1-Introduction
This article propose a new algorithm (LTSA) for nonlinear manifold
learning and nonlinear dimension reduction. it’s tite is
“PRINCIPAL MANIFOLDS AND NONLINEAR DIMENSION
REDUCTION VIA LOCAL TANGENT SPACE ALIGNMENT ”. Was Published in 2002 by
SIAM Journal of Scientific . The SIAM Journal on Scientific
Computing contains research articles on numerical methods and
techniques for scientific computation. Papers address computational
issues relevant to the solution of scientific or engineering problems
and include computational results demonstrating the effectiveness of the
proposed techniques. It was written by Zhenyue Zhang , et Hongyuan Zha .
hongyuan Zha was a faculty member of the Department of Computer
Science and Engineering at Pennsylvania State University from 1992 to
2006, and he worked from 1999 to 2001 at Inktomi Corporation. Zha’s
current research interests include computational mathematics and machine
learning applications. Zhenyue Zhang is a mathematical researcher in the Department of
Mathematics Research at Zhejiang • China University. The authors had presented, in this article manifold learning and
dimension reduction and he discusses the issue of learning local
geometry using tangent spaces, then he shows how to align those local
tangent spaces in order to learn the global coordinate system of the
underlying manifold, finaly he based for algorithm local tangent space
alignment (LTSA) algorithm and comparite with the local linear embedding
(LLE)in order to resume the article, I start with exposing the scientific context at first, secondly, I will try to highlight the most important existing works; and the position of the article. Thirdly, the contribution in the scientific world, and finally I end my resume with showing the results and experiments of the method
2 - Context of the
work
In recent years, a variety of nonlinear dimensionality reduction
techniques have been proposed that aim to overcome the limitations of
traditional techniques such as PCA and other dimension reduction .
The manifold learning is the approach how attempts to reduce dimension of the dataset to ease representation and interpretation (we called the approach representation learning) The main issue with the high dimensional dataset is more difficult it becomes to sample the space. This causes many problems. Algorithms that operate on high-dimensional data tend to have a very high time complexity. Many machine learning algorithms, for example, struggle with high-dimensional data. In this article the authors speak about principale manifolds and nonlinear dimention reduction via local tangent space alignmentso
what’s what’s local tangent aligment ??and why we use it?
LTSA is technique that describes local properties of the
high-dimensional data using the local tangent space of each datapoint
. LTSA is based on the observation that, if local linearity of
the manifold is assumed, there exists a linear mapping from a
high-dimensional datapoint to its local tangent space, and that there
exists a linear mapping from the corresponding low-dimensional data
point to the same local tangent space . LTSA attempts to align
these linear mappings in such a way, that they construct the local
tangent space of the manifold from the low-dimensional representation.
In other words, LTSA simultaneously searches for the coordinates of the
low-dimensional data representations, and for the linear mappings of the
low dimensional datapoints to the local tangent space of the
high-dimensional data.\ref{391140}
all manifold learning algorithms can be applied to data dimensionality reduction, producing a low-dimensional encoding from a high-dimensional one and among these algorithm LTSA .His principe is to to use the tangent space in the neighborhood of a
data point to represent the local geometry, and then align those local
tangent spaces to construct the global coordinate system for the
nonlinear manifold.