Associative memory and dead neurons

In ``Large Associative Memory Problem in Neurobiology and Machine Learning,'' Dmitry Krotov and John Hopfield introduced a general technique for the systematic construction of neural ordinary differential equations with non-increasing energy or Lyapunov function. We study this energy function and identify that it is vulnerable to the problem of dead neurons. Each point in the state space where the neuron dies is contained in a non-compact region with constant energy. In these flat regions, energy function alone does not completely determine all degrees of freedom and, as a consequence, can not be used to analyze stability or find steady states or basins of attraction. We perform a direct analysis of the dynamical system and show how to resolve problems caused by flat directions corresponding to dead neurons: (i) all information about the state vector at a fixed point can be extracted from the energy and Hessian matrix (of Lagrange function), (ii) it is enough to analyze stability in the range of Hessian matrix, (iii) if steady state touching flat region is stable the whole flat region is the basin of attraction. The analysis of the Hessian matrix can be complicated for realistic architectures, so we show that for a slightly altered dynamical system (with the same structure of steady states), one can derive a diverse family of Lyapunov functions that do not have flat regions corresponding to dead neurons. In addition, these energy functions allow one to use Lagrange functions with Hessian matrices that are not necessarily positive definite and even consider architectures with non-symmetric feedforward and feedback connections.