At one of the journal clubs I recently attended, we discussed “The Topology and Geometry of Neural Representations”. The motivation for the paper is that procedures like RSA, which capture the overlap of population representations of different stimuli, can be overly sensitive to some geometrical features of the representation the brain might not care about. For example, suppose stimuli A and B are very similar and different from C. In that case, a transformation that makes C even more dissimilar from A and B shouldn’t have much of a perceptual effect when it comes to discrimination but could have a large impact on RSA. On the other hand, if A and B are already quite similar, then a transformation that makes them even more similar would probably not have a large perceptual effect, but might still influence the RSA.
One way to address this geometric oversensitivity is to make our similarity measures more “topological.” For example, we can binarize the overlaps used in RSA, so that below a certain threshold the overlaps are set to 0, and above a certain threshold, to 1. But such a “topological” approach throws out most of the geometric information. Is there something in between? The paper addresses this by applying RSA to transformed distances. If the transformation is the identity, we get the original, geometric, comparison. If it’s a step function, we get the topological comparison. By moving between these extremes the authors explore “geometro-topological” measures that hopefully combine the best of both worlds.
Questions
When the presenter started talking about moving between geometry and topology, I actually thought what the paper was going to be about was something like this:
We say that two shapes $X$ and $Y$ are geometrically equivalent if there is a (generalized) rotation and translation i.e. an isometry, that can turn $X$ into $Y$. We say that they’re topologically equivalent if there’s a continuous function with continuous inverse (a homeomorphism) that can turn $X$ into $Y$. The set of isometries is a (small) subset of the set of homeomorphisms.
- Can we go between geometry and topology by finding a nesting of families of continuous functions, whose smallest element is the set of isometries and whose largest element is the set of homeomorphisms? Something like $$ \text{Isometries } = \mathcal{F}_0 \subset \mathcal{F}_1 \dots \mathcal{F}_{N-1} \subset \mathcal{F}_N = \text{Homeomorphisms}?$$
- Can we do this in a continuous way? I.e. index $\mathcal{F}_t$ with $t \in [0,1]$?
- Are their transformed distances one way of doing the continuous nesting? What are others?
An approach that seems useful for answering these equations is to measure equivalence as Alex Williams does by computing $$ d_\mathcal{F}(X,Y) = \min_{f \in \mathcal{F}} |Y – f(X)|.$$ Here $\mathcal F$ is the family that defines equivalence. For example, it could be the set of isometries, but could also be the set of homeomorphism.
- Can we describe the proposed updates to RSA in this way?
$$\begin{flalign*} && \phantom{a} & \hfill \square \end{flalign*}$$
Leave a Reply