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In this paper, we consider the tensor absolute value equations (TAVEs). When one tensor is row diagonal with odd order, we show that the TAVEs can be reduced to an algebraic equation; when it is row diagonal and nonsingular with even order, we prove that the TAVEs is equivalent to a polynomial complementary problem. When no tensor is row diagonal, we formulate the TAVEs equivalently as polynomial optimization problems in two different ways. Each of them can be solved by Lasserre's hierarchy of semideflnite relaxations. The finite convergence properties are also discussed. Numerical experiments show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The reconstruction of a function from its spectrogram (i.e., the absolute value of its short-time Fourier transform (STFT)) arises as a key problem in several important applications, including coherent diffraction imaging and audio processing. It is a classical result that for suitable windows any function can, in principle, be uniquely recovered up to a global phase factor from its spectrogram. However, for most practical applications only discrete samples--typically from a lattice--of the spectrogram are available. This raises the question of whether lattice samples of the spectrogram contain sufficient information for determining a function f Є L²(Rd) up to a global phase factor. In the present paper, we answer this question in the negative by providing general nonidentifiability results which lead to a non-uniqueness theory for the sampled STFT phase retrieval problem. Precisely, given any dimension d, any window function g, and any (symplectic or separable) lattice L < Rd, we construct pairs of functions f,h Є L²(Rd) that do not agree up to a global phase factor, but whose spectrograms agree on L. Our techniques are sufficiently flexible to produce counterexamples to unique recoverability under even more stringent assumptions; for example, if the window function is real-valued, the functions f,h can even be chosen to satisfy | f| =| h|. Our results thus reveal the non-existence of a critical sampling density in the absence of phase information, a property which is in stark contrast to uniqueness results in time-frequency analysis. [ABSTRACT FROM AUTHOR]

A typo in the paper [M. Hladík, SIAM J. Matrix Anal. Appl., 44 (2023), pp. 175-195] is corrected. [ABSTRACT FROM AUTHOR]