The MinSR equation was derived to address underdetermined linear equations in the context of variational Monte Carlo (VMC). The MinSR method aims to minimize higher-order effects, prevent overfitting, and improve stability by employing the least-squares minimum-norm condition. Two approaches, the Lagrangian multiplier method and the pseudo-inverse method, were used to derive the MinSR formula. The Lagrangian method involved minimizing the variational step under the constraint of minimum residual error, while the pseudo-inverse method simplified the notation to find the least-squares minimum-norm solution.
The MinSR solution was derived for complex neural networks, focusing on ResNet1 and ResNet2 architectures. ResNet1 consists of two convolutional layers in each residual block, while ResNet2 removes normalization layers and uses different activation functions. Sign structures, symmetry considerations, and zero-variance extrapolation were discussed to enhance the accuracy and stability of neural quantum states (NQS).
The Lanczos step was introduced as a method to improve variational accuracy by constructing new states orthogonal to the variational wavefunction. The Lanczos step involved computing an initial guess of the parameter alpha, estimating the energy variance, and performing an extrapolation to minimize errors. The Lanczos step aimed to reduce the variational error and enhance the accuracy of the ground-state energy estimation in VMC.
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