Manifold learning has become a hot issue in the research fields of machine learning and data mining. Current manifold learning algorithms assume that the observed data set has the high density. But, how to evaluate the denseness of the high dimensional observed data set? This paper proposes an algorithm based on the average geodesic distance as the preprocessing step of manifold learning. Moreo...

We introduce a boosting framework to solve a classification problem with added manifold and ambient regularization costs. It allows for a natural extension of boosting into both semisupervised problems and unsupervised problems. The augmented cost is minimized in a greedy, stagewise functional minimization procedure as in GradientBoost. Our method provides insights into generalization issues in...

Manifold learning have attracted considerable attention over the last decade. The most frequently used functional is the l-norm of the gradient of the function. In this paper, we consider the linear manifold learning problem by minimizing this functional with appropriate constraint. We provide theoretical analysis on both the functional and the constraint, which shows the affine hulls of the ma...

Biometric face data are essentially high dimensional data and as such are susceptible to the well-known problem of the curse of dimensionality when analyzed using machine learning techniques. Various dimensionality reduction methods have been proposed in the literature to represent high dimensional data in a lower dimensional space. Research has shown that biometric face data are non-linear in ...

Many natural image sets are samples of a low-dimensional manifold in the space of all possible images. Understanding this manifold is a key first step in understanding many sets of images, and manifold learning approaches have recently been used within many application domains, including face recognition, medical image segmentation, gait recognition and hand-written character recognition. This ...

Recognizing objects from different viewpoints is a challenging task. One approach for handling this task is to model the appearance of an object under different viewing conditions using a low dimensional subspace. Manifold learning describes the process by which this low dimensional embedding can be generated. However, manifold learning is an unsupervised method and thus gives poor results on c...

We propose a new algorithm that simultaneously estimates the intrinsic dimension and intrinsic entropy of random data sets lying on smooth manifolds. The method is based on asymptotic properties of entropic graph constructions. In particular, we compute the Euclidean -nearest neighbors ( NN) graph over the sample points and use its overall total edge length to estimate intrinsic dimension and e...

Many problems in the fields of Computer Vision deal with image data that is embedded in very high-dimensional spaces. However, it is typical that there are few variables, with a small number of degrees of freedom, that control the underlying process that generated the images. Therefore, a typical assumption behind many algorithms is that the data lie on a low-dimensional manifold. Modeling the ...

Visualization of high-dimensional data, such as text documents, is useful to map out the similarities among various data points. In the high-dimensional space, documents are commonly represented as bags of words, with dimensionality equal to the size of the vocabulary. Classical approaches to document visualization directly reduce this into visualizable two or three dimensions, using techniques...

We derive theoretical bounds on the performance of manifold learning algorithms, given access to a small number of random projections of the input dataset. We prove that with the number of projections only logarithmic in the size of the original space, we may reliably learn the structure of the nonlinear manifold, as compared to performing conventional manifold learning on the full dataset.