Nonlinear Supervised Dimensionality Reduction via Smooth Regular Embeddings

The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing methods primarily focus on the embedding of the training data, and the generalization of the embedding to initially unseen test data is rather ignored. In this work, we build on recent theoretical results on the generalization performance of supervised manifold learning algorithms. Motivated by these performance bounds, we propose a supervised manifold learning method that computes a nonlinear embedding while constructing a smooth and regular interpolation function that extends the embedding to the whole data space in order to achieve satisfactory generalization. The embedding and the interpolator are jointly learnt such that the Lipschitz regularity of the interpolator is imposed while ensuring the separation between different classes. Experimental results on several image data sets show that the proposed method outperforms traditional classifiers and the supervised dimensionality reduction algorithms in comparison in terms of classification accuracy in most settings

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

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

Efficient Nonlinear Dimensionality Reduction for Pixel-wise Classification of Hyperspectral Imagery

Classification, target detection, and compression are all important tasks in analyzing hyperspectral imagery (HSI). Because of the high dimensionality of HSI, it is often useful to identify low-dimensional representations of HSI data that can be used to make analysis tasks tractable. Traditional linear dimensionality reduction (DR) methods are not adequate due to the nonlinear distribution of HSI data. Many nonlinear DR methods, which are successful in the general data processing domain, such as Local Linear Embedding (LLE) [1], Isometric Feature Mapping (ISOMAP) [2] and Kernel Principal Components Analysis (KPCA) [3], run very slowly and require large amounts of memory when applied to HSI. For example, applying KPCA to the 512×217 pixel, 204-band Salinas image using a modern desktop computer (AMD FX-6300 Six-Core Processor, 32 GB memory) requires more than 5 days of computing time and 28GB memory! In this thesis, we propose two different algorithms for significantly improving the computational efficiency of nonlinear DR without adversely affecting the performance of classification task: Simple Linear Iterative Clustering (SLIC) superpixels and semi-supervised deep autoencoder networks (SSDAN). SLIC is a very popular algorithm developed for computing superpixels in RGB images that can easily be extended to HSI. Each superpixel includes hundreds or thousands of pixels based on spatial and spectral similarities and is represented by the mean spectrum and spatial position of all of its component pixels. Since the number of superpixels is much smaller than the number of pixels in the image, they can be used as input for nonlinearDR, which significantly reduces the required computation time and memory versus providing all of the original pixels as input. After nonlinear DR is performed using superpixels as input, an interpolation step can be used to obtain the embedding of each original image pixel in the low dimensional space. To illustrate the power of using superpixels in an HSI classification pipeline,we conduct experiments on three widely used and publicly available hyperspectral images: Indian Pines, Salinas and Pavia. The experimental results for all three images demonstrate that for moderately sized superpixels, the overall accuracy of classification using superpixel-based nonlinear DR matches and sometimes exceeds the overall accuracy of classification using pixel-based nonlinear DR, with a computational speed that is two-three orders of magnitude faster. Even though superpixel-based nonlinear DR shows promise for HSI classification, it does have disadvantages. First, it is costly to perform out-of-sample extensions. Second, it does not generalize to handle other types of data that might not have spatial information. Third, the original input pixels cannot approximately be recovered, as is possible in many DR algorithms.In order to overcome these difficulties, a new autoencoder network - SSDAN is proposed. It is a fully-connected semi-supervised autoencoder network that performs nonlinear DR in a manner that enables class information to be integrated. Features learned from SSDAN will be similar to those computed via traditional nonlinear DR, and features from the same class will be close to each other. Once the network is trained well with training data, test data can be easily mapped to the low dimensional embedding. Any kind of data can be used to train a SSDAN,and the decoder portion of the SSDAN can easily recover the initial input with reasonable loss.Experimental results on pixel-based classification in the Indian Pines, Salinas and Pavia images show that SSDANs can approximate the overall accuracy of nonlinear DR while significantly improving computational efficiency. We also show that transfer learning can be use to finetune features of a trained SSDAN for a new HSI dataset. Finally, experimental results on HSI compression show a trade-off between Overall Accuracy (OA) of extracted features and PeakSignal to Noise Ratio (PSNR) of the reconstructed image

Node Embedding over Temporal Graphs

In this work, we present a method for node embedding in temporal graphs. We propose an algorithm that learns the evolution of a temporal graph's nodes and edges over time and incorporates this dynamics in a temporal node embedding framework for different graph prediction tasks. We present a joint loss function that creates a temporal embedding of a node by learning to combine its historical temporal embeddings, such that it optimizes per given task (e.g., link prediction). The algorithm is initialized using static node embeddings, which are then aligned over the representations of a node at different time points, and eventually adapted for the given task in a joint optimization. We evaluate the effectiveness of our approach over a variety of temporal graphs for the two fundamental tasks of temporal link prediction and multi-label node classification, comparing to competitive baselines and algorithmic alternatives. Our algorithm shows performance improvements across many of the datasets and baselines and is found particularly effective for graphs that are less cohesive, with a lower clustering coefficient