Slide 34
Slide 34 text
Obstruction for Global P-Ł for Neural ODEs
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Linear ResNet, time-independant weights: 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x = ✓x 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
✓(x) = e✓x
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˜
f(z)
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Re(z)
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0
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⇡
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
For yj = xi, i.e. learning Id: 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
f(✓) , ||e✓ + Id||2
d
dt
θ(t) = − ∇f(θ(t))
Proposition: If θ(0) = Udiag(z(0)
1
, z(0)
2
, …)U*
then θ(t) = Udiag(z(t)
1
, z(t)
2
, …)U*
where is a gradient flow of
z(t) ∈ ℂ
˜
f(z) := |ez + 1|
Problem: does not satisfy a global P-Ł
˜
f
For Im(z(0)) = 0[2π], Re(z(t)) → − ∞
if (not invertible)
⇔ θ(0) = Id, eθ(t)
→ 0