Soft Matching Distance: A metric on neural representations that captures
  single-neuron tuning
Meenakshi Khosla Alex H. Williams
Common measures of neural representational (dis)similarity are designed to be
insensitive to rotations and reflections of the neural activation space.
Motivated by the premise that the tuning of individual units may be important,
there has been recent interest in developing stricter notions of
representational (dis)similarity that require neurons to be individually
matched across networks. When two networks have the same size (i.e. same number
of neurons), a distance metric can be formulated by optimizing over neuron
index permutations to maximize tuning curve alignment. However, it is not clear
how to generalize this metric to measure distances between networks with
different sizes. Here, we leverage a connection to optimal transport theory to
derive a natural generalization based on "soft" permutations. The resulting
metric is symmetric, satisfies the triangle inequality, and can be interpreted
as a Wasserstein distance between two empirical distributions. Further, our
proposed metric avoids counter-intuitive outcomes suffered by alternative
approaches, and captures complementary geometric insights into neural
representations that are entirely missed by rotation-invariant metrics.