Inourpaper theGrassmannian21 structure is utilized together with the RRC to analyze the convergence of the projected Riemannian subgradient22 method. Since33 both the robust subspace learning and dictionary learning problems are regular, their Riemannian subdifferentials34 computedinSection4arecorrect.

Thus a gap has arisen between theory and practice.

Least-absolute-deviations (LAD) line fitting is robust to outliers but computationally more involved than least squares regression. Although the literature includes linear and near-linear time algorithms for the LAD line fitting problem, these methods are difficult to implement and, to our knowledge, lack maintained public implementations. As a result, practitioners often resort to linear programming (LP) based methods such as the simplex-based Barrodale-Roberts method and interior-point methods, or on iteratively reweighted least squares (IRLS) approximation which does not guarantee exact solutions. To close this gap, we propose the Piecewise Affine Lower-Bounding (PALB) method, an exact algorithm for LAD line fitting. PALB uses supporting lines derived from subgradients to build piecewise-affine lower bounds, and employs a subdivision scheme involving minima of these lower bounds. We prove correctness and provide bounds on the number of iterations. On synthetic datasets with varied signal types and noise including heavy-tailed outliers as well as a real dataset from the NOAA's Integrated Surface Database, PALB exhibits empirical log-linear scaling. It is consistently faster than publicly available implementations of LP based and IRLS based solvers. We provide a reference implementation written in Rust with a Python API.

This paper investigates the partial linear model by Least Absolute Deviation (LAD) regression. We parameterize the nonparametric term using Deep Neural Networks (DNNs) and formulate a penalized LAD problem for estimation. Specifically, our model exhibits the following challenges. First, the regularization term can be nonconvex and nonsmooth, necessitating the introduction of infinite dimensional variational analysis and nonsmooth analysis into the asymptotic normality discussion. Second, our network must expand (in width, sparsity level and depth) as more samples are observed, thereby introducing additional difficulties for theoretical analysis. Third, the oracle of the proposed estimator is itself defined through a ultra high-dimensional, nonconvex, and discontinuous optimization problem, which already entails substantial computational and theoretical challenges. Under such the challenges, we establish the consistency, convergence rate, and asymptotic normality of the estimator. Furthermore, we analyze the oracle problem itself and its continuous relaxation. We study the convergence of a proximal subgradient method for both formulations, highlighting their structural differences lead to distinct computational subproblems along the iterations. In particular, the relaxed formulation admits significantly cheaper proximal updates, reflecting an inherent trade-off between statistical accuracy and computational tractability.

We provide counterexamples showing that uniform laws of large numbers do not hold for subdifferentials under natural assumptions. Our results apply to random Lipschitz functions and random convex functions with a finite number of smooth pieces. Consequently, they resolve the questions posed by Shapiro and Xu [J. Math. Anal. Appl., 325(2), 2007] in the negative and highlight the obstacles nonsmoothness poses to uniform results.

This work addresses the finite-time analysis of nonsmooth nonconvex stochastic optimization under Riemannian manifold constraints. We adapt the notion of Goldstein stationarity to the Riemannian setting as a performance metric for nonsmooth optimization on manifolds. We then propose a Riemannian Online to NonConvex (RO2NC) algorithm, for which we establish the sample complexity of $O(ε^{-3}δ^{-1})$ in finding $(δ,ε)$-stationary points. This result is the first-ever finite-time guarantee for fully nonsmooth, nonconvex optimization on manifolds and matches the optimal complexity in the Euclidean setting. When gradient information is unavailable, we develop a zeroth order version of RO2NC algorithm (ZO-RO2NC), for which we establish the same sample complexity. The numerical results support the theory and demonstrate the practical effectiveness of the algorithms.