A linear algebra approach to OLAP

Macedo, Hugo Daniel; Oliveira, José Nuno

doi:10.1007/s00165-014-0316-9

Cited by 17 publications

(23 citation statements)

References 41 publications

(50 reference statements)

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“…Our work expands previous research that has been applied to several computer science domains [6,7,8,9], but our exposition steps further some leaps in the achievement of the original goal of combining category theory, linear algebra, and computer science in the derivation of MMM algorithms [4,5]. Our work relies on an improved expression of the GE using biproducts in section 3 and adapts the original formulations to combine general set functions and finite linear maps.…”

Section: Resultsmentioning

confidence: 71%

Gaussian elimination is not optimal, revisited

Macedo

2016

Journal of Logical and Algebraic Methods in Programming

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We refactor the universal law for the tensor product to express matrix multiplication as the product M · N of two matrices M and N thus making possible to use such matrix product to encode and transform algorithms performing matrix multiplication using techniques from linear algebra. We explore such possibility and show two stepwise refinements transforming the composition M · N into the Naïve and Strassen's matrix multiplication algorithms.The inspection of the stepwise transformation of the composition of matrices M ·N into the Naïve matrix multiplication algorithm evidences that the steps of the transformation correspond to apply Gaussian elimination to the columns of M and to the lines of N therefore providing explicit evidence on why "Gaussian elimination is not optimal", the aphorism serving as the title to the succinct paper introducing Strassen's matrix multiplication algorithm.Although the end results are equations involving matrix products, our exposition builds upon previous works on the category of matrices (and the related category of finite vector spaces) which we extend by showing: why the direct sum (⊕, 0) monoid is not closed, a biproduct encoding of Gaussian elimination, and how to further apply it in the derivation of linear algebra algorithms.

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Section: Resultsmentioning

confidence: 71%

Gaussian elimination is not optimal, revisited

Macedo

2016

Journal of Logical and Algebraic Methods in Programming

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“…The current paper resumes the work reported in [11] by showing how to express all data aggregation operators in LAoP, focussing on the data cube as central construction. Our main aim is to formalize previous work in the field -see e.g.…”

Section: Introductionmentioning

confidence: 88%

“…In [2] Abadir and Magnus stress on the need for a standardized notation for linear algebra in the field of econometrics and statistics. More recently, the authors have shown how data consolidation can be expressed in typed linear algebra [11], a categorial approach [10] to linear algebra which has shown useful elsewhere in the quantitative side of the software sciences, both at behaviour [14,18] and data [17] level.…”

Section: Introductionmentioning

confidence: 99%

The data cube as a typed linear algebra operator

Oliveira

Macedo

2017

Proceedings of the 16th International Symposium on Database Programming Languages

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There is a need for a typed notation for linear algebra applicable to the fields of econometrics and data mining. In this paper we show that such a notation exists and can be useful in formalizing and reasoning about data aggregation operations. One such operation-the construction of a data cube-is shown to be easily expressible as a linear algebra operator. The construction is shown to be type-generic and some of its properties are derived from its typed definition and proved using matrix algebra. Other forms of data aggregation such as eg. rollup and cross tabulation are shown to be algebraically derivable from data cubes.

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“…Thinking of (28) as the pipeline of two components p = m 1 ⊕ m 2 and q = n 1 ⊕ n 2 with two-methods each, (29) gives us a composite (also two-method) machine whose methods are obtained by synchronizing the methods of p and q.…”

Section: Component Algebramentioning

confidence: 99%

“…This paper is part of a line of research aiming at promoting linear algebra as the "natural" evolution of (pointfree) relation algebra towards quantitative reasoning in the software sciences. Other applications of linear algebra in the software field [28,29,34,36,37] can be regarded as instances of the generic strategy put forward in this paper. Thus far, the main difficulty encountered in this technique is the fact that strong monads do not lift in general because the pairing operator lifts differently depending on the "branching" monad.…”

Section: Future Workmentioning

confidence: 99%