31,480 research outputs found
Kernel Spectral Clustering and applications
In this chapter we review the main literature related to kernel spectral
clustering (KSC), an approach to clustering cast within a kernel-based
optimization setting. KSC represents a least-squares support vector machine
based formulation of spectral clustering described by a weighted kernel PCA
objective. Just as in the classifier case, the binary clustering model is
expressed by a hyperplane in a high dimensional space induced by a kernel. In
addition, the multi-way clustering can be obtained by combining a set of binary
decision functions via an Error Correcting Output Codes (ECOC) encoding scheme.
Because of its model-based nature, the KSC method encompasses three main steps:
training, validation, testing. In the validation stage model selection is
performed to obtain tuning parameters, like the number of clusters present in
the data. This is a major advantage compared to classical spectral clustering
where the determination of the clustering parameters is unclear and relies on
heuristics. Once a KSC model is trained on a small subset of the entire data,
it is able to generalize well to unseen test points. Beyond the basic
formulation, sparse KSC algorithms based on the Incomplete Cholesky
Decomposition (ICD) and , , Group Lasso regularization are
reviewed. In that respect, we show how it is possible to handle large scale
data. Also, two possible ways to perform hierarchical clustering and a soft
clustering method are presented. Finally, real-world applications such as image
segmentation, power load time-series clustering, document clustering and big
data learning are considered.Comment: chapter contribution to the book "Unsupervised Learning Algorithms
Learning with Algebraic Invariances, and the Invariant Kernel Trick
When solving data analysis problems it is important to integrate prior
knowledge and/or structural invariances. This paper contributes by a novel
framework for incorporating algebraic invariance structure into kernels. In
particular, we show that algebraic properties such as sign symmetries in data,
phase independence, scaling etc. can be included easily by essentially
performing the kernel trick twice. We demonstrate the usefulness of our theory
in simulations on selected applications such as sign-invariant spectral
clustering and underdetermined ICA
Spectral analysis of the Gram matrix of mixture models
This text is devoted to the asymptotic study of some spectral properties of
the Gram matrix built upon a collection of random vectors (the columns of ), as both the number of
observations and the dimension of the observations tend to infinity and are
of similar order of magnitude. The random vectors are
independent observations, each of them belonging to one of classes
. The observations of each class
() are characterized by their distribution
, where are some non negative
definite matrices. The cardinality of class
and the dimension of the observations are such that () and stay bounded away from and . We provide
deterministic equivalents to the empirical spectral distribution of and to the matrix entries of its resolvent (as well as of the resolvent of
). These deterministic equivalents are defined thanks to the
solutions of a fixed-point system. Besides, we prove that has
asymptotically no eigenvalues outside the bulk of its spectrum, defined thanks
to these deterministic equivalents. These results are directly used in our
companion paper "Kernel spectral clustering of large dimensional data", which
is devoted to the analysis of the spectral clustering algorithm in large
dimensions. They also find applications in various other fields such as
wireless communications where functionals of the aforementioned resolvents
allow one to assess the communication performance across multi-user
multi-antenna channels.Comment: 25 pages, 1 figure. The results of this paper are directly used in
our companion paper "Kernel spectral clustering of large dimensional data",
which is devoted to the analysis of the spectral clustering algorithm in
large dimensions. To appear in ESAIM Probab. Statis
Spectral Clustering of Mixed-Type Data
Cluster analysis seeks to assign objects with similar characteristics into groups called clusters so that objects within a group are similar to each other and dissimilar to objects in other groups. Spectral clustering has been shown to perform well in different scenarios on continuous data: it can detect convex and non-convex clusters, and can detect overlapping clusters. However, the constraint on continuous data can be limiting in real applications where data are often of mixed-type, i.e., data that contains both continuous and categorical features. This paper looks at extending spectral clustering to mixed-type data. The new method replaces the Euclidean-based similarity distance used in conventional spectral clustering with different dissimilarity measures for continuous and categorical variables. A global dissimilarity measure is than computed using a weighted sum, and a Gaussian kernel is used to convert the dissimilarity matrix into a similarity matrix. The new method includes an automatic tuning of the variable weight and kernel parameter. The performance of spectral clustering in different scenarios is compared with that of two state-of-the-art mixed-type data clustering methods, k-prototypes and KAMILA, using several simulated and real data sets
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