Sampling theory, signal processing, and data analysis最新文献

Many empirical studies suggest that samples of continuous-time signals taken at locations randomly deviated from an equispaced grid (i.e., off-the-grid) can benefit signal acquisition, e.g., undersampling and anti-aliasing. However, explicit statements of such advantages and their respective conditions are scarce in the literature. This paper provides some insight on this topic when the sampling positions are known, with grid deviations generated i.i.d. from a variety distributions. By solving a square-root LASSO decoder with an interpolation kernel we demonstrate the capabilities of nonuniform samples for compressive sampling, an effective paradigm for undersampling and anti-aliasing. For functions in the Wiener algebra that admit a discrete s-sparse representation in some transform domain, we show that $O\left(s\mathrm{poly}\log N\right)$ random off-the-grid samples are sufficient to recover an accurate $\frac{N}{2}$-bandlimited approximation of the signal. For sparse signals (i.e., $s\ll N$), this sampling complexity is a great reduction in comparison to equispaced sampling where $O\left(N\right)$ measurements are needed for the same quality of reconstruction (Nyquist-Shannon sampling theorem). We further consider noise attenuation via oversampling (relative to a desired bandwidth), a standard technique with limited theoretical understanding when the sampling positions are non-equispaced. By solving a least squares problem, we show that $O(NlogN)$ i.i.d. randomly deviated samples provide an accurate $\frac{N}{2}$-bandlimited approximation of the signal with suppression of the noise energy by a factor $\sim \frac{1}{\sqrt{log\left(N\right)}}.$

Frames have been investigated frequently over the last few decades due to their valuable properties, which are desirable for various applications as well as interesting for theory. Some applications additionally require distributed processing techniques, which naturally leads to the concept of fusion frames and fusion frame systems. The latter consists of a system of subspaces, equipped with local frames on each of them, and a global frame. In this paper, we investigate the relations of the associated frame-related operators on all those three levels. For that we provide a detailed investigation on bounded block diagonal operators between Hilbert direct sums. We give the relation of the frame-related operators of the fusion frame and the corresponding frame systems in terms of operator identities. By applying these identities we prove further properties of fusion frame systems.

In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset ${\left\{x_i\right\}}_{i=1}^n$ and a set of noisy labels ${\left\{y_i\right\}}_{i=1}^n\subset R$ we let $u_n:{\left\{x_i\right\}}_{i=1}^n\to R$ be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When $y_i=g\left(x_i\right)+{\xi }_i$, for iid noise ${\xi }_i$, and using the geometric random graph, we identify (with high probability) the rate of convergence of $u_n$ to g in the large data limit $n\to \infty$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.