Low-power millimeter wave (mmWave) multi-input multi-output communication systems can be enabled with the use of one-bit analog-to-digital converters. Owing to the extreme quantization, conventional signal processing tasks such as channel estimation are challenging, making uplink (UL) multiuser receivers difficult to implement. To address this issue, we first reformulate the UL channel estimation problem, and then combine the idea of $\ell_{1}$ regularized logistic regression classification and Toeplitz matrix reconstruction in a properly designed optimization problem. Our new method is referred to as $\ell_{1}$ regularized logistic regression with Toeplitz matrix reconstruction (L1-RLR-TMR). In addition, we develop a computationally efficient alternating direction method of multi-pliers (ADMM)-based implementation for the L1-RLR-TMR method. Numerical results demonstrate the performance of the L1-RLR-TMR method in comparison with other existing methods.