| import OAI.LinearAlgebra.MatrixMultiplication.Duality.LawEntropy |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.CenterLaw |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.CompletionLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.HiddenCounts |
|
|
| namespace OAI |
|
|
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.DualHiddenRates |
|
|
| universe u |
|
|
| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels CompletionColorLaws |
| open DualCompletionLaws CompletionPartitions CompletionEntropyAlgebra |
| attribute [local instance 10000] Classical.propDecidable Classical.decEq |
| attribute [local instance 11000] instDecidableEqFin |
|
|
| variable {X Y Z : Type u} [Fintype X] [Fintype Y] [Fintype Z] |
|
|
| structure HiddenInvariant (S : FlaggedTensor X Y Z) (p : Laws S) |
| (s : DualWitness.State) : Prop where |
| B : finiteEntropy (p .B).mass = Real.log (activeCount S) + s.hB |
| A : finiteEntropy (p .A).mass = Real.log (inactiveCount S) + s.hA |
| C : finiteEntropy (p .C).mass = Real.log (inactiveCount S) + s.hC |
|
|
| variable [Nonempty X] |
|
|
| private theorem incident_minus_entropy (S : FlaggedTensor X Y Z) (output : Color) |
| (hne : Color.B ≠ output) |
| (pc : FiniteLaw (ColorSlice S .B)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (hm : 0 < m) (rho hb ho : ℝ) (h0 : 0 < rho) (h1 : rho < 1) |
| (hrho : rho = activeFraction S) |
| (hpc : finiteEntropy pc.mass = Real.log (activeCount S) + hb) |
| (hpo : finiteEntropy po.mass = Real.log (inactiveCount S) + ho) : |
| finiteEntropy (conditionalCompletionLaw S .B output pc po m hm rho h0.le h1).mass = |
| Real.log (inactiveCount (complete S .B m)) + |
| m * ((rho - rho ^ m) / (1 - rho ^ m) * hb + |
| (1 - rho) / (1 - rho ^ m) * ho) := by |
| have hc := counts_pos S (hrho ▸ h0) (hrho ▸ h1) |
| have hr : (activeCount S : ℝ) / ((activeCount S : ℝ) + inactiveCount S) = rho := |
| (activeFraction_ratio S).symm.trans hrho.symm |
| have he := conditionalMixEntropy_incident (activeCount S) (inactiveCount S) |
| hb ho m hc.1 hc.2 hm |
| dsimp only at he |
| rw [hr] at he |
| rw [DualLawEntropy.conditionalCompletionLaw_entropy S .B output hne, |
| hpc, hpo, inactiveCount_minus_real] |
| linarith only [he] |
|
|
| private theorem incident_plus_entropy (S : FlaggedTensor X Y Z) (center : Color) |
| (hc : center ≠ .B) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S .B)) |
| (m : ℕ) (hm : 0 < m) (rho hcEntropy hb : ℝ) (h0 : 0 < rho) (h1 : rho < 1) |
| (hrho : rho = activeFraction S) |
| (hpc : finiteEntropy pc.mass = Real.log (inactiveCount S) + hcEntropy) |
| (hpo : finiteEntropy po.mass = Real.log (activeCount S) + hb) : |
| finiteEntropy (conditionalCompletionLaw S center .B pc po m hm (1 - rho) |
| (sub_nonneg.mpr h1.le) (by linarith)).mass = |
| Real.log (activeCount (complete S center m)) + |
| m * (rho / (1 - (1 - rho) ^ m) * hb + |
| ((1 - (1 - rho) ^ m) - rho) / (1 - (1 - rho) ^ m) * hcEntropy) := by |
| have hcounts := counts_pos S (hrho ▸ h0) (hrho ▸ h1) |
| have hr : (inactiveCount S : ℝ) / ((inactiveCount S : ℝ) + activeCount S) = |
| 1 - rho := by rw [← inactiveFraction_ratio S, ← hrho] |
| have he := conditionalMixEntropy_incident (inactiveCount S) (activeCount S) |
| hcEntropy hb m hcounts.2 hcounts.1 hm |
| dsimp only at he |
| rw [hr] at he |
| have hcoeff : |
| (m : ℝ) * (((1 - rho) - (1 - rho) ^ m) / (1 - (1 - rho) ^ m) * hcEntropy + |
| (1 - (1 - rho)) / (1 - (1 - rho) ^ m) * hb) = |
| m * (rho / (1 - (1 - rho) ^ m) * hb + |
| ((1 - (1 - rho) ^ m) - rho) / (1 - (1 - rho) ^ m) * hcEntropy) := by ring |
| rw [hcoeff] at he |
| rw [DualLawEntropy.conditionalCompletionLaw_entropy S center .B hc, |
| hpc, hpo, activeCount_plus_real S center hc] |
| linarith only [he] |
|
|
| omit [Nonempty X] in |
| private theorem gibbs_entropy (S : FlaggedTensor X Y Z) (center output : Color) |
| (hc : center ≠ .B) (hne : center ≠ output) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (hm : 0 < m) (hcEntropy hoEntropy : ℝ) |
| (hpc : finiteEntropy pc.mass = Real.log (inactiveCount S) + hcEntropy) |
| (hpo : finiteEntropy po.mass = Real.log (inactiveCount S) + hoEntropy) : |
| finiteEntropy (conditionalCompletionLaw S center output pc po m hm |
| (gibbsWeight hcEntropy hoEntropy) (gibbsWeight_nonneg _ _) |
| (gibbsWeight_lt_one _ _)).mass = |
| Real.log (inactiveCount (complete S center m)) + |
| Real.log ((Real.exp hcEntropy + Real.exp hoEntropy) ^ m - |
| Real.exp ((m : ℝ) * hcEntropy)) := by |
| rw [DualLawEntropy.conditionalCompletionLaw_entropy S center output hne, |
| hpc, hpo, inactiveCount_plus S center hc, Nat.cast_pow, Real.log_pow] |
| have he := conditionalMixEntropy_gibbs (inactiveCount S) hcEntropy hoEntropy m hm |
| dsimp only [gibbsWeight] |
| linarith only [he] |
|
|
| theorem minus_hidden (S : FlaggedTensor X Y Z) (p : Laws S) (s : DualWitness.State) |
| (m : ℕ) (hm : 0 < m) (h0 : 0 < s.rho) (h1 : s.rho < 1) |
| (hrho : s.rho = activeFraction S) (h : HiddenInvariant S p s) : |
| HiddenInvariant (complete S .B m) (minusLaws S p m hm s.rho h0 h1) |
| (DualWitness.minus m s) := by |
| constructor |
| · dsimp only [minusLaws, DualWitness.minus] |
| rw [conditionalCompletionLaw_same_zero_entropy, h.B, |
| activeCount_minus, Nat.cast_pow, Real.log_pow] |
| ring |
| · exact incident_minus_entropy S .A (by intro hne; cases hne) (p .B) (p .A) m hm |
| s.rho s.hB s.hA h0 h1 hrho h.B h.A |
| · exact incident_minus_entropy S .C (by intro hne; cases hne) (p .B) (p .C) m hm |
| s.rho s.hB s.hC h0 h1 hrho h.B h.C |
| |
| theorem plusA_hidden (S : FlaggedTensor X Y Z) (p : Laws S) (s : DualWitness.State) |
| (m : ℕ) (hm : 0 < m) (h0 : 0 < s.rho) (h1 : s.rho < 1) |
| (hrho : s.rho = activeFraction S) (h : HiddenInvariant S p s) : |
| HiddenInvariant (complete S .A m) |
| (plusALaws S p m hm s.rho s.hA s.hC h0 h1) (DualWitness.plusA m s) := by |
| constructor |
| · exact incident_plus_entropy S .A (by intro hne; cases hne) (p .A) (p .B) m hm |
| s.rho s.hA s.hB h0 h1 hrho h.A h.B |
| · dsimp only [plusALaws, DualWitness.plusA] |
| rw [conditionalCompletionLaw_same_zero_entropy, h.A, |
| inactiveCount_plus S .A (by intro hne; cases hne), Nat.cast_pow, Real.log_pow] |
| ring |
| · exact gibbs_entropy S .A .C (by intro hne; cases hne) (by intro hne; cases hne) (p .A) (p .C) m hm |
| s.hA s.hC h.A h.C |
| |
| theorem plusC_hidden (S : FlaggedTensor X Y Z) (p : Laws S) (s : DualWitness.State) |
| (m : ℕ) (hm : 0 < m) (h0 : 0 < s.rho) (h1 : s.rho < 1) |
| (hrho : s.rho = activeFraction S) (h : HiddenInvariant S p s) : |
| HiddenInvariant (complete S .C m) |
| (plusCLaws S p m hm s.rho s.hA s.hC h0 h1) (DualWitness.plusC m s) := by |
| constructor |
| · exact incident_plus_entropy S .C (by intro hne; cases hne) (p .C) (p .B) m hm |
| s.rho s.hC s.hB h0 h1 hrho h.C h.B |
| · exact gibbs_entropy S .C .A (by intro hne; cases hne) (by intro hne; cases hne) (p .C) (p .A) m hm |
| s.hC s.hA h.C h.A |
| · dsimp only [plusCLaws, DualWitness.plusC] |
| rw [conditionalCompletionLaw_same_zero_entropy, h.C, |
| inactiveCount_plus S .C (by intro hne; cases hne), Nat.cast_pow, Real.log_pow] |
| ring |
| |
| end MatrixMultiplication.DualHiddenRates |
| |
| end |
| |
| end OAI |
| |
