We develop fast, reliable algorithms for recovering useful information from incomplete, noisy, or corrupted data, a challenge that sits at the heart of modern machine learning, communications, and data systems. By combining rigorous mathematical foundations with practical algorithmic design, our work produces tools for intelligent data compression, robust error-correction, efficient optimization, and secure distributed learning that are built to perform under real-world constraints. The result is a body of expertise directly applicable to problems in signal processing, large-scale model training, wireless communications, and data-efficient AI, delivering solutions that are not only theoretically grounded but engineered for efficiency and resilience at scale.