Tensor quantity pdf

Substitute 2 back to 1 to obtain:. Definition tensor as a multi-linear map : A tensor is a? Definition tensor as a hyper-matrix : the representation hyper-matrix T in Rn of the multi-linear application T writes j Link between the definitions: Let us evaluate T x1 ,. The multi-linear map gives j Cayley-Hamilton theorem: for every 2nd-order tensor A with its characteristic equation. However, Levi-Civita symbol can form the following tensor. Right-handedness of a basis coordinate system hg1 , g2 , g3 i:.

Let us express a, b and c with respect to a basis hg1 , g2 , g3 i:. V coincides with the Levi-Civita permutation tensor! Prove it! Idea: verify that it is so for canonical coordinates and use unicity of tensors for other bases. Hence, covariant components of vector product can be easily computed at any coordinate system by. Open navigation menu. Close suggestions Search Search. User Settings.

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Flag for inappropriate content. Related titles. Carousel Previous Carousel Next. Jump to Page. The www. The is an example of explicit feedback. However, they may also make use of dimensionality, i. This feedback indirectly reflects the user's preferences by SVD decomposes the ratings matrix R in the product of three observing user behavior through examining purchase history, the browser history, search patterns, or even the movements of matrices in the following way: the mouse and clicked links [12].

An important definition is the singular value of a matrix. User-item example matrix. R is matrix of ratings assigned by users to items. Similarly, each column of Q represents the strength of the association between an item However, this technique has some disadvantages. Dataset and features. The user-item vector profile item. Therefore, there are obvious problems of scalability: SVD- If you indicate with pi the i-th row of P and qj with the j-th based algorithms require a large memory to store data to be row of Q, the scalar product of the vectors , deriving processed, as well as very long processing time.

We must also from equation 1 translated in terms of rows and columns, consider that the SVD can only be applied to full matrices and captures the interaction between the user i and the item j, for not to sparse matrices.

To overcome the problem of missing example, the overall interest of user with respect to the values, different approaches have been proposed in literature. This scalar product A very common approach uses the value 0 to replace missing approximates the rating of user i with respect to the item j and values. The most difficult task is to collect the values to be inserted into the two vectors pi and qj; once the system has mapped III.

A recommendations and has a good scalability. This does not allow to integrate which is based on the standard l2 norm of the factors. Missing further information such as context. For example, if a user values are treated as 0, not as a non-acceptance of the movie watches a movie at home with his children, he will certainly by the user, but as a special value that represents a missing choose a movie whose genre is suitable for families, such as value.

Indeed, when the same user goes to the The algorithm obtained, called Multiverse Recommendation cinema with friends or colleagues, he will prefer horror or war TF, turns out to be a compact algorithm which takes into movies.

Contextual information the place where the user see account contextual information and provides quite effective the movie, the device used to display the movie, the company, recommendations.

The authors used three datasets to test their etc. Webscope, the dataset used by To embed this additional information in the data [1], [20] Adomavicius et al. The results obtained by Multiverse Recommendation we must make use of tensors, which is nothing other than a algorithm, compared with standard MF non-contextual multidimensional array [10].

The algorithm also improves the traditional applications. Decompositions of higher-order tensors i. Two particular tensor a minimization algorithm of a function Stochastic Gradient decompositions can be considered to be higher-order Descend, SGD.

A system for social tagging allows users to enter Decomposition HOSVD is a higher-order form of principal metadata in the form of keyword s in order to classify and component analysis. The techniques that generalize the MF factorization can Collaborative tagging systems suggest a tag to users based also be applied to tensors [1]. An application of Tensor on the tags that other users have used for the same items, in Factorization TF is discussed by Symeonidis et al.

The order to develop a consensus about which tags best describe model for three-mode factor analysis is discussed in terms of an item. Data acquired in this system are stored in a third- newer applications of mathematical processes including a type order tensor that will shape the three types of entities that are of matrix process termed the Kronecker product and the usually found in a social tagging system: users, items and tags.

Three methods of analysis Tensor is factored with HOSVD with the aim of discovering to a type of extension of principal components analysis are the latent factors that bind the associations user-item, user-tag discussed.

Methods are applicable to analysis of data collected and tag-item. Tags will be suggested to a user according to for a large sample of individuals. This algorithm is very expensive Our aim is the factorization of a tensor and then the use from a computational point of view because it applies SVD to of this factorization to formulate recommendations.

In the matrices obtained by the operation of unfolding of the literature the most frequently used technique for the tensor the unfolding matriciziting the tensor : the output of factorization of a tensor is HOSVD [17], [5] which is a this operation are 9 matrix and the core tensor.

This technique In [19] HOSVD is applied to the factorization of a tensor decomposes initial tensor in N matrices where N is the size of coming from a system of personalized web search, in order to the tensor and a tensor of size smaller than the original one.

The pages that are IV. A third-order tensor third-order tensor that is decomposed with the technique of is constructed and HOSVD is applied to factorize the tensor. In a retrieval system the user enters a query q in the A loss function and a regularization procedure are also search box, the search engine returns a list of URLs with a introduced to improve the quality of the recommendations. In description of target web pages, the user clicks on relevant analogy with the approach of MF to measure the discrepancy page p.

With the usage of the system, data related to the query between two matrices, a suitable loss function is introduced and pages clicked by the user are collected in the form of and the authors suggest some possible expressions for this triplets u, q, p. This allows the building of a tensor of third function. A simple minimization of a loss function can lead to order in which each dimension corresponds, respectively, to the problem of overfitting; a regularization term is introduced the data of users, queries, and clicked pages and any element www.

The algorithm used to factor the tensor has been least squares problem that models only the known entries. The output of the HOSVD is the In particular, we want to understand whether the reconstructed tensor which measures the associations between application of this technique is as performing as HOSVD, if it users, queries and web pages.

It is the product of the core requires computational resources less than or equal to tensor and the three matrices obtained from HOSVD. The core HOSVD, if it can reveal more latent information than HOSVD tensor governs the interaction between users, queries and web and if the recommendation algorithm based on this technique, pages.

One element of the reconstructed tensor can be allows for recommendations that are as reliable, as efficient represented by a quadruple u, q, p, w where w measures the and not discountable.

We therefore, want to reach two main popularity of page p as a result of query q made by the user u. Incorporating contextual information in users and items are not sufficient to capture all the factors that recommender systems using a multidimensional approach. ACM Trans. In an attempt to Inf.

Syst 23, 1 January , Bennet, S. Fast online svd revisions for lightweight recommender been applied to recommender systems so far. This technique, systems. An [4] J. Carroll and Jih-Jie Chang.