SamBaTen: Sampling-based Batch Incremental Tensor Decomposition

Tensor decompositions are invaluable tools in analyzing multimodal datasets. In many real-world scenarios, such datasets are far from being static, to the contrary they tend to grow over time. For instance, in an online social network setting, as we observe new interactions over time, our dataset gets updated in its "time" mode. How can we maintain a valid and accurate tensor decomposition of such a dynamically evolving multimodal dataset, without having to re-compute the entire decomposition after every single update? In this paper we introduce SaMbaTen, a Sampling-based Batch Incremental Tensor Decomposition algorithm, which incrementally maintains the decomposition given new updates to the tensor dataset. SaMbaTen is able to scale to datasets that the state-of-the-art in incremental tensor decomposition is unable to operate on, due to its ability to effectively summarize the existing tensor and the incoming updates, and perform all computations in the reduced summary space. We extensively evaluate SaMbaTen using synthetic and real datasets. Indicatively, SaMbaTen achieves comparable accuracy to state-of-the-art incremental and non-incremental techniques, while being 25-30 times faster. Furthermore, SaMbaTen scales to very large sparse and dense dynamically evolving tensors of dimensions up to 100K x 100K x 100K where state-of-the-art incremental approaches were not able to operate

Streaming Tensor Train Approximation

Tensor trains are a versatile tool to compress and work with high-dimensional data and functions. In this work we introduce the Streaming Tensor Train Approximation (STTA), a new class of algorithms for approximating a given tensor $\mathcal T$ in the tensor train format. STTA accesses $\mathcal T$ exclusively via two-sided random sketches of the original data, making it streamable and easy to implement in parallel -- unlike existing deterministic and randomized tensor train approximations. This property also allows STTA to conveniently leverage structure in $\mathcal T$, such as sparsity and various low-rank tensor formats, as well as linear combinations thereof. When Gaussian random matrices are used for sketching, STTA is admissible to an analysis that builds and extends upon existing results on the generalized Nystr\"om approximation for matrices. Our results show that STTA can be expected to attain a nearly optimal approximation error if the sizes of the sketches are suitably chosen. A range of numerical experiments illustrates the performance of STTA compared to existing deterministic and randomized approaches.Comment: 21 pages, code available at https://github.com/RikVoorhaar/tt-sketc