DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling

This paper explores a fully unsupervised deep learning approach for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We use the Siamese configuration to train a neural network to solve the problem of least squares multidimensional scaling for generating maps that approximately preserve geodesic distances. By training with only a few landmarks, we show a significantly improved local and nonlocal generalization of the isometric mapping as compared to analogous non-parametric counterparts. Importantly, the combination of a deep-learning framework with a multidimensional scaling objective enables a numerical analysis of network architectures to aid in understanding their representation power. This provides a geometric perspective to the generalizability of deep learning.Comment: 10 pages, 11 Figure

A comparative evaluation of nonlinear dynamics methods for time series prediction

A key problem in time series prediction using autoregressive models is to fix the model order, namely the number of past samples required to model the time series adequately. The estimation of the model order using cross-validation may be a long process. In this paper, we investigate alternative methods to cross-validation, based on nonlinear dynamics methods, namely Grassberger-Procaccia, K,gl, Levina-Bickel and False Nearest Neighbors algorithms. The experiments have been performed in two different ways. In the first case, the model order has been used to carry out the prediction, performed by a SVM for regression on three real data time series showing that nonlinear dynamics methods have performances very close to the cross-validation ones. In the second case, we have tested the accuracy of nonlinear dynamics methods in predicting the known model order of synthetic time series. In this case, most of the methods have yielded a correct estimate and when the estimate was not correct, the value was very close to the real one

A study of the classification of low-dimensional data with supervised manifold learning

Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of supervised manifold learning for classification. We consider nonlinear dimensionality reduction algorithms that yield linearly separable embeddings of training data and present generalization bounds for this type of algorithms. A necessary condition for satisfactory generalization performance is that the embedding allow the construction of a sufficiently regular interpolation function in relation with the separation margin of the embedding. We show that for supervised embeddings satisfying this condition, the classification error decays at an exponential rate with the number of training samples. Finally, we examine the separability of supervised nonlinear embeddings that aim to preserve the low-dimensional geometric structure of data based on graph representations. The proposed analysis is supported by experiments on several real data sets