3,020 research outputs found
Graph-based classification of multiple observation sets
We consider the problem of classification of an object given multiple
observations that possibly include different transformations. The possible
transformations of the object generally span a low-dimensional manifold in the
original signal space. We propose to take advantage of this manifold structure
for the effective classification of the object represented by the observation
set. In particular, we design a low complexity solution that is able to exploit
the properties of the data manifolds with a graph-based algorithm. Hence, we
formulate the computation of the unknown label matrix as a smoothing process on
the manifold under the constraint that all observations represent an object of
one single class. It results into a discrete optimization problem, which can be
solved by an efficient and low complexity algorithm. We demonstrate the
performance of the proposed graph-based algorithm in the classification of sets
of multiple images. Moreover, we show its high potential in video-based face
recognition, where it outperforms state-of-the-art solutions that fall short of
exploiting the manifold structure of the face image data sets.Comment: New content adde
On landmark selection and sampling in high-dimensional data analysis
In recent years, the spectral analysis of appropriately defined kernel
matrices has emerged as a principled way to extract the low-dimensional
structure often prevalent in high-dimensional data. Here we provide an
introduction to spectral methods for linear and nonlinear dimension reduction,
emphasizing ways to overcome the computational limitations currently faced by
practitioners with massive datasets. In particular, a data subsampling or
landmark selection process is often employed to construct a kernel based on
partial information, followed by an approximate spectral analysis termed the
Nystrom extension. We provide a quantitative framework to analyse this
procedure, and use it to demonstrate algorithmic performance bounds on a range
of practical approaches designed to optimize the landmark selection process. We
compare the practical implications of these bounds by way of real-world
examples drawn from the field of computer vision, whereby low-dimensional
manifold structure is shown to emerge from high-dimensional video data streams.Comment: 18 pages, 6 figures, submitted for publicatio
Geometric deep learning
The goal of these course notes is to describe the main mathematical ideas behind geometric deep learning and to provide implementation details for several applications in shape analysis and synthesis, computer vision and computer graphics. The text in the course materials is primarily based on previously published work. With these notes we gather and provide a clear picture of the key concepts and techniques that fall under the umbrella of geometric deep learning, and illustrate the applications they enable. We also aim to provide practical implementation details for the methods presented in these works, as well as suggest further readings and extensions of these ideas
Geometry-Aware Neighborhood Search for Learning Local Models for Image Reconstruction
Local learning of sparse image models has proven to be very effective to
solve inverse problems in many computer vision applications. To learn such
models, the data samples are often clustered using the K-means algorithm with
the Euclidean distance as a dissimilarity metric. However, the Euclidean
distance may not always be a good dissimilarity measure for comparing data
samples lying on a manifold. In this paper, we propose two algorithms for
determining a local subset of training samples from which a good local model
can be computed for reconstructing a given input test sample, where we take
into account the underlying geometry of the data. The first algorithm, called
Adaptive Geometry-driven Nearest Neighbor search (AGNN), is an adaptive scheme
which can be seen as an out-of-sample extension of the replicator graph
clustering method for local model learning. The second method, called
Geometry-driven Overlapping Clusters (GOC), is a less complex nonadaptive
alternative for training subset selection. The proposed AGNN and GOC methods
are evaluated in image super-resolution, deblurring and denoising applications
and shown to outperform spectral clustering, soft clustering, and geodesic
distance based subset selection in most settings.Comment: 15 pages, 10 figures and 5 table
NetLSD: Hearing the Shape of a Graph
Comparison among graphs is ubiquitous in graph analytics. However, it is a
hard task in terms of the expressiveness of the employed similarity measure and
the efficiency of its computation. Ideally, graph comparison should be
invariant to the order of nodes and the sizes of compared graphs, adaptive to
the scale of graph patterns, and scalable. Unfortunately, these properties have
not been addressed together. Graph comparisons still rely on direct approaches,
graph kernels, or representation-based methods, which are all inefficient and
impractical for large graph collections.
In this paper, we propose the Network Laplacian Spectral Descriptor (NetLSD):
the first, to our knowledge, permutation- and size-invariant, scale-adaptive,
and efficiently computable graph representation method that allows for
straightforward comparisons of large graphs. NetLSD extracts a compact
signature that inherits the formal properties of the Laplacian spectrum,
specifically its heat or wave kernel; thus, it hears the shape of a graph. Our
evaluation on a variety of real-world graphs demonstrates that it outperforms
previous works in both expressiveness and efficiency.Comment: KDD '18: The 24th ACM SIGKDD International Conference on Knowledge
Discovery & Data Mining, August 19--23, 2018, London, United Kingdo
Expanding the Family of Grassmannian Kernels: An Embedding Perspective
Modeling videos and image-sets as linear subspaces has proven beneficial for
many visual recognition tasks. However, it also incurs challenges arising from
the fact that linear subspaces do not obey Euclidean geometry, but lie on a
special type of Riemannian manifolds known as Grassmannian. To leverage the
techniques developed for Euclidean spaces (e.g, support vector machines) with
subspaces, several recent studies have proposed to embed the Grassmannian into
a Hilbert space by making use of a positive definite kernel. Unfortunately,
only two Grassmannian kernels are known, none of which -as we will show- is
universal, which limits their ability to approximate a target function
arbitrarily well. Here, we introduce several positive definite Grassmannian
kernels, including universal ones, and demonstrate their superiority over
previously-known kernels in various tasks, such as classification, clustering,
sparse coding and hashing
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