  Nonnegative tensors arise very naturally in many applications that involve
large and complex data flows. Due to the relatively small requirement in terms
of memory storage and number of operations per step, the (shifted) higher-order
power method is one of the most commonly used technique for the computation of
positive Z-eigenvectors of this type of tensors. However, unlike the matrix
case, the method may fail to converge even for irreducible tensors. Moreover,
when it converges, its convergence rate can be very slow. These two drawbacks
often make the computation of the eigenvectors demanding or unfeasible for
large problems. In this work we consider a particular class of nonnegative
tensors associated to the multilinear PageRank modification of higher-order
Markov chains. Based on the simplified topological ${\epsilon}$-algorithm in
its restarted form, we introduce an extrapolation-based acceleration of power
method type algorithms, namely the shifted fixed-point method and the
inner-outer method. The accelerated methods show remarkably better performance,
with faster convergence rates and reduced overall computational time. Extensive
numerical experiments on synthetic and real-world datasets demonstrate the
advantages of the introduced extrapolation techniques.
