1,536 research outputs found
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
Data based identification and prediction of nonlinear and complex dynamical systems
We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin
Time for dithering: fast and quantized random embeddings via the restricted isometry property
Recently, many works have focused on the characterization of non-linear
dimensionality reduction methods obtained by quantizing linear embeddings,
e.g., to reach fast processing time, efficient data compression procedures,
novel geometry-preserving embeddings or to estimate the information/bits stored
in this reduced data representation. In this work, we prove that many linear
maps known to respect the restricted isometry property (RIP) can induce a
quantized random embedding with controllable multiplicative and additive
distortions with respect to the pairwise distances of the data points beings
considered. In other words, linear matrices having fast matrix-vector
multiplication algorithms (e.g., based on partial Fourier ensembles or on the
adjacency matrix of unbalanced expanders) can be readily used in the definition
of fast quantized embeddings with small distortions. This implication is made
possible by applying right after the linear map an additive and random "dither"
that stabilizes the impact of the uniform scalar quantization operator applied
afterwards. For different categories of RIP matrices, i.e., for different
linear embeddings of a metric space
in with , we derive upper bounds on the
additive distortion induced by quantization, showing that it decays either when
the embedding dimension increases or when the distance of a pair of
embedded vectors in decreases. Finally, we develop a novel
"bi-dithered" quantization scheme, which allows for a reduced distortion that
decreases when the embedding dimension grows and independently of the
considered pair of vectors.Comment: Keywords: random projections, non-linear embeddings, quantization,
dither, restricted isometry property, dimensionality reduction, compressive
sensing, low-complexity signal models, fast and structured sensing matrices,
quantized rank-one projections (31 pages
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