Abstract
In visual perception, the brain faces a difficult dimensionality reduction problem, with many nerve fibers converging to relatively few perceptual features. This is possible because meaningful structure in visual input lies on a lower-dimensional psychological manifold extracted by the visual system. However, while many cognitive processes are hypothesized to use geometric relationships between stimuli, there is no general method to assign psychological manifolds a geometry with an independently-verified notion of distance. To this end, we propose a model of the geometry of psychological manifolds that combines perceptual models with an attentional process that modifies discriminability according to context. These constraints induce a metric tensor that measures the length of curves on the manifold, including those corresponding to stimuli that change continuously through time. To investigate whether curve length corresponds to perceptual distance, we conducted an experiment (N = 39) comparing model predictions to judgements of the rate of change for dynamic stimuli with a one second duration. We first showed participants static, Gabor-like stimuli varying in spatial frequency and orientation, instructing them to place stimuli on a line according to the value of each dimension. We then presented pairs of changing stimuli and participants reported their confidence in which was changing faster in frequency or orientation with a slider. Using responses to static stimuli and assuming 80% attention to the cued dimension, difference in curve length predicted judgements for 36 of 39 participants (logistic GLM, >95% posterior probability). We then fit our model to all data and found it predicted judgements better than a flat approximation with one fewer parameter for 18 participants (approximate leave-one-out cross validation). These results suggest that distance on a psychological manifold is a meaningful measure of subjective change. Additionally, our second analysis suggests that flat approximations of psychological manifolds are inappropriate for some participants.