This paper examines a matrix-regularized multiple kernel learning (MKL) technique based on a notion of (r, p) norms. For the problem of learning a linear combination in the support vector machine-based framework, model complexity is typically controlled using various regularization strategies on the combined kernel weights. Recent research has developed a generalized l(p)-norm MKL framework with tunable variable p(p >= 1) to support controlled intrinsic sparsity. Unfortunately, this "1-D" vector l(p)-norm hardly exploits potentially useful information on how the base kernels "interact." To allow for higher order kernel-pair relationships, we extend the "1-D" vector l(p)-MKL to the "2-D" matrix (r, p) norms (1 <= r, p < infinity). We develop a new formulation and an efficient optimization strategy for (r, p)-MKL with guaranteed convergence. A theoretical analysis and experiments on seven UCI data sets shed light on the superiority of (r, p)-MKL over l(p)-MKL in various scenarios.