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Gradient descent (GD) is crucial for generalization in machine learningmodels, as it induces implicit regularization, promoting compactrepresentations. In this work, we examine the role of GD in inducing implicitregularization for tensor optimization, particularly within the context of thelifted matrix sensing framework. This framework has been recently proposed toaddress the non-convex matrix sensing problem by transforming spurioussolutions into strict saddles when optimizing over symmetric, rank-1 tensors.We show that, with sufficiently small initialization scale, GD applied to thislifted problem results in approximate rank-1 tensors and critical points withescape directions. Our findings underscore the significance of the tensorparametrization of matrix sensing, in combination with first-order methods, inachieving global optimality in such problems.

Algorithmic Regularization in Tensor Optimization: Towards a Lifted Approach in Matrix Sensing