No abstract

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y = Φx of a k-sparse signal x ∈ R N , where Φ satisfies the restricted isometry property, then the approximate recoveryThe simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth order Σ∆ quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduction of the approximation error by a factor of (m/k) (r−1/2)α for any 0 < α < 1, if m r k(log N ) 1/(1−α) . The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x that satisfy a mild size condition on their supports.

SummarySynthetic biology is an increasingly high-profile area of research that can be understood as encompassing three broad approaches towards the synthesis of living systems: DNA-based device construction, genome-driven cell engineering and protocell creation. Each approach is characterized by different aims, methods and constructs, in addition to a range of positions on intellectual property and regulatory regimes. We identify subtle but important differences between the schools in relation to their treatments of genetic determinism, cellular context and complexity. These distinctions tie into two broader issues that define synthetic biology: the relationships between biology and engineering, and between synthesis and analysis. These themes also illuminate synthetic biology's connections to genetic and other forms of biological engineering, as well as to systems biology. We suggest that all these knowledge-making distinctions in synthetic biology raise fundamental questions about the nature of biological investigation and its relationship to the construction of biological components and systems.

Abstract. A new class of alternative dual frames is introduced in the setting of finite frames for R d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (Σ∆) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order O(N −r ) for a wide class of finite frames of size N . This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.

The second order Sigma-Delta (Σ∆) scheme with linear quantization rule is analyzed for quantizing finite unit-norm tight frame expansions for R d. Approximation error estimates are derived, and it is shown that for certain choices of frames the quantization error is of order 1/N 2 , where N is the frame size. However, in contrast to the setting of bandlimited functions there are many situations where the second order scheme only gives approximation error of order 1/N. For example, this is the case when quantizing harmonic frames of odd length in even dimensions. An important component of the error analysis involves extending existing stability results to yield smaller invariant sets for the linear second order Σ∆ scheme.