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import OAI.Probability.InvariantIsing.Fields.CascadeTilt
namespace OAI
/-! The actual finite tensor terminal induces ancestor-dependent retained mark transitions. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal ENNReal
namespace InvariantIsing
def tensorCascadeBackward {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (i : ℕ) :
(SpinTensorIndex I degree → ℝ) → ℝ :=
backwardValue n b (fun j => tensorGaussianLaw I degree (v j)) (fun _ p => p.1 + p.2)
(spinTensorTerminal eig U c I degree amplitude) i
def tensorCascadeStopped {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) :
ℕ → ((SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) →
(SpinTensorIndex I degree → ℝ) := stoppedUpdate n (fun _ p => p.1 + p.2)
def tensorCascadeMultiplier {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (i : ℕ)
(p : (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) : ℝ :=
Real.exp (tensorCascadeBackward eig U c I degree amplitude n b v (i + 1)
(tensorCascadeStopped I degree n i p) -
tensorCascadeBackward eig U c I degree amplitude n b v i p.1)
lemma measurable_tensorCascadeBackward {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (i : ℕ) :
Measurable (tensorCascadeBackward eig U c I degree amplitude n b v i) :=
measurable_backwardValue n b (fun j => tensorGaussianLaw I degree (v j))
(fun _ => measurable_fst.add measurable_snd)
(measurable_spinTensorTerminal eig U c I degree amplitude) i
lemma measurable_tensorCascadeStopped {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n i : ℕ) :
Measurable (tensorCascadeStopped I degree n i) :=
measurable_stoppedUpdate n (fun _ => measurable_fst.add measurable_snd) i
lemma measurable_tensorCascadeMultiplier {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (i : ℕ) :
Measurable (tensorCascadeMultiplier eig U c I degree amplitude n b v i) :=
(((measurable_tensorCascadeBackward eig U c I degree amplitude n b v (i + 1)).comp
(measurable_tensorCascadeStopped I degree n i)).sub
((measurable_tensorCascadeBackward eig U c I degree amplitude n b v i).comp measurable_fst)).exp
lemma tensorCascadeMultiplier_pos {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (i : ℕ)
(z a : SpinTensorIndex I degree → ℝ) :
0 < tensorCascadeMultiplier eig U c I degree amplitude n b v i (z, a) := Real.exp_pos _
/-- Each transition is normalized at its own cascade exponent, for every
ancestor state. The terminal is the actual finite Ising log partition. -/
lemma tensorCascadeMultiplier_moment {N m k : ℕ} (hN : 0 < 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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (i : ℕ) (z : SpinTensorIndex I degree → ℝ) :
(∫⁻ a, ENNReal.ofReal (tensorCascadeMultiplier eig U c I degree amplitude n b v i (z, a) ^ b i)
∂(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ))) = 1 := by
have hadm := backwardValue_linearGrowth_moments n b
(fun j => tensorGaussianLaw I degree (v j))
(fun j _ => tensorGaussianLaw_moments hN I degree (v j))
(measurable_spinTensorTerminal eig U c I degree amplitude)
(spinTensorTerminal_linearGrowth eig U c I degree amplitude) (fun j hj => (hb.1 j hj).1)
exact backwardValue_multiplier_moment_admissible n b
(fun j => tensorGaussianLaw I degree (v j)) (fun _ p => p.1 + p.2)
(spinTensorTerminal eig U c I degree amplitude) (fun j hj => (hb.1 j hj).1) hadm i z
private lemma measurable_tensorAncestorMarkMeasure {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (i : ℕ) :
Measurable (fun z => (tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ)).withDensity
(fun a => ENNReal.ofReal (tensorCascadeMultiplier eig U c I degree amplitude n b v i (z, a) ^ b i))) := by
let ν := fun _ : SpinTensorIndex I degree → ℝ =>
(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ))
let W := fun p : (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ) =>
ENNReal.ofReal (tensorCascadeMultiplier eig U c I degree amplitude n b v i p ^ b i)
have hW : Measurable W :=
((measurable_tensorCascadeMultiplier eig U c I degree amplitude n b v i).pow_const (b i)).ennreal_ofReal
exact measurable_random_withDensity (ν := ν) (W := W) measurable_const hW
/-- The ancestor-dependent mark transition, with density `c_i(z,a)^b_i`. -/
def tensorAncestorMarkKernel {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (i : ℕ) :
Kernel (SpinTensorIndex I degree → ℝ) (SpinTensorIndex I degree → ℝ) :=
⟨fun z => (tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ)).withDensity
(fun a => ENNReal.ofReal (tensorCascadeMultiplier eig U c I degree amplitude n b v i (z, a) ^ b i)),
measurable_tensorAncestorMarkMeasure eig U c I degree amplitude n b v i⟩
theorem tensorAncestorMarkKernel_markov {N m k : ℕ} (hN : 0 < 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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (hb : CascadeExponents n b) (i : ℕ) :
IsMarkovKernel (tensorAncestorMarkKernel eig U c I degree amplitude n b v i) := by
constructor
intro z
constructor
change ((tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ)).withDensity
(fun a => ENNReal.ofReal (tensorCascadeMultiplier eig U c I degree amplitude n b v i (z, a) ^ b i))) Set.univ = 1
rw [withDensity_apply _ MeasurableSet.univ, setLIntegral_univ]
exact tensorCascadeMultiplier_moment hN eig U c I degree amplitude n b v hb i z
lemma tensorAncestorMarkKernel_density {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(i : ℕ) (hi : i < n) (z : SpinTensorIndex I degree → ℝ) :
tensorAncestorMarkKernel eig U c I degree amplitude n b v i z =
(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ)).withDensity
(fun a => ENNReal.ofReal (Real.exp (b i *
(tensorCascadeBackward eig U c I degree amplitude n b v (i + 1) (z + a) -
tensorCascadeBackward eig U c I degree amplitude n b v i z)))) := by
change Measure.withDensity _ _ = _
congr 1
funext a
simp only [tensorCascadeMultiplier, tensorCascadeStopped, stoppedUpdate, hi, ite_true, ← Real.exp_mul]
rw [mul_comm]
def tensorRetainedCascade {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(z : SpinTensorIndex I degree → ℝ) :
NoiseTree (SpinTensorIndex I degree → ℝ) n → NoiseTree (SpinTensorIndex I degree → ℝ) n :=
noiseTreeKeep n b (fun i => tensorGaussianLaw I degree (v i))
(tensorCascadeMultiplier eig U c I degree amplitude n b v) (tensorCascadeStopped I degree n) z
lemma measurable_tensorRetainedCascade {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(z : SpinTensorIndex I degree → ℝ) :
Measurable (tensorRetainedCascade eig U c I degree amplitude n b v z) :=
(measurable_noiseTreeKeep n b (fun i => tensorGaussianLaw I degree (v i))
(measurable_tensorCascadeMultiplier eig U c I degree amplitude n b v)
(measurable_tensorCascadeStopped I degree n)).comp (measurable_const.prodMk measurable_id)
/-- Forgetting the retained marks gives exactly the original raw cascade law. -/
theorem tensorRetainedCascade_forget_law {N m k : ℕ} (hN : 0 < 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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (z : SpinTensorIndex I degree → ℝ) :
(tensorCascadeLaw I degree n b v).map
(fun T => noiseTreeForget (SpinTensorIndex I degree → ℝ) n
(tensorRetainedCascade eig U c I degree amplitude n b v z T)) = rawCascadeLaw n b := by
have he : (fun T => noiseTreeForget (SpinTensorIndex I degree → ℝ) n
(tensorRetainedCascade eig U c I degree amplitude n b v z T)) =
noiseTreeDisplace n b (fun i => tensorGaussianLaw I degree (v i))
(tensorCascadeMultiplier eig U c I degree amplitude n b v) (tensorCascadeStopped I degree n) z := by
funext T
exact noiseTreeKeep_forget n b (fun i => tensorGaussianLaw I degree (v i))
(measurable_tensorCascadeMultiplier eig U c I degree amplitude n b v)
(measurable_tensorCascadeStopped I degree n) z T
rw [he]
exact noiseTreeDisplace_law n b (fun i => tensorGaussianLaw I degree (v i))
(measurable_tensorCascadeMultiplier eig U c I degree amplitude n b v)
(measurable_tensorCascadeStopped I degree n)
(tensorCascadeMultiplier_pos eig U c I degree amplitude n b v)
(tensorCascadeMultiplier_moment hN eig U c I degree amplitude n b v hb) z
/-- The retained tree has the actual ancestor-dependent root mark kernel.
The descendant is transformed at the updated ancestor state; it is not
replaced by an independent level tilt. -/
theorem tensorRetainedCascade_law_succ {N m k : ℕ} (hN : 0 < 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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents (n + 1) b) (z : SpinTensorIndex I degree → ℝ) :
letI := tensorAncestorMarkKernel_markov hN eig U c I degree amplitude (n + 1) b v hb 0
let μ := fun i => tensorGaussianLaw I degree (v i)
let ct := tensorCascadeMultiplier eig U c I degree amplitude (n + 1) b v
let ut := tensorCascadeStopped I degree (n + 1)
let P := (noiseCascadeLaw (SpinTensorIndex I degree → ℝ) n
(fun i => b (i + 1)) (fun i => μ (i + 1)) : Measure (NoiseTree (SpinTensorIndex I degree → ℝ) n))
let G := fun p : (SpinTensorIndex I degree → ℝ) × NoiseTree (SpinTensorIndex I degree → ℝ) n =>
(p.1, noiseTreeKeep n (fun i => b (i + 1)) (fun i => μ (i + 1))
(fun i => ct (i + 1)) (fun i => ut (i + 1)) (ut 0 (z, p.1)) p.2)
(tensorCascadeLaw I degree (n + 1) b v).map
(tensorRetainedCascade eig U c I degree amplitude (n + 1) b v z) =
poissonLaw ((powerIntensity (b 0)).prod
(((tensorAncestorMarkKernel eig U c I degree amplitude (n + 1) b v 0 z).prod P).map G)) := by
let := tensorAncestorMarkKernel_markov hN eig U c I degree amplitude (n + 1) b v hb 0
intro μ ct ut P G
have h := noiseTreeKeep_law_succ n b μ
(measurable_tensorCascadeMultiplier eig U c I degree amplitude (n + 1) b v)
(measurable_tensorCascadeStopped I degree (n + 1))
(tensorCascadeMultiplier_pos eig U c I degree amplitude (n + 1) b v) z
have hd : Measurable (fun a : SpinTensorIndex I degree → ℝ => ENNReal.ofReal (ct 0 (z, a) ^ b 0)) :=
(((measurable_tensorCascadeMultiplier eig U c I degree amplitude (n + 1) b v 0).comp
(measurable_const.prodMk measurable_id)).pow_const (b 0)).ennreal_ofReal
change (tensorCascadeLaw I degree (n + 1) b v).map
(tensorRetainedCascade eig U c I degree amplitude (n + 1) b v z) =
poissonLaw ((powerIntensity (b 0)).prod
((((μ 0 : Measure (SpinTensorIndex I degree → ℝ)).withDensity
(fun a => ENNReal.ofReal (ct 0 (z, a) ^ b 0))).prod P).map G))
apply h.trans
apply poissonLaw_congr
exact congrArg (fun Q => (powerIntensity (b 0)).prod (Q.map G))
(prod_withDensity_left (ν := P) hd).symm
private lemma tensorLeafTerminal_last {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ)
(n : ℕ) (X : ℕ → (SpinTensorIndex I degree → ℝ) → ℝ)
(u : ℕ → ((SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) →
(SpinTensorIndex I degree → ℝ)) (F : (SpinTensorIndex I degree → ℝ) → ℝ)
(hX : X n = F) (hu : ∀ i < n, u i = fun p => p.1 + p.2)
(z : SpinTensorIndex I degree → ℝ) (leaf : NoiseLeaf (SpinTensorIndex I degree → ℝ) n) :
noiseLeafTerminal n X u z leaf = F (tensorLeafState I degree n z leaf) := by
induction n generalizing X u z with
| zero => simpa only [noiseLeafTerminal, tensorLeafState] using congrFun hX z
| succ n ih =>
rw [noiseLeafTerminal, hu 0 (by omega)]
exact ih (fun i => X (i + 1)) (fun i => u (i + 1))
(by simpa only [Nat.add_comm] using hX) (fun i hi => hu (i + 1) (by omega))
(z + leaf.2.1) leaf.2.2
def tensorTerminalLeafLaw {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) (z : SpinTensorIndex I degree → ℝ) (T : NoiseTree (SpinTensorIndex I degree → ℝ) n) :
Measure (NoiseLeaf (SpinTensorIndex I degree → ℝ) n) :=
normalizeMass ((noiseLeafKernel (SpinTensorIndex I degree → ℝ) n T).withDensity
(fun leaf => ENNReal.ofReal (Real.exp
(spinTensorTerminal eig U c I degree amplitude (tensorLeafState I degree n z leaf)))))
lemma tensorTerminalLeafLaw_eq_backward {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(z : SpinTensorIndex I degree → ℝ) (T : NoiseTree (SpinTensorIndex I degree → ℝ) n) :
tensorTerminalLeafLaw eig U c I degree amplitude n z T =
noiseTerminalGibbsLaw n (tensorCascadeBackward eig U c I degree amplitude n b v)
(tensorCascadeStopped I degree n) z T := by
have hX : tensorCascadeBackward eig U c I degree amplitude n b v n =
spinTensorTerminal eig U c I degree amplitude :=
backwardValue_terminal n b (fun i => tensorGaussianLaw I degree (v i))
(fun _ p => p.1 + p.2) (spinTensorTerminal eig U c I degree amplitude) (le_refl n)
have hu : ∀ i < n, tensorCascadeStopped I degree n i = fun p => p.1 + p.2 := by
intro i hi
funext p
simp only [tensorCascadeStopped, stoppedUpdate, hi, ite_true]
unfold tensorTerminalLeafLaw noiseTerminalGibbsLaw
congr 2
funext leaf
rw [tensorLeafTerminal_last I degree n _ _ _ hX hu]
/-- Sampling after the actual terminal tilt is transported to ordinary
sampling from the retained marked tree, with every mark kept. -/
theorem tensorTerminalLeafLaw_keep_ae {N m k : ℕ} (hN : 0 < 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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (z : SpinTensorIndex I degree → ℝ) :
∀ᵐ T ∂tensorCascadeLaw I degree n b v,
(tensorTerminalLeafLaw eig U c I degree amplitude n z T).map
(noiseLeafKeep n (tensorCascadeMultiplier eig U c I degree amplitude n b v)
(tensorCascadeStopped I degree n) z) =
noiseLeafKernel (SpinTensorIndex I degree → ℝ) n
(tensorRetainedCascade eig U c I degree amplitude n b v z T) := by
have hr := noiseGibbsRegular_ae n b hb (fun i => tensorGaussianLaw I degree (v i))
(measurable_tensorCascadeMultiplier eig U c I degree amplitude n b v)
(measurable_tensorCascadeStopped I degree n)
(tensorCascadeMultiplier_pos eig U c I degree amplitude n b v)
(tensorCascadeMultiplier_moment hN eig U c I degree amplitude n b v hb) z
filter_upwards [hr] with T hT
rw [tensorTerminalLeafLaw_eq_backward eig U c I degree amplitude n b v]
exact noiseTerminalGibbsLaw_keep n b (fun i => tensorGaussianLaw I degree (v i))
(measurable_tensorCascadeBackward eig U c I degree amplitude n b v)
(measurable_tensorCascadeStopped I degree n) z T hT
end InvariantIsing
end
end OAI
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