Optimization Algorithms for Tensor Decompositions with Parameter Sharing

Tensor decompositions provide us ways to work effectively with multidimensional data arrays. A special class of decompositions is decompositions with shared factors which reduce the number of parameters even further and turn out to be successful in the case of symmetries in data. An example of such decomposition is the recently introduced SF-Tucker decomposition, a generalization of the usual Tucker decomposition. Despite decent properties, it suffers from the curse of dimensionality. In this paper, we propose two architectures of combining the tensor train decomposition, which is freed from the curse of dimensionality, and the idea of shared factors. The issues of their existence and minimum ranks are investigated. For one of them, a rounding algorithm has been developed, and it turns out that this algorithm outperforms the conventional TT rounding in terms of data compression.