Algorithm 1.

Parameter estimation in rLDS

INPUT: Initialization Ω(0) = {A(0), C(0), Q(0), R(0), ξ(0), Ψ(0)}.
PROCEDURE:
1:	repeat
2:	E-step: estimate 𝔼[zt|y], $𝔼[{\mathbf{z}}_t{\mathbf{z}}_t^{\prime }|\mathbf{y}]$ and $𝔼[{\mathbf{z}}_t{\mathbf{z}}_{t-1}^{\prime }|\mathbf{y}]$ .
3:	M-step: M1:estimate C, R, Q, ξ, Ψ by eq.(10) – eq.(14)
4:	if rLDS𝒢 then
5:	M2:estimate A by SOCP solvers.
6:	end if
7:	if rLDSℛ then
8:	M2:estimate A by generalized gradient descent algorithm.
9:	end if
10:	until Convergence
OUTPUT: Learned LDS parameters: Ω̂ = {Â, Ĉ, Q̂, R̂, ξ̂, Ψ̂}.