121 research outputs found
Parametric Regression on the Grassmannian
We address the problem of fitting parametric curves on the Grassmann manifold
for the purpose of intrinsic parametric regression. As customary in the
literature, we start from the energy minimization formulation of linear
least-squares in Euclidean spaces and generalize this concept to general
nonflat Riemannian manifolds, following an optimal-control point of view. We
then specialize this idea to the Grassmann manifold and demonstrate that it
yields a simple, extensible and easy-to-implement solution to the parametric
regression problem. In fact, it allows us to extend the basic geodesic model to
(1) a time-warped variant and (2) cubic splines. We demonstrate the utility of
the proposed solution on different vision problems, such as shape regression as
a function of age, traffic-speed estimation and crowd-counting from
surveillance video clips. Most notably, these problems can be conveniently
solved within the same framework without any specifically-tailored steps along
the processing pipeline.Comment: 14 pages, 11 figure
Multivariate Regression on the Grassmannian for Predicting Novel Domains
This work was supported by EPSRC (EP/L023385/1), and the European Union’s Horizon 2020 research and innovation program under grant agreement No 640891
Learning Dictionaries with Bounded Self-Coherence
Sparse coding in learned dictionaries has been established as a successful
approach for signal denoising, source separation and solving inverse problems
in general. A dictionary learning method adapts an initial dictionary to a
particular signal class by iteratively computing an approximate factorization
of a training data matrix into a dictionary and a sparse coding matrix. The
learned dictionary is characterized by two properties: the coherence of the
dictionary to observations of the signal class, and the self-coherence of the
dictionary atoms. A high coherence to the signal class enables the sparse
coding of signal observations with a small approximation error, while a low
self-coherence of the atoms guarantees atom recovery and a more rapid residual
error decay rate for the sparse coding algorithm. The two goals of high signal
coherence and low self-coherence are typically in conflict, therefore one seeks
a trade-off between them, depending on the application. We present a dictionary
learning method with an effective control over the self-coherence of the
trained dictionary, enabling a trade-off between maximizing the sparsity of
codings and approximating an equiangular tight frame.Comment: 4 pages, 2 figures; IEEE Signal Processing Letters, vol. 19, no. 12,
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