9,008 research outputs found
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
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
megaman: Manifold Learning with Millions of points
Manifold Learning is a class of algorithms seeking a low-dimensional
non-linear representation of high-dimensional data. Thus manifold learning
algorithms are, at least in theory, most applicable to high-dimensional data
and sample sizes to enable accurate estimation of the manifold. Despite this,
most existing manifold learning implementations are not particularly scalable.
Here we present a Python package that implements a variety of manifold learning
algorithms in a modular and scalable fashion, using fast approximate neighbors
searches and fast sparse eigendecompositions. The package incorporates
theoretical advances in manifold learning, such as the unbiased Laplacian
estimator and the estimation of the embedding distortion by the Riemannian
metric method. In benchmarks, even on a single-core desktop computer, our code
embeds millions of data points in minutes, and takes just 200 minutes to embed
the main sample of galaxy spectra from the Sloan Digital Sky Survey ---
consisting of 0.6 million samples in 3750-dimensions --- a task which has not
previously been possible.Comment: 12 pages, 6 figure
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
Web image annotation by diffusion maps manifold learning algorithm
Automatic image annotation is one of the most challenging problems in machine
vision areas. The goal of this task is to predict number of keywords
automatically for images captured in real data. Many methods are based on
visual features in order to calculate similarities between image samples. But
the computation cost of these approaches is very high. These methods require
many training samples to be stored in memory. To lessen this burden, a number
of techniques have been developed to reduce the number of features in a
dataset. Manifold learning is a popular approach to nonlinear dimensionality
reduction. In this paper, we investigate Diffusion maps manifold learning
method for web image auto-annotation task. Diffusion maps manifold learning
method is used to reduce the dimension of some visual features. Extensive
experiments and analysis on NUS-WIDE-LITE web image dataset with different
visual features show how this manifold learning dimensionality reduction method
can be applied effectively to image annotation.Comment: 11 pages, 8 figure
Co-manifold learning with missing data
Representation learning is typically applied to only one mode of a data
matrix, either its rows or columns. Yet in many applications, there is an
underlying geometry to both the rows and the columns. We propose utilizing this
coupled structure to perform co-manifold learning: uncovering the underlying
geometry of both the rows and the columns of a given matrix, where we focus on
a missing data setting. Our unsupervised approach consists of three components.
We first solve a family of optimization problems to estimate a complete matrix
at multiple scales of smoothness. We then use this collection of smooth matrix
estimates to compute pairwise distances on the rows and columns based on a new
multi-scale metric that implicitly introduces a coupling between the rows and
the columns. Finally, we construct row and column representations from these
multi-scale metrics. We demonstrate that our approach outperforms competing
methods in both data visualization and clustering.Comment: 16 pages, 9 figure
Deep nets for local manifold learning
The problem of extending a function defined on a training data
on an unknown manifold to the entire manifold and a
tubular neighborhood of this manifold is considered in this paper. For
embedded in a high dimensional ambient Euclidean space
, a deep learning algorithm is developed for finding a local
coordinate system for the manifold {\bf without eigen--decomposition}, which
reduces the problem to the classical problem of function approximation on a low
dimensional cube. Deep nets (or multilayered neural networks) are proposed to
accomplish this approximation scheme by using the training data. Our methods do
not involve such optimization techniques as back--propagation, while assuring
optimal (a priori) error bounds on the output in terms of the number of
derivatives of the target function. In addition, these methods are universal,
in that they do not require a prior knowledge of the smoothness of the target
function, but adjust the accuracy of approximation locally and automatically,
depending only upon the local smoothness of the target function. Our ideas are
easily extended to solve both the pre--image problem and the out--of--sample
extension problem, with a priori bounds on the growth of the function thus
extended.Comment: Submitted on Sept. 17, 201
Manifold Learning: The Price of Normalization
We analyze the performance of a class of manifold-learning algorithms that
find their output by minimizing a quadratic form under some normalization
constraints. This class consists of Locally Linear Embedding (LLE), Laplacian
Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and
Diffusion maps. We present and prove conditions on the manifold that are
necessary for the success of the algorithms. Both the finite sample case and
the limit case are analyzed. We show that there are simple manifolds in which
the necessary conditions are violated, and hence the algorithms cannot recover
the underlying manifolds. Finally, we present numerical results that
demonstrate our claims.Comment: Submitted to JML
Constrained Manifold Learning for Hyperspectral Imagery Visualization
Displaying the large number of bands in a hyper- spectral image (HSI) on a
trichromatic monitor is important for HSI processing and analysis system. The
visualized image shall convey as much information as possible from the original
HSI and meanwhile facilitate image interpretation. However, most existing
methods display HSIs in false color, which contradicts with user experience and
expectation. In this paper, we propose a visualization approach based on
constrained manifold learning, whose goal is to learn a visualized image that
not only preserves the manifold structure of the HSI but also has natural
colors. Manifold learning preserves the image structure by forcing pixels with
similar signatures to be displayed with similar colors. A composite kernel is
applied in manifold learning to incorporate both the spatial and spectral
information of HSI in the embedded space. The colors of the output image are
constrained by a corresponding natural-looking RGB image, which can either be
generated from the HSI itself (e.g., band selection from the visible
wavelength) or be captured by a separate device. Our method can be done at
instance-level and feature-level. Instance-level learning directly obtains the
RGB coordinates for the pixels in the HSI while feature-level learning learns
an explicit mapping function from the high dimensional spectral space to the
RGB space. Experimental results demonstrate the advantage of the proposed
method in information preservation and natural color visualization
Implicit Manifold Learning on Generative Adversarial Networks
This paper raises an implicit manifold learning perspective in Generative
Adversarial Networks (GANs), by studying how the support of the learned
distribution, modelled as a submanifold , perfectly match
with , the support of the real data distribution. We show that
optimizing Jensen-Shannon divergence forces to perfectly
match with , while optimizing Wasserstein distance does not.
On the other hand, by comparing the gradients of the Jensen-Shannon divergence
and the Wasserstein distances ( and ) in their primal forms, we
conjecture that Wasserstein may enjoy desirable properties such as
reduced mode collapse. It is therefore interesting to design new distances that
inherit the best from both distances
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