Locally Linear Embedding (LLE) is an elegant nonlinear dimensionality-reduction technique recently introduced by Roweis and Saul [2]. It fails when the data is divided into separate groups. We study a variant of LLE that can simultaneously group the data and calculate local embedding of each group. An estimate for the upper bound on the intrinsic dimension of the data set is obtained automatica...

We study the following problem: Given k paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension d > 2, there is a set of d+ 1 paths that does not admit a monotone simultaneous geometric embedding.

Miyaji, Nakabayashi and Takano (MNT) gave families of group orders of ordinary elliptic curves with embedding degree suitable for pairing applications. In this paper we generalise their results by giving families corresponding to non-prime group orders. We also consider the case of ordinary abelian varieties of dimension 2. We give families of group orders with embedding degrees 5, 10 and 12.

We demonstrate the use of neural networks to model and analyze time series of nonlinear dynamical systems. Based on recent results concerning the embedding of attractors from scalar time series, we use the neural models to estimate the embedding dimension and the nonnegative Lyapunov exponents of the system.

Dimensionality reduction is a topic of recent interest. In this paper, we present the classification constrained dimensionality reduction (CCDR) algorithm to account for label information. The algorithm can account for multiple classes as well as the semi-supervised setting. We present an out-of-sample expressions for both labeled and unlabeled data. For unlabeled data, we introduce a method of...

Data analysis is critical to many (if not a ll) Homeland Security missions. Data to be fielded in this domain is typically immense in cardinality (number of records) and immense in dimensionality (features per record). Random Projections (RP) have been proposed as an effective technique for embedding a metric space of dimension d into one of dimension k while retaining bounds on the distortion ...

An lp oblivious subspace embedding is a distribution over r × n matrices Π such that for any fixed n× d matrix A, Pr Π [for all x, ‖Ax‖p ≤ ‖ΠAx‖p ≤ κ‖Ax‖p] ≥ 9/10, where r is the dimension of the embedding, κ is the distortion of the embedding, and for an n-dimensional vector y, ‖y‖p = ( ∑n i=1 |yi|) 1/p is the lp-norm. Another important property is the sparsity of Π, that is, the maximum numbe...

One of the key problems in non-commutative algebra is the classification of central simple algebras and more generally of separable algebras over fields, i.e., Azumaya-algebras whose center is étale over the given field. In this paper we fix a central simple F -algebra A of prime power degree and study seperable algebras over extensions K/F , which embed in AK . The type of such an embedding is...