Thermodynamic matrix inversion by an Ornstein-Uhlenbeck process is, to first order, exactly a gradient descent iteration on the residual, so the stochastic sampling is not needed in a digital implementation.
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Theoretical Analysis of Thermodynamic Matrix Inversion: First-order Equivalence to Preconditioned Gradient Descent and Implications for Analog Computing
Thermodynamic matrix inversion by an Ornstein-Uhlenbeck process is, to first order, exactly a gradient descent iteration on the residual, so the stochastic sampling is not needed in a digital implementation.