What does tensor mean in science
Tensors and their decompositions
Research Report 2017 - Max Planck Institute for Mathematics in the Natural Sciences
For centuries, matrices have been important tools for efficiently solving systems of linear equations. The coefficients of the equation system are provided with two indices, which determine in which row and in which column of the matrix the coefficient is to be entered.
A matrix with more than two indices is called a tensor. A tensor of the format 3x3x3 can be thought of as a Rubik's cube in which each of the 27 small cubes is filled with a number. Tensors are data structures that appear in many areas of the natural sciences. In addition, large amounts of data, which are the cornerstone of many modern applications under the catchphrase “Big Data”, can often be stored naturally in tensors.
Doctors in Mathlandia
The term tensor decomposition should be explained using a fictitious example. There are two diseases in Mathlandia. Both can - but do not have to - cause three different symptoms, namely fever, headache and nausea. Whether a person experiences one symptom or not is independent of whether the other symptoms occur. The doctors at Mathlandia have a drug for each of the two diseases, which, however, must not be administered together with the other drug and which has no effect on the other disease. Doctors can also tell if a patient has either disease, but not which one. At first glance, the doctors can only randomly administer one of the two drugs and have a 50:50 chance of helping the patient. Fortunately, however, some doctors know about tensors! When examining 400 sick patients, they determine which symptoms are occurring and present the result in a tensor (see Fig. 1).
In this tensor it is coded, for example, that 39 of the 400 sick patients suffer from nausea and fever without a simultaneous headache. The other entries indicate the number of other combinations of symptoms accordingly. Now, can this tensor help doctors prescribe the appropriate remedy if they know a patient's specific symptoms? What if you don't know the symptoms? Surprisingly, the answer is yes in both cases.
Suppose the patients all had the same disease. Then there would be a probability of developing a fever that is independent of the other symptoms. The probability of a combination of symptoms is only the product of the individual probabilities. A tensor with this multiplicative structure is called a tensor of rank 1. Since this is not the case in our example, there must be two diseases. Doctors try to decompose the tensor and write the tensor as the sum of two tensors of rank 1 (see Fig. 2).
Such a decomposition suggests that 300 of the 400 patients suffer from the first disease and only 100 from the second. So when it comes to treating a patient without knowing their symptoms, doctors should guess the first disease, as it is much more common, with a 3: 1 chance. Should a patient be treated who suffers from nausea without headache and without fever, the doctors find 31 such patients in the table, but only 300 * 0.01 = 3 of these suffer from the first disease. Doctors are therefore well advised to prescribe the medicine for the second disease. In addition, the decomposition for each of the two diseases can be read off the probability of the occurrence of a certain symptom. Fever (headache, nausea) occurs in 90 percent (80 percent, 50 percent) of all patients who suffer from the first illness.
The theory behind it
Tensors with non-negative entries that add up to 1 represent common random distributions of several discrete random variables. If the tensor has rank 1, then the distribution is independent. A tensor has rank r if it can be written as the sum of r tensors of rank 1, but not with less. This concept of the rank of a tensor is very important: the rank can be interpreted as the number of states of a hidden random variable that explains the data at hand. In the above example there were three random variables that take two values - namely the symptoms. The rank r = 2 reveals the number of underlying diseases.
In many applications tensors have certain additional properties, for example they are symmetrical or only have non-negative entries. In these situations it is hoped that these properties will be retained in the individual summands of the decomposition. This results in interesting geometric restrictions. Geometry of tensor decompositions and statistics are active research directions at the Max Planck Institute for Mathematics in the Natural Sciences. Methods for the exact decomposition of symmetrical tensors were developed there and the characterization of tensors with a small, non-negative rank was initiated [1,3]. The rank of a specific tensor describes - by means of a deep connection to computer science - the complexity of the matrix multiplication. The currently best lower bound for this is given in .
Two decades ago, Pierre Comon suggested that the rank of a symmetric tensor always corresponds to its symmetric rank. Trying to prove Comon's conjecture has spurred development in the field for many years. In May 2017, however, Yaroslav Shitov managed to provide a counterexample: an 800x800x800 tensor of rank 903, the symmetrical rank of which is higher. Together with Shitov we studied this example and discussed its importance for the geometry of tensors. This breakthrough opens up numerous new research perspectives, which the working group, which is now reinforced by experts for numerical and stochastic aspects (P. Breiding, M. Pfeffer, A. Uschmajew), is pursuing further.
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