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import OAI.Probability.InvariantIsing.Arrays.TensorLeafCoefficients
namespace OAI
/-! The flat countable Gaussian field has the actual root/forest mark law. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
def tensorNodeTag {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (w : Option (ForestVertex n))
(i : SpinTensorIndex I degree) : ℕ :=
Encodable.encode ((nodeAddress n w).length, nodeAddress n w, i)
lemma tensorNodeTag_injective {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) :
Function.Injective (fun p : Option (ForestVertex n) × SpinTensorIndex I degree =>
tensorNodeTag I degree n p.1 p.2) := by
rintro ⟨v, i⟩ ⟨w, j⟩ h
have he := Encodable.encode_inj.mp h
exact Prod.ext (nodeAddress_injective n (congrArg (fun p => p.2.1) he))
(congrArg (fun p => p.2.2) he)
lemma tensorNodeTag_nodeAt {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (leaf : LabeledLeaf n)
(level : Fin (n + 1)) (i : SpinTensorIndex I degree) :
tensorNodeTag I degree n (nodeAt n leaf level) i = treeFeatureTag n leaf (level, i) := by
unfold tensorNodeTag treeFeatureTag
rw [nodeAddress_nodeAt_length, nodeAddress_nodeAt]
def tensorStandardGaussianNodes {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (g : ℕ → ℝ)
(w : Option (ForestVertex n)) (i : SpinTensorIndex I degree) : ℝ :=
g (tensorNodeTag I degree n w i)
lemma measurable_tensorStandardGaussianNodes {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) : Measurable (tensorStandardGaussianNodes I degree n) := by
unfold tensorStandardGaussianNodes
fun_prop
theorem tensorStandardGaussianNodes_law {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) :
gaussianCoordinates.map (tensorStandardGaussianNodes I degree n) =
Measure.infinitePi (fun _ : Option (ForestVertex n) =>
Measure.pi (fun _ : SpinTensorIndex I degree => gaussianReal 0 1)) := by
let t := fun p : Option (ForestVertex n) × SpinTensorIndex I degree => tensorNodeTag I degree n p.1 p.2
have hp := gaussian_pullback_measurePreserving t (tensorNodeTag_injective I degree n)
change gaussianCoordinates.map
((MeasurableEquiv.curry (Option (ForestVertex n)) (SpinTensorIndex I degree) ℝ) ∘
(fun g => fun p => g (t p))) = _
rw [← Measure.map_map (MeasurableEquiv.curry _ _ _).measurable hp.measurable, hp.map_eq,
Measure.infinitePi_map_curry (fun (_ : Option (ForestVertex n)) (_ : SpinTensorIndex I degree) => gaussianReal 0 1)]
simp only [Measure.infinitePi_eq_pi]
private theorem tensorMarkScale_measurePreserving {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ)
(v : SpinTensorIndex I degree → ℝ≥0) :
MeasurePreserving (fun z : SpinTensorIndex I degree → ℝ => fun i => (NNReal.sqrt (v i) : ℝ) * z i)
(Measure.pi (fun _ : SpinTensorIndex I degree => gaussianReal 0 1))
(tensorGaussianLaw I degree v : Measure (SpinTensorIndex I degree → ℝ)) := by
change MeasurePreserving _ _ (Measure.pi (fun i : SpinTensorIndex I degree => gaussianReal 0 (v i)))
apply measurePreserving_pi
intro i
refine ⟨by fun_prop, ?_⟩
have hs : NNReal.mk ((NNReal.sqrt (v i) : ℝ) ^ 2) (sq_nonneg _) = v i := by
apply Subtype.ext
change (NNReal.sqrt (v i) : ℝ) ^ 2 = (v i : ℝ)
exact_mod_cast NNReal.sq_sqrt (v i)
simpa only [hs, mul_zero, mul_one] using
gaussianReal_map_const_mul (μ := 0) (v := 1) (NNReal.sqrt (v i) : ℝ)
def tensorGaussianNodes {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(g : ℕ → ℝ) (w : Option (ForestVertex n)) (i : SpinTensorIndex I degree) : ℝ :=
(NNReal.sqrt (v (nodeAddress n w).length i) : ℝ) * tensorStandardGaussianNodes I degree n g w i
lemma measurable_tensorGaussianNodes {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) :
Measurable (tensorGaussianNodes I degree n v) := by
unfold tensorGaussianNodes tensorStandardGaussianNodes
fun_prop
theorem tensorGaussianNodes_measurePreserving {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) :
MeasurePreserving (tensorGaussianNodes I degree n v) gaussianCoordinates
(Measure.infinitePi (fun w : Option (ForestVertex n) =>
(tensorGaussianLaw I degree (v (nodeAddress n w).length) : Measure (SpinTensorIndex I degree → ℝ)))) := by
have hstd : MeasurePreserving (tensorStandardGaussianNodes I degree n) gaussianCoordinates
(Measure.infinitePi (fun _ : Option (ForestVertex n) =>
Measure.pi (fun _ : SpinTensorIndex I degree => gaussianReal 0 1))) :=
⟨measurable_tensorStandardGaussianNodes I degree n, tensorStandardGaussianNodes_law I degree n⟩
have hscale : MeasurePreserving
(fun q : Option (ForestVertex n) → SpinTensorIndex I degree → ℝ =>
fun w i => (NNReal.sqrt (v (nodeAddress n w).length i) : ℝ) * q w i)
(Measure.infinitePi (fun _ : Option (ForestVertex n) =>
Measure.pi (fun _ : SpinTensorIndex I degree => gaussianReal 0 1)))
(Measure.infinitePi (fun w : Option (ForestVertex n) =>
(tensorGaussianLaw I degree (v (nodeAddress n w).length) : Measure (SpinTensorIndex I degree → ℝ)))) := by
refine ⟨by fun_prop, ?_⟩
rw [Measure.infinitePi_map_pi (f := fun w (z : SpinTensorIndex I degree → ℝ) =>
fun i => (NNReal.sqrt (v (nodeAddress n w).length i) : ℝ) * z i) _ (fun _ => by fun_prop)]
apply congrArg Measure.infinitePi
funext w
exact (tensorMarkScale_measurePreserving I degree (v (nodeAddress n w).length)).map_eq
exact hscale.comp hstd
/-- The root and all forest coordinates have their independent prescribed
Gaussian laws, including zero variances. -/
theorem tensorGaussianRootForest_measurePreserving {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ)
(n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) :
MeasurePreserving (fun g => (tensorGaussianNodes I degree n v g none,
fun w => tensorGaussianNodes I degree n v g (some w))) gaussianCoordinates
((tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ)).prod
(Measure.infinitePi (fun w : ForestVertex n =>
(tensorGaussianLaw I degree (v (forestVertexDepth n w + 1)) :
Measure (SpinTensorIndex I degree → ℝ))))) := by
refine ⟨by unfold tensorGaussianNodes tensorStandardGaussianNodes; fun_prop, ?_⟩
change gaussianCoordinates.map ((fun q : Option (ForestVertex n) → SpinTensorIndex I degree → ℝ =>
(q none, fun w => q (some w))) ∘ tensorGaussianNodes I degree n v) = _
rw [← Measure.map_map (by fun_prop) (measurable_tensorGaussianNodes I degree n v),
(tensorGaussianNodes_measurePreserving I degree n v).map_eq]
have he := infinitePi_option_split
(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))
(fun w : ForestVertex n => (tensorGaussianLaw I degree (v (forestVertexDepth n w + 1)) :
Measure (SpinTensorIndex I degree → ℝ)))
have hc : (fun w : Option (ForestVertex n) => w.elim
(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))
(fun w => (tensorGaussianLaw I degree (v (forestVertexDepth n w + 1)) :
Measure (SpinTensorIndex I degree → ℝ)))) =
(fun w => (tensorGaussianLaw I degree (v (nodeAddress n w).length) :
Measure (SpinTensorIndex I degree → ℝ))) := by
funext w
cases w with
| none => cases n <;> rfl
| some w => simp only [Option.elim_some, nodeAddress_length_some]
simpa only [hc] using he
lemma cylinderField_tensorLeaf_nodes {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(σ : Spin N) (leaf : LabeledLeaf n) (g : ℕ → ℝ) :
cylinderField (tensorLeafCoefficients U I degree amplitude n (fun i => v i) (σ, leaf)) g =
spinTensorEnergy U I degree amplitude (tensorGaussianNodes I degree n v g none) σ +
∑ i : Fin n, spinTensorEnergy U I degree amplitude
(tensorGaussianNodes I degree n v g (some (edgeAt n leaf i))) σ := by
classical
have he : cylinderField (tensorLeafCoefficients U I degree amplitude n (fun i => v i) (σ, leaf)) g =
∑ i : Fin (n + 1), spinTensorEnergy U I degree amplitude
(tensorGaussianNodes I degree n v g (nodeAt n leaf i)) σ := by
rw [tensorLeafCoefficients, cylinderField_feature, Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro i _
unfold spinTensorEnergy tensorGaussianNodes tensorStandardGaussianNodes
rw [nodeAddress_nodeAt_length]
apply Finset.sum_congr rfl
intro j _
rw [tensorNodeTag_nodeAt]
ring
rw [he, Fin.sum_univ_succ]
simp only [nodeAt_zero, nodeAt_succ]
def tensorFlatToCoordinates {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(p : LabeledTree n × (ℕ → ℝ)) :
(SpinTensorIndex I degree → ℝ) × TensorCoordinateData I degree n :=
(tensorGaussianNodes I degree n v p.2 none,
(p.1, fun w => tensorGaussianNodes I degree n v p.2 (some w)))
theorem tensorFlatToCoordinates_measurePreserving {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) :
MeasurePreserving (tensorFlatToCoordinates I degree n v)
((labeledCascadeLaw n b : Measure (LabeledTree n)).prod gaussianCoordinates)
((tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ)).prod
(tensorCoordinateLaw I degree n b (fun i => v (i + 1)))) := by
let R := (tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))
let T := (labeledCascadeLaw n b : Measure (LabeledTree n))
let F := Measure.infinitePi (fun w : ForestVertex n =>
(tensorGaussianLaw I degree (v (forestVertexDepth n w + 1)) : Measure (SpinTensorIndex I degree → ℝ)))
have hswap : MeasurePreserving Prod.swap (T.prod R) (R.prod T) := Measure.measurePreserving_swap
have hshuffle := (measurePreserving_prodAssoc R T F).comp
((hswap.prod (MeasurePreserving.id F)).comp
((measurePreserving_prodAssoc T R F).symm MeasurableEquiv.prodAssoc))
have hprod := (MeasurePreserving.id T).prod (tensorGaussianRootForest_measurePreserving I degree n v)
convert hshuffle.comp hprod using 1
all_goals rfl
def tensorFlatToLabeled {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (p : LabeledTree n × (ℕ → ℝ)) :
(SpinTensorIndex I degree → ℝ) × TensorLabeledData N n :=
let q := tensorFlatToCoordinates I degree n v p
(q.1, tensorEnergyCoordinates U I degree amplitude n q.2)
/-- The flat Gaussian spin/leaf model is exactly the independent Gaussian
root and tensor-energy forest model used by the cascade recursion. -/
theorem tensorFlatToLabeled_measurePreserving {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) :
MeasurePreserving (tensorFlatToLabeled U I degree amplitude n v)
((labeledCascadeLaw n b : Measure (LabeledTree n)).prod gaussianCoordinates)
((tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ)).prod
(tensorLabeledLaw U I degree amplitude n b (fun i => v (i + 1)))) := by
have h := ((MeasurePreserving.id
(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))).prod
(tensorEnergyCoordinates_measurePreserving U I degree amplitude n b (fun i => v (i + 1)))).comp
(tensorFlatToCoordinates_measurePreserving I degree n b v)
convert h using 1
all_goals rfl
theorem tensorLabeledEnergy_flat_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)
(n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(p : LabeledTree n × (ℕ → ℝ)) (x : Spin N × LabeledLeaf n) :
let q := tensorFlatToLabeled U I degree amplitude n v p
tensorLabeledEnergy eig U c I degree amplitude n q.1 q.2 x =
rotatedEnergy eig U x.1 + fieldEnergy c x.1 +
cylinderField (tensorLeafCoefficients U I degree amplitude n (fun i => v i) x) p.2 := by
rw [cylinderField_tensorLeaf_nodes]
dsimp only [tensorFlatToLabeled, tensorFlatToCoordinates, tensorLabeledEnergy,
cascadeCoordinateEnergy, tensorSpinBaseEnergy, tensorEnergyCoordinates]
ring
end InvariantIsing
end
end OAI
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