We develop a supervised factor-augmented matrix regression framework for high-dimensional analysis with weak factors. When factor loadings are concentrated on unknown subsets of rows and columns, conventional full-matrix factor extraction may suffer substantial signal dilution, with weakness in one matrix dimension propagating to estimation in the other. We characterize this two-way weak-loading phenomenon for both \(α\)-PCA and projected estimators and show that their loading and factor estimates may deteriorate under weak signals. To address this problem, we propose a sequential response-guided screening procedure that identifies informative rows and columns prior to factor extraction. We establish exact support recovery and show that post-screening factor re-estimation converts the weak-loading problem into a pervasive one relative to the retained dimensions, yielding sharper convergence rates under substantially weaker dimensional growth conditions and consistent factor-number determination. We further establish asymptotic distributions for both unrestricted and bilinear regression estimators based on the estimated matrix factors. Simulations demonstrate substantial improvements in support recovery and factor estimation under weak signals. We illustrate the gains obtained by the response-guided factor construction with two empirical applications.

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