Manifold learning techniques for dynamical systems and time series have shown their utility for a broad spectrum of applications in recent years. While these methods are effective at learning a low-dimensional representation, they are often insufficient for visualizing the global and local structure of the data. In this paper, we present DIG (Dynamical Information Geometry), a visualization method for multivariate time series data that extracts an information geometry from a diffusion framework. Specifically, we implement a novel group of distances in the context of diffusion operators, which may be useful to reveal structure in the data that may not be accessible by the commonly used diffusion distances. Finally, we present a case study applying our visualization tool to EEG data to visualize sleep stages.
Many dimensionality reduction methods exist, some of which have been used for visualization @cite_4 @cite_15 @cite_13 @cite_3 @cite_19 @cite_20 @cite_26 . Principal components analysis (PCA) @cite_20 and t-distributed stochastic neighborhood embedding (t-SNE) @cite_4 are two of the most commonly used methods for visualization. However, these and other methods are inadequate in many applications. First, these methods tend to favor one aspect of the data at the expense of the other. For example, when used for visualization, PCA typically shows the large scale global structure of the data while neglecting the finer, local structure. In contrast, t-SNE is explicitly designed to focus on the local structure and often distorts the global structure, potentially leading to misinterpretations @cite_1 . Second, PCA and t-SNE fail to explicitly denoise the data for visualization. Thus in noisy settings, the true structure of the data can be obscured. In addition, none of these methods are designed to exploit the structure present in dynamical systems.
@cite_26 @cite_4 @cite_1 @cite_3 @cite_19 @cite_15 @cite_13 @cite_20
2807933484 2467197661 1679894842 1485262465
Real world networks constructed from raw data are often characterized by complex community structures. Existing dimensionality reduction techniques, however, do not take such characteristics into account. This is especially important for problems with low number of samples where the curse of dimensionality is particularly significant. Therefore, in this paper, we propose FeatureNet, a novel community-based dimensionality reduction framework targeting small sample problems. To this end, we propose a new method to directly construct a network from high-dimensional raw data while explicitly revealing its hidden community structure; these communities are then used to learn low-dimensional features using a representation learning framework. We show the effectiveness of our approach on eight datasets covering application areas as diverse as handwritten digits, biology, physical sciences, NLP, and computational sustainability. Extensive experiments on the above datasets (with sizes mostly between 100 and 1500 samples) demonstrate that FeatureNet significantly outperforms (i.e., up to 40 improvement in classification accuracy) ten well-known dimensionality reduction methods like PCA, Kernel PCA, Isomap, SNE, t-SNE, etc. In recent years, dimensionality-reduction techniques have been developed and are widely used for hypothesis generation in Exploratory Data Analysis. However, these techniques are confronted with overcoming the trade-off between computation time and the quality of the provided dimensionality reduction. In this work, we address this limitation, by introducing Hierarchical Stochastic Neighbor Embedding (Hierarchical-SNE). Using a hierarchical representation of the data, we incorporate the well-known mantra of Overview-First, Details-On-Demand in non-linear dimensionality reduction. First, the analysis shows an embedding, that reveals only the dominant structures in the data (Overview). Then, by selecting structures that are visible in the overview, the user can filter the data and drill down in the hierarchy. While the user descends into the hierarchy, detailed visualizations of the high-dimensional structures will lead to new insights. In this paper, we explain how Hierarchical-SNE scales to the analysis of big datasets. In addition, we show its application potential in the visualization of Deep-Learning architectures and the analysis of hyperspectral images. The paper addresses large scale image retrieval with short vector representations. We study dimensionality reduction by Principal Component Analysis (PCA) and propose improvements to its different phases. We show and explicitly exploit relations between i) mean subtraction and the negative evidence, i.e., a visual word that is mutually missing in two descriptions being compared, and ii) the axis de-correlation and the co-occurrences phenomenon. Finally, we propose an effective way to alleviate the quantization artifacts through a joint dimensionality reduction of multiple vocabularies. The proposed techniques are simple, yet significantly and consistently improve over the state of the art on compact image representations. Complementary experiments in image classification show that the methods are generally applicable. We propose an image denoising method that ex- ploits nonlocal image modeling, principal component analysis (PCA), and local shape-adaptive anisotropic estimation. The nonlocal modeling is exploited by grouping similar image patches in 3-D groups. The denoising is performed by shrinkage of the spectrum of a 3-D transform applied on such groups. The effectiveness of the shrinkage depends on the ability of the transform to sparsely represent the true-image data, thus separating it from the noise. We propose to improve the sparsity in two aspects. First, we employ image patches (neighborhoods) which can have data-adaptive shape. Second, we propose PCA on these adaptive-shape neighborhoods as part of the employed 3-D transform. The PCA bases are obtained by eigenvalue decompo- sition of empirical second-moment matrices that are estimated from groups of similar adaptive-shape neighborhoods. We show that the proposed method is competitive and outperforms some of the current best denoising methods, especially in preserving image details and introducing very few artifacts.