Straightforward generalizations of the linear methods include the incorporation of parallelism, exploration of their utility in computing other matrix decompositions, e.g., CUR decompositions, and utilization of sparsifying techniques from fields such as compressed sensing for enhanced com-pression. Another natural generalization of many of the linear approaches presented in this work is to extend their application to higher order tensor decompositions such as the Tucker decompo-sition [240] and tensor interpolative decompodecompo-sition [20]. In the nonlinear setting, linear sketching enables online compression using autoencoders. Extending this approach to tensors could entail forming a core tensor, i.e., a sketched tensor, via, e.g., tucker decomposition, and then compressing the sketched tensor using, e.g., convolutional autoencoders.
Despite the known appeals of tensor-based decomposition methods, we note their limitations in certain applications. First, if the grid from which simulation data is obtained is non-Cartesian, the benefits of storing data in tensor format are significantly reduced. Second, many of the single-pass algorithms for tensor compression, e.g., [170], scale quite unfavorably with target rank and lack many of the global approximation guarantees that low-rank matrix methods provide. Generalizing our approaches to tensor-based settings must therefore be done with care and acceptance of the trade-offs inherent to approximating high order tensors, in particular in a pass-efficient manner.
Finally, it would be a significant break-through to identify single-pass approaches to introduce nonlinearity to the sketching/lifting procedures in the neural network based approaches discussed in Appendix A. If successful, such methods could allow the single-pass autoencoder compression scheme to break free of limitations related to Kolmogorov n-width decay. If doing so in one pass over
the input is not feasible, then building pass-efficient methods to improve our single-pass autoencoder approximation could be an acceptable compromise.
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