In aggregation-fragmentation processes, a steady state is usually reached in the long time limit. This indicates the existence of a fixed point in the underlying system of ordinary differential equations. The next simplest possibility is an asymptotically periodic motion. Never-ending oscillations have not been rigorously established so far, although oscillations have been recently numerically detected in a few systems. For a class of addition-shattering processes, we provide convincing numerical evidence for never-ending oscillations in a certain region U of the parameter space. The processes which we investigate admit a fixed point that becomes unstable when parameters belong to U and never-ending oscillations effectively emerge through a Hopf bifurcation.
We consider the low-rank tensor train completion problem when additional side information is available in the form of subspaces that contain the mode-k fiber spans. We propose an algorithm based on Riemannian optimization to solve the problem. Numerical experiments show that the proposed algorithm requires far fewer known entries to recover the tensor compared to standard tensor train completion methods.
In this work we estimate the number of randomly selected elements of a tensor that with high probability guarantees local convergence of Riemannian gradient descent for tensor train completion. We derive a new bound for the orthogonal projections onto the tangent spaces based on the harmonic mean of the unfoldings' singular values and introduce a notion of core coherence for tensor trains. We also extend the results to tensor train completion with side information and obtain the corresponding local convergence guarantees.
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