| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexWitness |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real |
|
|
| namespace OAI |
|
|
| /-! Finite entropy, rate estimates and ordered asymptotic limits. -/ |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace MatrixMultiplication.CurveLaws |
|
|
| open MatrixMultiplication.Foundation ComplexWitness |
|
|
| theorem fiberSize_real_pos (u : CurveCoordinate) : (0 : ℝ) < curveFiberSize u := by |
| exact_mod_cast curveFiberSize_pos u |
|
|
| def uniformLift (p : FiniteLaw CurveCoordinate) : FiniteLaw CurveLeaf where |
| mass s := p.mass (CW75LabelHierarchy.x s) / curveFiberSize (CW75LabelHierarchy.x s) |
| nonneg s := div_nonneg (p.nonneg _) (fiberSize_real_pos _).le |
| total := by |
| rw [sum_curveLeaves_by_X (fun u => p.mass u / (curveFiberSize u : ℝ))] |
| simpa only [mul_div_cancel₀ _ (fiberSize_real_pos _).ne'] using p.total |
| |
| theorem uniformLift_marginal (p : FiniteLaw CurveCoordinate) (u : CurveCoordinate) : |
| ((uniformLift p).map CW75LabelHierarchy.x).mass u = p.mass u := by |
| classical |
| rw [FiniteLaw.map_mass] |
| change (∑ s : CurveLeaf, |
| if CW75LabelHierarchy.x s = u then |
| p.mass (CW75LabelHierarchy.x s) / curveFiberSize (CW75LabelHierarchy.x s) |
| else 0) = p.mass u |
| rw [sum_curveLeaves_by_X |
| (fun v => if v = u then p.mass v / (curveFiberSize v : ℝ) else 0)] |
| simp only [mul_ite, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, ite_true] |
| exact mul_div_cancel₀ (p.mass u) (fiberSize_real_pos u).ne' |
| |
| theorem entropyTerm_div_fiber (p : ℝ) {d : ℝ} (hd : 0 < d) : |
| d * entropyTerm (p / d) = entropyTerm p + p * Real.log d := by |
| by_cases hp : p = 0 |
| · simp [hp] |
| · rw [entropyTerm, Real.log_div hp hd.ne', entropyTerm] |
| field_simp [hd.ne'] |
| ring |
| |
| theorem uniformLift_entropy (p : FiniteLaw CurveCoordinate) : |
| finiteEntropy (uniformLift p).mass = finiteEntropy p.mass + |
| ∑ u : CurveCoordinate, p.mass u * Real.log (curveFiberSize u) := by |
| change (∑ s : CurveLeaf, |
| entropyTerm (p.mass (CW75LabelHierarchy.x s) / |
| curveFiberSize (CW75LabelHierarchy.x s))) = _ |
| rw [sum_curveLeaves_by_X |
| (fun u => entropyTerm (p.mass u / (curveFiberSize u : ℝ)))] |
| simp_rw [entropyTerm_div_fiber _ (fiberSize_real_pos _)] |
| simp only [finiteEntropy, Finset.sum_add_distrib] |
| |
| def partition (t : ℝ) : ℝ := |
| ∑ u : CurveCoordinate, (curveFiberSize u : ℝ) ^ t |
| |
| theorem partition_pos (t : ℝ) : 0 < partition t := by |
| apply Finset.sum_pos |
| · intro u _ |
| exact Real.rpow_pos_of_pos (fiberSize_real_pos u) t |
| · exact Finset.univ_nonempty |
| |
| theorem partition_closed (t : ℝ) : |
| partition t = 8 + 12 * (2 : ℝ) ^ t + 6 * (4 : ℝ) ^ t + (19 : ℝ) ^ t := by |
| rw [partition, sum_curveFiberSizes (fun d => (d : ℝ) ^ t)] |
| simp |
| |
| def marginal (t : ℝ) : FiniteLaw CurveCoordinate where |
| mass u := (curveFiberSize u : ℝ) ^ t / partition t |
| nonneg u := div_nonneg (Real.rpow_pos_of_pos (fiberSize_real_pos u) t).le |
| (partition_pos t).le |
| total := by |
| rw [← Finset.sum_div] |
| exact div_self (partition_pos t).ne' |
|
|
| theorem marginal_pos (t : ℝ) (u : CurveCoordinate) : 0 < (marginal t).mass u := |
| div_pos (Real.rpow_pos_of_pos (fiberSize_real_pos u) t) (partition_pos t) |
|
|
| theorem marginal_lt_one (t : ℝ) (u : CurveCoordinate) : (marginal t).mass u < 1 := by |
| classical |
| obtain ⟨v, hv⟩ := exists_ne u |
| calc |
| (marginal t).mass u < ∑ w : CurveCoordinate, (marginal t).mass w := |
| Finset.single_lt_sum hv (Finset.mem_univ u) (Finset.mem_univ v) |
| (marginal_pos t v) (fun w _ _ => (marginal t).nonneg w) |
| _ = 1 := (marginal t).total |
|
|
| def leafLaw (t : ℝ) : FiniteLaw CurveLeaf := uniformLift (marginal t) |
|
|
| theorem leafLaw_pos (t : ℝ) (s : CurveLeaf) : 0 < (leafLaw t).mass s := |
| div_pos (marginal_pos t _) (fiberSize_real_pos _) |
|
|
| theorem marginal_of_leafLaw (t : ℝ) : |
| ((leafLaw t).map CW75LabelHierarchy.x).mass = (marginal t).mass := by |
| funext u |
| exact uniformLift_marginal (marginal t) u |
|
|
| theorem leafLaw_uniform_in_fibers (t : ℝ) (s r : CurveLeaf) |
| (h : CW75LabelHierarchy.x s = CW75LabelHierarchy.x r) : |
| (leafLaw t).mass s = (leafLaw t).mass r := by |
| change (marginal t).mass _ / _ = (marginal t).mass _ / _ |
| rw [h] |
|
|
| def hidden (t : ℝ) : ℝ := |
| ∑ u : CurveCoordinate, (marginal t).mass u * Real.log (curveFiberSize u) |
|
|
| def incident (t : ℝ) : ℝ := Real.log (partition t) - t * hidden t |
|
|
| theorem log_marginal (t : ℝ) (u : CurveCoordinate) : |
| Real.log ((marginal t).mass u) = |
| t * Real.log (curveFiberSize u) - Real.log (partition t) := by |
| change Real.log ((curveFiberSize u : ℝ) ^ t / partition t) = _ |
| rw [Real.log_div (Real.rpow_pos_of_pos (fiberSize_real_pos u) t).ne' |
| (partition_pos t).ne', Real.log_rpow (fiberSize_real_pos u)] |
|
|
| theorem incident_entropy (t : ℝ) : finiteEntropy (marginal t).mass = incident t := by |
| unfold finiteEntropy |
| simp_rw [entropyTerm, log_marginal] |
| calc |
| (∑ u : CurveCoordinate, |
| -(marginal t).mass u * |
| (t * Real.log (curveFiberSize u) - Real.log (partition t))) = |
| (∑ u : CurveCoordinate, (marginal t).mass u) * Real.log (partition t) - |
| t * ∑ u : CurveCoordinate, |
| (marginal t).mass u * Real.log (curveFiberSize u) := by |
| simp only [Finset.sum_mul, Finset.mul_sum, ← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro u _ |
| ring |
| _ = incident t := by rw [(marginal t).total]; simp [incident, hidden] |
|
|
| theorem incident_pos (t : ℝ) : 0 < incident t := by |
| rw [← incident_entropy] |
| apply Finset.sum_pos |
| · intro u _ |
| exact mul_pos_of_neg_of_neg (neg_neg_of_pos (marginal_pos t u)) |
| (Real.log_neg (marginal_pos t u) (marginal_lt_one t u)) |
| · exact Finset.univ_nonempty |
|
|
| theorem hidden_nonneg (t : ℝ) : 0 ≤ hidden t := by |
| apply Finset.sum_nonneg |
| intro u _ |
| have hd : (1 : ℝ) ≤ curveFiberSize u := by |
| exact_mod_cast (curveFiberSize_pos u : 1 ≤ curveFiberSize u) |
| exact mul_nonneg ((marginal t).nonneg u) (Real.log_nonneg hd) |
|
|
| theorem leaf_entropy (t : ℝ) : |
| finiteEntropy (leafLaw t).mass = incident t + hidden t := by |
| rw [leafLaw, uniformLift_entropy, incident_entropy] |
| rfl |
|
|
| theorem conditional_entropy (t : ℝ) : |
| finiteEntropy (leafLaw t).mass - |
| finiteEntropy ((leafLaw t).map CW75LabelHierarchy.x).mass = hidden t := by |
| rw [marginal_of_leafLaw, leaf_entropy, incident_entropy] |
| ring |
|
|
| theorem hidden_closed (t : ℝ) : |
| hidden t = |
| (12 * (2 : ℝ) ^ t * Real.log 2 + 6 * (4 : ℝ) ^ t * Real.log 4 + |
| (19 : ℝ) ^ t * Real.log 19) / partition t := by |
| unfold hidden |
| change (∑ u : CurveCoordinate, |
| ((curveFiberSize u : ℝ) ^ t / partition t) * Real.log (curveFiberSize u)) = _ |
| simp_rw [div_mul_eq_mul_div] |
| rw [← Finset.sum_div, sum_curveFiberSizes (fun d => (d : ℝ) ^ t * Real.log d)] |
| simp |
| ring |
|
|
| theorem incident_rate (t : ℝ) : |
| ConditionalLabels.lawPairingRate (leafLaw t) curveLabels 5 curveReaderPair .xy + |
| ConditionalLabels.lawPairingRate (leafLaw t) curveLabels 5 curveReaderPair .xz = |
| incident t := by |
| rw [curveIncidentRate_eq_entropy_X, marginal_of_leafLaw, incident_entropy] |
|
|
| theorem hidden_rate (t : ℝ) : |
| ConditionalLabels.lawPairingRate (leafLaw t) curveLabels 5 curveReaderPair .yz = |
| hidden t := by |
| rw [curveHiddenRate_eq_entropy_sub_X] |
| exact conditional_entropy t |
|
|
| end MatrixMultiplication.CurveLaws |
|
|
| end |
|
|
| end OAI |
|
|
