Regression is essential across many domains but remains challenging in high-dimensional settings, where existing methods often lose spatial structure or demand heavy storage. In this work, we address the problem of matrix-valued regression, where each sample is naturally represented as a matrix. We propose MELCOT, a hybrid model that integrates a classical machine–learning–based Marginal Estimation (ME) block with a deep-learning–based Learnable-Cost Optimal Transport (LCOT) block. The ME block estimates data marginals to preserve spatial information, while the LCOT block learns complex global features. This design enables MELCOT to inherit the strengths of both classical and deep learning methods. Extensive experiments across diverse datasets and domains demonstrate that MELCOT consistently outperforms all baselines while remaining highly efficient and provides potential applications in various domains, including high-content imaging (HCI).

Graph neural networks (GNNs) are susceptible to privacy inference attacks (PIAS) given their ability to learn joint representation from features and edges among nodes in graph data. To prevent privacy leakages in GNNs, we propose a novel heterogeneous randomized response (HeteroRR) mechanism to protect nodes’ features and edges against PIAS under differential privacy (DP) guarantees, without an undue cost of data and model utility in training GNNs. Our idea is to balance the importance and sensitivity of nodes’ features and edges in redistributing the privacy budgets since some features and edges are more sensitive or important to the model utility than others. As a result, we derive significantly better randomization probabilities and tighter error bounds at both levels of nodes’ features and edges departing from existing approaches, thus enabling us to maintain high data utility for training GNNs. An extensive theoretical and empirical analysis using benchmark datasets shows that HeteroRR significantly outperforms various baselines in terms of model utility under rigorous privacy protection for both nodes’ features and edges. That enables us to defend PIAs in DP-preserving GNNs effectively.