Adjusting Regularization Parameters in Group-Sparce User Activity Massive MIMO

User activity detection (UAD) is a critical procedure in massive machine-type communications (mMTC) massive MIMO systems (M-MIMO). Physical phenomena are continuously monitored by sensors in the real world, leading to well-defined activity patterns in the data. This work investigates adjustments in regularization weight parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda$</tex> of sparse group regularizer GLASSO under activity correlation scenarios. By adjusting the input parameters in such regularizer, one can improve the non-Bayesian UAD performance, where pilot hopping (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{(H H}$</tex>) sequences are deployed by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{m M T C}$</tex> sensors. Several input factors influence UAD performance, specifically the square cell side <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Q$</tex>, the number of sensor terminals (UT) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$U$</tex>, and path loss coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\xi$</tex>. Our work identifies how changes in these factors can be incorporated into the regularization parameter to improve the UAD performance. We present an analytical framework that links the optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda_{\text{opt}}$</tex> to the key system parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\xi, Q$</tex>, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$U$</tex> through power-law functions. We fit results curves and derive expressions that effectively quantify the relationships for three parameters, achieving high coefficients of determination and low mean squared errors in all three fits with high accuracy. Furthermore, we propose a joint fit expression for the optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda_{\text{opt}}$</tex>, enabling automatic parameter tuning based on system and channel configuration. This advancement enhances the efficiency and robustness of GLASSO-based UAD systems across various operational conditions and channel scenarios.