31,251 research outputs found
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
LASAGNE: Locality And Structure Aware Graph Node Embedding
In this work we propose Lasagne, a methodology to learn locality and
structure aware graph node embeddings in an unsupervised way. In particular, we
show that the performance of existing random-walk based approaches depends
strongly on the structural properties of the graph, e.g., the size of the
graph, whether the graph has a flat or upward-sloping Network Community Profile
(NCP), whether the graph is expander-like, whether the classes of interest are
more k-core-like or more peripheral, etc. For larger graphs with flat NCPs that
are strongly expander-like, existing methods lead to random walks that expand
rapidly, touching many dissimilar nodes, thereby leading to lower-quality
vector representations that are less useful for downstream tasks. Rather than
relying on global random walks or neighbors within fixed hop distances, Lasagne
exploits strongly local Approximate Personalized PageRank stationary
distributions to more precisely engineer local information into node
embeddings. This leads, in particular, to more meaningful and more useful
vector representations of nodes in poorly-structured graphs. We show that
Lasagne leads to significant improvement in downstream multi-label
classification for larger graphs with flat NCPs, that it is comparable for
smaller graphs with upward-sloping NCPs, and that is comparable to existing
methods for link prediction tasks
Shrunken Locally Linear Embedding for Passive Microwave Retrieval of Precipitation
This paper introduces a new Bayesian approach to the inverse problem of
passive microwave rainfall retrieval. The proposed methodology relies on a
regularization technique and makes use of two joint dictionaries of
coincidental rainfall profiles and their corresponding upwelling spectral
radiative fluxes. A sequential detection-estimation strategy is adopted, which
basically assumes that similar rainfall intensity values and their spectral
radiances live close to some sufficiently smooth manifolds with analogous local
geometry. The detection step employs a nearest neighborhood classification
rule, while the estimation scheme is equipped with a constrained shrinkage
estimator to ensure stability of retrieval and some physical consistency. The
algorithm is examined using coincidental observations of the active
precipitation radar (PR) and passive microwave imager (TMI) on board the
Tropical Rainfall Measuring Mission (TRMM) satellite. We present promising
results of instantaneous rainfall retrieval for some tropical storms and
mesoscale convective systems over ocean, land, and coastal zones. We provide
evidence that the algorithm is capable of properly capturing different storm
morphologies including high intensity rain-cells and trailing light rainfall,
especially over land and coastal areas. The algorithm is also validated at an
annual scale for calendar year 2013 versus the standard (version 7) radar
(2A25) and radiometer (2A12) rainfall products of the TRMM satellite
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