An Explicit Nonlinear Mapping for Manifold Learning

Manifold learning is a hot research topic in the field of computer science and has many applications in the real world. A main drawback of manifold learning methods is, however, that there is no explicit mappings from the input data manifold to the output embedding. This prohibits the application of manifold learning methods in many practical problems such as classification and target detection. Previously, in order to provide explicit mappings for manifold learning methods, many methods have been proposed to get an approximate explicit representation mapping with the assumption that there exists a linear projection between the high-dimensional data samples and their low-dimensional embedding. However, this linearity assumption may be too restrictive. In this paper, an explicit nonlinear mapping is proposed for manifold learning, based on the assumption that there exists a polynomial mapping between the high-dimensional data samples and their low-dimensional representations. As far as we know, this is the first time that an explicit nonlinear mapping for manifold learning is given. In particular, we apply this to the method of Locally Linear Embedding (LLE) and derive an explicit nonlinear manifold learning algorithm, named Neighborhood Preserving Polynomial Embedding (NPPE). Experimental results on both synthetic and real-world data show that the proposed mapping is much more effective in preserving the local neighborhood information and the nonlinear geometry of the high-dimensional data samples than previous work

Landmark Diffusion Maps (L-dMaps): Accelerated manifold learning out-of-sample extension

Diffusion maps are a nonlinear manifold learning technique based on harmonic analysis of a diffusion process over the data. Out-of-sample extensions with computational complexity $\mathcal{O}(N)$, where $N$ is the number of points comprising the manifold, frustrate applications to online learning applications requiring rapid embedding of high-dimensional data streams. We propose landmark diffusion maps (L-dMaps) to reduce the complexity to $\mathcal{O}(M)$, where $M \ll N$ is the number of landmark points selected using pruned spanning trees or k-medoids. Offering $(N/M)$ speedups in out-of-sample extension, L-dMaps enables the application of diffusion maps to high-volume and/or high-velocity streaming data. We illustrate our approach on three datasets: the Swiss roll, molecular simulations of a C$_{24}$H$_{50}$ polymer chain, and biomolecular simulations of alanine dipeptide. We demonstrate up to 50-fold speedups in out-of-sample extension for the molecular systems with less than 4% errors in manifold reconstruction fidelity relative to calculations over the full dataset.Comment: Submitte

A Unified Semi-Supervised Dimensionality Reduction Framework for Manifold Learning

We present a general framework of semi-supervised dimensionality reduction for manifold learning which naturally generalizes existing supervised and unsupervised learning frameworks which apply the spectral decomposition. Algorithms derived under our framework are able to employ both labeled and unlabeled examples and are able to handle complex problems where data form separate clusters of manifolds. Our framework offers simple views, explains relationships among existing frameworks and provides further extensions which can improve existing algorithms. Furthermore, a new semi-supervised kernelization framework called ``KPCA trick'' is proposed to handle non-linear problems.Comment: 22 pages, 9 figure

Incomplete Pivoted QR-based Dimensionality Reduction

High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of dimensionality' phenomenon. Therefore, dimensionality reduction methods are applied to the data prior to its analysis. Many of these methods are based on principal components analysis, which is statistically driven, namely they map the data into a low-dimension subspace that preserves significant statistical properties of the high-dimensional data. As a consequence, such methods do not directly address the geometry of the data, reflected by the mutual distances between multidimensional data point. Thus, operations such as classification, anomaly detection or other machine learning tasks may be affected. This work provides a dictionary-based framework for geometrically driven data analysis that includes dimensionality reduction, out-of-sample extension and anomaly detection. It embeds high-dimensional data in a low-dimensional subspace. This embedding preserves the original high-dimensional geometry of the data up to a user-defined distortion rate. In addition, it identifies a subset of landmark data points that constitute a dictionary for the analyzed dataset. The dictionary enables to have a natural extension of the low-dimensional embedding to out-of-sample data points, which gives rise to a distortion-based criterion for anomaly detection. The suggested method is demonstrated on synthetic and real-world datasets and achieves good results for classification, anomaly detection and out-of-sample tasks

Locality preserving projection on SPD matrix Lie group: algorithm and analysis

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data. For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.Comment: 15 pages, 3 table

Shamap: Shape-based Manifold Learning

For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric according to angular changes along a geodesic line, thereby reflecting the underlying shape-oriented information or a topological similarity between high- and low-dimensional representations of a data cloud. Our results demonstrate the feasibility and merits of the proposed dimensionality reduction scheme

Principal Polynomial Analysis

This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository

Curvature-aware Manifold Learning

Traditional manifold learning algorithms assumed that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, they divided manifold into a set of overlapping local patches which are locally isometric to linear subsets of Euclidean space. By analyzing the global or local isometry assumptions it can be shown that the learnt manifold is a flat manifold with zero Riemannian curvature tensor. In general, manifolds may not satisfy these hypotheses. One major limitation of traditional manifold learning is that it does not consider the curvature information of manifold. In order to remove these limitations, we present our curvature-aware manifold learning algorithm called CAML. The purpose of our algorithm is to break the local isometry assumption and to reduce the dimension of the general manifold which is not isometric to Euclidean space. Thus, our method adds the curvature information to the process of manifold learning. The experiments have shown that our method CAML is more stable than other manifold learning algorithms by comparing the neighborhood preserving ratios.Comment: 24 pages, 4 figure

Dimensionality Reduction has Quantifiable Imperfections: Two Geometric Bounds

In this paper, we investigate Dimensionality reduction (DR) maps in an information retrieval setting from a quantitative topology point of view. In particular, we show that no DR maps can achieve perfect precision and perfect recall simultaneously. Thus a continuous DR map must have imperfect precision. We further prove an upper bound on the precision of Lipschitz continuous DR maps. While precision is a natural measure in an information retrieval setting, it does not measure `how' wrong the retrieved data is. We therefore propose a new measure based on Wasserstein distance that comes with similar theoretical guarantee. A key technical step in our proofs is a particular optimization problem of the $L_2$-Wasserstein distance over a constrained set of distributions. We provide a complete solution to this optimization problem, which can be of independent interest on the technical side.Comment: 32nd Conference on Neural Information Processing Systems (NIPS 2018), Montreal, Canad

Linearly-Recurrent Autoencoder Networks for Learning Dynamics

This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and over-fitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of over-specified EDMD systems in feature space. Nonlinear reconstruction using partially linear multi-kernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto-Sivashinsky equation.Comment: 37 pages, 16 figure