提升常曲率流形上的精确PGA算法效率

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标题 "An Efficient Exact-PGA Algorithm for Constant Curvature Manifolds" 描述了在计算机视觉任务中广泛遇到的具有曲率的流形数据集处理方法。传统的Principal Geodesic Analysis (PGA)算法，作为非线性PCA的扩展，旨在适应Riemannian流形的数据结构。然而，由于其目标函数高度非线性和一般情况下难以高效求解，研究人员曾提出了一种线性近似方法，以简化计算过程。线性近似PGA虽然计算简便，但当数据存在较大方差时，其准确性往往不足。这限制了在处理复杂流形数据时的精度。为解决这个问题，近期出现了一种称为"精确PGA"（Exact PGA）的替代方案，它试图直接解决优化问题而无需进行线性化处理。与传统的线性化PGA相比，对于具有大方差的数据，精确PGA提供了更好的准确性。然而，尽管精确PGA在保持高精度方面有所提升，但针对一般Riemannian流形，它的优化过程可能会遇到挑战。这可能涉及到更复杂的数学模型和求解策略，因为它需要处理非线性的几何特性，如流形上的曲率保持不变这一关键假设。这种算法设计的目标是提高分析效率和结果的可靠性，尤其是在处理那些数据分布复杂且要求精确度高的应用场景。在具体实现上，精确PGA可能包括一系列高级的数值优化技术，如梯度下降、拟牛顿法或者借助于流形上的特定工具，如拉普拉斯型算子或辛坐标系统，来处理曲面的微分几何问题。此外，算法可能需要对数据预处理和正则化进行细致的考虑，以确保在求解过程中避免局部最优，并找到全局最优解。总结来说，"An Efficient Exact-PGA Algorithm for Constant Curvature Manifolds" 是一项旨在克服传统PGA在处理大方差数据时局限性的研究，通过精确地执行优化过程来提升分析的准确性和效率。这个算法的发展对于处理计算机视觉中的复杂几何数据具有重要意义，特别是在医学成像、机器学习中的特征提取以及计算机图形学等领域。

An eﬃcient Exact-PGA algorithm for constant curvature manifolds

Rudrasis Chakraborty

, Dohyung Seo

, and Baba C. Vemuri

Department of CISE, University of Florida, FL 32611, USA

U-Systems, A GE Healthcare Company, CA, USA

{rudrasis, vemuri}@cise.ufl.edu

{dhseo.118}@gmail.com

Abstract

Manifold-valued datasets are widely encountered in

many computer vision tasks. A non-linear analog of

the PCA algorithm, called the Principal Geodesic Anal-

ysis (PGA) algorithm suited for data lying on Rieman-

nian manifolds was reported in literature a decade ago.

Since the objective function in the PGA algorithm is

highly non-linear and hard to solve eﬃciently in gen-

eral, researchers have proposed a linear approximation.

Though this linear approximation is easy to compute, it

lacks accuracy especially when the data exhibits a large

variance. Recently, an alternative called the exact PGA

was proposed which tries to solve the optimization with-

out any linearization. For general Riemannian mani-

folds, though it yields a better accuracy than the orig-

inal (linearized) PGA, for data that exhibit large vari-

ance, the optimization is not computationally eﬃcient.

In this paper, we propose an eﬃcient exact PGA al-

gorithm for constant curvature Riemannian manifolds

(CCM-EPGA). The CCM-EPGA algorithm diﬀers sig-

niﬁcantly from existing PGA algorithms in two aspects,

(i) the distance between a given manifold-valued data

point and the principal submanifold is computed an-

alytically and thus no optimization is required as in

the existing methods. (ii) Unlike the existing PGA al-

gorithms, the descent into codimension-1 submanifolds

does not require any optimization but is accomplished

through the use of the Rimeannian inverse Exponential

map and the parallel transport operations. We present

theoretical and experimental results for constant cur-

vature Riemannian manifolds depicting favorable per-

formance of the CCM-EPGA algorithm compared to

existing PGA algorithms. We also present data recon-

struction from the principal components which has not

been reported in literature in this setting.

1. Introduction

Principal Component Analysis (PCA) is a widely

used dimensionality reduction technique in Science and

Engineering. PCA however requires the input data to

lie in a vector space. With the advent of new technolo-

gies and wide spread use of sophisticated feature ex-

traction methods, manifold-valued data have become

ubiquitous in many ﬁelds including but not limited

to, Computer Vision, Medical Imaging and Machine

Learning. A nonlinear version of PCA, called the Prin-

cipal Geodesic Analysis (PGA), for data lying on Rie-

mannian manifolds was introduced in [8].

Since the objective function of PGA is highly non-

linear and hard to solve in general, researchers pro-

posed a linearized version of the PGA [8]. Though this

linearized PGA, hereafter referred to as PGA, is com-

putationally eﬃcient, it lacks accuracy for data with

large spread/variance. In order to solve the objective

function exactly, Sommer et al., [25] proposed to solve

the original objective function (not the approximation)

and called it exact PGA . While exact PGA attempts

to solve this complex nonlinear optimization problem,

it is however computationally ineﬃcient. Though it

is not possible to eﬃciently and accurately solve this

optimization problem for a general manifold, however,

for manifolds with constant sectional curvature, we for-

mulate an eﬃcient and exact PGA algorithm, dubbed

CCM-EPGA. It is well known in geometry, by virtue

of the Killing-Hopf theorem [4], that any non-zero con-

stant curvature manifold is isomorphic to either the hy-

persphere (S

) or the hyperbolic space (H

), hence in

this work, we present the CCM-EPGA formulation for

) and (H

). Our formulation has several applica-

tions to Computer Vision and Statistics including di-

rectional data [21] and color spaces [19]. Several other

applications of hyperbolic geometry are, shape analy-

sis [30], Electrical Impedance Tomography, Geoscience

Imaging [28], Brain Morphometry [29], Catadiaoptric

Vision [3] etc.

In order to depict the eﬀectiveness of our proposed

2016 IEEE Conference on Computer Vision and Pattern Recognition

DOI 10.1109/CVPR.2016.431

3976

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