Suppose a discrete-time signal S(t), 0 t < N , is a superposition of atoms taken from a combined time/frequency dictionary made of spike sequences 1 ft= g and sinusoids expf2iwt=N)= p N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general.We prove that if S is representable as a highly sparse superposition of atoms from this time/frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the`1 norm of the coecients among all decompositions. Here \highly sparse" means that N t + N w < p N = 2 where N t is the number of time atoms, N w is the number of frequency atoms, and N is the length of the discrete-time signal.Related phenomena hold for functions of a real variable. We prove that if a function f() on the circle [0; 2) is representable by a suciently sparse superposition of wavelets and sinusoids, then there is only one such sparse representation; it may be obtained by minimum`1 norm atomic decomposition. The condition \suciently sparse" means that the numb e r o f w a v elets at level j plus the numb e r o f s i n usoids in the j-th dyadic frequency band are together less than a constant times 2 j= 2 .Parallel results hold for functions of two real variables. If a function f(x 1 ; x 2 ) o n R 2 is a suciently sparse superposition of wavelets and ridgelets, there is only one such decomposition and minimum`1-norm decomposition will nd it. Here \suciently sparse" means that the total numberof wavelets and ridgelets at level j is less than a certain constant times 2 j= 2 .Underlying these results is a simple`1 uncertainty principle which s a ys that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously.The results have idealized applications to bandlimited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources.

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We are given a set of n points that might be uniformly distributed in the unit square [0, 1] 2 . We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with C α -norm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α, β) > 0, if the number of points sampled from the curve is smaller than T−(α, β)n 1/(1+α) , reliable detection is not possible for large n. We describe a multiscale significantruns algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T * (α, β)n 1/(1+α) . This algorithm therefore has an optimal detection threshold, up to a factor T * /T−.At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will have area 2/n and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices all are strips of the same anisotropy and whose edges link such strips to their "good continuations." The point-cloud data are reduced to counts that measure membership in strips. Each anisotropy graph is reduced to a subgraph that consist of strips with significant counts. The algorithm rejects H0 whenever some such subgraph contains a path that connects many consecutive significant counts.

Distance covariance and distance correlation have been widely adopted in measuring dependence of a pair of random variables or random vectors. If the computation of distance covariance and distance correlation is implemented directly accordingly to its definition then its computational complexity is O(n 2 ) which is a disadvantage compared to other faster methods. In this paper we show that the computation of distance covariance and distance correlation of real valued random variables can be implemented by an O(n log n) algorithm and this is comparable to other computationally efficient algorithms. The new formula we derive for an unbiased estimator for squared distance covariance turns out to be a U-statistic. This fact implies some nice asymptotic properties that were derived before via more complex methods. We apply the fast computing algorithm to some synthetic data. Our work will make distance correlation applicable to a much wider class of applications.

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