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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 | import OAI.Probability.InvariantIsing.Arrays.NSpinTensorPairPaths
import OAI.Probability.InvariantIsing.Pressure.ThermalPressure
namespace OAI
/-! Terminal derivatives and the projected magnetization form of pair paths. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal ENNReal
namespace InvariantIsing
lemma integral_uniformSpinPrior {N : ℕ} (f : Spin N → ℝ) :
(∫ σ, f σ ∂(uniformSpinPrior N : Measure (Spin N))) =
(Fintype.card (Spin N) : ℝ)⁻¹ * ∑ σ, f σ := by
change (∫ σ, f σ ∂(PMF.uniformOfFintype (Spin N)).toMeasure) = _
rw [integral_fintype (Integrable.of_finite)]
simp only [measureReal_def,
PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _),
PMF.uniformOfFintype_apply, ENNReal.toReal_inv, ENNReal.toReal_natCast, smul_eq_mul]
rw [Finset.mul_sum]
lemma integral_uniformSpinGibbs_eq {N : ℕ} (H A : Spin N → ℝ) :
(∫ σ, A σ ∂gibbsProbability (uniformSpinPrior N : Measure (Spin N)) H) =
gibbsAverage (fun _ => 1) H A := by
rw [gibbsProbability_eq_tilted _ _ Integrable.of_finite, integral_tilted,
integral_uniformSpinPrior]
have hp : referencePartition (uniformSpinPrior N : Measure (Spin N)) H =
(Fintype.card (Spin N) : ℝ)⁻¹ * finitePartition (fun _ => 1) H := by
rw [referencePartition, integral_uniformSpinPrior]
simp only [finitePartition, one_mul]
change (Fintype.card (Spin N) : ℝ)⁻¹ *
∑ σ, (Real.exp (H σ) / referencePartition (uniformSpinPrior N : Measure (Spin N)) H) * A σ = _
rw [hp, Finset.mul_sum]
unfold gibbsAverage finiteGibbs
apply Finset.sum_congr rfl
intro σ _
have hc : (Fintype.card (Spin N) : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
have hZ : finitePartition (fun _ : Spin N => 1) H ≠ 0 :=
ne_of_gt (finitePartition_pos GibbsReference_one H)
field_simp
def tensorCoordinateVariation {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (z : SpinTensorIndex I degree → ℝ)
(j : SpinTensorIndex I degree) (t : ℝ) : SpinTensorIndex I degree → ℝ :=
fun i => z i + if i = j then t else 0
lemma spinTensorEnergy_coordinateVariation {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(z : SpinTensorIndex I degree → ℝ) (j : SpinTensorIndex I degree) (t : ℝ) (σ : Spin N) :
spinTensorEnergy U I degree amplitude (tensorCoordinateVariation I degree z j t) σ =
spinTensorEnergy U I degree amplitude z σ + t * spinTensorFeature U I degree amplitude σ j := by
classical
simp only [spinTensorEnergy, tensorCoordinateVariation, add_mul, Finset.sum_add_distrib]
congr 1
simp only [ite_mul, zero_mul]
simp
def tensorTerminalCoordinateMean {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(z : SpinTensorIndex I degree → ℝ) (j : SpinTensorIndex I degree) : ℝ :=
gibbsAverage (fun _ => 1) (tensorSpinBaseEnergy eig U c I degree amplitude z)
(fun σ => spinTensorFeature U I degree amplitude σ j)
def tensorFeatureCoordinateCap {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(j : SpinTensorIndex I degree) : ℝ :=
∑ σ : Spin N, |spinTensorFeature U I degree amplitude σ j|
lemma tensorTerminalCoordinateMean_abs_le {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(z : SpinTensorIndex I degree → ℝ) (j : SpinTensorIndex I degree) :
|tensorTerminalCoordinateMean eig U c I degree amplitude z j| ≤
tensorFeatureCoordinateCap U I degree amplitude j := by
apply abs_gibbsAverage_le GibbsReference_one
intro σ
exact Finset.single_le_sum (fun τ _ => abs_nonneg (spinTensorFeature U I degree amplitude τ j))
(Finset.mem_univ σ)
lemma measurable_tensorTerminalCoordinateMean {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(j : SpinTensorIndex I degree) :
Measurable (fun z => tensorTerminalCoordinateMean eig U c I degree amplitude z j) := by
apply measurable_gibbsAverage (fun _ => 1)
· intro σ
exact (((measurable_pi_apply σ).comp (measurable_spinTensorEnergyMark U I degree amplitude))).const_add
(rotatedEnergy eig U σ + fieldEnergy c σ)
· intro _
exact measurable_const
/-- An actual coordinate derivative of the finite Ising terminal. -/
theorem hasDerivAt_spinTensorTerminal_coordinate {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(z : SpinTensorIndex I degree → ℝ) (j : SpinTensorIndex I degree) (t : ℝ) :
HasDerivAt (fun s => spinTensorTerminal eig U c I degree amplitude
(tensorCoordinateVariation I degree z j s))
(tensorTerminalCoordinateMean eig U c I degree amplitude
(tensorCoordinateVariation I degree z j t) j) t := by
have hH (σ : Spin N) : HasDerivAt (fun s => tensorSpinBaseEnergy eig U c I degree amplitude
(tensorCoordinateVariation I degree z j s) σ) (spinTensorFeature U I degree amplitude σ j) t := by
simp only [tensorSpinBaseEnergy, spinTensorEnergy_coordinateVariation]
convert ((hasDerivAt_id t).mul_const (spinTensorFeature U I degree amplitude σ j)).const_add
(rotatedEnergy eig U σ + fieldEnergy c σ + spinTensorEnergy U I degree amplitude z σ) using 1
· funext s
simp only [id_eq]
ring
· simp
have hd := (hasDerivAt_log_finitePartition GibbsReference_one hH).sub_const
(Real.log (Fintype.card (Spin N)))
convert hd using 1
· funext s
simp only [spinTensorTerminal, logPartition_eq, finitePartition, one_mul, tensorSpinBaseEnergy]
· rfl
def tensorProjectedMagnetization {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(z : SpinTensorIndex I degree → ℝ) (a : Fin N) : ℝ :=
gibbsAverage (fun _ => 1) (tensorSpinBaseEnergy eig U c I degree amplitude z)
(fun σ => U (spinVector σ) a)
lemma tensorProjectedMagnetization_eq {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(z : SpinTensorIndex I degree → ℝ) (a : Fin N) :
tensorProjectedMagnetization eig U c I degree amplitude z a =
U (WithLp.toLp 2 (fun i => tensorTerminalCoordinateMean eig U c I degree amplitude z (Sum.inl i))) a := by
let H := tensorSpinBaseEnergy eig U c I degree amplitude z
have hm : WithLp.toLp 2 (fun i => tensorTerminalCoordinateMean eig U c I degree amplitude z (Sum.inl i)) =
∑ σ : Spin N, finiteGibbs (fun _ => 1) H σ • spinVector σ := by
ext i
change tensorTerminalCoordinateMean eig U c I degree amplitude z (Sum.inl i) =
(WithLp.ofLp (∑ σ : Spin N, finiteGibbs (fun _ => 1) H σ • spinVector σ)) i
rw [WithLp.ofLp_sum, Finset.sum_apply]
simp only [tensorTerminalCoordinateMean, gibbsAverage, spinTensorFeature,
WithLp.ofLp_smul, Pi.smul_apply, smul_eq_mul, spinVector, H]
rw [hm, map_sum]
change gibbsAverage (fun _ => 1) H (fun σ => U (spinVector σ) a) =
(WithLp.ofLp (∑ σ : Spin N, U (finiteGibbs (fun _ => 1) H σ • spinVector σ))) a
rw [WithLp.ofLp_sum, Finset.sum_apply]
simp only [map_smul, WithLp.ofLp_smul, Pi.smul_apply, smul_eq_mul, gibbsAverage]
lemma tensorPairPathSpinMean_overlap {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (p : TensorPairPathData I degree n) (J : Finset (Fin N)) :
tensorPairPathSpinMean eig U c I degree amplitude n p
(fun σ => projectedOverlap U J (σ 0) (σ 1)) =
(N : ℝ)⁻¹ * ∑ a ∈ J,
tensorProjectedMagnetization eig U c I degree amplitude (tensorPairPathState I degree n p 0) a *
tensorProjectedMagnetization eig U c I degree amplitude (tensorPairPathState I degree n p 1) a := by
let μ := fun j : Fin 2 => gibbsProbability (uniformSpinPrior N : Measure (Spin N))
(tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n p j))
change (∫ σ : Fin 2 → Spin N, (N : ℝ)⁻¹ *
∑ a ∈ J, U (spinVector (σ 0)) a * U (spinVector (σ 1)) a ∂Measure.pi μ) = _
rw [integral_const_mul, integral_finsetSum _ (fun _ _ => Integrable.of_finite)]
congr 1
apply Finset.sum_congr rfl
intro a ha
have h := integral_fin_nat_prod_eq_prod (μ := μ) (fun (_ : Fin 2) (σ : Spin N) => U (spinVector σ) a)
simpa only [Fin.prod_univ_two, μ, integral_uniformSpinGibbs_eq, tensorProjectedMagnetization] using h
/-- The actual projected overlap along two paths is the product of the
projected terminal Gibbs gradients. -/
theorem tensorPairPathSpinMean_overlap_gradient {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (p : TensorPairPathData I degree n) (J : Finset (Fin N)) :
tensorPairPathSpinMean eig U c I degree amplitude n p
(fun σ => projectedOverlap U J (σ 0) (σ 1)) =
(N : ℝ)⁻¹ * ∑ a ∈ J,
U (WithLp.toLp 2 (fun i => tensorTerminalCoordinateMean eig U c I degree amplitude
(tensorPairPathState I degree n p 0) (Sum.inl i))) a *
U (WithLp.toLp 2 (fun i => tensorTerminalCoordinateMean eig U c I degree amplitude
(tensorPairPathState I degree n p 1) (Sum.inl i))) a := by
simp only [tensorPairPathSpinMean_overlap, tensorProjectedMagnetization_eq]
end InvariantIsing
end
end OAI
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