Shape Constrained Tensor Decompositions using Sparse Representations in Over-Complete Libraries
This provides a more comprehensible technique for analyzing multitudes of data in fields like data-driven discovery, though it appears incremental as it builds upon existing CP decomposition methods.
The paper tackles the problem of extracting interpretable spatio-temporal modes from high-dimensional tensor data by proposing a Shape Constrained Tensor Decomposition (SCTD) method that constrains temporal vectors to known analytic forms from an over-complete library, achieving more interpretable decompositions and discovering analytic time dependencies.
We consider $N$-way data arrays and low-rank tensor factorizations where the time mode is coded as a sparse linear combination of temporal elements from an over-complete library. Our method, Shape Constrained Tensor Decomposition (SCTD) is based upon the CANDECOMP/PARAFAC (CP) decomposition which produces $r$-rank approximations of data tensors via outer products of vectors in each dimension of the data. By constraining the vector in the temporal dimension to known analytic forms which are selected from a large set of candidate functions, more readily interpretable decompositions are achieved and analytic time dependencies discovered. The SCTD method circumvents traditional {\em flattening} techniques where an $N$-way array is reshaped into a matrix in order to perform a singular value decomposition. A clear advantage of the SCTD algorithm is its ability to extract transient and intermittent phenomena which is often difficult for SVD-based methods. We motivate the SCTD method using several intuitively appealing results before applying it on a number of high-dimensional, real-world data sets in order to illustrate the efficiency of the algorithm in extracting interpretable spatio-temporal modes. With the rise of data-driven discovery methods, the decomposition proposed provides a viable technique for analyzing multitudes of data in a more comprehensible fashion.