ESTIMATING RIEMANNIAN METRIC WITH NOISE-CONTAMINATED INTRINSIC DISTANCE

Abstract

We extend metric learning by studying the Riemannian manifold structure of the underlying data space induced by dissimilarity measures between data points. The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold. Being able to estimate the Riemannian metric allows us to gain insights into the underlying manifold and compute geometric features such as the geodesic curves. We model the observed dissimilarity measures as noisy responses generated from a function of the intrinsic geodesic distance between data points. A new local regression approach is proposed to learn the Riemannian metric tensor and its derivatives based on a Taylor expansion for the squared geodesic distances. Our framework is general and accommodates different types of responses, whether they are continuous, binary, or comparative, extending the existing works which consider a single type of response at a time. We develop theoretical foundation for our method by deriving the rates of convergence for the asymptotic bias and variance of the estimated metric tensor. The proposed method is shown to be versatile in simulation studies and real data applications involving taxi trip time in New York City and MNIST digits.

1. INTRODUCTION

The estimation of distance metric, also known as metric learning, has attracted great interest since its introduction for classification (Hastie & Tibshirani, 1996) and clustering (Xing et al., 2002) . A global Mahalanobis distance is commonly used to obtain the best distance for discriminating two classes (Xing et al., 2002; Weinberger & Saul, 2009) . While a global metric is often the focus of earlier works, multiple local metrics (Frome et al., 2007; Weinberger & Saul, 2009; Ramanan & Baker, 2011; Chen et al., 2019) are found to be useful because they better capture the data space geometry. There is a great body of work on distance metric learning; see, e.g., Bellet et al. (2015) ; Suárez et al. (2021) for recent reviews. Metric learning is intimately connected with learning on Riemannian manifolds. Hauberg et al. ( 2012) connects multi-metric learning to learning the geometric structure of a Riemannian manifold, and advocates its benefits in regression and dimensional reduction tasks. Lebanon (2002; 2006) ; Le & Cuturi (2015) discuss Riemannian metric learning by utilizing a parametric family of metric, and demonstrate applications in text and image classification. Like in these works, our target is to learn the Riemannian metric instead of the distance metric, which fundamentally differentiates our approach from most existing works in metric learning. We focus on the nonparametric estimation of the data geometry as quantified by the Riemannian metric tensor. Contrary to distance metric learning, where the coefficient matrix for the Mahalanobis distance is constant in a neighborhood, the Riemannian metric is a smooth tensor field that allows analysis of finer structures. Our emphasis is in inference, namely learning how differences in the response measure are explained by specific differences in the predictor coordinates, rather than obtaining a metric optimal for a supervised learning task. A related field is manifold learning which attempts to find low-dimensional nonlinear representations of apparent high-dimensional data sampled from an underlying manifold (Roweis & Saul, 2000; Tenenbaum et al., 2000; Coifman & Lafon, 2006) . Those embedding methods generally start by assuming the local geometry is given, e.g., by the Euclidean distance between ambient data

