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e5a97c6 | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.Combinatorics.PerfectMatching.SecondVariation
import OAI.Combinatorics.PerfectMatching.Face
namespace OAI
/-!
Boundary lower semicontinuity of maximum entropy and attainment of the entropy defect.
-/
noncomputable section
open scoped BigOperators Topology
open Filter
namespace MatchingEntropy
variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
omit [Fintype V] [DecidableEq V] in
/-- Concavity of H follows from mixing genuine optimizing probability laws. -/
theorem maxMatchingEntropy_mix_ge (G : LooplessGraph V E) {x y : E → ℝ}
(hx : x ∈ G.polytope) (hy : y ∈ G.polytope) {t : ℝ}
(ht : t ∈ Set.Icc (0 : ℝ) 1) :
(1-t)*maxMatchingEntropy G x+t*maxMatchingEntropy G y ≤
maxMatchingEntropy G ((1-t) • x+t • y) := by
obtain ⟨p,hp,hpe⟩ := maxMatchingEntropy_attained G hx
obtain ⟨q,hq,hqe⟩ := maxMatchingEntropy_attained G hy
rw [← hpe,← hqe]
apply (entropy_mix_ge hp.1.1 hq.1.1 ht).trans
apply entropy_le_maxMatchingEntropy
refine ⟨convex_probabilitySimplex _ hp.1 hq.1 (sub_nonneg.mpr ht.2) ht.1 (by ring),?_⟩
change G.meanLinear ((1-t) • p+t • q) = _
rw [map_add,map_smul,map_smul]
change (1-t) • G.mean p+t • G.mean q = _
rw [hp.2,hq.2]
/-- Boundary lower semicontinuity, using the finite matching-polytope constraints.
A fixed small mixing coefficient suffices for each prescribed entropy tolerance. -/
theorem maxMatchingEntropy_lowerSemicontinuousOn (G : LooplessGraph V E)
(heven : Even (Fintype.card V)) :
LowerSemicontinuousOn (maxMatchingEntropy G) G.polytope := by
intro x hx r hr
have hn := maxMatchingEntropy_nonneg G hx
have hden : 0 < maxMatchingEntropy G x+1 := by linarith
have ha : 0 < min (1:ℝ) ((maxMatchingEntropy G x-r)/(maxMatchingEntropy G x+1)) :=
lt_min (by norm_num) (div_pos (sub_pos.mpr hr) hden)
obtain ⟨t,ht0,htt⟩ := exists_between ha
have ht1 : t<1 := htt.trans_le (min_le_left _ _)
have htb : t*(maxMatchingEntropy G x+1) < maxMatchingEntropy G x-r :=
(lt_div_iff₀ hden).mp (htt.trans_le (min_le_right _ _))
filter_upwards [G.polytope_eventually_expand heven hx ht0] with y hy
have hh := maxMatchingEntropy_mix_ge G hx hy ⟨ht0.le,ht1.le⟩
have he : (1-t) • x+t • (x+t⁻¹ • (y-x))=y := by
ext e
simp only [Pi.add_apply,Pi.smul_apply,Pi.sub_apply,smul_eq_mul]
field_simp
ring
rw [he] at hh
have hz := mul_nonneg ht0.le (maxMatchingEntropy_nonneg G hy)
nlinarith
/-- The actual entropy defect attains a maximum on the closed matching polytope. -/
theorem entropyDefect_attains_max (G : LooplessGraph V E)
(heven : Even (Fintype.card V)) (hPM : Nonempty G.Matching) (m : ℝ) :
∃ x ∈ G.polytope, ∀ y ∈ G.polytope, entropyDefect G m y ≤ entropyDefect G m x := by
have hc : Continuous (fun y : E → ℝ => entropyBarrier m (entropy y)) := by
have hb : Continuous (entropyBarrier m) := by unfold entropyBarrier; fun_prop
exact hb.comp continuous_entropy
have hs : UpperSemicontinuousOn (entropyDefect G m) G.polytope := by
change UpperSemicontinuousOn (fun y => entropyBarrier m (entropy y)-maxMatchingEntropy G y) G.polytope
simpa only [sub_eq_add_neg, Pi.neg_apply] using
hc.continuousOn.upperSemicontinuousOn.add
(maxMatchingEntropy_lowerSemicontinuousOn G heven).neg
have hn : G.polytope.Nonempty := by
obtain ⟨M⟩ := hPM
exact ⟨G.indicator M,subset_convexHull ℝ _ (Set.mem_range_self M)⟩
exact UpperSemicontinuousOn.exists_isMaxOn hn G.compact_polytope hs
end MatchingEntropy
end
end OAI
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