Novel Algorithms for Exact and Efficient L1-NORM-BASED Tucker2 Decomposition

We consider corruption-resistant L1-norm-based TUCKER2 (L1-TUCKER2) decomposition of a D×M×N 3-way tensor, treated (with no loss of generality) as a collection of N D×M matrices. Our contributions are as follows. First, we show that rank-1 L1-TUCKER2 can be cast as a combinatorial problem over N antipodal-binary variables; accordingly, we provide the first exact algorithm for its solution. Then, we develop an efficient (quadratic-cost/near-exact) algorithm that approximates the solution to rank-1 L1- TUCKER2 by means of a converging sequence of optimal single-bit flips; the algorithm is accompanied by formal convergence proof and complexity analysis. Finally, by means of the standard deflation technique, we generalize the developed bit - flipping algorithm for solving L1- TUCKER2 decomposition problems of general rank. Our extensive numerical studies show that the bit-flipping algorithm returns the exact L1- TUCKER2 solution with very high frequency. Moreover, the developed exact and efficient algorithms exhibit remarkable outlier resistance, outperforming some of the most popular L2-norm-based and L1-norm-based counterparts.