The success of semi-supervised manifold learning is highly dependent on the quality of the labeled samples. Active manifold learning aims to select and label representative landmarks on a manifold from a given set of samples to improve semi-supervised manifold learning. In this paper, we propose a novel active manifold learning method based on a unified framework of manifold landmarking. In par...

Dimensionality reduction is required to produce visualizations of high dimensional data. In this framework, one of the most straightforward approaches to visualising high dimensional data is based on reducing complexity and applying linear projections while tumbling the projection axes in a defined sequence which generates a Grand Tour of the data. We propose using smooth nonlinear topographic ...

The goal of Manifold Learning (ML) is to find a description of low-dimensional structure of an unknown q-dimensional manifold embedded in high-dimensional ambient Euclidean space R p , q < p, from their finite samples. There are a variety of formulations of the problem. The methods of Manifold Approximation (MA) reconstruct (estimate) the manifold but don’t find a low-dimensional parameterizati...

We consider an alignment algorithm for reconstructing global coordinates from local coordinates constructed for sections of manifolds. We show that, under certain conditions, the alignment algorithm can successfully recover global coordinates even when local neighborhoods have different dimensions. Our results generalize an earlier analysis to allow alignment of sections of different dimensions...

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality, at the dimension of the true underlying manifold. This observation suggests to explore non-local manifold learning algorithms which attempt to discover shared structure in the tang...

the present article serves the purpose of pursuing geometrization of heat flow on volumetrically isothermal manifold by means of rf approach. in this article, we have analyzed the evolution of heat equation in a 3-dimensional smooth isothermal manifold bearing characteristics of riemannian manifold and fundamental properties of thermodynamic systems. by making use of the notions of various curva...

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality, at the dimension of the true underlying manifold. This observation invites an exploration of nonlocal manifold learning algorithms that attempt to discover shared structure in the ...

Manifold learning is the process of estimating a low-dimensional structure which underlies a collection of high-dimensional data. Here we review two popular methods for nonlinear dimensionality reduction, locally linear embedding (LLE, [1]) and IsoMap [2]. We also discuss their roots in principal component analysis and multidimensional scaling, and provide a brief comparison of the underlying a...

In many biomedical imaging applications, video sequences are captured with low resolution and low contrast challenging conditions in which to detect, segment, or track features. When image deformations have just a few underlying causes, such as continuously captured cardiac MRI without breath-holds or gating, the captured images lie on a lowdimensional, non-linear manifold. The manifold structu...

In this paper we consider a sequential, coarse-to-fine estimation of a piecewise constant function with smooth boundaries. Accurate detection and localization of the boundary (a manifold) is the key aspect of this problem. In general, algorithms capable of achieving optimal performance require exhaustive searches over large dictionaries that grow exponentially with the dimension of the observat...