We develop new tools to study landscapes in nonconvex optimization. Given one
optimization problem, we pair it with another by smoothly parametrizing the domain. This is …

We consider the problem of provably finding a stationary point of a smooth function to be
minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly …

This paper considers the global geometry of general low-rank minimization problems via the
Burer-Monterio factorization approach. For the rank-$1 $ case, we prove that there is no …

We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of
decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank …

Empirical evidence suggests that for a variety of overparameterized nonlinear models, most
notably in neural network training, the growth of the loss around a minimizer strongly …

We study the convergence rate of gradient-based local search methods for solving low-rank
matrix recovery problems with general objectives in both symmetric and asymmetric cases …

This paper considers the projected gradient descent (PGD) algorithm for the problem of
minimizing a continuously differentiable function on a nonempty closed subset of a …

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 …

Many fundamental low-rank optimization problems, such as matrix completion, phase
retrieval, and robust PCA, can be formulated as the matrix sensing problem. Two main …

Most existing methodologies of estimating low-rank matrices rely on Burer-Monteiro
factorization, but these approaches can suffer from slow convergence, especially when …