Algorithmic Regularization in Tensor Optimization: Towards a Lifted
  Approach in Matrix Sensing
Ziye Ma Javad Lavaei Somayeh Sojoudi
Gradient descent (GD) is crucial for generalization in machine learning
models, as it induces implicit regularization, promoting compact
representations. In this work, we examine the role of GD in inducing implicit
regularization for tensor optimization, particularly within the context of the
lifted matrix sensing framework. This framework has been recently proposed to
address the non-convex matrix sensing problem by transforming spurious
solutions into strict saddles when optimizing over symmetric, rank-1 tensors.
We show that, with sufficiently small initialization scale, GD applied to this
lifted problem results in approximate rank-1 tensors and critical points with
escape directions. Our findings underscore the significance of the tensor
parametrization of matrix sensing, in combination with first-order methods, in
achieving global optimality in such problems.