Understanding Deflation Process in Over-parametrized Tensor Decomposition
This provides insight into implicit regularization in over-parametrized models for low-rank tensors, which is incremental as it extends known matrix results to tensors.
The paper tackles the training dynamics of gradient flow on over-parametrized tensor decomposition, showing that for orthogonally decomposable tensors, a modified gradient flow follows a deflation process and recovers all tensor components, similar to greedy low-rank learning in matrices.
In this paper we study the training dynamics for gradient flow on over-parametrized tensor decomposition problems. Empirically, such training process often first fits larger components and then discovers smaller components, which is similar to a tensor deflation process that is commonly used in tensor decomposition algorithms. We prove that for orthogonally decomposable tensor, a slightly modified version of gradient flow would follow a tensor deflation process and recover all the tensor components. Our proof suggests that for orthogonal tensors, gradient flow dynamics works similarly as greedy low-rank learning in the matrix setting, which is a first step towards understanding the implicit regularization effect of over-parametrized models for low-rank tensors.