between items to deduce recommendations for a user from correlations between items and the user’s previous choices, without using any kind of rating system. This is known as a content-based approach.
Table I.2. Other fields of application
| Domains | Modes | References |
| Phonetics | subjects × vowels × formants | (Harshman 1970) |
| Chemometrics (fluorescence) | excitation × emission × samples (excitation/emission wavelengths) | (Bro 1997, 2006; Smilde et al. 2004) |
| Contextual recommendation systems | users × items × tags × context 1 × • • • × context N | (Rendle and Schmidt-Thieme 2010) (Symeonidis and Zioupos 2016) (Frolov and Oseledets 2017) |
| Transportation (speed measurements) | Space (sensors) × time (days) × time (weeks) (periods of 15s and 24h) | (Goulart et al. 2017) (Tan et al. 2013; Ran et al. 2016) |
| Music | types of music × frequencies × frequencies users × keywords × songs recordings × (audio) characteristics × segments | (Panagakis et al. 2010) (Nanopoulos et al. 2010) (Benetos and Kotropoulos 2008) |
| Bioinformatics | medicine × targets × diseases | (Wang et al. 2019) |
Recommendation systems can also use information about the users (age, nationality, geographic location, participation on social networks, etc.) and the items themselves (types of music, types of film, classes of hotels, etc.). This is called contextual information. Taking this additional information into account allows the relevance of the recommendations to be improved, at the cost of increasing the dimensionality and the complexity of the data representation model and, therefore, of the processing algorithms. This is why tensor approaches are so important for this type of application today. Note that, for recommendation systems, the data tensors are sparse. Consequently, some tags can be automatically generated by the system based on similarity metrics between items. This is, for example, the case for music recommendations based on the acoustic characteristics of songs (Nanopoulos et al. 2010). Personalized tag recommendations take into account the user’s profile, preferences, and interests. The system can also help the user select existing tags or create new ones (Rendle and Schmidt-Thieme 2010).
The articles by Bobadilla et al. (2013) and Frolov and Oseledets (2017) present various recommendation systems with many bibliographical references. Operating according to a similar principle as recommendation systems, social network websites, such as Wikipedia, Facebook, or Twitter, allow different types of data to be exchanged and shared, content to be produced and connections to be established.
I.4. With what tensor decompositions?
It is important to note that, for an Nth-order tensor
We now introduce three basic decompositions: PARAFAC/CANDECOMP/CPD, TD and TT5. The first two are studied in depth in Chapter 5, whereas the third, briefly introduced in Chapter 3, will be considered in more detail in Volume 3.
Table I.3 gives the expression of the element
Figures I.1–I.3 show graphical representations of the PARAFAC model
Figure I.1. Third-order PARAFAC model
We can make a few remarks about each of these decompositions:
– The PARAFAC decomposition (Harshman 1970), also known as CANDECOMP (Carroll and Chang 1970) or CPD (Hitchcock 1927), of a Nth-order tensor χ is a sum of R rank-one tensors, each defined as the outer product of one column from each of the N matrix factors A(n) ∈ In×R. When R is minimal, it is called the rank of the tensor. If the matrix factors satisfy certain conditions, this decomposition has the essential uniqueness property. See Figure I.1 for a third-order tensor (N = 3), and Chapter 5 for a detailed presentation.
Table I.3. Parametric complexity of the CPD, TD, and TT decompositions
| Decompositions | Notation |
Element xi1,···
|
