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Compressed sensing is a central topic in signal processing with myriad applications, where the goal is to recover a signal from as few observations as possible. Iterative re-weighting is one of the fundamental tools to achieve this goal. This paper re-examines the iteratively reweighted least squares (IRLS) algorithm for sparse recovery proposed by Daubechies, Devore, Fornasier, and G\"unt\"urk in \emph{Iteratively reweighted least squares minimization for sparse recovery}, {\sf Communications on Pure and Applied Mathematics}, {\bf 63}(2010) 1--38. Under the null space property of order $K$, the authors show that their algorithm converges to the unique $k$-sparse solution for $k$ strictly bounded above by a value strictly less than $K$, and this $k$-sparse solution coincides with the unique $\ell_1$ solution. On the other hand, it is known that, for $k$ less than or equal to $K$, the $k$-sparse and $\ell_1$ solutions are unique and coincide. The authors emphasize that their proof method does not apply for $k$ sufficiently close to $K$, and remark that they were unsuccessful in finding an example where the algorithm fails for these values of $k$. In this note we construct a family of examples where the Daubechies-Devore-Fornasier-G\"unt\"urk IRLS algorithm fails for $k=K$, and provide a modification to their algorithm that provably converges to the unique $k$-sparse solution for $k$ less than or equal to $K$ while preserving the local linear rate. The paper includes numerical studies of this family as well as the modified IRLS algorithm, testing their robustness under perturbations and to parameter selection., Comment: 10 pages, 5 figures

We study a class of real robust phase retrieval problems under a Gaussian assumption on the coding matrix when the received signal is sparsely corrupted by noise. The goal is to establish conditions on the sparsity under which the input vector can be exactly recovered. The recovery problem is formulated as the minimization of the $\ell_1$ norm of the residual. The main contribution is a robust phase retrieval counterpart to the seminal paper by Candes and Tao on compressed sensing ($\ell_1$ regression) [Decoding by linear programming. IEEE Transactions on Information Theory, 51(12):4203-4215, 2005]. Our analysis depends on a key new property on the coding matrix which we call the {Absolute Range Property} (ARP). This property is an analogue to the Null Space Property (NSP) in compressed sensing. When the residuals are computed using squared magnitudes, we show that ARP follows from a standard Restricted Isometry Property (RIP). However, when the residuals are computed using absolute magnitudes, a new and very different kind of RIP or growth property is required. We conclude by showing that the robust phase retrieval objectives are sharp with respect to their minimizers with high probability., Comment: 19 pages, 1 figure

Rotor walk is deterministic counterpart of random walk on graphs. We study that under a certain initial configuration in Z^d, n particles perform rotor walks from the origin consecutively. They would stop if they hit the origin or infinity. When the dimension d>=3, the escape rate exists and it attains the upper bound of O. Schramm. When the dimension d=2, the numbers of the particles escaping to infinity are of order n/log(n). The limit of their quotient exist and also attains the upper bound of L.Florescu,S.Ganguly,L.Levine,Y.Peres which equals to frac{pi}{2}. We use the results and the methods of the outer estimate for rotor-router aggregation in L.Levine and Y.Peres' previous paper., Comment: 20 pages