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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Fields.SpinProjectionEntropy
namespace OAI
/-! A trial measure obtained by pushing canonical spins through a finite
map. The log-partition loss consists of the expected energy loss and the
entropy of the projection. -/
noncomputable section
open scoped BigOperators
namespace InvariantIsing
theorem log_sum_exp_projection_bound {X Y : Type*}
[Fintype X] [Fintype Y] [Nonempty X] [Nonempty Y] [DecidableEq Y]
(f : X → Y) (H : X → ℝ) (K : Y → ℝ) (C : X → ℝ)
(Z : ℝ) (hZ : 0 < Z)
(hkernel : ∀ y, (∑ x ∈ Finset.univ.filter (fun x => f x = y),
Real.exp (-C x)) ≤ Z) :
Real.log (∑ x, Real.exp (H x)) ≤ Real.log (∑ y, Real.exp (K y)) +
(∑ x, finiteCanonicalWeights H x * (H x - K (f x))) +
(∑ x, finiteCanonicalWeights H x * C x) + Real.log Z := by
let p := finiteCanonicalWeights H
let q := finitePushforwardWeights f p
have htrial := log_sum_exp_ge_trial K q
(finitePushforwardWeights_nonneg f p (fun x => (finiteCanonicalWeights_pos H x).le))
((finitePushforwardWeights_sum f p).trans (finiteCanonicalWeights_sum H))
have he := finite_pushforward_entropy_bound f p C
(finiteCanonicalWeights_pos H) (finiteCanonicalWeights_sum H) Z hZ hkernel
have hcan := log_sum_exp_eq_canonical H
have hav := finitePushforwardWeights_average f p K
have hsub : (∑ x, p x * (H x - K (f x))) =
(∑ x, p x * H x) - ∑ x, p x * K (f x) := by
simp_rw [mul_sub]
rw [Finset.sum_sub_distrib]
change Real.log (∑ x, Real.exp (H x)) ≤ _
change Real.log (∑ x, Real.exp (H x)) = (∑ x, p x * H x) + finiteShannonEntropy p at hcan
change (∑ y, q y * K y) = ∑ x, p x * K (f x) at hav
change finiteShannonEntropy p - finiteShannonEntropy q ≤ _ at he
change _ ≤ Real.log (∑ y, Real.exp (K y)) at htrial
change Real.log (∑ x, Real.exp (H x)) ≤ Real.log (∑ y, Real.exp (K y)) +
(∑ x, p x * (H x - K (f x))) + (∑ x, p x * C x) + Real.log Z
rw [hsub]
linarith
end InvariantIsing
end
end OAI
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