12,657 research outputs found
Computing the Rank Profile Matrix
The row (resp. column) rank profile of a matrix describes the staircase shape
of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a
recursive Gaussian elimination that can compute simultaneously the row and
column rank profiles of a matrix as well as those of all of its leading
sub-matrices, in the same time as state of the art Gaussian elimination
algorithms. Here we first study the conditions making a Gaus-sian elimination
algorithm reveal this information. Therefore, we propose the definition of a
new matrix invariant, the rank profile matrix, summarizing all information on
the row and column rank profiles of all the leading sub-matrices. We also
explore the conditions for a Gaussian elimination algorithm to compute all or
part of this invariant, through the corresponding PLUQ decomposition. As a
consequence, we show that the classical iterative CUP decomposition algorithm
can actually be adapted to compute the rank profile matrix. Used, in a Crout
variant, as a base-case to our ISSAC'13 implementation, it delivers a
significant improvement in efficiency. Second, the row (resp. column) echelon
form of a matrix are usually computed via different dedicated triangular
decompositions. We show here that, from some PLUQ decompositions, it is
possible to recover the row and column echelon forms of a matrix and of any of
its leading sub-matrices thanks to an elementary post-processing algorithm
How to use the Kohonen algorithm to simultaneously analyse individuals in a survey
The Kohonen algorithm (SOM, Kohonen,1984, 1995) is a very powerful tool for
data analysis. It was originally designed to model organized connections
between some biological neural networks. It was also immediately considered as
a very good algorithm to realize vectorial quantization, and at the same time
pertinent classification, with nice properties for visualization. If the
individuals are described by quantitative variables (ratios, frequencies,
measurements, amounts, etc.), the straightforward application of the original
algorithm leads to build code vectors and to associate to each of them the
class of all the individuals which are more similar to this code-vector than to
the others. But, in case of individuals described by categorical (qualitative)
variables having a finite number of modalities (like in a survey), it is
necessary to define a specific algorithm. In this paper, we present a new
algorithm inspired by the SOM algorithm, which provides a simultaneous
classification of the individuals and of their modalities.Comment: Special issue ESANN 0
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