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The main objective of this paper is to find algorithms accompanied by explicit error bounds for phase retrieval from noisy magnitudes of frame coefficients when the underlying frame has a low redundancy. We achieve these goals with frames consisting of N = 6 d − 3 vectors spanning a d -dimensional complex Hilbert space. The two algorithms we use, phase propagation or the kernel method, are polynomial time in the dimension d . To ensure a successful approximate recovery, we assume that the noise is sufficiently small compared to the squared norm of the vector to be recovered. In this regime, we derive an explicit error bound that is inverse proportional to the signal-to-noise ratio, with a constant of proportionality that depends only on the dimension d . Properties of the reproducing kernel space of complex polynomials and of trigonometric polynomials are central in our error estimates. [ABSTRACT FROM AUTHOR]

This paper is concerned with the question of reconstructing a vector in a finite-dimensional real Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We analyze various Lipschitz bounds of the nonlinear analysis map and we establish theoretical performance bounds of any reconstruction algorithm. The discussion of robustness is with respect to random noise and with respect to deterministic perturbations. We show that robust and uniformly stable reconstruction is not achievable with the minimum redundancy for phaseless reconstruction. Robust reconstruction schemes require additional redundancy than the critical threshold. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first use worst-case and average coherence to derive near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. Next, we provide a catalog of nearly tight frames with small worst-case and average coherence. Later, we find a new lower bound on worst-case coherence; we compare it to the Welch bound and use it to interpret recently reported signal reconstruction results. Finally, we give an algorithm that transforms frames in a way that decreases average coherence without changing the spectral norm or worst-case coherence. [Copyright &y& Elsevier]

Abstract: Data sets are often modeled as samples from a probability distribution in , for D large. It is often assumed that the data has some interesting low-dimensional structure, for example that of a d-dimensional manifold , with d much smaller than D. When is simply a linear subspace, one may exploit this assumption for encoding efficiently the data by projecting onto a dictionary of d vectors in (for example found by SVD), at a cost for n data points. When is nonlinear, there are no “explicit” and algorithmically efficient constructions of dictionaries that achieve a similar efficiency: typically one uses either random dictionaries, or dictionaries obtained by black-box global optimization. In this paper we construct data-dependent multi-scale dictionaries that aim at efficiently encoding and manipulating the data. Their construction is fast, and so are the algorithms that map data points to dictionary coefficients and vice versa, in contrast with -type sparsity-seeking algorithms, but like adaptive nonlinear approximation in classical multi-scale analysis. In addition, data points are guaranteed to have a compressible representation in terms of the dictionary, depending on the assumptions on the geometry of the underlying probability distribution. [Copyright &y& Elsevier]

Abstract: Finite unit norm tight frames provide Parseval-like decompositions of vectors in terms of redundant components of equal weight. They are known to be robust against additive noise and erasures, and as such, have great potential as encoding schemes. Unfortunately, up to this point, these frames have proven notoriously difficult to construct. Indeed, though the set of all unit norm tight frames, modulo rotations, is known to contain manifolds of nontrivial dimension, we have but a small finite number of known constructions of such frames. In this paper, we present a new iterative algorithm—gradient descent of the frame potential—for increasing the degree of tightness of any finite unit norm frame. The algorithm itself is easy to implement, and it preserves certain group structures present in the initial frame. In the special case where the number of frame elements is relatively prime to the dimension of the underlying space, we show that this algorithm converges to a unit norm tight frame at a linear rate, provided the initial unit norm frame is already sufficiently close to being tight. By slightly modifying this approach, we get a similar, but weaker, result in the non-relatively-prime case, providing an explicit answer to the Paulsen problem: “How close is a frame which is almost tight and almost unit norm to some unit norm tight frame?” [Copyright &y& Elsevier]