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67aee1d | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | import OAI.Analysis.FreeEntropy.GUEBounds
namespace OAI
universe u1
noncomputable section
open MeasureTheory ProbabilityTheory Finset
open scoped Classical
namespace FiniteEntropySeparation
variable {α : Type u1} {n d : ℕ}
theorem basis_trace_contract (P Q : Matrix (Fin d) (Fin d) ℂ) :
(∑ j, ∑ k, Matrix.trace (hermitianBasis j k*P*hermitianBasis j k*Q)) = Matrix.trace P*Matrix.trace Q := by
simp_rw [← Matrix.trace_sum, ← sum_mul]
rw [hermitianBasis_contract, smul_mul_assoc, one_mul, Matrix.trace_smul, smul_eq_mul]
theorem fixedTraceWord_polynomial (A : α → Matrix (Fin d) (Fin d) ℂ)
(B : Matrix (Fin d) (Fin d) ℂ) (w : List (α ⊕ Fin n)) :
IsComplexPolynomial (fun x : MatrixIndex n d → ℝ => Matrix.trace (B*matrixWord (jointMatrix A x) w)) :=
((IsMatrixPolynomial.const B).mul (IsMatrixPolynomial.word (jointMatrix_polynomial A) w)).trace
theorem fixedTraceWord_derivative (A : α → Matrix (Fin d) (Fin d) ℂ)
(B : Matrix (Fin d) (Fin d) ℂ) (w : List (α ⊕ Fin n)) (q : MatrixIndex n d) (x : MatrixIndex n d → ℝ) :
complexCoordinateDerivative (fun y => Matrix.trace (B*matrixWord (jointMatrix A y) w)) q x =
(Real.sqrt d:ℂ)⁻¹ * ∑ r : Fin w.length, if w[r.val]=Sum.inr q.1 then
Matrix.trace (B*matrixWord (jointMatrix A x) (w.take r.val)*hermitianBasis q.2.1 q.2.2*matrixWord (jointMatrix A x) (w.drop (r.val+1))) else 0 := by
have hm := hasDerivAt_matrix_mul (fun j k => hasDerivAt_const (x q) (B j k)) (hasDerivAt_matrixWord (jointMatrix_hasDeriv A x q) w)
have ht := HasDerivAt.sum (u := univ) (fun j _ => hm j j)
have htrace : HasDerivAt (fun y => Matrix.trace (B*matrixWord (jointMatrix A (Function.update x q y)) w))
(Matrix.trace (B*matrixWordVariation (jointMatrix A (Function.update x q (x q))) (noiseDirection q) w)) (x q) := by
simpa only [Finset.sum_fn, Matrix.trace, Matrix.diag, show (fun _ _ : Fin d => (0:ℂ)) = (0:Matrix (Fin d) (Fin d) ℂ) from rfl, Matrix.zero_mul, zero_add] using ht
have he := htrace.deriv
change complexCoordinateDerivative (fun y => Matrix.trace (B*matrixWord (jointMatrix A y) w)) q x = _ at he
simp only [Function.update_eq_self] at he
rw [he]
change Matrix.trace (B*matrixWordVariation (jointMatrix A x) (noiseDirection q) w) = _
simp only [matrixWordVariation, mul_sum, Matrix.trace_sum]
apply sum_congr rfl
intro r _
simp only [noiseDirection]
split_ifs <;> simp [Matrix.trace_smul, mul_assoc]
theorem gueTraceWord_expand (A : α → Matrix (Fin d) (Fin d) ℂ) (i : Fin n) (w : List (α ⊕ Fin n)) (x : MatrixIndex n d → ℝ) :
matrixTraceWord (jointMatrix A x) (Sum.inr i::w) =
((d:ℂ)⁻¹*(Real.sqrt d:ℂ)⁻¹)*∑ j, ∑ k, (x (i,j,k):ℂ)*Matrix.trace (hermitianBasis j k*matrixWord (jointMatrix A x) w) := by
simp only [matrixTraceWord, matrixWord_cons, jointMatrix, Sum.elim_inr, gueMatrix, coordinateMatrix_expand,
smul_mul_assoc, sum_mul, Matrix.trace_sum, Matrix.trace_smul, smul_eq_mul, mul_sum]
ring_nf
theorem gueTraceWord_stein (_hd : 0 < d) (A : α → Matrix (Fin d) (Fin d) ℂ) (i : Fin n) (w : List (α ⊕ Fin n)) :
(∫ x, matrixTraceWord (jointMatrix A x) (Sum.inr i::w) ∂productGaussian (MatrixIndex n d)) =
∑ r : Fin w.length, if w[r.val]=Sum.inr i then
∫ x, matrixTraceWord (jointMatrix A x) (w.take r.val)*matrixTraceWord (jointMatrix A x) (w.drop (r.val+1)) ∂productGaussian (MatrixIndex n d) else 0 := by
let F := fun (j k : Fin d) (x : MatrixIndex n d → ℝ) => Matrix.trace (hermitianBasis j k*matrixWord (jointMatrix A x) w)
have hF (j k : Fin d) : IsComplexPolynomial (F j k) := fixedTraceWord_polynomial A _ w
have hi (j k : Fin d) := ((IsComplexPolynomial.coord (i,j,k)).mul (hF j k)).integrable
have hdF (j k : Fin d) := ((hF j k).derivative (i,j,k)).integrable
simp_rw [gueTraceWord_expand]
rw [integral_const_mul, integral_finsetSum _ (fun j _ => integrable_finsetSum _ (fun k _ => hi j k))]
simp_rw [integral_finsetSum _ (fun k _ => hi _ k)]
change ((d:ℂ)⁻¹*(Real.sqrt d:ℂ)⁻¹)*(∑ j, ∑ k, ∫ x, (x (i,j,k):ℂ)*F j k x ∂productGaussian (MatrixIndex n d)) = _
simp_rw [(hF _ _).stein]
simp_rw [← integral_finsetSum _ (fun k _ => hdF _ k)]
rw [← integral_finsetSum _ (fun j _ => integrable_finsetSum _ (fun k _ => hdF j k))]
have hcalc (x : MatrixIndex n d → ℝ) : (∑ j, ∑ k, complexCoordinateDerivative (F j k) (i,j,k) x) =
(Real.sqrt d:ℂ)⁻¹ * ∑ r : Fin w.length, if w[r.val]=Sum.inr i then
Matrix.trace (matrixWord (jointMatrix A x) (w.take r.val))*Matrix.trace (matrixWord (jointMatrix A x) (w.drop (r.val+1))) else 0 := by
simp only [F, fixedTraceWord_derivative]
simp_rw [← mul_sum]
congr 1
have hs : ∀ j : Fin d, (∑ k : Fin d, ∑ r : Fin w.length, if w[r.val]=Sum.inr i then
Matrix.trace (hermitianBasis j k*matrixWord (jointMatrix A x) (w.take r.val)*hermitianBasis j k*matrixWord (jointMatrix A x) (w.drop (r.val+1))) else 0) =
∑ r : Fin w.length, ∑ k : Fin d, if w[r.val]=Sum.inr i then
Matrix.trace (hermitianBasis j k*matrixWord (jointMatrix A x) (w.take r.val)*hermitianBasis j k*matrixWord (jointMatrix A x) (w.drop (r.val+1))) else 0 := fun _ => sum_comm
simp_rw [hs]
rw [sum_comm]
apply sum_congr rfl
intro r _
simp only [sum_ite_irrel, sum_const_zero]
split_ifs
· exact basis_trace_contract _ _
· rfl
simp_rw [hcalc]
rw [integral_const_mul, ← mul_assoc]
have hc : (d:ℂ)⁻¹*(Real.sqrt d:ℂ)⁻¹*(Real.sqrt d:ℂ)⁻¹=(d:ℂ)⁻¹*(d:ℂ)⁻¹ := by
rw [mul_assoc, ← pow_two, inv_pow]
have hs : (Real.sqrt d:ℂ)^2=(d:ℂ) := by norm_cast; exact Real.sq_sqrt (Nat.cast_nonneg _)
rw [hs]
rw [hc]
have hp (r : Fin w.length) := ((IsMatrixPolynomial.word (jointMatrix_polynomial A) (w.take r.val)).trace.mul
(IsMatrixPolynomial.word (jointMatrix_polynomial A) (w.drop (r.val+1))).trace).integrable
rw [integral_finsetSum _ (fun r _ => by split_ifs <;> first | exact hp r | exact integrable_const 0)]
rw [mul_sum]
apply sum_congr rfl
intro r _
split_ifs with h
· simp only [matrixTraceWord]
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards [] with x
ring
· simp
end FiniteEntropySeparation
end
end OAI
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