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cdec422 | 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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | 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
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