Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.
There have been other attempts to address the problem of finding good coordinate representations of simple non-Euclidean data spaces. One approach @cite_7 is to use modified versions of multidimensional scaling specifically devised to find the best embedding of a data set into the cylinder, the sphere and so on. The target space has to be chosen in advance. Another class of approaches @cite_11 @cite_8 involves cutting the data manifold along arcs and curves until it has trivial topology. The resulting configuration can then be embedded in Euclidean space in the usual way. In our approach, the number of circular coordinates is not fixed in advance, but is determined experimentally after a persistent homology calculation. Moreover, there is no cutting involved; the coordinate functions respect the original topology of the data.
@cite_8 @cite_7 @cite_11
2020141201 2169504875 2810912957 2106890447
We introduce a novel projection-based visualization method for high-dimensional data sets by combining concepts from MDS and the geometry of the hyperbolic spaces. This approach hyperbolic multi-dimensional scaling (H-MDS) is a synthesis of two important concepts for explorative data analysis and visualization: (i) multi-dimensional scaling uses proximity or pair distance data to generate a low-dimensional, spatial presentation of the data; (ii) previous work on the "hyperbolic tree browser" demonstrated the extraordinary advantages for an interactive display of graph-like data in the two-dimensional hyperbolic space (H2).In the new approach, H-MDS maps proximity data directly into the H2. This removes the restriction to "quasihierarchical", graph-based data--a major limitation of (ii). Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data, this type of data can be considered the most general.We review important properties of the hyperbolic space and, in particular, the circular Poincare model of the H2. It enables effective human-computer interaction: by mouse dragging the "focus", the user can navigate in the data without loosing the context. In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.Several high-dimensional mapping examples including synthetic and real-world data are presented. Since high-dimensional data produce "ring"-shaped displays, we present methods to enhance the display by modulating the dissimilarity contrast. This is demonstrated for an application for unstructured text: i.e., by using multiple film critiques from news:rec.art.movies.reviews and www.imdb.com, each movie is placed within the H2--creating a "space of movies" for interactive exploration. We propose a novel projection based visualization method for high-dimensional datasets by combining concepts from MDS and the geometry of the hyperbolic spaces. Our approach Hyperbolic Multi-Dimensional Scaling (H-MDS) extends earlier work [7] using hyperbolic spaces for visualization of tree structures data ( "hyperbolic tree browser" ).By borrowing concepts from multi-dimensional scaling we map proximity data directly into the 2-dimensional hyperbolic space (H2). This removes the restriction to "quasihierarchical", graph-based data -- limiting previous work. Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data this type of data can be considered the most general.We used the circular Poincare model of the H2 which allows effective human-computer interaction: by moving the "focus" via mouse the user can navigate in the data without loosing the "context". In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.We present several high-dimensional mapping examples including synthetic and real world data and a successful application for unstructured text. By analyzing and integrating multiple film critiques from news:rec.art.movies.reviews and the internet movie database, each movie becomes placed within the H2. Here the idea is, that related films share more words in their reviews than unrelated. Their semantic proximity leads to a closer arrangement. The result is a kind of high-level content structured display allowing the user to explore the "space of movies". In this paper, we propose a novel space-time geometric representation of human landmark configurations and derive tools for comparison and classification. We model the temporal evolution of landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the benefit to bring naturally a second desirable quantity when comparing shapes — the spatial covariance — in addition to the conventional affine-shape representation. We derived then geometric and computational tools for rate-invariant analysis and adaptive re-sampling of trajectories, grounding on the Riemannian geometry of the underlying manifold. Specifically, our approach involves three steps: (1) landmarks are first mapped into the Riemannian manifold of positive semidefinite matrices of fixed-rank to build time-parameterized trajectories; (2) a temporal warping is performed on the trajectories, providing a geometry-aware (dis-)similarity measure between them; (3) finally, a pairwise proximity function SVM is used to classify them, incorporating the (dis-)similarity measure into the kernel function. We show that such representation and metric achieve competitive results in applications as action recognition and emotion recognition from 3D skelet al data, and facial expression recognition from videos. Experiments have been conducted on several publicly available up-to-date benchmarks. We address the problem of metric learning for multi-view data, namely the construction of embedding projections from data in different representations into a shared feature space, such that the Euclidean distance in this space provides a meaningful within-view as well as between-view similarity. Our motivation stems from the problem of cross-media retrieval tasks, where the availability of a joint Euclidean distance function is a prerequisite to allow fast, in particular hashing-based, nearest neighbor queries. We formulate an objective function that expresses the intuitive concept that matching samples are mapped closely together in the output space, whereas non-matching samples are pushed apart, no matter in which view they are available. The resulting optimization problem is not convex, but it can be decomposed explicitly into a convex and a concave part, thereby allowing efficient optimization using the convex-concave procedure. Experiments on an image retrieval task show that nearest-neighbor based cross-view retrieval is indeed possible, and the proposed technique improves the retrieval accuracy over baseline techniques.