9,429 research outputs found
Nearly Optimal Sparse Polynomial Multiplication
In the sparse polynomial multiplication problem, one is asked to multiply two
sparse polynomials f and g in time that is proportional to the size of the
input plus the size of the output. The polynomials are given via lists of their
coefficients F and G, respectively. Cole and Hariharan (STOC 02) have given a
nearly optimal algorithm when the coefficients are positive, and Arnold and
Roche (ISSAC 15) devised an algorithm running in time proportional to the
"structural sparsity" of the product, i.e. the set supp(F)+supp(G). The latter
algorithm is particularly efficient when there not "too many cancellations" of
coefficients in the product. In this work we give a clean, nearly optimal
algorithm for the sparse polynomial multiplication problem.Comment: Accepted to IEEE Transactions on Information Theor
Fast Exact Bayesian Inference for Sparse Signals in the Normal Sequence Model
We consider exact algorithms for Bayesian inference with model selection
priors (including spike-and-slab priors) in the sparse normal sequence model.
Because the best existing exact algorithm becomes numerically unstable for
sample sizes over n=500, there has been much attention for alternative
approaches like approximate algorithms (Gibbs sampling, variational Bayes,
etc.), shrinkage priors (e.g. the Horseshoe prior and the Spike-and-Slab LASSO)
or empirical Bayesian methods. However, by introducing algorithmic ideas from
online sequential prediction, we show that exact calculations are feasible for
much larger sample sizes: for general model selection priors we reach n=25000,
and for certain spike-and-slab priors we can easily reach n=100000. We further
prove a de Finetti-like result for finite sample sizes that characterizes
exactly which model selection priors can be expressed as spike-and-slab priors.
The computational speed and numerical accuracy of the proposed methods are
demonstrated in experiments on simulated data, on a differential gene
expression data set, and to compare the effect of multiple hyper-parameter
settings in the beta-binomial prior. In our experimental evaluation we compute
guaranteed bounds on the numerical accuracy of all new algorithms, which shows
that the proposed methods are numerically reliable whereas an alternative based
on long division is not
Interactive certificate for the verification of Wiedemann's Krylov sequence: application to the certification of the determinant, the minimal and the characteristic polynomials of sparse matrices
Certificates to a linear algebra computation are additional data structures
for each output, which can be used by a-possibly randomized- verification
algorithm that proves the correctness of each output. Wiede-mann's algorithm
projects the Krylov sequence obtained by repeatedly multiplying a vector by a
matrix to obtain a linearly recurrent sequence. The minimal polynomial of this
sequence divides the minimal polynomial of the matrix. For instance, if the
input matrix is sparse with n 1+o(1) non-zero entries, the
computation of the sequence is quadratic in the dimension of the matrix while
the computation of the minimal polynomial is n 1+o(1), once that projected
Krylov sequence is obtained. In this paper we give algorithms that compute
certificates for the Krylov sequence of sparse or structured
matrices over an abstract field, whose Monte Carlo verification complexity can
be made essentially linear. As an application this gives certificates for the
determinant, the minimal and characteristic polynomials of sparse or structured
matrices at the same cost
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