We present a robust multiple manifolds structure learning (RMMSL) scheme to robustly estimate data structures under the multiple low intrinsic dimensional manifolds assumption. In the local learning stage, RMMSL efficiently estimates local tangent space by weighted low-rank matrix factorization. In the global learning stage, we propose a robust manifold clustering method based on local structur...

In this paper, we present a method for isometric correction of manifold learning techniques. We first present an isometric nonlinear dimension reduction method. Our proposed method overcomes the issues associated with well-known isometric embedding techniques such as ISOMAP and maximum variance unfolding (MVU), i.e., computational complexity and the geodesic convexity requirement. Based on the ...

We propose methods for outlier handling and noise reduction using weighted local linear smoothing for a set of noisy points sampled from a nonlinear manifold. The methods can be used by manifold learning methods such as Isomap, LLE and LTSA as a preprocessing step to obtain a more accurate reconstruction of the underlying nonlinear manifolds. Weighted PCA is used as a building block for our met...

Over the past few years, a large family of manifold learning algorithms have been proposed, and applied to various applications. While designing new manifold learning algorithms has attracted much research attention, fewer research efforts have been focused on out-ofsample extrapolation of learned manifold. In this paper, we propose a novel algorithm of manifold learning. The proposed algorithm...

Abstract- Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the prerequisites for applying standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to high dimensional feature space without considering the intrinsic geometry of data points. ...

Most real data sets contain atypical observations, often referred to as outliers. Their presence may have a negative impact in data modeling using machine learning. This is particularly the case in data density estimation approaches. Manifold learning techniques provide low-dimensional data representations, often oriented towards visualization. The visualization provided by density estimation m...

For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. Our ...

We propose a novel method for linear dimensionality reduction of manifold modeled data. First, we show that with a small number M of random projections of sample points in R belonging to an unknown K-dimensional Euclidean manifold, the intrinsic dimension (ID) of the sample set can be estimated to high accuracy. Second, we rigorously prove that using only this set of random projections, we can ...

This senior thesis project explores and generalizes some fundamental machine learning algorithms from the Euclidean space to the statisticalmanifold, an abstract space in which each point is a probability distribution. In this thesis, we adapt the optimal separating hyperplane, the k-means clusteringmethod, and the hierarchical clustering method for classifying and clustering probability distri...

The Locally Linear Embedding (LLE) algorithm is an unsupervised nonlinear dimensionality-reduction method, which reports a low recognition rate in classification because it gives no consideration to the label information of sample distribution. In this paper, a classification method of supervised LLE (SLLE) based on Linear Discriminant Analysis (LDA) is proposed. First, samples are classified a...