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 points. Not all existing methods are isometric, so the geometry obtained this way can be distorted. Perraul-Joncas & Meila (2013) uses the Laplacian operator to obtain pushforward metric for the lowdimensional representations. Instead of specifying the geometry of the ambient space, our focus is to learn the geometry from noisy measures of intrinsic distances. Fefferman et al. ( 2020) discusses an abstract setting of this task, while our work proposes a practical estimation of the Riemannian metric tensor when coordinates are also available, and we show that our approach is numerically sound. We suppose that data are generated from an unknown Riemannian manifold, and we have available the coordinates of the data objects. The Euclidean distance between the coordinates may not reflect the underlying geometry. Instead, we assume that we further observe similarity measures between objects, modeled as noise-contaminated intrinsic distances, that are used to characterize the intrinsic geometry on the Riemannian manifold. The targeted Riemannian metric is estimated in a datadriven fashion, which enables estimating geodesics (straight lines and locally shortest paths) and performing calculus on the manifold. To formulate the problem, let (M, G) be a Riemannian manifold with Riemannian metric G, and dist (•, •) be the geodesic distance induced by G which measures the true intrinsic difference between points. The coordinates of data points x 0 , x 1 ∈ M are assumed known, identifying each point via a tuple of real numbers. Also observed are noisy measurements y of the intrinsic distance between data points, which we refer to as similarity measurements (equivalently dissimilarity). The response is modeled flexibly, and we consider the following common scenarios: (i) noisy distance, where y = dist (x 0 , x 1 ) 2 + for error , (ii) similarity/dissimilarity, where y = 0 if the two points x 0 , x 1 are considered similar and y = 1 otherwise, and (iii) relative comparison, where a triplet of points (x 0 , x 1 , x 2 ) are given and y = 1 if x 0 is more similar to x 1 than to x 2 and y = 0 otherwise. The binary similarity measurement is common in computer vision (e.g. Chopra et al., 2005) , while the relative comparison could be useful for perceptional tasks and recommendation system (e.g. Schultz & Joachims, 2003; Berenzweig et al., 2004) . We aim to estimate the Riemannian metric G and its derivatives using the coordinates and similarity measures among the data points. The major contribution of this paper is threefold. First, we formulate a framework for probabilistic modeling of similarity measurements among data on manifold via intrinsic distances. Based on a Taylor expansion for the spread of geodesic curves in differential geometry, the local regression procedure successfully estimates the Riemannian metric and its derivatives. Second, a theoretical foundation is developed for the proposed method including asymptotic consistency. Last and most importantly, the proposed method provides a geometric interpretation for the structure of the data space induced by the similarity measurements, as demonstrated in the numerical examples that include a taxi travel and an MNIST digit application.

2. BACKGROUND IN RIEMANNIAN GEOMETRY

For brevity, metric now refers to Riemannian metric while distance metric is always spelled out. Throughout the paper, M denotes a d-dimensional manifold endowed with a coordinate chart (U, ϕ), where ϕ : U → R d maps a point p ∈ U ⊂ M on the manifold to its coordinate ϕ(p) = ϕ 1 (p), . . . , ϕ d (p) ∈ R d . Without loss of generality, we identify a point by its coordinate as p 1 , . . . , p d , suppressing ϕ for the coordinate chart. Upper-script Roman letters denote the components of a coordinate, e.g., p i is the i-th entry in the coordinate of the point p, and γ i is the i-th component function of a curve γ : R ⊃ [a, b] → M when expressed on chart U . The tangent space T p M is a vector space consisting of velocities of the form v = γ (0) where γ is any curve satisfying γ(0) = p. The coordinate chart induces a basis on the tangent space T p M, as ∂ i | p = ∂/∂x i | p for i = 1, . . . , d, so that a tangent vector v ∈ T p M is represented as v = d i=1 v i ∂ i for some v i ∈ R, suppressing the subscript p in the basis. We adopt the Einstein summation convention unless otherwise specified, namely v i ∂ i denotes d i=1 v i ∂ i , where common pairs of upper-and lower-indices denotes a summation from 1 to d (see e.g., Lee, 2013, pp.18-19) . The Riemannian metric G on a d-dimensional manifold M is a smooth tensor field acting on the tangent vectors. At any p ∈ M, G(p) : T p M × T p M → R is a symmetric bi-linear tensor/function satisfying G(p)(v, v) ≥ 0 for any v ∈ T p M and G(p)(v, v) = 0 if and only if v = 0. On a chart ϕ, the metric is represented as a d-by-d positive definite matrix that quantifies the distance traveled

