We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding singular vectors up to an effective rank, adaptively determined based on the data. We establish instance-specific lower bounds for the sample complexity of such algorithms, uncovering fundamental trade-offs in selecting the effective rank: balancing the precision of estimating a subset of singular values against the approximation cost incurred for the remaining ones. Our analysis identifies how the optimal effective rank depends on the matrix being estimated, the sample size, and the noise level. We propose an algorithm that combines a Least-Squares estimator with a universal singular value thresholding procedure. We provide finite-sample error bounds for this algorithm and demonstrate that its performance nearly matches the derived fundamental limits. Our results rely on an enhanced analysis of matrix denoising methods based on singular value thresholding. We validate our findings with applications to multivariate regression and linear dynamical system identification.

翻译：我们研究从线性测量中估计高维矩阵的问题，重点关注最优秩自适应算法的设计。这类算法通过基于数据自适应确定的有效秩，估计矩阵的奇异值及对应奇异向量，从而推断矩阵。我们建立了此类算法样本复杂度的实例特定下界，揭示了有效秩选择中的基本权衡：在估计部分奇异值的精度与其余奇异值产生的近似代价之间寻求平衡。我们的分析确定了最优有效秩如何依赖于被估计矩阵、样本量及噪声水平。我们提出一种算法，将最小二乘估计器与通用奇异值阈值化过程相结合。我们给出了该算法的有限样本误差界，并证明其性能近乎匹配所导出的基本极限。我们的结果依赖于基于奇异值阈值化的矩阵去噪方法的增强分析。我们通过多元回归和线性动力系统识别的应用验证了所得结论。