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14
Random projections of smooth manifolds
 Foundations of Computational Mathematics
, 2006
"... We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N ..."
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Cited by 144 (25 self)
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We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N → R M, M < N, on a smooth wellconditioned Kdimensional submanifold M ⊂ R N. As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on M are wellpreserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in R N. Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifoldmodeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.
Manifoldbased signal recovery and parameter estimation from compressive measurements
"... A field known as Compressive Sensing (CS) has recently emerged to help address the growing challenges of capturing and processing highdimensional signals and data sets. CS exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressiv ..."
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Cited by 21 (6 self)
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A field known as Compressive Sensing (CS) has recently emerged to help address the growing challenges of capturing and processing highdimensional signals and data sets. CS exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive (or random) linear measurements of that signal. Strong theoretical guarantees have been established on the accuracy to which sparse or nearsparse signals can be recovered from noisy compressive measurements. In this paper, we address similar questions in the context of a different modeling framework. Instead of sparse models, we focus on the broad class of manifold models, which can arise in both parametric and nonparametric signal families. Building upon recent results concerning the stable embeddings of manifolds within the measurement space, we establish both deterministic and probabilistic instanceoptimal bounds in ℓ2 for manifoldbased signal recovery and parameter estimation from noisy compressive measurements. In line with analogous results for sparsitybased CS, we conclude that much stronger bounds are possible in the probabilistic setting. Our work supports the growing empirical evidence that manifoldbased models can be used with high accuracy in compressive signal processing.
New analysis of manifold embeddings and signal recovery from compressive measurements. arXiv:1306.4748
"... Compressive Sensing (CS) exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive, often random linear measurements of that signal. Strong theoretical guarantees have been established concerning the embedding of a sparse signal ..."
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Cited by 13 (1 self)
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Compressive Sensing (CS) exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive, often random linear measurements of that signal. Strong theoretical guarantees have been established concerning the embedding of a sparse signal family under a random measurement operator and on the accuracy to which sparse signals can be recovered from noisy compressive measurements. In this paper, we address similar questions in the context of a different modeling framework. Instead of sparse models, we focus on the broad class of manifold models, which can arise in both parametric and nonparametric signal families. Using tools from the theory of empirical processes, we improve upon previous results concerning the embedding of lowdimensional manifolds under random measurement operators. We also establish both deterministic and probabilistic instanceoptimal bounds in `2 for manifoldbased signal recovery and parameter estimation from noisy compressive measurements. In line with analogous results for sparsitybased CS, we conclude that much stronger bounds are possible in the probabilistic setting. Our work supports the growing evidence that manifoldbased models can be used with high accuracy in compressive signal processing.
Signal Recovery on Incoherent Manifolds
"... Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear submanifold of a highdimensional ambient space. We introduce SPIN, a firstorder projected gradient method to recover the signal com ..."
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Cited by 6 (1 self)
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Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear submanifold of a highdimensional ambient space. We introduce SPIN, a firstorder projected gradient method to recover the signal components. Despite the nonconvex nature of the recovery problem and the possibility of underdetermined measurements, SPIN provably recovers the signal components, provided that the signal manifolds are incoherent and that the measurement operator satisfies a certain restricted isometry property. SPIN significantly extends the scope of current recovery models and algorithms for lowdimensional linear inverse problems and matches (or exceeds) the current state of the art in terms of performance. 1
Discriminative kmetrics
"... The kqflats algorithm is a generalization of the popular kmeans algorithm where q dimensional best fit affine sets replace centroids as the cluster prototypes. In this work, a modification of the k qflats framework for pattern classification is introduced. The basic idea is to replace the origina ..."
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Cited by 5 (1 self)
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The kqflats algorithm is a generalization of the popular kmeans algorithm where q dimensional best fit affine sets replace centroids as the cluster prototypes. In this work, a modification of the k qflats framework for pattern classification is introduced. The basic idea is to replace the original reconstruction only energy, which is optimized to obtain the k affine spaces, by a new energy that incorporates discriminative terms. This way, the actual classification task is introduced as part of the design and optimization. The presentation of the proposed framework is complemented with experimental results, showing that the method is computationally very efficient and gives excellent results on standard supervised learning benchmarks.
http://www.nps.gov/cave/planyourvisit/maps.htm [Accessed on
, 2008
"... Research article The complete mitochondrial genome of the sea spider Achelia bituberculata (Pycnogonida, Ammotheidae): arthropod ground pattern of gene arrangement ..."
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Research article The complete mitochondrial genome of the sea spider Achelia bituberculata (Pycnogonida, Ammotheidae): arthropod ground pattern of gene arrangement
A Tutorial on Recovery Conditions for Compressive System Identification of Sparse Channels
"... Abstract — In this tutorial, we review some of the recent ..."
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Abstract — In this tutorial, we review some of the recent
IEEE TRANSACTIONS ON SIGNAL PROCESSING 0 1 IEEE TRANSACTIONS ON SIGNAL PROCESSING 1 Topological localization via signals of opportunity
"... Abstract—We consider the problems of localization, disambiguation, and mapping in a domain filled with signalsofopportunity generated by transmitters. One or more (static or mobile) receivers utilize these signals and from them characterize the domain, localize, disambiguate, etc. The tools we deve ..."
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Abstract—We consider the problems of localization, disambiguation, and mapping in a domain filled with signalsofopportunity generated by transmitters. One or more (static or mobile) receivers utilize these signals and from them characterize the domain, localize, disambiguate, etc. The tools we develop are topological in nature, and rely on interpreting the problem as one of embedding the domain into a sufficiently highdimensional space of signals via a signal profile function. Varying kinds of signal processing (TOA, TDOA, DOA, etc.) and discretization are addressed. Finally, we describe experiments that demonstrate the feasibility of these ideas in practice. Index Terms—sensor networks, opportunistic signals, mapping, localization
IEEE TRANSACTIONS ON SIGNAL PROCESSING 1 Topological
"... localization via signals of opportunity ..."
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