31 research outputs found
Robust Principal Component Analysis on Graphs
Principal Component Analysis (PCA) is the most widely used tool for linear
dimensionality reduction and clustering. Still it is highly sensitive to
outliers and does not scale well with respect to the number of data samples.
Robust PCA solves the first issue with a sparse penalty term. The second issue
can be handled with the matrix factorization model, which is however
non-convex. Besides, PCA based clustering can also be enhanced by using a graph
of data similarity. In this article, we introduce a new model called "Robust
PCA on Graphs" which incorporates spectral graph regularization into the Robust
PCA framework. Our proposed model benefits from 1) the robustness of principal
components to occlusions and missing values, 2) enhanced low-rank recovery, 3)
improved clustering property due to the graph smoothness assumption on the
low-rank matrix, and 4) convexity of the resulting optimization problem.
Extensive experiments on 8 benchmark, 3 video and 2 artificial datasets with
corruptions clearly reveal that our model outperforms 10 other state-of-the-art
models in its clustering and low-rank recovery tasks
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
Linear Shape Deformation Models with Local Support Using Graph-based Structured Matrix Factorisation
Representing 3D shape deformations by linear models in high-dimensional space
has many applications in computer vision and medical imaging, such as
shape-based interpolation or segmentation. Commonly, using Principal Components
Analysis a low-dimensional (affine) subspace of the high-dimensional shape
space is determined. However, the resulting factors (the most dominant
eigenvectors of the covariance matrix) have global support, i.e. changing the
coefficient of a single factor deforms the entire shape. In this paper, a
method to obtain deformation factors with local support is presented. The
benefits of such models include better flexibility and interpretability as well
as the possibility of interactively deforming shapes locally. For that, based
on a well-grounded theoretical motivation, we formulate a matrix factorisation
problem employing sparsity and graph-based regularisation terms. We demonstrate
that for brain shapes our method outperforms the state of the art in local
support models with respect to generalisation ability and sparse shape
reconstruction, whereas for human body shapes our method gives more realistic
deformations.Comment: Please cite CVPR 2016 versio
Compressive PCA for Low-Rank Matrices on Graphs
We introduce a novel framework for an approxi- mate recovery of data matrices
which are low-rank on graphs, from sampled measurements. The rows and columns
of such matrices belong to the span of the first few eigenvectors of the graphs
constructed between their rows and columns. We leverage this property to
recover the non-linear low-rank structures efficiently from sampled data
measurements, with a low cost (linear in n). First, a Resrtricted Isometry
Property (RIP) condition is introduced for efficient uniform sampling of the
rows and columns of such matrices based on the cumulative coherence of graph
eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is
suggested for the sampled data. Finally, several efficient, parallel and
parameter-free decoders are presented along with their theoretical analysis for
decoding the low-rank and cluster indicators for the full data matrix. Thus, we
overcome the computational limitations of the standard linear low-rank recovery
methods for big datasets. Our method can also be seen as a major step towards
efficient recovery of non- linear low-rank structures. For a matrix of size n X
p, on a single core machine, our method gains a speed up of over Robust
Principal Component Analysis (RPCA), where k << p is the subspace dimension.
Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times
faster than Robust PCA