The Readout Decides Which Hopfield Networks Have a Constant-Metric Energy

A modern Hopfield network retrieves a memory by iterating a map that scores the stored patterns against a query, weights them by a softmax, and forms the next state with a readout. Its iteration is backed by an energy that never increases from one step to the next. However, the frameworks that generalise its score and weighting keep the readout a weighted average of the stored patterns, which we call the pattern readout. We show that the readout decides which scores make each step a concave-convex step on an energy in a constant metric. We call a map with this property admissible. In particular, we consider scores whose gradient splits into a fixed per-pattern part and a part shared by all patterns. Among them, and for patterns in general position, the pattern readout is admissible in the Euclidean metric essentially only for the dot product. In contrast, the gradient readout takes the gradient of a potential of the scores. It is admissible for every score that is convex up to an added quadratic. Among 32 published retrieval maps, 8 are shown not to be admissible. For 5 of them, one reason is that their readout is not the gradient of their score's potential. One of them is the adaptive Hopfield network, which can enter cycles of period two as a multiple-instance pooling layer. We keep its score on squared distances, switch to the gradient readout, and constrain its learned weights to be non-negative and non-increasing. These changes give an exact energy that is bounded below throughout training. In our multiple-instance experiments, non-negative weights alone already stop the cycling under the pattern readout, so the energy adds a guarantee that no step raises it. Overall, our results identify the readout as the component that decides which scores give an admissible map, and they show how to test and design admissible retrieval maps.