| import OAI.Combinatorics.MatchingEntropy.Face |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators Topology |
| open Filter |
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| universe uV uE uI uJ uA uB uC uW uF uD uι |
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| namespace MatchingEntropyBounds |
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| variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
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| def marginalEntropy (y : E → ℝ) : ℝ := entropy y |
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|
| theorem entropy_eq {ι : Type uι} [Fintype ι] (p : ι → ℝ) : |
| entropy p = -(∑ i, p i * Real.log (p i)) := by |
| simp [entropy, Real.negMulLog, Finset.sum_neg_distrib, neg_mul] |
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|
| theorem continuous_entropy {ι : Type uι} [Fintype ι] : |
| Continuous (entropy : (ι → ℝ) → ℝ) := by |
| unfold entropy |
| fun_prop |
|
|
| theorem entropy_nonneg {ι : Type uι} [Fintype ι] {p : ι → ℝ} |
| (hp : p ∈ probabilitySimplex ι) : 0 ≤ entropy p := by |
| apply Finset.sum_nonneg |
| intro i _ |
| exact Real.negMulLog_nonneg (mem_Icc_of_mem_probabilitySimplex hp i).1 |
| (mem_Icc_of_mem_probabilitySimplex hp i).2 |
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| theorem entropy_le_log_card {ι : Type uι} [Fintype ι] {p : ι → ℝ} |
| (hp : p ∈ probabilitySimplex ι) : entropy p ≤ Real.log (Fintype.card ι) := by |
| have hn : 0 < Fintype.card ι := by |
| by_contra h |
| have hz : Fintype.card ι = 0 := Nat.eq_zero_of_not_pos h |
| have : IsEmpty ι := Fintype.card_eq_zero_iff.mp hz |
| simpa using hp.2 |
| have hnR : (0 : ℝ) < Fintype.card ι := by exact_mod_cast hn |
| have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ ↦ |
| Real.negMulLog_le_one_sub_self (mul_nonneg hnR.le (hp.1 i))) |
| simp_rw [Real.negMulLog_mul] at h |
| rw [Finset.sum_add_distrib, ← Finset.sum_mul, ← Finset.mul_sum, |
| hp.2, one_mul, Finset.sum_sub_distrib, ← Finset.mul_sum, hp.2] at h |
| simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one, sub_self] at h |
| change Real.negMulLog (Fintype.card ι) + (Fintype.card ι : ℝ) * entropy p ≤ 0 at h |
| rw [Real.negMulLog] at h |
| nlinarith |
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| namespace LooplessGraph |
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| omit [Fintype V] [DecidableEq V] in |
| theorem coordinate_mem_Icc (G : LooplessGraph V E) {y : E → ℝ} |
| (hy : y ∈ G.polytope) (e : E) : y e ∈ Set.Icc (0 : ℝ) 1 := by |
| obtain ⟨p, hp, rfl⟩ := (G.mem_polytope_iff y).mp hy |
| constructor |
| · exact Finset.sum_nonneg (fun M _ ↦ mul_nonneg (hp.1 M) (G.indicator_nonneg M e)) |
| · calc |
| ∑ M, p M * G.indicator M e ≤ ∑ M, p M * 1 := |
| Finset.sum_le_sum (fun M _ ↦ mul_le_mul_of_nonneg_left (G.indicator_le_one M e) (hp.1 M)) |
| _ = 1 := by simpa using hp.2 |
|
|
| omit [Fintype E] [DecidableEq E] in |
|
|
| theorem incident_vertex_sum (G : LooplessGraph V E) (e : E) : |
| (∑ v, if G.Incident v e then (1 : ℕ) else 0) = 2 := by |
| classical |
| have he (v : V) : (if G.Incident v e then (1 : ℕ) else 0) = |
| (if G.left e = v then 1 else 0) + (if G.right e = v then 1 else 0) := by |
| by_cases hl : G.left e = v <;> by_cases hr : G.right e = v |
| · exact False.elim (G.loopless e (hl.trans hr.symm)) |
| · simp [Incident, hl, hr] |
| · simp [Incident, hl, hr] |
| · simp [Incident, hl, hr] |
| simp_rw [he, Finset.sum_add_distrib] |
| simp |
|
|
| omit [Fintype V] [Fintype E] [DecidableEq E] in |
| theorem matching_incident_sum (G : LooplessGraph V E) (M : G.Matching) (v : V) : |
| (∑ e ∈ M.val, if G.Incident v e then (1 : ℕ) else 0) = 1 := by |
| classical |
| obtain ⟨e, ⟨he, hv⟩, hu⟩ := M.property v |
| rw [Finset.sum_eq_single e] |
| · simp [hv] |
| · intro e' he' hne |
| have hn : ¬ G.Incident v e' := fun hv' ↦ hne (hu e' ⟨he', hv'⟩) |
| simp [hn] |
| · simp [he] |
| |
| omit [Fintype E] [DecidableEq E] in |
| theorem matching_card_twice (G : LooplessGraph V E) (M : G.Matching) : |
| 2 * M.val.card = Fintype.card V := by |
| classical |
| calc |
| 2 * M.val.card = ∑ e ∈ M.val, ∑ v, if G.Incident v e then (1 : ℕ) else 0 := by |
| simp_rw [G.incident_vertex_sum] |
| simp [Nat.mul_comm] |
| _ = ∑ v, ∑ e ∈ M.val, if G.Incident v e then (1 : ℕ) else 0 := Finset.sum_comm |
| _ = Fintype.card V := by |
| simp_rw [G.matching_incident_sum] |
| simp |
| |
| omit [Fintype E] [DecidableEq E] in |
| theorem matching_card (G : LooplessGraph V E) {m : ℕ} |
| (hV : Fintype.card V = 2 * m) (M : G.Matching) : M.val.card = m := by |
| have h := G.matching_card_twice M |
| omega |
| |
| theorem indicator_sum (G : LooplessGraph V E) {m : ℕ} |
| (hV : Fintype.card V = 2 * m) (M : G.Matching) : |
| ∑ e, G.indicator M e = (m : ℝ) := by |
| simp [indicator, G.matching_card hV M] |
| |
| theorem marginal_sum (G : LooplessGraph V E) {m : ℕ} |
| (hV : Fintype.card V = 2 * m) {y : E → ℝ} (hy : y ∈ G.polytope) : |
| ∑ e, y e = (m : ℝ) := by |
| obtain ⟨p, hp, rfl⟩ := (G.mem_polytope_iff y).mp hy |
| simp_rw [mean] |
| rw [Finset.sum_comm] |
| simp_rw [← Finset.mul_sum, G.indicator_sum hV] |
| rw [← Finset.sum_mul, hp.2, one_mul] |
| |
| end LooplessGraph |
| |
| omit [Fintype V] [DecidableEq V] in |
| |
| theorem maxMatchingEntropy_attained (G : LooplessGraph V E) {y : E → ℝ} |
| (hy : y ∈ G.polytope) : |
| ∃ p ∈ G.feasibleLaws y, entropy p = maxMatchingEntropy G y := by |
| have hc := (G.compact_feasibleLaws y).image continuous_entropy |
| have hn := ((G.mem_polytope_iff y).mp hy).image entropy |
| obtain ⟨p, hp, hpe⟩ := hc.sSup_mem hn |
| exact ⟨p, hp, hpe⟩ |
| |
| omit [Fintype V] [DecidableEq V] in |
| theorem maxMatchingEntropy_nonneg (G : LooplessGraph V E) {y : E → ℝ} |
| (hy : y ∈ G.polytope) : 0 ≤ maxMatchingEntropy G y := by |
| obtain ⟨p, hp, he⟩ := maxMatchingEntropy_attained G hy |
| rw [← he] |
| exact entropy_nonneg hp.1 |
| |
| omit [Fintype V] [DecidableEq V] in |
| theorem maxMatchingEntropy_le_log_card (G : LooplessGraph V E) {y : E → ℝ} |
| (hy : y ∈ G.polytope) : |
| maxMatchingEntropy G y ≤ Real.log (Fintype.card G.Matching) := by |
| obtain ⟨p, hp, he⟩ := maxMatchingEntropy_attained G hy |
| rw [← he] |
| exact entropy_le_log_card hp.1 |
| |
| omit [Fintype V] [DecidableEq V] in |
| theorem entropy_le_maxMatchingEntropy (G : LooplessGraph V E) {y : E → ℝ} |
| {p : G.Matching → ℝ} (hp : p ∈ G.feasibleLaws y) : |
| entropy p ≤ maxMatchingEntropy G y := by |
| apply le_csSup |
| · exact ((G.compact_feasibleLaws y).image continuous_entropy).bddAbove |
| · exact ⟨p, hp, rfl⟩ |
| |
| end MatchingEntropyBounds |
| |
| end |
| |
| end OAI |
| |
