Abstract
This chapter introduces the notion of instance optimality. While uniform ℓ 1-instance optimality is a practical concept in compressive sensing, uniform ℓ 2-instance optimality is shown not to be. A new property, called quotient property, is then developed to analyze measurement–reconstruction schemes. This property of the measurement matrix, coupled with equality-constrained ℓ 1-minimization, guarantees robustness of the scheme under measurement errors as well as nonuniform ℓ 2-instance optimality. The quotient property is proved to hold with high probability for Gaussian matrices and, modulo a slight modification, for subgaussian matrices.
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Foucart, S., Rauhut, H. (2013). Instance Optimality and Quotient Property. In: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4948-7_11
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DOI: https://doi.org/10.1007/978-0-8176-4948-7_11
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