Visualizing the effects of a changing distance using continuous embeddings

Most ML methods, from clustering to nearest-neighbour classification, rely on a distance function to describe relationships between datapoints. For complex datasets it is often hard to avoid making some arbitrary choices when defining a distance function. To compare images, one must choose a spatial scale. To compare signals, one must choose a temporal scale. The right scale is hard to pin down and it is preferable when results do not depend too tightly on the exact value that one picked. Topological data analysis seeks to address this issue by focusing on the notion of neighbourhood instead of that of distance. Here, we show that in some cases a simpler solution is available. One can check how strongly distance relationships depend on a hyperparameter using dimensionality reduction. We formulate a variant of dynamical multi-dimensional scaling (MDS), which embeds datapoints as timecurves. The resulting algorithm provides a simple and efficient way of visualizing changes and invariances in distance patterns as a hyperparameter is varied. We apply it to challenging brain connectivity datasets, which are difficult to average over subjects in a non-arbitrary way. We also show how the algorithm can be used to visualize the effects of changes in a weighted metric, when changing the relative weight of two groups of variables.