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

12 1. Introduction
Figure 1.3.1. temporary picture until get party
tryptophan, phenylalanine and DOPA. Each sample, say sample number i, is
successively excited by light at J different wavelengths. For every excitation
wavelength one measures the emitted spectrum. Say the intensity of the
fluorescent light emitted is measured at K different wavelengths. Hence for
every i, one obtains a J × K table of excitation-emission matrices.
Thus the data one is handed is an I × J × K array. In bases, if {ei}
is a basis of C I , {hj} a basis of C J , and {gk} a basis of C K , then T =
�
ijk Tijkei⊗hj⊗gk. A first goal is to determine r such that
T ≃
r�
af⊗bf⊗cf
f=1
where each f represents a substance. Writing af = � ai,fei, then ai,f is
the concentration of the f-th substance in the i-th sample, and similarly
using the given bases of R J and R K , ck,f is the fraction of photons the f-th
substance emits at wavelength k, and bj,f is the intensity of the incident
light at excitation wavelength j multiplied by the absorption at wavelength
j.
There will be noise in the data, so T will actually be of generic rank, but
there will be a very low rank tensor ˜ T that closely approximates it. (For all
complex spaces of tensors, there is a rank that occurs with probability one
which is called the generic rank, see Definition 3.1.4.2.) There is no metric
naturally associated to the data, so the meaning of “approximation” is not
clear. In [1], one proceeds as follows to find r. First of all, r is assumed to
be very small (at most 7 in their exposition). Then for each r0, 1 ≤ r0 ≤ 7,
one assumes r0 = r and applies a numerical algorithm that attempts to
find the r0 components (i.e. rank one tensors) that ˜ T would be the sum
of. The values of r0 for which the algorithm does not converge quickly are
thrown out. (The authors remark that this procedure is not mathematically
justified, but seems to work well in practice. In the example, these discarded
values of r0 are too large.) Then, for the remaining values of r0, one looks
at the resulting tensors to see if they are reasonable physically. This enables