| import OAI.Combinatorics.MatchingEntropy.MatchingCovariance |
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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 binaryEntropy (x : E → ℝ) : ℝ := entropy x + entropy (fun e => 1 - x e) |
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| def separableLower (x : E → ℝ) : ℝ := entropy x - 3 * entropy (fun e => 1 - x e) |
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| def structuralBase (k : ℕ) : ℝ := |
| if k = 1 then 1 else (k : ℝ) * (1 - 1 / (k : ℝ)) ^ (3 * (k - 1)) |
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| omit [Fintype V] [DecidableEq V] [DecidableEq E] in |
| theorem continuous_separableLower : Continuous (separableLower (E := E)) := by |
| exact continuous_entropy.sub (continuous_const.mul (continuous_entropy.comp (by fun_prop))) |
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| def separableLineD (x v : E → ℝ) (t : ℝ) : ℝ := |
| entropyLineD x v t - 3 * entropyLineD (fun e => 1-x e) (fun e => -v e) t |
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| omit [Fintype V] [DecidableEq V] [DecidableEq E] in |
| theorem separable_line_eventually_hasDerivAt (x v : E → ℝ) |
| (hzero : ∀ e, x e = 0 → v e = 0) (hone : ∀ e, x e = 1 → v e = 0) : |
| ∀ᶠ t : ℝ in 𝓝 0, |
| HasDerivAt (fun t => separableLower (fun e => x e+t*v e)) |
| (separableLineD x v t) t := by |
| have hbar : ∀ e, 1-x e=0 → -v e=0 := by |
| intro e he |
| rw [hone e (by linarith), neg_zero] |
| filter_upwards [entropy_line_eventually_hasDerivAt x v hzero, |
| entropy_line_eventually_hasDerivAt (fun e => 1-x e) (fun e => -v e) hbar] |
| with t ht hb |
| have he : (fun t => separableLower (fun e => x e+t*v e)) = |
| (fun t => entropy (fun e => x e+t*v e) - |
| 3 * entropy (fun e => (1-x e)+t*(-v e))) := by |
| funext t |
| unfold separableLower |
| congr 2 |
| congr 1 |
| funext e |
| ring |
| rw [he] |
| exact ht.sub (hb.const_mul 3) |
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| omit [Fintype V] [DecidableEq V] [DecidableEq E] in |
| theorem separableLineD_hasDerivAt (x v : E → ℝ) |
| (hzero : ∀ e, x e = 0 → v e = 0) (hone : ∀ e, x e = 1 → v e = 0) : |
| HasDerivAt (separableLineD x v) |
| (-(∑ e, (v e)^2/x e) + 3*(∑ e, (v e)^2/(1-x e))) 0 := by |
| have hbar : ∀ e, 1-x e=0 → -v e=0 := by |
| intro e he |
| rw [hone e (by linarith), neg_zero] |
| have hd := (entropyLineD_hasDerivAt x v hzero).sub |
| ((entropyLineD_hasDerivAt (fun e => 1-x e) (fun e => -v e) hbar).const_mul 3) |
| exact hd.congr_deriv (by simp only [neg_sq]; ring) |
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| omit [Fintype V] [DecidableEq V] [DecidableEq E] in |
| theorem separable_second_variation {A : Type uA} [Fintype A] |
| (x v : E → ℝ) (hzero : ∀ e, x e=0 → v e=0) (hone : ∀ e, x e=1 → v e=0) |
| {p : A → ℝ} (hp : p ∈ probabilitySimplex A) (a : A → ℝ) |
| (hcenter : ∑ i, p i*a i=0) (c₀ c₁ : ℝ) |
| (hlocal : IsLocalMax (fun t => separableLower (fun e => x e+t*v e) + |
| c₀+t*c₁+t^2*(∑ i, p i*(a i)^2)-Real.log (partition p a t)) 0) : |
| -(∑ e, (v e)^2/x e) + 3*(∑ e, (v e)^2/(1-x e)) + |
| (∑ i, p i*(a i)^2) ≤ 0 := by |
| have hf := separable_line_eventually_hasDerivAt x v hzero hone |
| have hg : ∀ᶠ t : ℝ in 𝓝 0, HasDerivAt (fun t => Real.log (partition p a t)) |
| (partition (fun i => p i*a i) a t / partition p a t) t := |
| Filter.Eventually.of_forall (logPartition_hasDerivAt hp a) |
| have hd := second_deriv_quadratic_adjustment (c₀ := c₀) (c₁ := c₁) |
| (v := ∑ i, p i*(a i)^2) hf hg (separableLineD_hasDerivAt x v hzero hone) |
| (logPartition_second hp a) |
| have hc : ContinuousAt (fun t => separableLower (fun e => x e+t*v e) + |
| c₀+t*c₁+t^2*(∑ i, p i*(a i)^2)-Real.log (partition p a t)) 0 := by |
| apply ContinuousAt.sub |
| · apply ContinuousAt.add |
| · apply ContinuousAt.add |
| · exact hf.self_of_nhds.continuousAt.add_const c₀ |
| · fun_prop |
| · fun_prop |
| · exact hg.self_of_nhds.continuousAt |
| have hh := second_deriv_nonpos_of_local_max hlocal hc |
| rw [hd.deriv, hcenter] at hh |
| nlinarith |
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| omit [Fintype V] [DecidableEq V] in |
| theorem mean_score_one (G : LooplessGraph V E) {p : G.Matching → ℝ} |
| (hp : p ∈ probabilitySimplex G.Matching) (a : G.Matching → ℝ) |
| (hcenter : ∑ M, p M*a M=0) : |
| ∀ e, G.mean p e=1 → G.mean (fun M => p M*a M) e=0 := by |
| intro e he |
| have hz : ∑ M, p M*(1-G.indicator M e)=0 := by |
| simp only [mul_sub, mul_one, Finset.sum_sub_distrib, hp.2] |
| change 1-G.mean p e=0 |
| rw [he, sub_self] |
| have hterms : ∀ M, p M*(1-G.indicator M e)=0 := by |
| exact fun M => (Finset.sum_eq_zero_iff_of_nonneg |
| (fun M (_ : M ∈ Finset.univ) => |
| mul_nonneg (hp.1 M) (sub_nonneg.mpr (G.indicator_le_one M e)))).mp hz M (by simp) |
| calc |
| _ = ∑ M, p M*a M := by |
| apply Finset.sum_congr rfl |
| intro M _ |
| have hh := hterms M |
| have hh' : a M*(p M*(1-G.indicator M e))=0 := by rw [hh, mul_zero] |
| nlinarith [hh'] |
| _ = 0 := hcenter |
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| omit [Fintype V] [DecidableEq V] in |
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| theorem matching_separable_second_variation (G : LooplessGraph V E) {x : E → ℝ} |
| {p : G.Matching → ℝ} (hp : p ∈ G.feasibleLaws x) |
| (hopt : entropy p=maxMatchingEntropy G x) |
| (hmax : ∀ y ∈ G.polytope, separableLower y-maxMatchingEntropy G y ≤ |
| separableLower x-maxMatchingEntropy G x) |
| (u : E → ℝ) (c : ℝ) (hcenter : ∑ M, p M*linearScore G u c M=0) : |
| -(∑ e, (G.mean (fun M => p M*linearScore G u c M) e)^2/x e) + |
| 3*(∑ e, (G.mean (fun M => p M*linearScore G u c M) e)^2/(1-x e)) + |
| (∑ M, p M*(linearScore G u c M)^2) ≤ 0 := by |
| let a := linearScore G u c |
| let v := G.mean (fun M => p M*a M) |
| let c₀ := ∑ M, p M*Real.log (p M) |
| let c₁ := ∑ M, (p M*a M)*Real.log (p M) |
| have hzero : ∀ e, x e=0 → v e=0 := by |
| rw [← hp.2] |
| exact mean_score_support G hp.1 a |
| have hone : ∀ e, x e=1 → v e=0 := by |
| rw [← hp.2] |
| exact mean_score_one G hp.1 a hcenter |
| apply separable_second_variation x v hzero hone hp.1 a hcenter c₀ c₁ |
| filter_upwards [score_law_eventually_simplex hp.1 a hcenter] with t ht |
| let r := fun M => p M+t*(p M*a M) |
| have hr : r ∈ probabilitySimplex G.Matching := ht |
| have hrmean : G.mean r=fun e => x e+t*v e := by |
| change G.meanLinear (p+t • (fun M => p M*a M)) = _ |
| rw [map_add, map_smul] |
| change G.mean p+t • v = _ |
| rw [hp.2] |
| rfl |
| have hdual := matching_entropy_dual_bound G hp hopt hr |
| (fun M hM => by dsimp [r]; simp [hM]) u c t |
| have hscore : (∑ M, r M*a M)=t*(∑ M, p M*(a M)^2) := by |
| dsimp [r] |
| simp_rw [add_mul, mul_assoc t, Finset.sum_add_distrib, ← Finset.mul_sum] |
| rw [hcenter, zero_add] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro M _ |
| ring |
| have hlog : (∑ M, r M*Real.log (p M))=c₀+t*c₁ := by |
| dsimp [r, c₀, c₁] |
| simp_rw [add_mul, mul_assoc t, Finset.sum_add_distrib, ← Finset.mul_sum] |
| change maxMatchingEntropy G (G.mean r) ≤ |
| Real.log (partition p a t) - (∑ M, r M*Real.log (p M)) - t*(∑ M, r M*a M) at hdual |
| rw [hlog, hscore] at hdual |
| have hmaxr := hmax (G.mean r) (G.mean_mem_polytope hr) |
| have hpart := partition_zero hp.1 a |
| simp only [zero_mul, zero_pow (by omega : 2 ≠ 0), add_zero, hpart, |
| Real.log_one, sub_zero] |
| rw [← hopt, entropy_eq p] at hmaxr |
| rw [← hrmean] |
| dsimp only [c₀] at * |
| nlinarith |
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| end MatchingEntropyBounds |
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| end |
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| end OAI |
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