Efficient estimation of high-dimensional matrices--including covariance and precision matrices--is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators (e.g., consistency and sparsity), while largely overlooking the computational challenges inherent in high-dimensional settings. Theoretically, we first prove the convergence of LADMM, and then establish the convergence, convergence rate, and monotonicity of its reparameterized counterpart; importantly, we show that the reparameterized LADMM enjoys a faster convergence rate. Notably, the proposed reparameterization theory and methodology are applicable to the estimation of both high-dimensional covariance and precision matrices. Keywords: ADMM; High-dimensional; Learning-based optimization; Matrix estimation. 1. Introduction High-dimensional matrix estimation--covering both covariance and precision matrix estimation--constitutes a cornerstone of modern statistics and data science [1, 2, 3]. Accurate covariance estimation enables the characterization of dependence structures among a large number of variables [4, 5, 6], which is indispensable in diverse domains such as genomics [7, 8], neuroscience [9], finance [10, 11, 12], and climate science [13, 14]. Over the past two decades, substantial progress has been made in the statistical theory of high-dimensional matrix estimation, particularly with respect to the accuracy of estimators, including properties such as sparsistency and consistency [5, 15, 16]. However, in empirical studies, the dimensionality is often only on the order of tens to hundreds, and in many cases is comparable to the sample size [21, 22, 23, 24]. This observation highlights a notable gap between the statistical theory of estimators and the practical challenges of their computational implementation.

Implicit bias induced by gradient-based algorithms is essential to the generalization of overparameterized models, yet its mechanisms can be subtle. This work leverages the Normalized Steepest Descent} (NSD) framework to investigate how optimization geometry shapes solutions on multiclass separable data. We introduce NucGD, a geometry-aware optimizer designed to enforce low rank structures through nuclear norm constraints. Beyond the algorithm itself, we connect NucGD with emerging low-rank projection methods, providing a unified perspective. To enable scalable training, we derive an efficient SVD-free update rule via asynchronous power iteration. Furthermore, we empirically dissect the impact of stochastic optimization dynamics, characterizing how varying levels of gradient noise induced by mini-batch sampling and momentum modulate the convergence toward the expected maximum margin solutions.Our code is accessible at: https://github.com/Tsokarsic/observing-the-implicit-bias-on-multiclass-seperable-data.

Coupled norms have emerged as a convex method to solve coupled tensor completion. A limitation with coupled norms is that they only induce low-rankness using the multilinear rank of coupled tensors. In this paper, we introduce a new set of coupled norms known as coupled nuclear norms by constraining the CP rank of coupled tensors. We propose new coupled completion models using the coupled nuclear norms as regularizers, which can be optimized using computationally efficient optimization methods. We derive excess risk bounds for proposed coupled completion models and show that proposed norms lead to better performance. Through simulation and real-data experiments, we demonstrate that proposed norms achieve better performance for coupled completion compared to existing coupled norms.

A major hurdle in this study is that implicit regularization in deep learning seems to kick in only withcertain types ofdata(notwithrandom dataforexample), andwelackmathematical tools for reasoning about real-life data. Thus one needs a simple test-bed for the investigation, where data admits a crisp mathematical formulation. Following earlier works, we focus on the problem of matrix completion: given a randomly chosen subset of entries from an unknown matrixW, the taskistorecovertheunseen entries. Tocastthisasaprediction problem, wemayvieweach entry inW as a data point: observed entries constitute the training set, and the average reconstruction error over the unobserved entries is the test error,quantifying generalization. Fitting the observed entries is obviously an underdetermined problem with multiple solutions.

Unfortunately, this common choice tends to cause two serious issues.

A fundamental structural model for data is that the data points lie close to an unknown subspace, meaning that the matrix created by concatenating the data vectors has low rank. We address a particular low-rank matrix recovery problem where we wish to recover a set of vectors from a low-dimensional subspace after they have been individually compressed (or "sketched").

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Clearly, due to nonconvexity, the task of finding a minimizer of (1) canbequite challenging andpotentially limits what canbeguaranteed theoretically about solving problems with form(1).