Spread representations

Fuchs, Jean–Jacques

doi:10.1109/acssc.2011.6190120

Cited by 43 publications

(40 citation statements)

References 5 publications

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“…These observations motivate a recent approach [12] for a better encoding strategy based on spread representations [20]. It reduces the quantization error underpinning the sign function.…”

Section: Spread Representationsmentioning

confidence: 97%

Beyond “project and sign” for cosine estimation with binary codes

Balu

Furon

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Many nearest neighbor search algorithms rely on encoding real vectors into binary vectors. The most common strategy projects the vectors onto random directions and takes the sign to produce so-called sketches. This paper discusses the sub-optimality of this choice, and proposes a better encoding strategy based on the quantization and reconstruction points of view. Our second contribution is a novel asymmetric estimator for the cosine similarity. Similar to previous asymmetric schemes, the query is not quantized and the similarity is computed in the compressed domain.Both our contribution leads to improve the quality of nearest neighbor search with binary codes. Its efficiency compares favorably against a recent encoding technique.

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“…These observations motivate a recent approach [12] for a better encoding strategy based on spread representations [20]. It reduces the quantization error underpinning the sign function.…”

Section: Spread Representationsmentioning

confidence: 97%

Beyond “project and sign” for cosine estimation with binary codes

Balu

Furon

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…Specifically, it was shown in [6] that for most matrices D, a dominant portion of the magnitudes of the solutionẋ to (P ∞ ) correspond to ẋ ∞ , whereas only a small fraction of the entries have smaller magnitude (see also [12]). …”

Section: B Relevant Prior Artmentioning

confidence: 98%

“…Note that in (12) we assumed that D H y 1 > 0. Since D is a frame with lower frame bound A > 0, we have…”

Section: Appendix B Proof Of Lemmamentioning

confidence: 99%

Signal representations with minimum ℓ<inf>∞</inf>-norm

Studer

Yin

Baraniuk

2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Maximum (or ∞) norm minimization subject to an underdetermined system of linear equations finds use in a large number of practical applications, such as vector quantization, peak-to-average power ratio (PAPR) (or "crest factor") reduction in wireless communication systems, approximate neighbor search, robotics, and control. In this paper, we analyze the fundamental properties of signal representations with minimum ∞-norm. In particular, we develop bounds on the maximum magnitude of such representations using the uncertainty principle (UP) introduced by Lyubarskii and Vershynin, 2010, and we characterize the limits of ∞-normbased PAPR reduction. Our results show that matrices satisfying the UP, such as randomly subsampled Fourier or i.i.d. Gaussian matrices, enable the efficient computation of so-called democratic representations, which have both provably small ∞-norm and low PAPR.

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“…An approach proposed in [138] for realizing this binary quantification is to compute these vectors as solutions of (P y,λ ) for J = · ∞ and a random Φ. A study of this regularization is done in [108], where an homotopy-like algorithm is provided. The use of this ℓ ∞ regularization is also connected to Kashin's representation [157], which is known to be useful in stabilizing the quantization error for instance.…”

Section: Literature Reviewmentioning

confidence: 99%

Low Complexity Regularization of Linear Inverse Problems

Vaiter

Peyré

Fadili

2015

Sampling Theory, a Renaissance

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Inverse problems and regularization theory is a central theme in imaging sciences, statistics and machine learning. The goal is to reconstruct an unknown vector from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown vector is to solve a convex optimization problem that enforces some prior knowledge about its structure. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of ℓ 2 -stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem.

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Spread representations

Cited by 43 publications

References 5 publications

Beyond “project and sign” for cosine estimation with binary codes

Beyond “project and sign” for cosine estimation with binary codes

Signal representations with minimum ℓ<inf>∞</inf>-norm

Low Complexity Regularization of Linear Inverse Problems

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Spread representations

Cited by 43 publications

References 5 publications

Beyond &#x201C;project and sign&#x201D; for cosine estimation with binary codes

Beyond &#x201C;project and sign&#x201D; for cosine estimation with binary codes

Signal representations with minimum &#x2113;<inf>&#x221E;</inf>-norm

Low Complexity Regularization of Linear Inverse Problems

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Beyond “project and sign” for cosine estimation with binary codes

Beyond “project and sign” for cosine estimation with binary codes

Signal representations with minimum ℓ<inf>∞</inf>-norm