Many manifold learning algorithms utilize graphs of local neighborhoods to estimate manifold topology. When neighborhood connections short-circuit between geodesically distant regions of the manifold, poor results are obtained due to the compromises that the manifold learner must make to satisfy the erroneous criteria. Also, existing manifold learning algorithms have difficulty unfolding manifo...

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these are graph-based: they associate vertex each point weighted edge pair. Existing theory shows that the Laplacian matrix graph converges to Laplace–Beltrami operator manifold, under assumption pairwise affinities...

The need to reduce the dimensionality of a dataset whilst retaining its inherent manifold structure is key to many pattern recognition, machine learning, and computer vision problems. This process is often referred to as manifold learning since the structure is preserved during dimensionality reduction by learning the intrinsic low-dimensional manifold that the data lies upon. In this paper a h...

Manifold learning aims to find a low dimensional parameterization for data sets which lie on nonlinear manifolds in a high-dimensional space. Applications of manifold learning include face recognition, image retrieval, machine learning, classification, visualization, and so on. By studying the existing manifold learning algorithms and geometric properties of local tangent space of a manifold, w...

In Non-Linear Dimensionality Reduction (NLDR) algorithms, one is given a collection of N objects, each a vector in RD, and the objective is to find an embedding in a low-dimensional Euclidean space Rd while preserving the geometry as faithfully as possible. In traditional NLDR methods like classical MDS, space complexity turns out to be O(N2) which for large N becomes unaffordable with reasonab...

Two multilevel frameworks for manifold learning algorithms are discussed which are based on an affinity graph whose goal is to sketch the neighborhood of each sample point. One framework is geometric and is suitable for methods aiming to find an isometric or a conformal mapping, such as isometric feature mapping (Isomap) and semidefinite embedding (SDE). The other is algebraic and can be incorp...

Analysis of CT scans for studying Chronic Obstructive Pulmonary Disease (COPD) is generally limited to mean scores of disease extent. However, the evolution of local pulmonary damage may vary between patients with discordant effects on lung physiology. This limits the explanatory power of mean values in clinical studies. We present local disease and deformation distributions to address this lim...

Manifold learning is a popular recent approach to nonlinear dimensionality reduction. Algorithms for this task are based on the idea that the dimensionality of many data sets is only artificially high; though each data point consists of perhaps thousands of features, it may be described as a function of only a few underlying parameters. That is, the data points are actually samples from a low-d...