The paper by McEliece, Posner, Rodemich, and Venkatesh rigorously analyzes the capacity of Hopfield associative memory, a neural network model for storing and retrieving information. The memory stores $n$-tuples of $\pm 1$ values, and the state of each neuron changes based on a hard-limited version of linear functions of all other neurons. The authors focus on the sum-of-outer products connection matrix construction, where the connection matrix $T$ is the sum of outer products of $m$ fundamental memories. They derive the maximum asymptotic value of $m$ for which most of the $m$ original memories can be exactly recovered, which is $n/(2 \log n)$ when no restrictions are placed on the fundamental memories. If each fundamental memory must be recoverable exactly, the maximum value is $n/(4 \log n)$. The paper also discusses extensions, such as quantizing the outer-product connection matrix, and relates this to the capacity of the quantized Gaussian channel. The authors provide a heuristic derivation of the capacity and rigorous proofs for certain cases, showing that the capacity is asymptotically $n/(2 \log n)$ or $n/(4 \log n)$ depending on the constraints.The paper by McEliece, Posner, Rodemich, and Venkatesh rigorously analyzes the capacity of Hopfield associative memory, a neural network model for storing and retrieving information. The memory stores $n$-tuples of $\pm 1$ values, and the state of each neuron changes based on a hard-limited version of linear functions of all other neurons. The authors focus on the sum-of-outer products connection matrix construction, where the connection matrix $T$ is the sum of outer products of $m$ fundamental memories. They derive the maximum asymptotic value of $m$ for which most of the $m$ original memories can be exactly recovered, which is $n/(2 \log n)$ when no restrictions are placed on the fundamental memories. If each fundamental memory must be recoverable exactly, the maximum value is $n/(4 \log n)$. The paper also discusses extensions, such as quantizing the outer-product connection matrix, and relates this to the capacity of the quantized Gaussian channel. The authors provide a heuristic derivation of the capacity and rigorous proofs for certain cases, showing that the capacity is asymptotically $n/(2 \log n)$ or $n/(4 \log n)$ depending on the constraints.