Out-of-sample generalizations for supervised manifold learning for classification

Supervised manifold learning methods for data classification map data samples residing in a high-dimensional ambient space to a lower-dimensional domain in a structure-preserving way, while enhancing the separation between different classes in the learned embedding. Most nonlinear supervised manifold learning methods compute the embedding of the manifolds only at the initially available training points, while the generalization of the embedding to novel points, known as the out-of-sample extension problem in manifold learning, becomes especially important in classification applications. In this work, we propose a semi-supervised method for building an interpolation function that provides an out-of-sample extension for general supervised manifold learning algorithms studied in the context of classification. The proposed algorithm computes a radial basis function (RBF) interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples, defined as their distance to the embeddings of the manifolds of their own class, as well as a regularization term that controls the smoothness of the interpolation function in a direction-dependent way. The class labels of test data and the interpolation function parameters are estimated jointly with a progressive procedure. Experimental results on face and object images demonstrate the potential of the proposed out-of-sample extension algorithm for the classification of manifold-modeled data sets

Contribution to Graph-based Manifold Learning with Application to Image Categorization.

122 pLos algoritmos de aprendizaje de variedades basados en grafos (Graph,based manifold) son técnicas que han demostrado ser potentes herramientas para la extracción de características y la reducción de la dimensionalidad en los campos de reconomiento de patrones, visión por computador y aprendizaje automático. Estos algoritmos utilizan información basada en las similitudes de pares de muestras y del grafo ponderado resultante para revelar la estructura geométrica intrínseca de la variedad

Efficient Computation of K-Nearest Neighbor Graphs for Large High-Dimensional Data Sets on GPU Clusters

The k-Nearest Neighbor Graph (k-NNG) and the related k-Nearest Neighbor (k-NN) methods have a wide variety of applications in areas such as bioinformatics, machine learning, data mining, clustering analysis, and pattern recognition. Our application of interest is manifold embedding. Due to the large dimensionality of the input data (\u3c15k), spatial subdivision based techniques such OBBs, k-d tree, BSP etc., are not viable. The only alternative is the brute-force search, which has two distinct parts. The first finds distances between individual vectors in the corpus based on a pre-defined metric. Given the distance matrix, the second step selects k nearest neighbors for each member of the query data set. This thesis presents the development and implementation of a distributed exact k-Nearest Neighbor Graph (k-NNG) construction method. The proposed method uses Graphics Processing Units (GPUs) and exploits multiple levels of parallelism for distributed computational systems using GPUs. It is scalable for different cluster sizes, with each compute node in the cluster containing multiple GPUs. The distance computation is formulated as a basic matrix multiplication and reduction operation. The optimized CUBLAS matrix multiplication library is used for this purpose. Various distance metrics such as Euclidian, cosine, and Pearson are supported. For k-NNG construction, two different methods are presented. The first is based on an approach called batch index sorting to build the k-NNG with three sorting operations. This method uses the optimized radix sort implementation in the Thrust library for GPU. The second is an efficient implementation using the latest GPU functionalities of a variant of the quick select algorithm. Overall, the batch index sorting based k-NNG method is approximately 13x faster than a distributed MATLAB implementation. The quick select algorithm itself has a 5x speedup over state-of-the art GPU methods. This has enabled the processing of k-NNG construction on a data set containing 20 million image vectors, each with dimension 15,000, as part of a manifold embedding technique for analyzing the conformations of biomolecules

Contribution to Graph-based Manifold Learning with Application to Image Categorization.

122 pLos algoritmos de aprendizaje de variedades basados en grafos (Graph,based manifold) son técnicas que han demostrado ser potentes herramientas para la extracción de características y la reducción de la dimensionalidad en los campos de reconomiento de patrones, visión por computador y aprendizaje automático. Estos algoritmos utilizan información basada en las similitudes de pares de muestras y del grafo ponderado resultante para revelar la estructura geométrica intrínseca de la variedad