Frédéric Zheng

Frédéric Zheng, Yassir Jedra, Alexandre Proutiere

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.

Frédéric Zheng, Alexandre Proutiere

We study the split Conformal Prediction method when applied to Markovian data. We quantify the gap in terms of coverage induced by the correlations in the data (compared to exchangeable data). This gap strongly depends on the mixing properties of the underlying Markov chain, and we prove that it typically scales as $\sqrt{t_\mathrm{mix}\ln(n)/n}$ (where $t_\mathrm{mix}$ is the mixing time of the chain). We also derive upper bounds on the impact of the correlations on the size of the prediction set. Finally we present $K$-split CP, a method that consists in thinning the calibration dataset and that adapts to the mixing properties of the chain. Its coverage gap is reduced to $t_\mathrm{mix}/(n\ln(n))$ without really affecting the size of the prediction set. We finally test our algorithms on synthetic and real-world datasets.