Regression-Based Elastic Metric Learning on Shape Spaces of Cell Curves Adele Myers

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Regression-Based Elastic Metric Learning on Shape

Spaces of Cell Curves

Adele Myers

University of California, Santa Barbara

Santa Barbara, CA 93106

adele@ucsb.edu

Nina Miolane

University of California, Santa Barbara

Santa Barbara, CA 93106

ninamiolane@ucsb.edu

Abstract

We propose a metric learning paradigm, Regression-based Elastic Metric Learning

(REML), which optimizes the elastic metric for geodesic regression on the manifold

of discrete curves. Geodesic regression is most accurate when the chosen metric

models the data trajectory close to a geodesic on the discrete curve manifold. When

tested on cell shape trajectories, regression with REML’s learned metric has better

predictive power than with the conventionally used square-root-velocity (SRV)

metric. The code is publicly available here.

1 Introduction

Cell shape is strongly representative of cell function and can help diagnose many conditions including

cancer [14]. However, most cell analyses do not consider the cell’s shape in its entirety and instead

only consider coarse global attributes like area, perimeter, or convexity [6]. Analyses that do use

the cell’s entire shape are often “static”, as they disregard cell shape as it evolves over time [1, 13].

Because advances in live imaging are poised to provide an increasing amount of cell video data,

statistical tools that describe the time evolution of cell shape are timely and necessary to precisely

assess cell health and potential pathological conditions.

We propose a metric learning paradigm, Regression-based Elastic Metric Learning (REML). REML

learns the optimal elastic metric for geodesic regression on the manifold of discrete curves: here,

the manifold of cell shapes. Our work expands on the framework [3, 10] which parameterizes cell

shapes from microscopy images with a 2D array of discrete points that trace a cell’s outline. Then,

we use the elastic metric implemented in Geomstats [11] to analyze these shapes on the manifold of

discrete curves. The outline of each cell is itself a point on the manifold of discrete curves

, and

changes in cell shape over time form a trajectory on

. Our method analytically solves the geodesic

regression problem using different metrics and selects the metric that maximizes the coefﬁcient of

determination

on the validation set Fig. 1. The elastic metric is particularly meaningful when

analyzing cell shape because it quantiﬁes “stretching” and “bending” between curves, which provides

a biological measure of stretching and bending properties of the cells in a cell shape trajectory. We

validate our approach on synthetic trajectories between real osteosarcoma cells. The experimental

results conﬁrm that our paradigm (i) learns biological parameters of cell shape evolution, and (ii)

provides a mathematically-grounded approach to enhanced characterization of cell shape evolution.

2 Background

This section reviews the tools of Riemannian geometry that will support our metric learning method.

Additional background can be found in [12, 2]. We model cell outlines as elements of the space of

Published at the Workshop Learning Meaningful Representations of Life (LMRL 2022) of the 36th Conference

on Neural Information Processing Systems (NeurIPS 2022).

arXiv:2210.01932v2 [cs.LG] 22 Nov 2022

Data Trajectory SRV Geodesic

REML Geodesic Data Trajectory Predicted by REML (a*)

Predicted by SRV

Theory

Real Results

Figure 1: Summary and results of the proposed approach. Left: A trajectory may follow a geodesic

as calculated by one metric but not follow a geodesic as calculated by another metric. Our paradigm

learns the elastic metric (parameterized by

) that best represents the data trajectory as a geodesic

on the manifold of discrete curves. Right: true cell trajectory overlaid with 1) cells predicted by

a regression which utilizes our paradigm’s learned metric parameter (

a∗

) 2) cells predicted by a

square-root-velocity (SRV) regression. Regression predictions using the SRV metric (red) do not

match the data trajectory (blue), but our algorithm’s

a∗

predicts the data trajectory perfectly: our

prediction (green) perfectly overlays the data trajectory (blue).

regular planar curves

C=C1([0,1],R2)

. Our metric learning approach will require us to calculate

distances between cell shapes in C. To do so, we introduce the concept of elastic metrics.

Elastic metrics

ARiemannian metric

on a manifold

is a set of smoothly varying inner products

deﬁned on tangent spaces of

. These deﬁne geometric measurements on

, including distances,

geodesics and exponential maps. Geodesics on manifolds generalize straight lines on vector spaces:

they are curves that locally minimize the distance between points. Exponential maps generalize

addition on vector spaces.

We equip Cwith a family of elastic metrics ga,b parameterized by a, b > 0[9]:

ga,b

c(h, k) = a2Z1

hDsh, NihDsk, Nids+b2Z1

hDsh, T ihDsk, T ids, ∀c∈ C, h, k ∈TcC,(1)

where

is a cell outline (a point in

h, k

are inﬁnitesimal deformations of

(tangent vectors in

TcC

is the derivative with respect to arc-length

along the curve

(T, N )

denote the unit

tangent and the unit normal to

, respectively, and

<, >

is the canonical Euclidean inner-product of

. An elastic metric

ga,b

depends on two parameters: a “bending” parameter

and a “stretching”

parameter a, which respectively evaluate whether two curves are “bent” or “stretched” compared to

one another and then deﬁne how far apart these curves should be on

. For example, when

is large,

two curves that differ by a stretching operation will be far apart. When

is large, two curves that

differ by a bending operation will be far apart. Elastic metrics are invariant under shape-preserving

transformations, i.e., elements in the group

of 2D translations, 2D rotations, and re-scaling of

cell outlines [12]. As such, elastic metrics are well-deﬁned on the space of curve shapes, which is

formally given by the quotient M=C1([0,1],R2)/G. The elastic metric with a= 1 and b= 0.5is

called the square-root-velocity (SRV) metric, and is often used as the “default metric” on the manifold

of discrete curves. For this reason, we will consider the statistical analysis with the SRV metric as a

baseline in our experiments.

Geodesic regression

Because

is a manifold, we consider geodesic regression, which is the

linear regression equivalent for manifolds. Geodesic regression on

(M, ga,b)

solves a least-square

ﬁtting problem [5]:

min

(p,v)∈TM

i=1

d2Expp(tiv), ci,(2)

where

and

Exp

are the Riemannian distance and exponential maps associated with the metric

ga,b

When the metric is Euclidean, this expression simpliﬁes to the usual linear regression with intercept

and slope

. In geodesic regression, choice of metric affects the goodness of ﬁt. Because metrics

deﬁne distances between points in shape space, a data trajectory may follow the geodesic calculated

by one metric but not another. We design an algorithm that ﬁnds the "optimal" metric for regression

of a particular trajectory —the metric where the trajectory is closest to a geodesic, as judged by

the value of the coefﬁcient of determination

. This optimal metric gives the regression ﬁt more

predictive power, which in the case of microscopy image analysis in cell biology can thus provide a

more accurate prediction of future cell shapes. We will perform metric learning across all possible

elastic metrics and compare our results to the commonly used SRV metric.

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Regression-BasedElasticMetricLearningonShapeSpacesofCellCurvesAdeleMyersUniversityofCalifornia,SantaBarbaraSantaBarbara,CA93106adele@ucsb.eduNinaMiolaneUniversityofCalifornia,SantaBarbaraSantaBarbara,CA93106ninamiolane@ucsb.eduAbstractWeproposeametriclearningparadigm,Regression-basedElasticMetricLea...

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Regression-Based Elastic Metric Learning on Shape Spaces of Cell Curves Adele Myers

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