We propose a flexible and multi-scale method for organizing, visualizing, and understanding point cloud datasets sampled from or near stratified spaces. The first part of the algorithm produces a cover tree for a dataset using an adaptive threshold that is based on multi-scale local principal component analysis. The resulting cover tree nodes reflect the local geometry of the space and are organized via a scaffolding graph. In the second part of the algorithm, the goals are to uncover the strata that make up the underlying stratified space using a local dimension estimation procedure and topological data analysis, as well as to ultimately visualize the results in a simplified spine graph. We demonstrate our technique on several synthetic examples and then use it to visualize song structure in musical audio data.
MLPCA has been used before ( @cite_15 , @cite_0 , @cite_9 ) to analyze data, but not, to our knowledge, in combination with the cover tree.
@cite_0 @cite_9 @cite_15
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Nonlinear principal component analysis is a novel technique for multivariate data analysis, similar to the well-known method of principal component analysis. NLPCA, like PCA, is used to identify and remove correlations among problem variables as an aid to dimensionality reduction, visualization, and exploratory data analysis. While PCA identifies only linear correlations between variables, NLPCA uncovers both linear and nonlinear correlations, without restriction on the character of the nonlinearities present in the data. NLPCA operates by training a feedforward neural network to perform the identity mapping, where the network inputs are reproduced at the output layer. The network contains an internal “bottleneck” layer (containing fewer nodes than input or output layers), which forces the network to develop a compact representation of the input data, and two additional hidden layers. The NLPCA method is demonstrated using time-dependent, simulated batch reaction data. Results show that NLPCA successfully reduces dimensionality and produces a feature space map resembling the actual distribution of the underlying system parameters. Principal component analysis (PCA), as a key component in statistical learning, has been adopted in a wide variety of applications in computer vision and machine learning. From a different angle, weakly supervised learning, more specifically multiple instance learning (MIL), allows fine-grained information to be exploited from coarsely-grained label information. In this paper, we propose an algorithm using the robust PCA (RPCA) [1] in a iterative way to perform simultaneous common object discovery and model learning under a one-class multiple instance learning setting. We show the advantage of our method on common object discovery and model learning, which needs no fine coarse alignment in the input data; in addition, it achieves comparable results with standard two-class MIL learning algorithms but our method is learning from one-class data only. We consider the problem of using a large unlabeled sample to boost performance of a learning algorit,hrn when only a small set of labeled examples is available. In particular, we consider a problem setting motivated by the task of learning to classify web pages, in which the description of each example can be partitioned into two distinct views. For example, the description of a web page can be partitioned into the words occurring on that page, and the words occurring in hyperlinks t,hat point to that page. We assume that either view of the example would be sufficient for learning if we had enough labeled data, but our goal is to use both views together to allow inexpensive unlabeled data to augment, a much smaller set of labeled examples. Specifically, the presence of two distinct views of each example suggests strategies in which two learning algorithms are trained separately on each view, and then each algorithm’s predictions on new unlabeled examples are used to enlarge the training set of the other. Our goal in this paper is to provide a PAC-style analysis for this setting, and, more broadly, a PAC-style framework for the general problem of learning from both labeled and unlabeled data. We also provide empirical results on real web-page data indicating that this use of unlabeled examples can lead to significant improvement of hypotheses in practice. *This research was supported in part by the DARPA HPKB program under contract F30602-97-1-0215 and by NSF National Young investigator grant CCR-9357793. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. TO copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. COLT 98 Madison WI USA Copyright ACM 1998 l-58113-057--0 98 7... 5.00 92 Tom Mitchell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 mitchell+@cs.cmu.edu Representing documents by vectors that are independent of language enhances machine translation and multilingual text categorization. We use discriminative training to create a projection of documents from multiple languages into a single translingual vector space. We explore two variants to create these projections: Oriented Principal Component Analysis (OPCA) and Coupled Probabilistic Latent Semantic Analysis (CPLSA). Both of these variants start with a basic model of documents (PCA and PLSA). Each model is then made discriminative by encouraging comparable document pairs to have similar vector representations. We evaluate these algorithms on two tasks: parallel document retrieval for Wikipedia and Europarl documents, and cross-lingual text classification on Reuters. The two discriminative variants, OPCA and CPLSA, significantly outperform their corresponding baselines. The largest differences in performance are observed on the task of retrieval when the documents are only comparable and not parallel. The OPCA method is shown to perform best.