Shape Constrained Tensor Decompositions using Sparse Representations in Over-Complete Libraries

Abstract--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 T ensor Decomposition (SCTD) is based upon the CANDECOMP/PARAF AC (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 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 spatiotemporal 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. A TRIX decompositions are critically enabling algorithms for scientific computing and data analysis applications across every field of the engineering, social, biological, and physical sciences. Of particular importance is the singular value decomposition (SVD), which provides a principled method for dimensionality reduction and computation of interpretable subspaces within which the data reside. So widespread is the usage of the algorithm, and minor modifications thereof, that it has generated a myriad of names across various communities, including Principal Component Analysis (PCA) [1], the Karhunen-Lo eve (KL) decomposition, Hotelling transform [2], [3], Empirical Orthogonal Functions (EOFs) [4] and Proper Orthogonal Decomposition (POD) [5], [6]. However, in order to use the SVD, data, which generally may be of N distinct dimensions, must be flattened into a matrix form, potentially compromising the statistical accuracy of the subspaces computed. B. Lusch and J. N. Kutz are with the Department of Applied Mathematics, University of Washington, Seattle, W A 98195-3925 USA email: herwaldt@uw.edu, E. Chi is with the Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203 USA email: eric chi@ncsu.edu. It is often the case that one of the dimensions considered in the tensor is the time variable.