Abstract. Compressed sensing investigates the recovery of sparse signals from linear measurements. But often, in a wide range of applications, one is given only the absolute values (squared) of the linear measurements. Recovering such signals (not necessarily sparse) is known as the phase retrieval problem. We consider this problem in the case when the measurements are time-frequency shifts of a suitably chosen generator, i.e. coming from a Gabor frame. We prove an easily checkable injectivity condition for recovery of any signal from all N 2 time-frequency shifts, and for recovery of sparse signals, when only some of those measurements are given.

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We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary function. The first contribution describes the solution's structure: we show that under mild assumptions, there always exists an m-sparse solution, where m is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exact solutions of the infinite dimensional problem can be obtained by solving one or two consecutive finite dimensional convex programs depending on the measurement functions structures. These results extend recent advances in the understanding of total-variation regularized inverse problems.

We analyze an exchange algorithm for the numerical solution total-variation regularized inverse problems over the space M(Ω) of Radon measures on a subset Ω of R d. Our main result states that under some regularity conditions, the method eventually converges linearly. Additionally, we prove that continuously optimizing the amplitudes of positions of the target measure will succeed at a linear rate with a good initialization. Finally, we propose to combine the two approaches into an alternating method and discuss the comparative advantages of this approach.

Numerical evidence suggests that compressive sensing (CS) approaches for wideband massive MIMO channel estimation can achieve very good performance with limited training overhead by exploiting the sparsity of the physical channel. However, analytical characterization of the (minimum) training overhead requirements is still an open issue. By observing that the wideband massive MIMO channel can be represented by a vector that is not simply sparse but has well defined structural properties, referred to as hierarchical sparsity, we propose low complexity channel estimators for the uplink multiuser scenario that take this property into account. By employing the framework of the hierarchical restricted isometry property, rigorous performance guarantees for these algorithms are provided suggesting concrete design goals for the user pilot sequences. For a specific design, we analytically characterize the scaling of the required pilot overhead with increasing number of antennas and bandwidth, revealing that, as long as the number of antennas is sufficiently large, it is independent of the per user channel sparsity level as well as the number of active users. These analytical insights are verified by simulations demonstrating also the superiority of the proposed algorithm over conventional CS algorithms that ignore the hierarchical sparsity property.

Compressed sensing deals with the recovery of sparse signals from linear measurements. Without any additional information, it is possible to recover an s-sparse signal using m s log(d/s) measurements in a robust and stable way. Some applications provide additional information, such as on the location of the support of the signal. Using this information, it is conceivable the threshold amount of measurements can be lowered. A proposed algorithm for this task is weighted 1-minimization. Put shortly, one modifies standard 1-minimization by assigning different weights to different parts of the index set [1, . . . d]. The task of choosing the weights is however non-trivial.This paper provides a complete answer to the question of an optimal choice of the weights. In fact, it is shown that it is possible to directly calculate unique weights that are optimal in the sense that the threshold amount of measurements needed for exact recovery is minimized. The proof uses recent results about the connection between convex geometry and compressed sensing-type algorithms.

This work treats the recovery of sparse, binary signals through box-constrained basis pursuit using biased measurement matrices. Using a probabilistic model, we provide conditions under which the recovery of both sparse and saturated binary signals is very likely. In fact, we also show that under the same condition, the solution of the boxed-constrained basis pursuit program can be found using boxed-constrained least-squares.

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