2,063 research outputs found
A Graph-Based Semi-Supervised k Nearest-Neighbor Method for Nonlinear Manifold Distributed Data Classification
Nearest Neighbors (NN) is one of the most widely used supervised
learning algorithms to classify Gaussian distributed data, but it does not
achieve good results when it is applied to nonlinear manifold distributed data,
especially when a very limited amount of labeled samples are available. In this
paper, we propose a new graph-based NN algorithm which can effectively
handle both Gaussian distributed data and nonlinear manifold distributed data.
To achieve this goal, we first propose a constrained Tired Random Walk (TRW) by
constructing an -level nearest-neighbor strengthened tree over the graph,
and then compute a TRW matrix for similarity measurement purposes. After this,
the nearest neighbors are identified according to the TRW matrix and the class
label of a query point is determined by the sum of all the TRW weights of its
nearest neighbors. To deal with online situations, we also propose a new
algorithm to handle sequential samples based a local neighborhood
reconstruction. Comparison experiments are conducted on both synthetic data
sets and real-world data sets to demonstrate the validity of the proposed new
NN algorithm and its improvements to other version of NN algorithms.
Given the widespread appearance of manifold structures in real-world problems
and the popularity of the traditional NN algorithm, the proposed manifold
version NN shows promising potential for classifying manifold-distributed
data.Comment: 32 pages, 12 figures, 7 table
Nonlinear Supervised Dimensionality Reduction via Smooth Regular Embeddings
The recovery of the intrinsic geometric structures of data collections is an
important problem in data analysis. Supervised extensions of several manifold
learning approaches have been proposed in the recent years. Meanwhile, existing
methods primarily focus on the embedding of the training data, and the
generalization of the embedding to initially unseen test data is rather
ignored. In this work, we build on recent theoretical results on the
generalization performance of supervised manifold learning algorithms.
Motivated by these performance bounds, we propose a supervised manifold
learning method that computes a nonlinear embedding while constructing a smooth
and regular interpolation function that extends the embedding to the whole data
space in order to achieve satisfactory generalization. The embedding and the
interpolator are jointly learnt such that the Lipschitz regularity of the
interpolator is imposed while ensuring the separation between different
classes. Experimental results on several image data sets show that the proposed
method outperforms traditional classifiers and the supervised dimensionality
reduction algorithms in comparison in terms of classification accuracy in most
settings
A study of the classification of low-dimensional data with supervised manifold learning
Supervised manifold learning methods learn data representations by preserving
the geometric structure of data while enhancing the separation between data
samples from different classes. In this work, we propose a theoretical study of
supervised manifold learning for classification. We consider nonlinear
dimensionality reduction algorithms that yield linearly separable embeddings of
training data and present generalization bounds for this type of algorithms. A
necessary condition for satisfactory generalization performance is that the
embedding allow the construction of a sufficiently regular interpolation
function in relation with the separation margin of the embedding. We show that
for supervised embeddings satisfying this condition, the classification error
decays at an exponential rate with the number of training samples. Finally, we
examine the separability of supervised nonlinear embeddings that aim to
preserve the low-dimensional geometric structure of data based on graph
representations. The proposed analysis is supported by experiments on several
real data sets
Active Semi-Supervised Learning Using Sampling Theory for Graph Signals
We consider the problem of offline, pool-based active semi-supervised
learning on graphs. This problem is important when the labeled data is scarce
and expensive whereas unlabeled data is easily available. The data points are
represented by the vertices of an undirected graph with the similarity between
them captured by the edge weights. Given a target number of nodes to label, the
goal is to choose those nodes that are most informative and then predict the
unknown labels. We propose a novel framework for this problem based on our
recent results on sampling theory for graph signals. A graph signal is a
real-valued function defined on each node of the graph. A notion of frequency
for such signals can be defined using the spectrum of the graph Laplacian
matrix. The sampling theory for graph signals aims to extend the traditional
Nyquist-Shannon sampling theory by allowing us to identify the class of graph
signals that can be reconstructed from their values on a subset of vertices.
This approach allows us to define a criterion for active learning based on
sampling set selection which aims at maximizing the frequency of the signals
that can be reconstructed from their samples on the set. Experiments show the
effectiveness of our method.Comment: 10 pages, 6 figures, To appear in KDD'1
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