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A note on reductions between compressed sensing guarantees
Information Theory · Submitted 2 Jun 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim As a corollary of this theorem, by reducing from well-known results in the compressed sensing literature, we obtain an efficient $\ell_p/\ell_p$ scheme for any $0<p<1$ with the fewest number of measurements currently known amongst efficient schemes, improving recent bounds of [SomaY16].
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The abstract, sentence by sentence
In compressed sensing, one wishes to acquire an approximately sparse high-dimensional signal $x\in\mathbb{R}^n$ via $m\ll n$ noisy linear measurements, then later approximately recover $x$ given only those measurement outcomes. Various guarantees have been studied in terms of the notion of approximation in recovery, and some isolated folklore results are known stating that some forms of recovery are stronger than others, via black-box reductions. In this note we provide a general theorem concerning the hierarchy of strengths of various recovery guarantees. As a corollary of this theorem, by reducing from well-known results in the compressed sensing literature, we obtain an efficient $\ell_p/\ell_p$ scheme for any $0<p<1$ with the fewest number of measurements currently known amongst efficient schemes, improving recent bounds of [SomaY16].
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In compressed sensing, one wishes to acquire an approximately sparse high-dimensional signal $x\in\mathbb{R}^n$ via $m\ll n$ noisy linear measurements, then later approximately recover $x$ given only those measurement outcomes. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s1 | Various guarantees have been studied in terms of the notion of approximation in recovery, and some isolated folklore results are known stating that some forms of recovery are stronger than others, via black-box reductions. | background 0.97 | 0.04 | 0.03 | asserted 0.96 |
| s2 | In this note we provide a general theorem concerning the hierarchy of strengths of various recovery guarantees. | contribution 1.00 | 0.94 | 0.26 | asserted 1.00 |
| s3 | As a corollary of this theorem, by reducing from well-known results in the compressed sensing literature, we obtain an efficient $\ell_p/\ell_p$ scheme for any $0<p<1$ with the fewest number of measurements currently known amongst efficient schemes, improving recent bounds of [SomaY16]. | result 0.89 | 0.94 | 0.67 | asserted 1.00 |
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Original abstract
In compressed sensing, one wishes to acquire an approximately sparse high-dimensional signal $x\in\mathbb{R}^n$ via $m\ll n$ noisy linear measurements, then later approximately recover $x$ given only those measurement outcomes. Various guarantees have been studied in terms of the notion of approximation in recovery, and some isolated folklore results are known stating that some forms of recovery are stronger than others, via black-box reductions. In this note we provide a general theorem concerning the hierarchy of strengths of various recovery guarantees. As a corollary of this theorem, by reducing from well-known results in the compressed sensing literature, we obtain an efficient $\ell_p/\ell_p$ scheme for any $0<p<1$ with the fewest number of measurements currently known amongst efficient schemes, improving recent bounds of [SomaY16].
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