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namespace OAI
/-! Coordinate derivatives through the actual finite Gaussian cascade recursion. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped BigOperators Topology NNReal ENNReal
namespace InvariantIsing
/-- The bounded coordinate derivative gives a local exponential dominator,
even when the terminal itself is unbounded. -/
lemma hasDerivAt_logMean_of_bounded_derivative
{A : Type*} [MeasurableSpace A] (μ : Measure A) [IsProbabilityMeasure μ]
(F D : ℝ → A → ℝ) (hF : ∀ s, Measurable (F s)) (hD : ∀ s, Measurable (D s))
{K : ℝ} (hK : 0 ≤ K) (hbound : ∀ s a, |D s a| ≤ K)
(hd : ∀ s a, HasDerivAt (fun q => F q a) (D s a) s)
{b : ℝ} (hb : b ≠ 0) (t : ℝ)
(hi : Integrable (fun a => Real.exp (b * F t a)) μ) :
HasDerivAt (fun s => logMean b μ (F s))
(∫ a, D t a ∂μ.tilted (fun a => b * F t a)) t := by
have hdiff (a : A) (s : ℝ) : |F s a - F t a| ≤ K * |s - t| := by
simpa only [Real.norm_eq_abs] using
Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
(fun q (_ : q ∈ (Set.univ : Set ℝ)) => (hd q a).hasDerivWithinAt)
(fun q _ => by simpa only [Real.norm_eq_abs] using hbound q a)
convex_univ (Set.mem_univ t) (Set.mem_univ s)
have hloc (s : ℝ) (hs : s ∈ Ioo (t - 1) (t + 1)) (a : A) :
b * F s a ≤ b * F t a + |b| * K := by
have hst : |s - t| ≤ 1 := abs_le.mpr ⟨by linarith [hs.1], by linarith [hs.2]⟩
have h1 := (hdiff a s).trans (mul_le_of_le_one_right hK hst)
have h2 : b * (F s a - F t a) ≤ |b| * K := (le_abs_self (b * (F s a - F t a))).trans
(by rw [abs_mul]; exact mul_le_mul_of_nonneg_left h1 (abs_nonneg b))
nlinarith
let C := Real.exp (|b| * K) * (|b| * K)
have hdom : Integrable (fun a => C * Real.exp (b * F t a)) μ := hi.const_mul C
have hnorm (s : ℝ) (hs : s ∈ Ioo (t - 1) (t + 1)) (a : A) :
‖Real.exp (b * F s a) * (b * D s a)‖ ≤ C * Real.exp (b * F t a) := by
rw [Real.norm_eq_abs, abs_mul, abs_of_pos (Real.exp_pos _), abs_mul]
calc
_ ≤ Real.exp (b * F t a + |b| * K) * (|b| * K) :=
mul_le_mul (Real.exp_le_exp.mpr (hloc s hs a))
(mul_le_mul_of_nonneg_left (hbound s a) (abs_nonneg b))
(mul_nonneg (abs_nonneg _) (abs_nonneg _)) (Real.exp_pos _).le
_ = C * Real.exp (b * F t a) := by rw [Real.exp_add]; dsimp only [C]; ring
obtain ⟨_, hI⟩ := hasDerivAt_integral_of_dominated_loc_of_deriv_le
(Ioo_mem_nhds (by linarith : t - 1 < t) (by linarith : t < t + 1))
(Eventually.of_forall fun s => ((hF s).const_mul b).exp.aestronglyMeasurable) hi
(((hF t).const_mul b).exp.mul ((hD t).const_mul b)).aestronglyMeasurable
(ae_of_all _ fun a s hs => hnorm s hs a) hdom
(ae_of_all _ fun a s _ => ((hd s a).const_mul b).exp)
have hpos := IsingPerceptron.integral_exp_pos μ (F t) hi
convert (hI.log hpos.ne').div_const b using 1
· rfl
· rw [integral_tilted_eq_div]
have he : (∫ a, Real.exp (b * F t a) * (b * D t a) ∂μ) =
b * ∫ a, Real.exp (b * F t a) * D t a ∂μ := by
rw [← integral_const_mul]
congr 1
funext a
ring
rw [he]
field_simp
lemma tensorCascadeBackward_step {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) :
tensorCascadeBackward eig U c I degree amplitude n b v i z =
logMean (b i) (tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ))
(fun a => tensorCascadeBackward eig U c I degree amplitude n b v (i + 1) (z + a)) :=
backwardValue_step n b (fun j => tensorGaussianLaw I degree (v j))
(fun _ p => p.1 + p.2) (spinTensorTerminal eig U c I degree amplitude) hi z
/-- The normalized retained mark transition is exactly the Gaussian tilt
by the next actual backward value. -/
theorem tensorAncestorMarkKernel_eq_tilted {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) (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)).tilted
(fun a => b i * tensorCascadeBackward eig U c I degree amplitude n b v (i + 1) (z + a)) := by
have hint := 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) i hi z
have hp := IsingPerceptron.integral_exp_pos
(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ))
(fun a => tensorCascadeBackward eig U c I degree amplitude n b v (i + 1) (z + a)) hint
rw [tensorAncestorMarkKernel_density eig U c I degree amplitude n b v i hi z,
tensorCascadeBackward_step eig U c I degree amplitude n b v i hi z]
unfold Measure.tilted
congr 1
funext a
congr 1
have he (x y : ℝ) : b i * (x - Real.log y / b i) = b i * x - Real.log y := by
field_simp [(hb.1 i hi).1.ne']
rw [logMean, he, Real.exp_sub, Real.exp_log hp]
def tensorCascadeCoordinateMean {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) → (ℕ → ℝ) → (ℕ → SpinTensorIndex I degree → ℝ≥0) →
(SpinTensorIndex I degree → ℝ) → SpinTensorIndex I degree → ℝ
| 0, _, _, z, j => tensorTerminalCoordinateMean eig U c I degree amplitude z j
| n + 1, b, v, z, j =>
∫ a, tensorCascadeCoordinateMean eig U c I degree amplitude n
(fun i => b (i + 1)) (fun i => v (i + 1)) (z + a) j
∂tensorAncestorMarkKernel eig U c I degree amplitude (n + 1) b v 0 z
lemma measurable_tensorCascadeCoordinateMean {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) (j : SpinTensorIndex I degree) :
Measurable (fun z => tensorCascadeCoordinateMean eig U c I degree amplitude n b v z j) := by
induction n generalizing b v with
| zero => exact measurable_tensorTerminalCoordinateMean eig U c I degree amplitude j
| succ n ih =>
let := tensorAncestorMarkKernel_markov hN eig U c I degree amplitude (n + 1) b v hb 0
exact (((ih _ _ hb.tail).comp (measurable_fst.add measurable_snd)).stronglyMeasurable
|>.integral_kernel_prod_right').measurable
lemma tensorCascadeCoordinateMean_abs_le {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) (j : SpinTensorIndex I degree) :
|tensorCascadeCoordinateMean eig U c I degree amplitude n b v z j| ≤
tensorFeatureCoordinateCap U I degree amplitude j := by
induction n generalizing b v z with
| zero => exact tensorTerminalCoordinateMean_abs_le eig U c I degree amplitude z j
| succ n ih =>
let := tensorAncestorMarkKernel_markov hN eig U c I degree amplitude (n + 1) b v hb 0
simpa only [tensorCascadeCoordinateMean, Real.norm_eq_abs, probReal_univ, mul_one] using
norm_integral_le_of_norm_le_const
(μ := tensorAncestorMarkKernel eig U c I degree amplitude (n + 1) b v 0 z)
(f := fun a => tensorCascadeCoordinateMean eig U c I degree amplitude n
(fun i => b (i + 1)) (fun i => v (i + 1)) (z + a) j)
(ae_of_all _ fun a => by simpa only [Real.norm_eq_abs] using ih _ _ hb.tail (z + a))
lemma tensorCoordinateVariation_add {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (z a : SpinTensorIndex I degree → ℝ)
(j : SpinTensorIndex I degree) (s : ℝ) :
tensorCoordinateVariation I degree z j s + a = tensorCoordinateVariation I degree (z + a) j s := by
funext i
simp only [tensorCoordinateVariation, Pi.add_apply]
ring
/-- Every coordinate derivative of the actual backward value is the
corresponding mean propagated through the ancestor kernels. -/
theorem hasDerivAt_tensorCascadeValue_coordinate {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)
(j : SpinTensorIndex I degree) (t : ℝ) :
HasDerivAt (fun s => tensorCascadeValue eig U c I degree amplitude n b v
(tensorCoordinateVariation I degree z j s))
(tensorCascadeCoordinateMean eig U c I degree amplitude n b v
(tensorCoordinateVariation I degree z j t) j) t := by
induction n generalizing b v z t with
| zero => exact hasDerivAt_spinTensorTerminal_coordinate eig U c I degree amplitude z j t
| succ n ih =>
let bs := fun i => b (i + 1)
let vs := fun i => v (i + 1)
let G := tensorCascadeValue eig U c I degree amplitude n bs vs
let D := fun x => tensorCascadeCoordinateMean eig U c I degree amplitude n bs vs x j
have hm : Measurable G := measurable_cascadeRecursion n bs
(fun i => tensorGaussianLaw I degree (vs i)) (fun _ => measurable_fst.add measurable_snd)
(measurable_spinTensorTerminal eig U c I degree amplitude)
have hDm : Measurable D := measurable_tensorCascadeCoordinateMean hN eig U c I degree amplitude n bs vs hb.tail j
have hg : HasLinearGrowth G := cascadeRecursion_linearGrowth n bs
(fun i => tensorGaussianLaw I degree (vs i))
(fun i _ => tensorGaussianLaw_moments hN I degree (vs i))
(measurable_spinTensorTerminal eig U c I degree amplitude)
(spinTensorTerminal_linearGrowth eig U c I degree amplitude) (fun i hi => (hb.tail.1 i hi).1)
have hint : Integrable (fun a => Real.exp (b 0 * G (tensorCoordinateVariation I degree z j t + a)))
(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ)) :=
integrable_exp_of_linearGrowth _ (tensorGaussianLaw_moments hN I degree (v 0))
(hm.comp (measurable_const.add measurable_id))
(hg.add_left (tensorCoordinateVariation I degree z j t)) (b 0)
have hd (s : ℝ) (a : SpinTensorIndex I degree → ℝ) :
HasDerivAt (fun q => G (tensorCoordinateVariation I degree z j q + a))
(D (tensorCoordinateVariation I degree z j s + a)) s := by
simpa only [tensorCoordinateVariation_add, G, D] using ih bs vs hb.tail (z + a) s
have h := hasDerivAt_logMean_of_bounded_derivative
(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))
(fun s a => G (tensorCoordinateVariation I degree z j s + a))
(fun s a => D (tensorCoordinateVariation I degree z j s + a))
(fun s => hm.comp (measurable_const.add measurable_id))
(fun s => hDm.comp (measurable_const.add measurable_id))
(K := tensorFeatureCoordinateCap U I degree amplitude j)
(Finset.sum_nonneg fun _ _ => abs_nonneg _)
(fun s a => tensorCascadeCoordinateMean_abs_le hN eig U c I degree amplitude n bs vs hb.tail _ j)
hd (hb.1 0 (by omega)).1.ne' t hint
have hk := tensorAncestorMarkKernel_eq_tilted hN eig U c I degree amplitude
(n + 1) b v hb 0 (by omega) (tensorCoordinateVariation I degree z j t)
have hv : tensorCascadeBackward eig U c I degree amplitude (n + 1) b v 1 = G := by
dsimp only [G, tensorCascadeValue, tensorCascadeBackward, backwardValue, bs, vs]
simp only [Nat.add_sub_cancel, Nat.add_comm]
rfl
rw [hv] at hk
simpa only [tensorCascadeValue, cascadeRecursion, tensorCascadeCoordinateMean,
hk, G, D, bs, vs] using h
def tensorBackwardCoordinateMean {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 : SpinTensorIndex I degree → ℝ) (j : SpinTensorIndex I degree) : ℝ :=
tensorCascadeCoordinateMean eig U c I degree amplitude (n - i)
(fun q => b (i + q)) (fun q => v (i + q)) z j
/-- The same derivative identity at every backward level, with the
manuscript's ancestor indexing. -/
theorem hasDerivAt_tensorCascadeBackward_coordinate {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)
(j : SpinTensorIndex I degree) (t : ℝ) :
HasDerivAt (fun s => tensorCascadeBackward eig U c I degree amplitude n b v i
(tensorCoordinateVariation I degree z j s))
(tensorBackwardCoordinateMean eig U c I degree amplitude n b v i
(tensorCoordinateVariation I degree z j t) j) t := by
have hbs : CascadeExponents (n - i) (fun q => b (i + q)) := by
constructor
· intro q hq
exact hb.1 (i + q) (by omega)
· intro q r hqr hr
exact hb.2 (i + q) (i + r) (by omega) (by omega)
exact hasDerivAt_tensorCascadeValue_coordinate hN eig U c I degree amplitude (n - i)
(fun q => b (i + q)) (fun q => v (i + q)) hbs z j t
lemma measurable_tensorBackwardCoordinateMean {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) (j : SpinTensorIndex I degree) :
Measurable (fun z => tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z j) := by
have hbs : CascadeExponents (n - i) (fun q => b (i + q)) :=
⟨fun q hq => hb.1 (i + q) (by omega),
fun q r hqr hr => hb.2 (i + q) (i + r) (by omega) (by omega)⟩
exact measurable_tensorCascadeCoordinateMean hN eig U c I degree amplitude (n - i)
(fun q => b (i + q)) (fun q => v (i + q)) hbs j
lemma tensorBackwardCoordinateMean_abs_le {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)
(j : SpinTensorIndex I degree) :
|tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z j| ≤
tensorFeatureCoordinateCap U I degree amplitude j := by
have hbs : CascadeExponents (n - i) (fun q => b (i + q)) :=
⟨fun q hq => hb.1 (i + q) (by omega),
fun q r hqr hr => hb.2 (i + q) (i + r) (by omega) (by omega)⟩
exact tensorCascadeCoordinateMean_abs_le hN eig U c I degree amplitude (n - i)
(fun q => b (i + q)) (fun q => v (i + q)) hbs z j
/-- The backward gradient is a martingale under the actual normalized
ancestor mark transitions. -/
theorem tensorBackwardCoordinateMean_step {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) (hi : i < n)
(z : SpinTensorIndex I degree → ℝ) (j : SpinTensorIndex I degree) :
tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z j =
∫ a, tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1) (z + a) j
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z := by
have hm := measurable_tensorCascadeBackward eig U c I degree amplitude n b v (i + 1)
have hDm := measurable_tensorBackwardCoordinateMean hN eig U c I degree amplitude n b v hb (i + 1) j
have hint := backwardValue_linearGrowth_moments n b
(fun q => tensorGaussianLaw I degree (v q))
(fun q _ => tensorGaussianLaw_moments hN I degree (v q))
(measurable_spinTensorTerminal eig U c I degree amplitude)
(spinTensorTerminal_linearGrowth eig U c I degree amplitude)
(fun q hq => (hb.1 q hq).1) i hi z
have hd (s : ℝ) (a : SpinTensorIndex I degree → ℝ) :
HasDerivAt (fun q => tensorCascadeBackward eig U c I degree amplitude n b v (i + 1)
(tensorCoordinateVariation I degree z j q + a))
(tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1)
(tensorCoordinateVariation I degree z j s + a) j) s := by
simpa only [tensorCoordinateVariation_add] using
hasDerivAt_tensorCascadeBackward_coordinate hN eig U c I degree amplitude n b v hb (i + 1) (z + a) j s
have hz : tensorCoordinateVariation I degree z j 0 = z := by
ext a
simp [tensorCoordinateVariation]
have hh := hasDerivAt_logMean_of_bounded_derivative
(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ))
(fun s a => tensorCascadeBackward eig U c I degree amplitude n b v (i + 1)
(tensorCoordinateVariation I degree z j s + a))
(fun s a => tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1)
(tensorCoordinateVariation I degree z j s + a) j)
(fun s => hm.comp (measurable_const.add measurable_id))
(fun s => hDm.comp (measurable_const.add measurable_id))
(K := tensorFeatureCoordinateCap U I degree amplitude j)
(Finset.sum_nonneg fun _ _ => abs_nonneg _)
(fun s a => tensorBackwardCoordinateMean_abs_le hN eig U c I degree amplitude n b v hb (i + 1) _ j)
hd (hb.1 i hi).1.ne' 0 (by simpa only [hz, tensorCascadeBackward] using hint)
have hf : (fun s => logMean (b i)
(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ))
(fun a => tensorCascadeBackward eig U c I degree amplitude n b v (i + 1)
(tensorCoordinateVariation I degree z j s + a))) =
(fun s => tensorCascadeBackward eig U c I degree amplitude n b v i
(tensorCoordinateVariation I degree z j s)) := by
funext s
exact (tensorCascadeBackward_step eig U c I degree amplitude n b v i hi _).symm
rw [hf, hz, ← tensorAncestorMarkKernel_eq_tilted hN eig U c I degree amplitude n b v hb i hi z] at hh
have hc := hasDerivAt_tensorCascadeBackward_coordinate hN eig U c I degree amplitude n b v hb i z j 0
rw [hz] at hc
exact hc.unique hh
end InvariantIsing
end
end OAI
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