The existing nonlinear local methods for dimensionality reduction yield impressive results in data embedding and manifold visualization. However, they also open up the problem of how to define a unified projection from new data to the embedded subspace constructed by the training samples. Thinking globally and fitting locally, we present a new linear embedding approach, called Locally Embedded ...

Locally linear embedding (LLE) is a recently proposed unsupervised procedure for mapping high-dimensional data nonlinearly to a lower-dimensional space. In this paper, a supervised variation on LLE is proposed. This mapping, when combined with simple classifiers such as the nearest mean classifier, is shown to yield remarkably good classification results in experiments. Furthermore, a number of...

Recently, a new method intended to realize conformal mappings has been published. Called Locally Linear Embedding (LLE), this method can map high-dimensional data lying on a manifold to a representation of lower dimensionality that preserves the angles. Although LLE is claimed to solve problems that are usually managed by neural networks like Kohonen’s Self-Organizing Maps (SOMs), the method re...

A number of manifold learning algorithms have been recently proposed, including locally linear embedding (LLE). These algorithms not only merely reduce data dimensionality, but also attempt to discover a true low dimensional structure of the data. The common feature of the most of these algorithms is that they operate in a batch or offline mode. Hence, when new data arrive, one needs to rerun t...

Linear support vector machines (svms) have become popular for solving classification tasks due to their fast and simple online application to large scale data sets. However, many problems are not linearly separable. For these problems kernel-based svms are often used, but unlike their linear variant they suffer from various drawbacks in terms of computational and memory efficiency. Their respon...

We present locally-linear learning machines (L3M) for multi-class classification. We formulate a global convex risk function to jointly learn linear feature space partitions and region-specific linear classifiers. L3M’s features such as: (1) discriminative power similar to Kernel SVMs and Adaboost; (2) tight control on generalization error; (3) low training time cost due to on-line training; (4...

We propose a method for non-linear data projection that combines Generative Topographic Mapping and Coordinated PCA. We extend the Generative Topographic Mapping by using more complex nodes in the network: each node provides a linear map between the data space and the latent space. The location of a node in the data space is given by a smooth nonlinear function of its location in the latent spa...

We study linear time temporal logics of multiple agents, where the temporal modalities are local. These modalities not only refer to local next-instants and local eventuality, but also global views of agents at any local instant, which are updated due to communication from other agents. Thus agentsalso reason about the future, present and past of other agents in the system. The models for these...

In this report, the student presents her study on a multivariate visualization task. Specifically, the student would like to learn the use of Locally Linear Embedding (LLE) for visualization. While the experiment is using Wisconsin Breast Cancer dataset, the method is more generally applicable to other high-dimensional data as well. Three experiments were run to visualize the dataset. The three...