File size: 6,354 Bytes
b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Probability.FreeIsing.Posterior.GaussianTimeDerivative
namespace OAI
noncomputable section
namespace Problem367.FinitePosterior
variable {S V : Type*} [Fintype S] [Nonempty S] [Fintype V]
def fieldSquareGradient (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (v : V) : ℝ :=
2 * fieldMean energy spin observable x * fieldCovariance energy spin observable x v
def fieldSquareHessianDiagonal (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (v : V) : ℝ :=
2 * fieldCovariance energy spin observable x v ^ 2 -
4 * fieldMean energy spin observable x * fieldMean energy spin (fun s => spin s v) x *
fieldCovariance energy spin observable x v
theorem continuous_fieldSquareGradient (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (v : V) :
Continuous (fun x => fieldSquareGradient energy spin observable x v) :=
((continuous_fieldMean energy spin observable).const_mul 2).mul
(continuous_fieldCovariance energy spin observable v)
theorem continuous_fieldSquareHessianDiagonal (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (v : V) :
Continuous (fun x => fieldSquareHessianDiagonal energy spin observable x v) := by
exact (((continuous_fieldCovariance energy spin observable v).pow 2).const_mul 2).sub
((((continuous_fieldMean energy spin observable).const_mul 4).mul
(continuous_fieldMean energy spin (fun s => spin s v))).mul
(continuous_fieldCovariance energy spin observable v))
theorem abs_fieldSquareGradient_le (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (v : V) (A : ℝ)
(hcoord : ∀ s, spin s v = -1 ∨ spin s v = 1)
(hbound : ∀ s, |observable s| ≤ A) :
|fieldSquareGradient energy spin observable x v| ≤ 4 * A ^ 2 := by
have hm := abs_fieldMean_le energy spin observable x A hbound
have hc := abs_fieldCovariance_le energy spin observable x v A hcoord hbound
have hA : 0 ≤ A := (abs_nonneg _).trans hm
unfold fieldSquareGradient
rw [abs_mul, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)]
calc
2 * |fieldMean energy spin observable x| * |fieldCovariance energy spin observable x v| ≤
2 * A * (2 * A) := by gcongr
_ = 4 * A ^ 2 := by ring
variable [DecidableEq V]
omit [Nonempty S] in
lemma fieldCovariance_coordinate_tilt (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (v : V) (t : ℝ) :
fieldCovariance energy spin observable (fun u => x u + if u = v then t else 0) v =
tiltCovariance (fieldEnergy energy spin x) (fun s => spin s v) observable t := by
simp only [fieldCovariance, fieldMean_coordinate_tilt, tiltCovariance]
theorem hasDerivAt_fieldSquareGradient_coordinate (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (v : V) (t : ℝ)
(hcoord : ∀ s, spin s v = -1 ∨ spin s v = 1) :
HasDerivAt (fun r => fieldSquareGradient energy spin observable
(fun u => x u + if u = v then r else 0) v)
(fieldSquareHessianDiagonal energy spin observable
(fun u => x u + if u = v then t else 0) v) t := by
have hd := ((hasDerivAt_mean_covariance (fieldEnergy energy spin x)
(fun s => spin s v) observable t).const_mul 2).mul
(hasDerivAt_tiltCovariance (fieldEnergy energy spin x) (fun s => spin s v)
observable t hcoord)
simp only [fieldSquareGradient, fieldSquareHessianDiagonal,
fieldMean_coordinate_tilt, fieldCovariance_coordinate_tilt]
change HasDerivAt (fun r => 2 * mean (fieldEnergy energy spin x)
(fun s => spin s v) observable r *
tiltCovariance (fieldEnergy energy spin x) (fun s => spin s v) observable r) _ t at hd
apply hd.congr_deriv
ring
theorem abs_fieldSquareHessianDiagonal_le (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (v : V) (A : ℝ)
(hcoord : ∀ s, spin s v = -1 ∨ spin s v = 1)
(hbound : ∀ s, |observable s| ≤ A) :
|fieldSquareHessianDiagonal energy spin observable x v| ≤ 16 * A ^ 2 := by
have hd := (hasDerivAt_fieldSquareGradient_coordinate energy spin observable x v 0 hcoord).deriv
have hfirst : (fun r => fieldSquareGradient energy spin observable
(fun u => x u + if u = v then r else 0) v) =
deriv (fun r => fieldMean energy spin observable
(fun u => x u + if u = v then r else 0) ^ 2) := by
funext r
simp only [fieldSquareGradient, fieldMean_coordinate_tilt, fieldCovariance_coordinate_tilt]
exact (deriv_mean_square _ _ _ _).symm
rw [hfirst] at hd
simp only [ite_self, add_zero] at hd
rw [← hd]
exact abs_deriv_deriv_fieldMean_square_coordinate_le energy spin observable x v 0 A hcoord hbound
omit [Fintype V] in
lemma update_eq_zero_shift (x : V → ℝ) (v : V) (t : ℝ) :
Function.update x v t =
(fun u => Function.update x v 0 u + if u = v then t else 0) := by
funext u
by_cases hu : u = v
· subst u
simp
· simp [hu]
theorem hasDerivAt_fieldSquareGradient_update (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (v : V) (t : ℝ)
(hcoord : ∀ s, spin s v = -1 ∨ spin s v = 1) :
HasDerivAt (fun r => fieldSquareGradient energy spin observable (Function.update x v r) v)
(fieldSquareHessianDiagonal energy spin observable (Function.update x v t) v) t := by
simpa only [← update_eq_zero_shift] using
hasDerivAt_fieldSquareGradient_coordinate energy spin observable (Function.update x v 0) v t hcoord
theorem hasDerivAt_fieldSquareGradient_affine_update
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(x : V → ℝ) (v : V) (a b t : ℝ)
(hcoord : ∀ s, spin s v = -1 ∨ spin s v = 1) :
HasDerivAt (fun r => fieldSquareGradient energy spin observable
(Function.update x v (a + b * r)) v)
(b * fieldSquareHessianDiagonal energy spin observable
(Function.update x v (a + b * t)) v) t := by
have hd := (hasDerivAt_fieldSquareGradient_update energy spin observable x v (a + b * t)
hcoord).comp t (((hasDerivAt_id t).const_mul b).const_add a)
simpa only [Function.comp_def, id_eq, mul_one, mul_comm] using hd
end Problem367.FinitePosterior
end
end OAI
|