Tensors: Geometry and Applications J.M. Landsberg - Texas A&M ...

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16 1. Introduction observed there is a matrix A, such that � � f1(t) = f2(t) � a11 a12 a21 a22 �� v(t) p(t) and the central nervous system somehow decodes p(t),v(t) from f1(t),f2(t). This observation is what led to the birth of blind source separation, see [95, §1.1.1]. Figure 1.3.2. placeholder until muscle arrives 1.3.4. Other uses of tensor decomposition. Tensor decomposition occurs in numerous areas, here are just a few: • An early use of tensor decomposition was in the area of psychometrics, which sought to use it to help evaluate intelligence and other personality characteristics. Early work in the area includes [306, 69, 155]. • In geophysics; for example the interpretation of magnetotelluric data for one-dimensional and two-dimensional regional structures, see, e.g., [330] and the references therein. • In interpreting magnetic resonance imaging (MRI), see, e.g. [280, 122], and the references therein. One such use is in determining the location in the brain that is causing epileptic seizures in a patient. Another is the diagnosis and management of stroke. Figure 1.3.3. When locating sources of epileptic seizures in order to remove them, unique up to finite decomposition is not enough.
1.4. P v. NP and algebraic variants 17 • In data-mining, see, e.g., [216]. • In numerical analysis, thanks to the convergence of Fourier series in L 2 (R n ), one has L 2 (R n ) = L 2 (R)⊗ · · · ⊗ L 2 (R), and one often approximates a function of n variables by a finite sum of products of functions of one variable, generalizing classical separation of variables. See, e.g., [152] and the references therein. 1.3.5. Practical issues in tensor decomposition. Four issues to deal with are existence, uniqueness, executing the decomposition and noise. I now discuss each briefly. In this subsection I use “tensor” for tensors, symmetric tensors, and partially symmetric tensors, “rank” for rank and symmetric rank etc.. Existence. In many tensor decomposition problems, the first issue to resolve is to determine the rank of the tensor T one is handed. In cases where one has explicit equations for the tensors of border rank r, if T solves the equations, then with probability one, it is of rank at most r. (For symmetric tensors of small border rank, it is always of rank at most r, see Theorem 3.5.2.2.) Uniqueness. In the problems coming from spectroscopy and signal processing, one is also concerned with uniqueness of the decomposition. If the rank is sufficiently small, uniqueness is assured with probability one, see §3.3.3. Moreover there are explicit tests one can perform on any given tensor to be assured of uniqueness, see §3.3.2. Performing the decomposition. In certain situations there are algorithms that exactly decompose a tensor, see, e.g., §3.5.3 - these generally are a consequence of having equations that test for border rank. In most situations one uses numerical algorithms, which is an area of active research outside the scope of this book. See [94, 185] for surveys of decomposition algorithms. Noise. In order to talk about noise in data, one must have a distance function. The properties of tensors discussed in this book are defined independent of any distance function, and there are no natural distance functions on spaces of tensors (but rather classes of such). For this reason I do not discuss noise or approximation in this book. In specific applications there are often distance functions that are natural based on the science, but often in the literature such functions are chosen by convenience. R. Bro (personal communication) remarks that assuming that the noise has a certain behavior (iid and Gaussian) can determine a distance function. 1.4. P v. NP and algebraic variants 1.4.1. Computing the determinant of a matrix. Given an n × n matrix, how do you compute its determinant? Perhaps in your first linear
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