/ Caitlin Lienkaemper: Estimating rank despite monotone distortion

Caitlin Lienkaemper: Estimating rank despite monotone distortion

October 4, 2022
4:30 pm - 5:30 pm

Abstract: In many biological settings, measurements are distorted by unknown nonlinear transformations. Often, these nonlinearities are monotone, preserving the ordering of elements. While these nonlinearities make detecting low-dimensional structure using traditional matrix analysis impossible, we can recover some of this hidden structure using combinatorial and topological techniques. Here, we explore the underlying rank of a matrix, defined as the smallest rank possible given the ordering of matrix entries. We give several methods for estimating underlying rank.To derive these bounds, we decompose matrices as pairs of point configurations, and use the order of matrix entries to extract information about this point configuration. We introduce extremal nodes as tool for estimating the underlying rank of a random matrix, and introduce Radon rank as a lower bound for underlying rank.  We also show that underlying rank can exceed these bounds. We apply our techniques to re-analyze recordings of a large population of neurons in the mouse visual cortex from (Stringer et al, 2019). We show that in addition to the reported finding of high linear rank, this data set also has high underlying rank, even when visual input is low-dimensional.