Implicit Regularization in Deep Tensor Factorization
This work addresses a foundational issue in machine learning theory for researchers studying optimization dynamics, but it is incremental as it extends prior matrix completion findings to tensors.
The paper tackles the problem of understanding implicit regularization in gradient descent for tensor completion, showing that gradient descent promotes low-rank solutions and supporting the need for a dynamical perspective.
Attempts of studying implicit regularization associated to gradient descent (GD) have identified matrix completion as a suitable test-bed. Late findings suggest that this phenomenon cannot be phrased as a minimization-norm problem, implying that a paradigm shift is required and that dynamics has to be taken into account. In the present work we address the more general setup of tensor completion by leveraging two popularized tensor factorization, namely Tucker and TensorTrain (TT). We track relevant quantities such as tensor nuclear norm, effective rank, generalized singular values and we introduce deep Tucker and TT unconstrained factorization to deal with the completion task. Experiments on both synthetic and real data show that gradient descent promotes solution with low-rank, and validate the conjecture saying that the phenomenon has to be addressed from a dynamical perspective.