Selecting a small, diverse subset from a large candidate pool often means balancing several incompatible notions of diversity. In trademark curation, for instance, a subset should cover both the language used to describe marks and the visual space of their logos. A single determinantal point process (\DPP) kernel can hide failure in one view, and averaging kernels replaces the multi-view relaxation by an ordinary single-kernel spectral problem. We formulate \emph{fair multi-view determinant selection}: maximize the weakest per-view log determinant of a size-$k$ subset. We smooth this nonsmooth objective and relax it to the Stiefel manifold. The relaxation embeds every discrete subset exactly, but unlike its single-view counterpart it has no closed-form spectral solution in general. Its stationarity condition is a gauge-invariant nonlinear eigenvalue problem with eigenvector-dependent, view-adaptive weights. We derive an adaptive self-consistent-field (\SCF) solver with damping and level shifting, and round the resulting subspace by leverage-score screening followed by fair local refinement. The solver needs only feature-map products for each view. We report conflicting-view synthetic experiments and specify a multimodal USPTO protocol; the real-data multimodal results require aligned logo embeddings and are not claimed in this version.
Selecting a fixed-size subset that maximizes the determinant of a positive semidefinite kernel is the MAP problem for a size-constrained determinantal point process and the classical maximum-entropy sampling problem. Although this discrete problem is NP-hard, a classical spectral bound gives an efficiently computable ceiling using the leading eigenvalues. The same ceiling is the exact optimum of the associated Stiefel relaxation, so the continuous problem is already solved by the leading eigenspace. We study what this eigenspace implies for discrete rounding. The leading eigenvectors induce a projection determinantal point process whose probability for a subset equals its squared coordinate volume. We prove that the gap between the determinant of any subset and the spectral ceiling is at most its negative log-probability under this distribution. Consequently, the integrality gap is bounded by the min-entropy and equals it when the kernel rank matches the subset size. Projection-DPP rounding also has an expected gap bounded by the Shannon entropy and admits a high-probability additive guarantee. These results identify leading-subspace localization, rather than eigenvalue decay alone, as the geometry controlling roundability. This analysis yields CertDPP, a matrix-free pipeline that computes the leading eigenspace, draws projection-DPP samples, optionally improves them by determinant-increasing swaps, and reports the gap from a verified spectral ceiling. Controlled experiments validate the entropy identities, compare with exact MAP on small instances, and demonstrate linear scaling in the ground-set size for the rounding stage.
LLM judges are used to reduce the need for costly human labor in evaluating open-ended text generation. However, the reliability of these judges depends critically on their alignment with human raters -- a property that itself depends on costly human annotations. In this work, we develop a method (Metric Match) for estimating correlation-based reliability metrics of LLM judges from limited annotations. Metric Match selects a subset of samples for human annotation such that the subset matches the population reliability metric with respect to acquired synthetic labels. We empirically show that Metric Match achieves a win-rate of 0.838 against random subset selection across four different correlation metrics and 15 datasets, with an 18.7% decrease in average estimation error and reduces annotation needs by 32.5%. We provide a cost model and highlight a medical case study where our method saves $1,041.67 compared to random selection for expert annotation. Further, we shift our task from reliability estimation to reliability classification of whether a given judge is above a deployment threshold, outperforming random selection with Metric Match. All project code is publicly available, and we additionally provide an installable package for ease of use.
Scalable data attribution methods typically assign isolated utility scores to individual training examples. This prevalent additive assumption fundamentally fails to capture critical subset dynamics, including data redundancy and complementary coverage. In this work, we reframe attribution as subset-level counterfactual utility prediction and introduce GRASP, an interaction-aware surrogate. Grounded in a theoretical smoothness lower bound, GRASP explicitly models subset interactions through a quadratic geometric penalty. To achieve pretraining-scale efficiency without relying on hidden oracle tuning, we couple low-dimensional feature sketches with a strictly finite lower-confidence bound selection protocol. Extensive subset-retraining evaluations demonstrate that GRASP decisively outperforms existing scalable baselines. It more than doubles the task-level rank correlation for counterfactual subset fidelity while reducing upfront artifact construction costs by nearly an order of magnitude. Downstream diagnostics further show that this scoring mechanism transfers to language model curation and cross-domain vision selection, establishing a robust foundation for optimizing massive pretraining corpora.
Lucas D. Konrad, Nikolas Kuschnigstat.ML cs.LG econ.EM stat.CO
Identifying most influential sets (MIS) - size-$k$ subsets whose removal maximally changes a target estimand - is typically infeasible because it requires searching over $\binom{n}{k}$ subsets. For estimands with linear-fractional leave-set-out effects, we show that MIS selection reduces to a one-parameter sequence of top-$k$ problems. Dinkelbach's method yields an algorithm with $\mathcal{O}(n)$ cost per iteration and finite termination. For fixed residualized inputs, the algorithm returns a globally optimal set for the univariate ratio objective, including the oracle-residualized partial linear model. With estimated nuisance functions, uniform denominator and generated-score stability imply approximation to the first-order oracle orthogonal-score objective; exact set recovery follows under a separation condition. Simulations and applications show that the method recovers exact MIS that were previously computationally inaccessible.