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Symmetric tensor operations arise in a wide variety of computations. However,the benefits of exploiting symmetry in order to reduce storage and computationis in conflict with a desire to simplify memory access patterns. In this paper,we propose a blocked data structure (Blocked Compact Symmetric Storage) whereinwe consider the tensor by blocks and store only the unique blocks of asymmetric tensor. We propose an algorithm-by-blocks, already shown of benefitfor matrix computations, that exploits this storage format by utilizing aseries of temporary tensors to avoid redundant computation. Further, partialsymmetry within temporaries is exploited to further avoid redundant storage andredundant computation. A detailed analysis shows that, relative to storing andcomputing with tensors without taking advantage of symmetry and partialsymmetry, storage requirements are reduced by a factor of $ O\left( m! \right)$and computational requirements by a factor of $O\left( (m+1)!/2^m \right)$,where $ m $ is the order of the tensor. However, as the analysis shows, caremust be taken in choosing the correct block size to ensure these storage andcomputational benefits are achieved (particularly for low-order tensors). Animplementation demonstrates that storage is greatly reduced and the complexityintroduced by storing and computing with tensors by blocks is manageable.Preliminary results demonstrate that computational time is also reduced. Thepaper concludes with a discussion of how insights in this paper point toopportunities for generalizing recent advances in the domain of linear algebralibraries to the field of multi-linear computation.

Exploiting Symmetry in Tensors for High Performance: Multiplication with Symmetric Tensors