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namespace OAI
/-! Collapse the two independent residual paths to actual backward gradients. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal ENNReal
namespace InvariantIsing
def tensorPairCoordinatePayoff {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)
(q : ℕ) (z₁ z₂ : SpinTensorIndex I degree → ℝ) (j₁ j₂ : SpinTensorIndex I degree)
(w : Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) : ℝ :=
tensorTerminalCoordinateMean eig U c I degree amplitude (z₁ + ∑ i, (w i).1) j₁ *
tensorTerminalCoordinateMean eig U c I degree amplitude (z₂ + ∑ i, (w i).2) j₂
lemma tensorPairCoordinatePayoff_cons {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)
(q : ℕ) (z₁ z₂ a₁ a₂ : SpinTensorIndex I degree → ℝ) (j₁ j₂ : SpinTensorIndex I degree)
(w : Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) :
tensorPairCoordinatePayoff eig U c I degree amplitude (q + 1) z₁ z₂ j₁ j₂ (Fin.cons (a₁, a₂) w) =
tensorPairCoordinatePayoff eig U c I degree amplitude q (z₁ + a₁) (z₂ + a₂) j₁ j₂ w := by
simp only [tensorPairCoordinatePayoff, Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ, add_assoc]
lemma tensorPairCoordinatePayoff_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)
(q : ℕ) (z₁ z₂ : SpinTensorIndex I degree → ℝ) (j₁ j₂ : SpinTensorIndex I degree)
(w : Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) :
|tensorPairCoordinatePayoff eig U c I degree amplitude q z₁ z₂ j₁ j₂ w| ≤
tensorFeatureCoordinateCap U I degree amplitude j₁ * tensorFeatureCoordinateCap U I degree amplitude j₂ := by
rw [tensorPairCoordinatePayoff, abs_mul]
exact mul_le_mul (tensorTerminalCoordinateMean_abs_le eig U c I degree amplitude _ j₁)
(tensorTerminalCoordinateMean_abs_le eig U c I degree amplitude _ j₂) (abs_nonneg _)
(Finset.sum_nonneg fun _ _ => abs_nonneg _)
/-- Once the two paths have branched, each terminal gradient averages
independently to its actual backward gradient. -/
theorem tensorAncestorPairPathMean_independent_coordinates {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) (d q i : ℕ) (hqi : i + q = n) (hdi : d ≤ i)
(z₁ z₂ : SpinTensorIndex I degree → ℝ) (j₁ j₂ : SpinTensorIndex I degree) :
tensorAncestorPairPathMean eig U c I degree amplitude n b v d q i z₁ z₂
(tensorPairCoordinatePayoff eig U c I degree amplitude q z₁ z₂ j₁ j₂) =
tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z₁ j₁ *
tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z₂ j₂ := by
induction q generalizing i z₁ z₂ with
| zero =>
have hi : i = n := by omega
subst i
simp only [tensorAncestorPairPathMean, tensorPairCoordinatePayoff, Finset.univ_eq_empty,
Finset.sum_empty, add_zero, tensorBackwardCoordinateMean, Nat.sub_self, tensorCascadeCoordinateMean]
| succ q ih =>
have hi : i < n := by omega
have hqi' : i + 1 + q = n := by omega
have hdi' : d ≤ i + 1 := by omega
dsimp only [tensorAncestorPairPathMean]
rw [ite_eq_right (by omega : ¬i < d)]
calc
_ = ∫ a₁, ∫ a₂,
tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1) (z₁ + a₁) j₁ *
tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1) (z₂ + a₂) j₂
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₂
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₁ := by
apply integral_congr_ae
apply ae_of_all
intro a₁
apply integral_congr_ae
apply ae_of_all
intro a₂
dsimp only
have hF : (fun w => tensorPairCoordinatePayoff eig U c I degree amplitude (q + 1)
z₁ z₂ j₁ j₂ (Fin.cons (a₁, a₂) w)) =
tensorPairCoordinatePayoff eig U c I degree amplitude q (z₁ + a₁) (z₂ + a₂) j₁ j₂ :=
funext (tensorPairCoordinatePayoff_cons eig U c I degree amplitude q z₁ z₂ a₁ a₂ j₁ j₂)
rw [hF]
exact ih (i + 1) hqi' hdi' (z₁ + a₁) (z₂ + a₂)
_ = (∫ a₁, tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1) (z₁ + a₁) j₁
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₁) *
(∫ a₂, tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1) (z₂ + a₂) j₂
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₂) := by
simp_rw [integral_const_mul]
rw [integral_mul_const]
_ = _ := by
rw [← tensorBackwardCoordinateMean_step hN eig U c I degree amplitude n b v hb i hi z₁ j₁,
← tensorBackwardCoordinateMean_step hN eig U c I degree amplitude n b v hb i hi z₂ j₂]
/-- Only the shared ancestor increments remain after the independent
suffixes have been integrated. At the branching level the integrand is
the product of actual backward gradients. -/
def tensorSharedCoordinateProduct {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) (d : ℕ)
(j₁ j₂ : SpinTensorIndex I degree) :
(q : ℕ) → ℕ → (SpinTensorIndex I degree → ℝ) → (SpinTensorIndex I degree → ℝ) → ℝ
| 0, i, z₁, z₂ =>
tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z₁ j₁ *
tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z₂ j₂
| q + 1, i, z₁, z₂ =>
if i < d then
∫ a, tensorSharedCoordinateProduct eig U c I degree amplitude n b v d j₁ j₂ q
(i + 1) (z₁ + a) (z₂ + a) ∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₁
else
tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z₁ j₁ *
tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z₂ j₂
lemma tensorSharedCoordinateProduct_nonneg {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)
(d : ℕ) (j : SpinTensorIndex I degree) (q i : ℕ) (z : SpinTensorIndex I degree → ℝ) :
0 ≤ tensorSharedCoordinateProduct eig U c I degree amplitude n b v d j j q i z z := by
induction q generalizing i z with
| zero => exact mul_self_nonneg _
| succ q ih =>
dsimp only [tensorSharedCoordinateProduct]
split_ifs
· exact integral_nonneg fun a => ih (i + 1) (z + a)
· exact mul_self_nonneg _
/-- The finite pair-path expectation is an expectation over the common
ancestor path of the backward gradient product at branching. -/
theorem tensorAncestorPairPathMean_coordinates {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) (d q i : ℕ) (hqi : i + q = n)
(z₁ z₂ : SpinTensorIndex I degree → ℝ) (j₁ j₂ : SpinTensorIndex I degree) :
tensorAncestorPairPathMean eig U c I degree amplitude n b v d q i z₁ z₂
(tensorPairCoordinatePayoff eig U c I degree amplitude q z₁ z₂ j₁ j₂) =
tensorSharedCoordinateProduct eig U c I degree amplitude n b v d j₁ j₂ q i z₁ z₂ := by
induction q generalizing i z₁ z₂ with
| zero =>
have hi : i = n := by omega
subst i
simp only [tensorAncestorPairPathMean, tensorPairCoordinatePayoff, Finset.univ_eq_empty,
Finset.sum_empty, add_zero, tensorSharedCoordinateProduct, tensorBackwardCoordinateMean,
Nat.sub_self, tensorCascadeCoordinateMean]
| succ q ih =>
by_cases hid : i < d
· dsimp only [tensorAncestorPairPathMean, tensorSharedCoordinateProduct]
rw [ite_eq_left hid, ite_eq_left hid]
apply integral_congr_ae
apply ae_of_all
intro a
dsimp only
have hF : (fun w => tensorPairCoordinatePayoff eig U c I degree amplitude (q + 1)
z₁ z₂ j₁ j₂ (Fin.cons (a, a) w)) =
tensorPairCoordinatePayoff eig U c I degree amplitude q (z₁ + a) (z₂ + a) j₁ j₂ :=
funext (tensorPairCoordinatePayoff_cons eig U c I degree amplitude q z₁ z₂ a a j₁ j₂)
rw [hF]
exact ih (i + 1) (by omega) (z₁ + a) (z₂ + a)
· rw [tensorSharedCoordinateProduct, ite_eq_right hid]
exact tensorAncestorPairPathMean_independent_coordinates hN eig U c I degree amplitude
n b v hb d (q + 1) i hqi (by omega) z₁ z₂ j₁ j₂
lemma tensorAncestorPairPathMean_const_mul {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)
(d q i : ℕ) (z₁ z₂ : SpinTensorIndex I degree → ℝ)
(F : (Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) → ℝ) (C : ℝ) :
tensorAncestorPairPathMean eig U c I degree amplitude n b v d q i z₁ z₂ (fun w => C * F w) =
C * tensorAncestorPairPathMean eig U c I degree amplitude n b v d q i z₁ z₂ F := by
induction q generalizing i z₁ z₂ with
| zero => rfl
| succ q ih =>
dsimp only [tensorAncestorPairPathMean]
split_ifs <;> simp_rw [ih, integral_const_mul]
lemma tensorPairPathSpinMean_features {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) (j₁ j₂ : SpinTensorIndex I degree)
(w : Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) :
tensorPairPathSpinMean eig U c I degree amplitude n (z, w)
(fun σ => spinTensorFeature U I degree amplitude (σ 0) j₁ *
spinTensorFeature U I degree amplitude (σ 1) j₂) =
tensorPairCoordinatePayoff eig U c I degree amplitude n z z j₁ j₂ w := by
let μ := fun j : Fin 2 => gibbsProbability (uniformSpinPrior N : Measure (Spin N))
(tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n (z, w) j))
change (∫ σ : Fin 2 → Spin N, spinTensorFeature U I degree amplitude (σ 0) j₁ *
spinTensorFeature U I degree amplitude (σ 1) j₂ ∂Measure.pi μ) = _
have h := integral_fin_nat_prod_eq_prod (μ := μ)
(fun j σ => spinTensorFeature U I degree amplitude σ (if j = 0 then j₁ else j₂))
have h10 : (1 : Fin 2) ≠ 0 := by decide
simpa only [Fin.prod_univ_two, ite_eq_left rfl, ite_true, ite_eq_right h10, μ,
integral_uniformSpinGibbs_eq, tensorTerminalCoordinateMean, tensorPairCoordinatePayoff,
tensorPairPathState] using h
/-- Actual finite tensor feature pair tests are expectations of backward
gradient products along the shared ancestor path. -/
theorem tensorFeaturePair_backward_identity (hpub : PanchenkoTalagrandTensorPairInput)
{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)
(ψ : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B)
(j₁ j₂ : SpinTensorIndex I degree) :
tensorSpinPairAverage eig U c I degree amplitude n b v z ψ
(fun σ => spinTensorFeature U I degree amplitude (σ 0) j₁ *
spinTensorFeature U I degree amplitude (σ 1) j₂) =
∫ α : ℕ → LabeledLeaf n, ψ (labeledCommonDepth n (α 0) (α 1)) *
tensorSharedCoordinateProduct eig U c I degree amplitude n b v
(labeledCommonDepth n (α 0) (α 1)) j₁ j₂ n 0 z z ∂cascadeReplicaLaw n b := by
have hf (σ : Spin N) (j : SpinTensorIndex I degree) :
|spinTensorFeature U I degree amplitude σ j| ≤ tensorFeatureCoordinateCap U I degree amplitude j := by
exact Finset.single_le_sum (fun τ _ => abs_nonneg (spinTensorFeature U I degree amplitude τ j))
(Finset.mem_univ σ)
rw [tensorSpinPair_markov_identity hpub hN eig U c I degree amplitude n b v hb z ψ _ hB hψ
(C := tensorFeatureCoordinateCap U I degree amplitude j₁ * tensorFeatureCoordinateCap U I degree amplitude j₂)
(fun σ => by
rw [abs_mul]
exact mul_le_mul (hf (σ 0) j₁) (hf (σ 1) j₂) (abs_nonneg _)
(Finset.sum_nonneg fun _ _ => abs_nonneg _))]
apply integral_congr_ae
apply ae_of_all
intro α
simp only [tensorPairSpinPathPayoff, tensorPairPathSpinMean_features]
rw [tensorAncestorPairPathMean_const_mul,
tensorAncestorPairPathMean_coordinates hN eig U c I degree amplitude n b v hb _ n 0 (by omega)]
theorem tensorFeaturePair_nonneg (hpub : PanchenkoTalagrandTensorPairInput)
{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)
(ψ : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) (hψ0 : ∀ d, 0 ≤ ψ d)
(j : SpinTensorIndex I degree) :
0 ≤ tensorSpinPairAverage eig U c I degree amplitude n b v z ψ
(fun σ => spinTensorFeature U I degree amplitude (σ 0) j *
spinTensorFeature U I degree amplitude (σ 1) j) := by
rw [tensorFeaturePair_backward_identity hpub hN eig U c I degree amplitude n b v hb z ψ hB hψ j j]
exact integral_nonneg fun α => mul_nonneg (hψ0 _)
(tensorSharedCoordinateProduct_nonneg eig U c I degree amplitude n b v _ j n 0 z)
end InvariantIsing
end
end OAI
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