Low rank matrix recovery with adversarial sparse noise

Many problems in data science can be treated as recovering a low-rank matrix from a small number of random linear measurements, possibly corrupted with adversarial noise and dense noise. Recently, a bunch of theories on variants of models have been developed for different noises, but with fewer theories on the adversarial noise. In this paper, we study low-rank matrix recovery problem from linear measurements perturbed by $\ell_1$-bounded noise and sparse noise that can arbitrarily change an adversarially chosen $\omega$-fraction of the measurement vector. For Gaussian measurements with nearly optimal number of measurements, we show that the nuclear-norm constrained least absolute deviation (LAD) can successfully estimate the ground-truth matrix for any $\omega<0.239$. Similar robust recovery results are also established for an iterative hard thresholding algorithm applied to the rank-constrained LAD considering geometrically decaying step-sizes, and the unconstrained LAD based on matrix factorization as well as its subgradient descent solver.

Multivariate Genomewide Association Analysis with IHT

In genome-wide association studies (GWAS), analyzing multiple correlated traits is potentially superior to conducting multiple univariate analyses. Standard methods for multivariate GWAS operate marker-by-marker and are computationally intensive. We present a penalized regression algorithm for multivariate GWAS based on iterative hard thresholding (IHT) and implement it in a convenient Julia package MendelIHT.jl (https://github.com/OpenMendel/MendelIHT.jl). In simulation studies with up to 100 traits, IHT exhibits similar true positive rates, smaller false positive rates, and faster execution times than GEMMA's linear mixed models and mv-PLINK's canonical correlation analysis. As evidence of its scalability, our IHT code completed a trivariate trait analysis on the UK Biobank with 200,000 samples and 500,000 SNPs in 20 hours on a single machine.

Iterative Hard Thresholding with Combined Variable Step Size & Momentum-Based Estimator for Wireless Communication Systems with Dynamic Sparse Channels

The channel of the broadband wireless communications system can be modeled as a dynamic sparse channel. Such a channel is difficult to reconstruct by using linear channel estimators that are normally employed for dense channels’ estimation because of their lack of capacity to use the inherent channel’s sparsity. This paper focuses on reconstructing this type of time-varying sparse channel by extending a recently proposed dynamic channel estimator. Specifically, variable step size’s mechanism and variable momentum parameter are incorporated into traditional Iterative Hard Thresholding-based channel estimator to develop the proposed Iterative Hard Thresholding with Combined Variable Step Size and Momentum (IHT-wCVSSnM)-based estimator. Computer simulations carried out in the context of a wireless communication system operating in a dynamic sparse channel, show that the proposed IHT-wCVSSnM-based estimator performs better than all the other estimators significantly. However, the computational complexity cost of the proposed estimator is slightly higher than the closely performing channel estimator. Nevertheless, the inherent complexity cost of the proposed estimator could be compromised in a situation where the system’s performance is of higher priority when compared with the computational complexity cost.