Our approach is based on the 'Optimization-on-a-Manifold' framework proposed by Krishnan et al. [33], to which we contribute with both systemic and computational improvements.
We conclude with a few remarks concerning practical and theoretical aspects of the manifold lifting framework.
Finally, another open question in the manifold lifting framework is what could be said about the uniqueness of ℳ x) given samples of Φℳ x).
Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g−2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult.
Here we employ a data-dependent manifold regularization framework which uses the geometry of the probability distribution [ 9].
Then, we can use the following two-step procedure to extend our DSD method to this manifold-based framework: (1) Given {y i}, form a data-dependent dictionary D y i = f ~ i ( 1 ), …, f ~ i ( m ) corresponding to each y i by finding its nearest-neighbor in each manifold: f ~ i = arg max f ∈ ℳ P y i f i ∗ = f, α i, A for ℓ∈{1,…,m} and i=1,…,M.
We discuss the Ornstein Uhlenbeck process over a compact Riemannian manifold in the framework of a martingale problem.
In this paper, we propose to unify various dimensionality reduction algorithms by interpreting the Manifold Regularization (MR) framework in a new way.
Our primary contributions in this paper are (1) a formulation for modeling gait subspaces on the Grassmann manifold, (2) a framework to integrate supervised and unsupervised GE techniques in the Grassmann manifold, (3) a method to incorporate sparse representation in the learning algorithms, and (4) extensive experiment to corroborate the proposed approach.
The Grassmann manifold provides a robust framework for measuring data similarity in a subspace context.
