| 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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|
