Tensorized Random Projections
This work addresses computational efficiency for machine learning tasks involving high-dimensional tensor data, representing an incremental improvement over classical random projection methods.
The paper tackles the problem of efficiently reducing the dimension of high-dimensional tensors by introducing tensorized random projection techniques based on CP and TT decompositions, showing that the TT format requires smaller projections to achieve the same distortion ratio as validated in experiments.
We introduce a novel random projection technique for efficiently reducing the dimension of very high-dimensional tensors. Building upon classical results on Gaussian random projections and Johnson-Lindenstrauss transforms~(JLT), we propose two tensorized random projection maps relying on the tensor train~(TT) and CP decomposition format, respectively. The two maps offer very low memory requirements and can be applied efficiently when the inputs are low rank tensors given in the CP or TT format. Our theoretical analysis shows that the dense Gaussian matrix in JLT can be replaced by a low-rank tensor implicitly represented in compressed form with random factors, while still approximately preserving the Euclidean distance of the projected inputs. In addition, our results reveal that the TT format is substantially superior to CP in terms of the size of the random projection needed to achieve the same distortion ratio. Experiments on synthetic data validate our theoretical analysis and demonstrate the superiority of the TT decomposition.