690 research outputs found
Efficient Interpolation in the Guruswami-Sudan Algorithm
A novel algorithm is proposed for the interpolation step of the
Guruswami-Sudan list decoding algorithm. The proposed method is based on the
binary exponentiation algorithm, and can be considered as an extension of the
Lee-O'Sullivan algorithm. The algorithm is shown to achieve both asymptotical
and practical performance gain compared to the case of iterative interpolation
algorithm. Further complexity reduction is achieved by integrating the proposed
method with re-encoding. The key contribution of the paper, which enables the
complexity reduction, is a novel randomized ideal multiplication algorithm.Comment: Submitted to IEEE Transactions on Information Theor
On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS)
codes by using Gr\"obner bases over modules and the Euclidean algorithm (EA) as
the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We
present a novel method for constructing the interpolation polynomial fast. We
give a new application of the Wu list decoder by decoding irreducible binary
Goppa codes up to the binary Johnson radius. Finally, we point out a connection
between the governing equations of the Wu algorithm and the Guruswami-Sudan
algorithm (GSA), immediately leading to equality in the decoding range and a
duality in the choice of parameters needed for decoding, both in the case of
GRS codes and in the case of Goppa codes.Comment: To appear in IEEE Transactions of Information Theor
Polynomial-Division-Based Algorithms for Computing Linear Recurrence Relations
Sparse polynomial interpolation, sparse linear system solving or modular
rational reconstruction are fundamental problems in Computer Algebra. They come
down to computing linear recurrence relations of a sequence with the
Berlekamp-Massey algorithm. Likewise, sparse multivariate polynomial
interpolation and multidimensional cyclic code decoding require guessing linear
recurrence relations of a multivariate sequence.Several algorithms solve this
problem. The so-called Berlekamp-Massey-Sakata algorithm (1988) uses polynomial
additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on
linear algebra operations on a multi-Hankel matrix, a multivariate
generalization of a Hankel matrix. The Artinian Gorenstein border basis
algorithm (2017) uses a Gram-Schmidt process.We propose a new algorithm for
computing the Gr{\"o}bner basis of the ideal of relations of a sequence based
solely on multivariate polynomial arithmetic. This algorithm allows us to both
revisit the Berlekamp-Massey-Sakata algorithm through the use of polynomial
divisions and to completely revise the Scalar-FGLM algorithm without linear
algebra operations.A key observation in the design of this algorithm is to work
on the mirror of the truncated generating series allowing us to use polynomial
arithmetic modulo a monomial ideal. It appears to have some similarities with
Pad{\'e} approximants of this mirror polynomial.As an addition from the paper
published at the ISSAC conferance, we give an adaptive variant of this
algorithm taking into account the shape of the final Gr{\"o}bner basis
gradually as it is discovered. The main advantage of this algorithm is that its
complexity in terms of operations and sequence queries only depends on the
output Gr{\"o}bner basis.All these algorithms have been implemented in Maple
and we report on our comparisons
Computing syzygies in finite dimension using fast linear algebra
We consider the computation of syzygies of multivariate polynomials in afinite-dimensional setting: for a -module of finite dimension as a -vector space, andgiven elements in , the problem is to computesyzygies between the 's, that is, polynomials in such that in. Assuming that the multiplication matrices of the variables with respect to some basis of are known, we give analgorithm which computes the reduced Gr\"obner basis of the module of thesesyzygies, for any monomial order, using operations in the base field , where is theexponent of matrix multiplication. Furthermore, assuming that is itself given as ,under some assumptions on we show that these multiplicationmatrices can be computed from a Gr\"obner basis of within thesame complexity bound. In particular, taking , and in, this yields a change of monomial order algorithm along thelines of the FGLM algorithm with a complexity bound which is sub-cubic in
A polynomial-division-based algorithm for computing linear recurrence relations
International audienceSparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidi-mensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp– Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one
50 Years of Test (Un)fairness: Lessons for Machine Learning
Quantitative definitions of what is unfair and what is fair have been
introduced in multiple disciplines for well over 50 years, including in
education, hiring, and machine learning. We trace how the notion of fairness
has been defined within the testing communities of education and hiring over
the past half century, exploring the cultural and social context in which
different fairness definitions have emerged. In some cases, earlier definitions
of fairness are similar or identical to definitions of fairness in current
machine learning research, and foreshadow current formal work. In other cases,
insights into what fairness means and how to measure it have largely gone
overlooked. We compare past and current notions of fairness along several
dimensions, including the fairness criteria, the focus of the criteria (e.g., a
test, a model, or its use), the relationship of fairness to individuals,
groups, and subgroups, and the mathematical method for measuring fairness
(e.g., classification, regression). This work points the way towards future
research and measurement of (un)fairness that builds from our modern
understanding of fairness while incorporating insights from the past.Comment: FAT* '19: Conference on Fairness, Accountability, and Transparency
(FAT* '19), January 29--31, 2019, Atlanta, GA, US
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