Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO

We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss--Newton (RGN) methods. For an $n\times n$ matrix of rank $r$ with incoherence parameter $μ$ and condition number $κ$, the two methods achieve exact recovery with high probability from $O(μnr\log n\log(nκ))$ and $O(μnr\log n\log(2μrκ))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.

We study the optimization landscape of low-tubal-rank tensor sensing through a balanced factorization. Under a tubal restricted isometry condition, we establish a quantitative strict-saddle landscape with no spurious local minima for arbitrary Fourier multi-rank profiles. We further show that the local geometry depends on the Fourier-slice ranks rather than the tubal rank alone. Uniform ranks yield quadratic growth transverse to the solution orbit, whereas nonuniform ranks produce quartically flat directions through hidden frequency-wise overparameterization, even when the factor width equals the exact tubal rank. Numerical experiments illustrate the global optimization behavior and the contrasting local geometries.

Low-Rank Adaptation (LoRA) has become a standard approach for parameter-efficient fine-tuning, yet a fundamental practical question remains unresolved: how should the adapter rank be chosen? An overly small rank may lead to a poorly conditioned optimization landscape, whereas an unnecessarily large rank sacrifices the efficiency that motivates LoRA in the first place. Existing theoretical analyses provide only limited guidance on this trade-off, and their guarantees are typically established under restrictive theoretical settings. We address this gap by developing a substantially sharper landscape theory for LoRA, building on modern results from nonconvex low-rank matrix sensing. Our central insight is that the appropriate adapter rank should depend on the quality of the data-induced optimization geometry, rather than on the model alone. To formalize this connection, we introduce LoRA-RIP, a data-dependent restricted-isometry metric that characterizes the conditioning of the cross-entropy (CE) objective along LoRA-relevant low-rank directions. We prove that sufficient rank over-parameterization, with the required rank explicitly determined by the LoRA-RIP constant, eliminates spurious local minima, thereby extending existing RIP-based guarantees beyond the classical 1/3 regime. This characterization further enables principled data selection under a fixed rank budget. Experiments across language and vision tasks support these theoretical predictions, showing that rank and data quality are two coupled resources that should be jointly considered for more efficient and reliable LoRA fine-tuning.