扩展的流形聚类：多交嵌入数据的分类与参数化

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流形聚类方法是一种在现代数据驱动领域中的关键技术，特别是在处理视频、运动捕捉和手写字符数据时，这些数据通常位于低维、非线性的流形上。传统上，流形学习主要用于揭示数据在高维空间中的潜在结构，通过降维来理解和分析复杂的数据分布。然而，本文提出了一种新的流形学习扩展，旨在对未标记数据进行分类和参数化，即使这些数据分布在多个相交的流形上。 Richard Souvenir 和 Robert Pless，来自华盛顿大学圣路易斯分校计算机科学与工程系，他们的研究将流形学习的适用范围大大扩展，使得这种方法可以处理更为复杂的几何结构，如环状（figure eights）和相交路径，这些都是自然数据集中常见的模式。这种扩展的关键在于开发了针对多流形场景的算法，能够有效地处理具有混合拓扑和维度的相交流形。其中的技术贡献包括： 1. 节点加权多维尺度：传统多维尺度（MDS）方法通常假设数据均匀分布，而节点加权MDS引入了根据节点间距离或相似性自适应调整权重的概念，更好地适应非均匀数据分布于流形上。 2. 加权低秩近似：对于具有特定权重矩阵的流形，研究人员设计了一种快速算法来求解低秩近似，特别是针对秩一权重矩阵的情况。这种技术不仅提高了计算效率，还保证了结果的精度。在实际应用中，作者展示了多种相交流形的实例，涵盖了不同维度和拓扑结构，以及在人类运动数据上的具体应用。这些例子表明，流形聚类方法在处理复杂数据集时展现出强大的潜力，有助于揭示数据背后的深层次结构，从而支持更准确的数据理解、分类和分析。这篇论文不仅提升了流形学习的实用价值，还推动了相关理论和技术的发展，对于计算机视觉、机器学习和模式识别等领域具有重要意义。它为研究者提供了一套适用于多流形场景的工具，为未来的数据挖掘和分析打开了新的可能性。

Manifold Clustering

Richard Souvenir and Robert Pless

Washington University in St. Louis

Department of Computer Science and Engineering

Campus Box 1045, One Brookings Drive, St. Louis, MO 63130

{rms2, pless}@cse.wustl.edu

Abstract

Manifold learning has become a vital tool in data driven

methods for interpretation of video, motion capture, and

handwritten character data when they lie on a low dimen-

sional, non-linear manifold. This work extends manifold

learning to classify and parameterize unlabeled data which

lie on multiple, intersecting manifolds. This approach sig-

niﬁcantly increases the domain to which manifold learn-

ing methods can be applied, allowing parameterization of

example manifolds such as ﬁgure eights and intersecting

paths which are quite common in natural data sets. This

approach introduces several technical contributions which

may be of broader interest, including node-weighted multi-

dimensional scaling and a fast algorithm for weighted low-

rank approximation for rank-one weight matrices. We show

examples for intersecting manifolds of mixed topology and

dimension and demonstrations on human motion capture

data.

1. Introduction

Data-driven modeling is a powerful approach for non-

rigid motion analysis. Manifold learning approaches have

been applied to automatically parameterize image data sets

including head pose, facial expressions, bird ﬂight, MR im-

agery, and handwritten characters. Each of these data sets

lies on low-dimensional manifolds that are not linear sub-

spaces of the (high-dimensional) input data space. Manifold

learning approaches seek to explicitly or implicitly deﬁne

a low-dimensional embedding of the data points that pre-

serves some properties (such as geodesic distance or local

relationships) of the high-dimensional point set.

When the input data points are drawn from multiple

(low-dimensional) manifolds, many manifold learning ap-

proaches suffer. In the case where the multiple mani-

folds are separated by a gap, techniques such as isomet-

ric feature mapping (Isomap) [15] may discover the dif-

Figure 1. For high-dimensional data points

which lie on intersecting low-dimensional

manifolds, manifold embedding techniques

beneﬁt from ﬁrst separating points into dis-

tinct classes. These toy problems illustrate

relevant cases. (left) The spirals data set,

which can be embedded in two dimensions

with minimal error, can also be embedded

as two one-dimensional manifolds. (right)

The circle-plane data set includes component

manifolds of different dimension. The seg-

mentations shown here by the different sym-

bols are automatically determined by the ap-

proach developed in this paper.

ferent manifolds as different connected components in the

local neighborhood graph, and spectral clustering tech-

niques may identify and cluster each manifold based on

the optimization of certain objective functions. However,

if there is signiﬁcant overlap in the manifolds, prior meth-

ods fail in one of two ways: either, in the case of Isomap,

from the non-existence of a low-dimensional embedding

which exactly (or nearly) preserves properties of the high-

dimensional manifold, or, in the case of Locally Linear Em-

bedding (LLE) [13] or Semideﬁnite Embedding (SDE) [16],

from the fact that additional work is necessary to interpret

R. Souvenir and R. Pless, Manifold Clustering, in Proceedings of the 10th International Conference on Computer Vision (ICCV 2005),

Beijing, China, October 2005, pp. 648 – 653.

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