File size: 2,219 Bytes
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 | import OAI.Combinatorics.MatchingEntropy.Covariance
namespace OAI
noncomputable section
open scoped BigOperators Topology
open Filter Matrix
universe uV uE uI uJ uA uB uC uW uF uD uι
namespace MatchingEntropyBounds
variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
omit [Fintype V] [DecidableEq V] in
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]
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
end MatchingEntropyBounds
end
end OAI
|