A probabilistic numerical algorithm based on Terracini’s lemma for proving nondefectivity of Segre varieties

The CANDECOMP/PARAFAC (CP) tensor decomposition is a crucial data analysis and processing technique in applications and sciences. Tensors admitting a CP decomposition of rank at most s can be considered points of the sth order secant variety of a Segre variety, yet the most basic invariant of this variety—its dimension— is not yet fully understood. A conjecture was nevertheless proposed in [H. Abo, G. Ottaviani, and C. Peterson, Induction for secant varieties of Segre varieties. Trans. Amer. Math. Soc., 361:767–792, 2009], proved to be correct for s ≤ 6. We propose a memory efficient probabilistic numerical algorithm based on Terracini’s Lemma to verify with high probability whether this conjecture is true. The proposed method can geometrically decrease the probability of incorrectly classifying a defective Segre variety as nondefective by performing more statistically independent checks. While increasing the precision of the calculations is not required to decrease the probability of an incorrect classification, a meticulous numerical analysis illustrates that the computations must, nevertheless, be performed with sufficiently high precision. The numerical experiments in this paper establish that the Segre varieties PC1 × PC2 × · · · × PCd embedded in PCn1×···×nd with generic rank s ≤ 55 obey the conjecture; the probability that the obtained results are due to chance is less than 10−55.