arXiv research

Survey of Locally Linear Embedding and its variants.

Local Linear embedding (LLE) is a popular dimension reduction method. In this paper, we first show LLE with nonnegative constraint is equivalent to the widely used Laplacian embedding. We further propose to iterate the two steps in LLE repeatedly to improve the results. Thirdly, we relax the kNN constraint of LLE and p…

Generative LLE modifies LLE to generate stochastic embeddings.

The local linear embedding algorithm (LLE) is a non-linear dimension-reducing technique, widely used due to its computational simplicity and intuitive approach. LLE first linearly reconstructs each input point from its nearest neighbors and then preserves these neighborhood relations in the low-dimensional embedding. W…

This work connects LLE, factor analysis, and probabilistic PCA through a stochastic perspective.

A new method simplifies HLLE for better robustness.

Study proposes Local Linear Encoding for better feature discretization.

Recently manifold learning algorithm for dimensionality reduction attracts more and more interests, and various linear and nonlinear, global and local algorithms are proposed. The key step of manifold learning algorithm is the neighboring region selection. However, so far for the references we know, few of which propos…

LNPE enhances local connections in embeddings using extended neighbor propagation.

New methods for visualizing multi-view data improve clustering accuracy.

LLE produces unwanted results without regularization, which can be prevented with regularization.

Existing approaches to analyzing the asymptotics of graph Laplacians typically assume a well-behaved kernel function with smoothness assumptions. We remove the smoothness assumption and generalize the analysis of graph Laplacians to include previously unstudied graphs including kNN graphs. We also introduce a kernel-fr…

We introduce a new unsupervised representation learning and visualization using deep convolutional networks and self organizing maps called Deep Neural Maps (DNM). DNM jointly learns an embedding of the input data and a mapping from the embedding space to a two-dimensional lattice. We compare visualizations of DNM with…

In this paper, we propose a semi-supervised dictionary learning method that uses both the information in labelled and unlabelled data and jointly trains a linear classifier embedded on the sparse codes. The manifold structure of the data in the sparse code space is preserved using the same approach as the Locally Linea…

We present the Procrustes measure, a novel measure based on Procrustes rotation that enables quantitative comparison of the output of manifold-based embedding algorithms (such as LLE (Roweis and Saul, 2000) and Isomap (Tenenbaum et al, 2000)). The measure also serves as a natural tool when choosing dimension-reduction …

We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and Diffusion maps.…

Most of existing manifold learning methods rely on Mean Squared Error (MSE) or $\ell_2$ norm. However, for the problem of image quality assessment, these are not promising measure. In this paper, we introduce the concept of an image structure manifold which captures image structure features and discriminates image dist…

The ability to characterize the color content of natural imagery is an important application of image processing. The pixel by pixel coloring of images may be viewed naturally as points in color space, and the inherent structure and distribution of these points affords a quantization, through clustering, of the color i…

We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on a set of neighbourhood relationships. The model allows straightforward variatio…

We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…

Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of structural network properties. The present paper advocates two graph embedding appro…

We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps) focus on mining massive data sets using w…

Proposes a boundary detection method inspired by LLE for high-dimensional data.

ProbDR framework interprets DR algorithms as probabilistic inference.

AI-enabled precision medicine promises a transformational improvement in healthcare outcomes by enabling data-driven personalized diagnosis, prognosis, and treatment. However, the well-known "curse of dimensionality" and the clustered structure of biomedical data together interact to present a joint challenge in the hi…

Generalizes machine learning models using localization kernels and local means.

For many machine learning tasks, the input data lie on a low-dimensional manifold embedded in a high dimensional space and, because of this high-dimensional structure, most algorithms are inefficient. The typical solution is to reduce the dimension of the input data using standard dimension reduction algorithms such as…

Predictability enables efficient parallelization of nonlinear models.

In the era of big data, reducing data dimensionality is critical in many areas of science. Widely used Principal Component Analysis (PCA) addresses this problem by computing a low dimensional data embedding that maximally explain variance of the data. However, PCA has two major weaknesses. Firstly, it only considers li…