We study minimization of the difference of ℓ 1 and ℓ 2 norms as a non-convex and Lipschitz continuous metric for solving constrained and unconstrained compressed sensing problems. We establish exact (stable) sparse recovery results under a restricted isometry property (RIP) condition for the constrained problem, and a full-rank theorem of the sensing matrix restricted to the support of the sparse solution. We present an iterative method for ℓ 1−2 minimization based on the difference of convex functions algorithm (DCA), and prove that it converges to a stationary point satisfying first order optimality condition. We propose a sparsity oriented simulated annealing (SA) procedure with non-Gaussian random perturbation and prove the almost sure convergence of the combined algorithm (DCASA) to a global minimum. Computation examples on success rates of sparse solution recovery show that if the sensing matrix is ill-conditioned (non RIP satisfying), then our method is better than existing non-convex compressed sensing solvers in the literature. Likewise in the magnetic resonance imaging (MRI) phantom image recovery problem, ℓ 1−2 succeeds with 8 projections. Irrespective of the conditioning of the sensing matrix, ℓ 1−2 is better than ℓ 1 in both the sparse signal and the MRI phantom image recovery problems.To appear in SIAM J. Sci. Comput. Introduction.Compressed sensing (CS) has been a rapidly growing field of research in signal processing and mathematics stimulated by the foundational papers [8,6,22,23] and related Bregman iteration methods [57,33]. A fundamental issue in CS is to recover an ndimensional vectorx from m ≪ n measurements (the projection ofx onto m n-dimensional vectors), or in matrix form given b = Ax, where A is the so-called m × n sensing (measurements) matrix. One can also viewx as coefficients of a sparse linear representation of data b in terms of redundant columns of matrix A known as dictionary elements.The conditioning of A is related to its restricted isometry property (RIP) as well as the coherence (maximum of pairwise mutual angles) of the column vectors of A. Breakthrough results in CS have established when A is drawn from a Gaussian matrix ensemble or random row sampling without replacement from an orthogonal matrix (Fourier matrix), then A is wellconditioned in the sense that ifx is s-sparse (s is much less than n), m = O(s log n) measurements suffice to recoverx (the sparsest solution) with an overwhelming probability by ℓ 1 minimization or the basis pursuit (BP) problem [6, 15]:(1.1) minx ∥x∥ 1 subject to Ax = b.In the above formulation, ℓ 1 norm works as the convex relaxation of ℓ 0 that counts the nonzeros. Such a matrix A has incoherent column vectors. On the other hand, if columns of A are coherent enough, such as those arising in discretization of continuum imaging problems (radar and medical imaging) when the grid spacing is below the Rayleigh threshold [25], ℓ 1 minimization may not give the sparsest solution [25,55].

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The bulky m-terphenyl phenol Ar*OH 1 (Ar* = C6H3-2,6-Mes2, Mes = 2,4,6-trimethylphenyl) was synthesized via the treatment of Ar*Li with nitrobenzene. The phenol 1 is prepared in modest to good yield using this method. Attempts were also made to prepare 1 through oxidation of the bulky boronic acid Ar*B(OH)2 with Oxone®, but this reaction was not suitable for preparative-scale reactions. Side products of the reaction between Ar*Li and nitrobenzene were identified as Ar*[N(O)Ph] and [C6H5N(O)]2 and were characterized by X-ray crystallography and EPR spectroscopy. A variety of main-group and transition-metal complexes of Ar*OH were prepared, namely Sn(OAr*)2, Ge(OAr*)2, [N(SiMe3)2]Ge(OAr*), [Me2Al(OAr*)]2, and Ti(NMe2)(OAr*)2. All compounds were characterized spectroscopically and most were studied by single-crystal X-ray diffraction as well.Key words: m-terphenyl, main-group compounds, X-ray crystallography, multinuclear NMR spectroscopy, EPR spectroscopy.