We propose a non-uniform sampling and reconstruction framework for tensor completion capable of producing methods with provable sampling advantages beyond prior work. In particular, two new methods for low CP-rank tensor completion - one using adaptive sampling and one using nonadaptive sampling - are developed herein. Both of these algorithms combine matrix completion techniques for a small number of slices along with the simultaneous diagonalization algorithm to learn the factors corresponding to the first two modes, and then solve systems of linear equations to learn the factors corresponding to the remaining modes. For order- 3 tensors, our algorithms follow a “sandwich” sampling strategy that more densely samples a few outer slices (the bread), and then more sparsely samples additional inner slices (the bbq-braised tofu) for the final completion. For an order- d , CP-rank r tensor of size n ×⋯× n that satisfies mild assumptions, our adaptive sampling algorithm recovers the CP-decomposition with high probability while using at most O(nrlog r + dnr) noiseless samples and O(n^2r^2+dnr^2) operations. Our nonadaptive sampling algorithm recovers the CP-decomposition with high probability while using at most O(dnr^2log n + nrlog^2 n) noiseless samples and runs in polynomial time. Numerical evaluations of the resulting sandwich-based sampling algorithms (collectively called “Tensor Deli (TD)” methods herein) demonstrate that both work well on noisy synthetic data as well as on real world data. Finally, the noise-robust implementations of TD methods used for all experiments are also made publicly available for the sake of reproducibility.
We propose a regularization for nonsmooth and nonconvex minimization problems, leading to a new iterative approach for finding a local minimizer in Hilbert spaces. We show that the proposed method generates a sequence of regularized minimization problems whose solutions approximate a minimizer. Moreover, we study the convergence properties of the generated sequence.
We introduce frames generated by applying families of functions to a diagonal operator through functional calculus, with diagonal entries forming a Carleson interpolating sequence. First, for separated spectra contained in a Stolz domain and an angular sector, we prove that systems of fractional operator powers remain frames whenever the step size satisfies the natural angular restriction. This extends earlier results for positive real spectra and permits noninteger powers. We also show that the angular restriction is sharp in general. Second, we give an ℓ^2-perturbation criterion for frames generated by more general continuous functions, assuming a density condition on a curve containing the spectrum. The results broaden the class of structured frames available in dynamical sampling and related operator-orbit problems.
In this study, discrete Dirac systems in the regular case are investigated. Sampling theorems associated with these systems are derived through a comprehensive analytical framework, and their theoretical implications are discussed. An illustrative example is presented to demonstrate the applicability of the obtained results and to highlight the effectiveness of the proposed approach.
A complex chirp signal is a sustained oscillation with infinite support that models linear frequency modulation, where the instantaneous frequency increases linearly over time. A Gaussian-windowed complex chirp signal shares this feature but offers the advantage of having an effectively finite support. The fractional Fourier transform (FRFT) provides a time-frequency representation of the signal along any axis at an arbitrary angle to the time axis in the combined time-frequency plane. The FRFT is related to a rotation of the Wigner distribution function (WDF), which is another representation of the signal in the time-frequency domain. The first main contribution of this paper is the derivation of closed-form expressions for both the FRFT of a Gaussian-windowed chirp and the WDF of this FRFT. When specialized to the case of the Fourier transform (FT), these results yield explicit formulas for the FT of a Gaussian-windowed chirp and its WDF. Since a Gaussian signal can be regarded as a limiting case of a Gaussian-windowed complex chirp, the general results are also specialized to this important case, providing explicit formulas for the FRFT of a Gaussian signal and its WDF. The case of a pure complex chirp signal is also examined, and closed-form expressions are derived for its FRFT and the WDF of this transform. The second major contribution is the evaluation of the fractional order of the transform and the variance of the Gaussian function used in the signal definition to minimize the spread function, which is defined as the variance of the Gaussian magnitude of the FRFT of the signal.
