13,381 research outputs found
& Maximum Variance Embedding: Measuring and Optimizing Connectivity of A Graph Metric
Bobkov, Houdr\'e, and the last author [2000] introduced a Poincar\'e-type
functional parameter, , of a graph and related it to
connectivity of the graph via Cheeger-type inequalities. A work by the second
author, Raghavendra, and Vempala [2013] related the complexity of
to the so-called small-set expansion (SSE) problem and further
set forth the desiderata for NP-hardness of this optimization problem. We
confirm the conjecture that computing is NP-hard for weighted
trees.
Beyond measuring connectivity in many applications we want to optimize it.
This, via convex duality, leads to a problem in machine learning known as the
Maximum Variance Embedding (MVE). The output is a function from vertices to a
low dim Euclidean space, subject to bounds on Euclidean distances between
neighbors. The objective is to maximize output variance. Special cases of MVE
into and dims lead to absolute algebraic connectivity [1990] and spread
constant [1998], that measure connectivity of the graph and its Cartesian
-power, respectively. MVE has other applications in measuring diffusion
speed and robustness of networks, clustering, and dimension reduction.
We show that computing MVE in tree-width dims is NP-hard, while only one
additional dim beyond width of a given tree-decomposition makes the problem in
P. We show that MVE of a tree in 2 dims defines a non-convex yet benign
optimization landscape, i.e., local=global optima. We further develop a linear
time combinatorial algorithm for this case. Finally, we denote approximate
Maximum Variance Embedding is tractable in significantly lower dims. For trees
and general graphs, for which Maximum Variance Embedding cannot be solved in
less than and dims, we provide approximation
algorithms for embedding into and dims,
respectively
Masking Strategies for Image Manifolds
We consider the problem of selecting an optimal mask for an image manifold,
i.e., choosing a subset of the pixels of the image that preserves the
manifold's geometric structure present in the original data. Such masking
implements a form of compressive sensing through emerging imaging sensor
platforms for which the power expense grows with the number of pixels acquired.
Our goal is for the manifold learned from masked images to resemble its full
image counterpart as closely as possible. More precisely, we show that one can
indeed accurately learn an image manifold without having to consider a large
majority of the image pixels. In doing so, we consider two masking methods that
preserve the local and global geometric structure of the manifold,
respectively. In each case, the process of finding the optimal masking pattern
can be cast as a binary integer program, which is computationally expensive but
can be approximated by a fast greedy algorithm. Numerical experiments show that
the relevant manifold structure is preserved through the data-dependent masking
process, even for modest mask sizes
Structural Variability from Noisy Tomographic Projections
In cryo-electron microscopy, the 3D electric potentials of an ensemble of
molecules are projected along arbitrary viewing directions to yield noisy 2D
images. The volume maps representing these potentials typically exhibit a great
deal of structural variability, which is described by their 3D covariance
matrix. Typically, this covariance matrix is approximately low-rank and can be
used to cluster the volumes or estimate the intrinsic geometry of the
conformation space. We formulate the estimation of this covariance matrix as a
linear inverse problem, yielding a consistent least-squares estimator. For
images of size -by- pixels, we propose an algorithm for calculating this
covariance estimator with computational complexity
, where the condition number
is empirically in the range --. Its efficiency relies on the
observation that the normal equations are equivalent to a deconvolution problem
in 6D. This is then solved by the conjugate gradient method with an appropriate
circulant preconditioner. The result is the first computationally efficient
algorithm for consistent estimation of 3D covariance from noisy projections. It
also compares favorably in runtime with respect to previously proposed
non-consistent estimators. Motivated by the recent success of eigenvalue
shrinkage procedures for high-dimensional covariance matrices, we introduce a
shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We
evaluate our methods on simulated datasets and achieve classification results
comparable to state-of-the-art methods in shorter running time. We also present
results on clustering volumes in an experimental dataset, illustrating the
power of the proposed algorithm for practical determination of structural
variability.Comment: 52 pages, 11 figure
Topological data analysis of contagion maps for examining spreading processes on networks
Social and biological contagions are influenced by the spatial embeddedness
of networks. Historically, many epidemics spread as a wave across part of the
Earth's surface; however, in modern contagions long-range edges -- for example,
due to airline transportation or communication media -- allow clusters of a
contagion to appear in distant locations. Here we study the spread of
contagions on networks through a methodology grounded in topological data
analysis and nonlinear dimension reduction. We construct "contagion maps" that
use multiple contagions on a network to map the nodes as a point cloud. By
analyzing the topology, geometry, and dimensionality of manifold structure in
such point clouds, we reveal insights to aid in the modeling, forecast, and
control of spreading processes. Our approach highlights contagion maps also as
a viable tool for inferring low-dimensional structure in networks.Comment: Main Text and Supplementary Informatio
Conditional t-SNE: Complementary t-SNE embeddings through factoring out prior information
Dimensionality reduction and manifold learning methods such as t-Distributed
Stochastic Neighbor Embedding (t-SNE) are routinely used to map
high-dimensional data into a 2-dimensional space to visualize and explore the
data. However, two dimensions are typically insufficient to capture all
structure in the data, the salient structure is often already known, and it is
not obvious how to extract the remaining information in a similarly effective
manner. To fill this gap, we introduce \emph{conditional t-SNE} (ct-SNE), a
generalization of t-SNE that discounts prior information from the embedding in
the form of labels. To achieve this, we propose a conditioned version of the
t-SNE objective, obtaining a single, integrated, and elegant method. ct-SNE has
one extra parameter over t-SNE; we investigate its effects and show how to
efficiently optimize the objective. Factoring out prior knowledge allows
complementary structure to be captured in the embedding, providing new
insights. Qualitative and quantitative empirical results on synthetic and
(large) real data show ct-SNE is effective and achieves its goal
Spread spectrum-based video watermarking algorithms for copyright protection
Merged with duplicate record 10026.1/2263 on 14.03.2017 by CS (TIS)Digital technologies know an unprecedented expansion in the last years. The consumer can
now benefit from hardware and software which was considered state-of-the-art several years
ago. The advantages offered by the digital technologies are major but the same digital
technology opens the door for unlimited piracy. Copying an analogue VCR tape was certainly
possible and relatively easy, in spite of various forms of protection, but due to the analogue
environment, the subsequent copies had an inherent loss in quality. This was a natural way of
limiting the multiple copying of a video material. With digital technology, this barrier
disappears, being possible to make as many copies as desired, without any loss in quality
whatsoever. Digital watermarking is one of the best available tools for fighting this threat.
The aim of the present work was to develop a digital watermarking system compliant with the
recommendations drawn by the EBU, for video broadcast monitoring. Since the watermark
can be inserted in either spatial domain or transform domain, this aspect was investigated and
led to the conclusion that wavelet transform is one of the best solutions available. Since
watermarking is not an easy task, especially considering the robustness under various attacks
several techniques were employed in order to increase the capacity/robustness of the system:
spread-spectrum and modulation techniques to cast the watermark, powerful error correction
to protect the mark, human visual models to insert a robust mark and to ensure its invisibility.
The combination of these methods led to a major improvement, but yet the system wasn't
robust to several important geometrical attacks. In order to achieve this last milestone, the
system uses two distinct watermarks: a spatial domain reference watermark and the main
watermark embedded in the wavelet domain. By using this reference watermark and techniques
specific to image registration, the system is able to determine the parameters of the attack and
revert it. Once the attack was reverted, the main watermark is recovered. The final result is a
high capacity, blind DWr-based video watermarking system, robust to a wide range of attacks.BBC Research & Developmen
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