半年工作小结
报告人：吕小惠
2011年8月25日
报告提纲
一．学习了Non-negative Matrix
Factorization convergence proofs
二．学习了Sparse Non-negative Matrix
Factorization算法
三．学习了线性代数中有关子空间等基础知
识
四．学习了Ma yi等人提出的Generalized
PCA的相关理论
GPCA论文信息
• Generalized Principal Component
Analysis(GPCA)
• René Vidal,Member IEEE;Yi Ma,Member
IEEE;Shankar Sastry,Fellow IEEE
• IEEE TRANSACTIONS ON PATTERN ANALYSIS
AND MACHINE INTELLIGENCE,VOL. 27,NO. 12,
Page 1945-1959,December 2005
GPCA从数据中寻找基向量
GPCA Abstract
• This paper presents an algebro-geometric
solution to the problem of segmenting an
unknown number of subspaces of unknown
and varying dimensions from sample data
points.
GPCA Abstract
• We represent the subspaces with a set of
homogeneous polynomials whose degree is
the number of the subspaces and the
derivatives at a data point give normal vectors
to the subspace passing through the point.
GPCA Abstract
• When the number of subspaces is known,we
show that these polynomials can be estimated
linearly from data ,hence subspace
segmentation is reduced to classifying one
point per subspace.
GPCA Abstract
• We select these points optimally from the
data set by minimizing certain distance
function,thus dealing automatically with
moderate noise in the data.
• A basis for the complement of each subspace
is then recovered by applying standard PCA to
the collection of derivatives(normal vectors).
GPCA Abstract
• Extensions of GPCA that deal with data in a highdimensional space and with an unknown number of
subspaces are also presented.
• Our experiments on low-dimensional data show that
GPCA out-performs existing algebraic algorithms
based on polynomials factorization and provides a
good initialization to iterative techniques such as Ksubspace and Expectation Maximization.
GPCA Abstract
• We also present applications of GPCA to
computer vision problems such as face
clustering, temporal video segmentation, and
3-D motion segmentation from point
correspondences in multiple affine views.
Subspace Segmentation
• We consider the following alternative
extension of PCA to the case of data lying a
union of subspaces as illustrated in Figure 1
for two subspaces in R3.
Subspace Segmentation
GPCA
• Theorem 1(Generalized Principal Component
Analysis): A union of n subspaces of RD,can be
represented with a set of homogeneous
polynomials of degree n in D variables.
• These polynomials can be estimated linearly
given enough sample points in general
position in the subspaces.
GPCA
• A basis for the complement of each subspace
can be obtained from the derivatives of these
polynomials at a point in each of the
subspaces.
• Such points can be recursively selected via
polynomial division.
• Therefore, the subspace segmentation
problem is mathematically equivalent to
fitting,differentiating,and dividing a set of
homogeneous polynomials.拟合，微分，分
解
Thank you!
报告人：吕小惠
2011年8月25日