relabel (Equiv.sumComm Unit Unit) hgg y (hb k hk))
have he : (fun z => matrixArray d ((f z).comp (g z)))=
fun z => singleContract (matrixArray d (f z))
(arrayRelabel (Equiv.sumComm Unit Unit) (matrixArray d (g z))) :=
funext (fun z => matrixArray_comp (f z) (g z))
rw [he]
exact hh
end LogConcaveSampling.TensorEnergy
end
end
end
end OAI
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