| Summary: | Recently, great strides in sparse approximation theory and its application have been made. Many of these developments were spurred by the emerging area of compressed sensing. Compressed sensing is a signal acquisition paradigm that entails recovering estimates of sparse and compressible signals from n linear measurements, many fewer than the signal ambient dimension N. In general, these measurements are assumed to be imperfect. For example, they may be noisy or quantized (or both). Thus, the associated sparse recovery problem requires the existence of stable and robust “decoders” to reconstruct the signal. In this thesis, we first address the theoretical properties of ∆p, a class of compressed sensing decoders that rely on ℓp minimization with p ∈ (0, 1). In particular, we extend the known results regarding the decoder ∆₁, based on ℓ₁ minimization, to ∆p with p ∈ (0, 1). Our results are two-fold. We show that under sufficient conditions that are weaker than the analogous sufficient conditions for ∆₁, the decoders ∆p are robust to noise and stable. In particular, they are (2, p) instance optimal. We also show that, like ∆₁, the decoders ∆p are (2, 2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution. Second, we address quantization of compressed sensing measurements. Typically, the most commonly assumed approach (called PCM) entails quantizing each measurement independently, using a quantization alphabet with step-size d. Subsequently, by using a stable and robust decoder one can guarantee that the estimate produced by the decoder is within O(d) of the signal. As a more effective alternative, we propose using sigma-delta schemes to quantize compressed sensing measurements of a k-sparse signal. We show that there is an associated two-stage recovery method which guarantees a reduction of the approximation error by a factor of (n/k){α(r−¹/²)}, for any α < 1 if n ≳ k(log N)¹{¹−α)} (with high probability on the initial draw of the measurement matrix). In particular, the first recovery stage employs a stable decoder to estimate the support of the signal, while the second stage employs Sobolev dual frames to approximate the signal.
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