Training data is central to machine learning, yet it remains challenging to efficiently appraise dataset value for a learning process. Neural scaling laws appraise data through dataset size, while the recently proposed Vendi Score uses Von Neumann (quantum) entropy to measure dataset value. In this paper, we show both that common neural-scaling-law objectives and the Vendi Score are submodular. We further show that the Vendi Score is a special case of a broader class of submodular objectives that we call matrix spectral functions. This class also includes determinantal (DPP) objectives, which are known to be log-submodular, as well as many others, including neural-scaling-law-inspired matrix spectral hybrids. We also introduce weakly matrix monotone functions and show how they lead to weakly submodular matrix spectral functions, yielding a broad family of practical objectives for data selection and appraisal. A key challenge is scale. We thus develop secular-equation-based updates that avoid repeated eigendecompositions during greedy optimization, reducing marginal-gain evaluation for m-dimensional embeddings by an O(m) factor relative to oracle queries. In our implementation, this yields an average empirical speedup of ≈ 35,000×, making direct optimization of the Vendi Score feasible on ImageNet-1K-scale datasets. Thus enabled, we compare how well several objectives predict the value of training subsets for heldout test performance under fixed-size, class-balanced, and fixed training-budget regimes, including the Vendi Score, DPPs, facility location, and three new matrix spectral variants. Across multiple datasets, facility location performs the best. Direct optimization also reveals that, while the Vendi Score is predictive over moderate score ranges, pushing the objective to higher values can make it a poor downstream performance proxy. Other matrix spectral variants, including weakly submodular ones, perform better, though still below facility location. We also find that uniformly at random fixed-size subsets, both unconstrained and class-balanced, are remarkably concentrated in both appraisal scores and held-out performance, indicating that random sampling fails to expose much variety. Finally, we show that size, class balance, and training budget do not alone determine data value: even when controlling for these factors, performance ranges smoothly from good to bad.