Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1

Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (l_2) metric, but much less for the Manhattan (l_1) metric. Our primary motivation is the approximate nearest neighbor problem in l_1. We exploit its reduction to the decision-with-witness version, called approximate near neighbor, which incurs a roughly logarithmic overhead. In 2007, Indyk and Naor, in the context of approximate nearest neighbors, introduced the notion of nearest neighbor-preserving embeddings. These are randomized embeddings between two metric spaces with guaranteed bounded distortion only for the distances between a query point and a point set. Such embeddings are known to exist for both l_2 and l_1 metrics, as well as for doubling subsets of l_2. The case that remained open were doubling subsets of l_1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of l_1. Our approach is to represent the pointset with a carefully chosen covering set, then randomly project the latter. We study two types of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension. We employ Cauchy variables: certain concentration bounds derived should be of independent interest

Efficient Nearest Neighbor Classification Using a Cascade of Approximate Similarity Measures

Nearest neighbor classification using shape context can yield highly accurate results in a number of recognition problems. Unfortunately, the approach can be too slow for practical applications, and thus approximation strategies are needed to make shape context practical. This paper proposes a method for efficient and accurate nearest neighbor classification in non-Euclidean spaces, such as the space induced by the shape context measure. First, a method is introduced for constructing a Euclidean embedding that is optimized for nearest neighbor classification accuracy. Using that embedding, multiple approximations of the underlying non-Euclidean similarity measure are obtained, at different levels of accuracy and efficiency. The approximations are automatically combined to form a cascade classifier, which applies the slower approximations only to the hardest cases. Unlike typical cascade-of-classifiers approaches, that are applied to binary classification problems, our method constructs a cascade for a multiclass problem. Experiments with a standard shape data set indicate that a two-to-three order of magnitude speed up is gained over the standard shape context classifier, with minimal losses in classification accuracy.National Science Foundation (IIS-0308213, IIS-0329009, EIA-0202067); Office of Naval Research (N00014-03-1-0108

Parametric t-Distributed Stochastic Exemplar-centered Embedding

Parametric embedding methods such as parametric t-SNE (pt-SNE) have been widely adopted for data visualization and out-of-sample data embedding without further computationally expensive optimization or approximation. However, the performance of pt-SNE is highly sensitive to the hyper-parameter batch size due to conflicting optimization goals, and often produces dramatically different embeddings with different choices of user-defined perplexities. To effectively solve these issues, we present parametric t-distributed stochastic exemplar-centered embedding methods. Our strategy learns embedding parameters by comparing given data only with precomputed exemplars, resulting in a cost function with linear computational and memory complexity, which is further reduced by noise contrastive samples. Moreover, we propose a shallow embedding network with high-order feature interactions for data visualization, which is much easier to tune but produces comparable performance in contrast to a deep neural network employed by pt-SNE. We empirically demonstrate, using several benchmark datasets, that our proposed methods significantly outperform pt-SNE in terms of robustness, visual effects, and quantitative evaluations.Comment: fixed typo