Aran Nayebi and Daniel Yamins derived a theorem that defines when two neural networks solving the same difficult task match at the level of individual computations

Aran Nayebi and Daniel Yamins derived a theorem that defines when two neural networks solving the same difficult task match at the level of individual computations

In their preprint, the authors specify conditions for networks with ReLU and softplus activation functions. If two networks solve the same sufficiently difficult task and their representations are related by a linear transformation, individual computational directions may also correspond. A second theorem describes how this correspondence can extend from the final layer to earlier layers.

In a preprint published on July 9, Nayebi and Yamins examine a longstanding observation in NeuroAI, a field that compares artificial neural networks with brain function. Researchers show the same images to an animal and a model, then test whether the model’s internal signals predict neuronal responses.

In a 2014 study, models were trained to recognize objects across changes in pose, size, and background. The more accurately they performed this task, the more accurately they predicted responses in two regions of the macaque visual cortex, V4 and IT. However, a linear map can connect large sets of signals while mixing the contributions of individual units. On its own, it therefore cannot show whether the model and the brain perform the same individual computations.

Weak correspondence means that the activity of one layer can be mapped onto the activity of another using a linear rule. Strong correspondence means that individual directions within a layer match, such as a filter that responds to an image edge at a particular angle.

Nayebi and Yamins’ theorem applies to networks with ReLU and softplus activation functions. ReLU maps a negative input to zero, so its response develops a kink when the input crosses the threshold. The softplus transition is smooth but retains curvature. If the representations remain linearly related both before and after this nonlinear transformation, the map between them must preserve traces of the individual axes being used. An arbitrary mixing of axes would not preserve those traces. Therefore, provided that the task and the next layer genuinely use these directions, the directions can be matched up to permutation and scaling.

“The harder the task, the fewer solutions it has and the more likely those solutions are to resemble one another,” Daniel Yamins writes in an explanation of the theory.

The authors define a task as difficult if it cannot be solved to a specified level of accuracy using only a few nonlinear axes in the relevant layer. They call the minimum required number the used axis budget. The larger the fraction of the layer required by the task, the larger the fraction of axes for which the theorem guarantees correspondence between the networks.

A second theorem describes how the hierarchy can be “zipped up.” It requires minimality, meaning that the hidden functions at each step must make distinguishable contributions to the next layer. Under this condition, weak correspondence at the final layer being compared allows correspondences in earlier layers to be recovered for almost all weight configurations. The authors exclude certain special cases from the result. In the approximate version, transformations between layers must be regular, meaning that they must not collapse distinct signal changes into the same change. Under this condition, a small error at the final layer being compared limits the fraction of axes that can diverge at earlier layers.

The paper proposes testing whether natural tasks engage many axes in brain data, whether the minimality condition holds, and whether training leads to exceptional weight configurations. A simple task can be solved in many different ways. The authors suggest that tasks involving images and situations natural to an animal constrain the set of possible solutions more strongly and may therefore increase the chance that a model and the brain match at the level of individual computations.