Fair Multi-View Determinantal Coresets via Adaptive NEPv

Richard Yi Da Xu

stat.MLAug 18, 2026

Fair Multi-View Determinantal Coresets via Adaptive NEPv

Authors: Richard Yi Da Xu

Organizations: Hong Kong Baptist University and TadReamk Limited

Abstract

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 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.

Figures & tables

Figure 1: One-sided diversity creates blind spots. (a) A text-only selector can keep diverse wording while repeating nearly identical logos. (b) A vision-only selector can keep diverse images while repeating the same language. (c) Each candidate may carry $M\geq 2$ aligned views that share one index; explicitly scoring the weakest view targets this blind spot.

Figure 2: Fairness prefers coverage in both views. Illustrative per-view log-determinants $d_{q}(S)$ for two size- $k$ subsets. (a) $S_{A}$ is strong in text but collapsed in vision, so an arithmetic average of per-view log-scores can hide the failure. (b) $S_{B}$ is preferred by $\min_{q}d_{q}(S)$ . FairNEPv optimizes the smooth soft-min $F_{\tau}$ in ( D-Fair ), which makes the same choice.

Figure 3: Quality-conflict fairness stress test ( $k=5$ , $\tau=0.15$ ). Both view kernels are diagonal here, so candidate coordinates carry no information and are not plotted. (a) Quality of each candidate in the two views: specialists are strong in one view and negligible in the other, balanced candidates are moderate in both. (b) Per-view normalized log-determinant $d_{q}(S)$ of each selector’s subset. The outlined bar is the weaker view, which the soft-min $F_{\tau}$ of ( D-Fair ) tracks, and both bars are outlined when the two views are tied. The single-view selectors collapse the other view. Both aggregate baselines are indifferent among the ten specialist candidates; the reported deterministic tie-breaking returns a one-sided averaged-kernel subset and a two-versus-three specialist split for the fixed-weight baseline.

Figure 4: The task: a registered mark and its description of the mark , filed by the applicant and accepted by the examining attorney, which must name every significant literal and design element so that a reader can identify the mark from the words alone. Here the description reads: “The mark consists of a cross design with an arc shape between the cross design. The words ‘MIDWEST INSTITUTE’ appear below the design. The words ‘FOR HEART’ appear below ‘MIDWEST INSTITUTE’ and the words ‘AT SILVER CROSS HOSPITAL’ appear below ‘FOR HEART’.”

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

	per view		$d_{\min}\leq F_{\tau}\leq d_{\min}\!+\!\tau\log M$
Method/subset	$d_{\mathrm{text}}$	$d_{\mathrm{vis}}$	$d_{\min}$	$F_{\tau}$	$d_{\min}\!+\!\tau\log M$
Exact enum.	$1.141$	$1.141$	$1.141$	$\mathbf{1.141}$	$1.245$
FairNEPv	$1.120$	$1.120$	$1.120$	$\mathbf{1.120}$	$1.224$
Text-only	$4.605$	$-2.303$	$-2.303$	$-2.199$	$\mathbf{-2.199}$
Vision-only	$-2.303$	$4.605$	$-2.303$	$-2.199$	$\mathbf{-2.199}$
Avg. kernel	$4.605$	$-2.303$	$-2.303$	$-2.199$	$\mathbf{-2.199}$

Appendix

Table 1: The fair score against both ends of ( 18 ), for the synthetic instance (

k=5

\tau=0.15

); an exact lower endpoint or an upper endpoint equal after displayed rounding is bold. The exact enumeration benchmark and FairNEPv have tied views; the latter is

0.021

below the former because the reported one-swap refinement is local. A tie drives

F_{\tau}

to the lower bound because the slack

\tau\log[\sum_{q}e^{-\Delta_{q}/\tau}]

is largest,

\tau\log M=0.104

, when

\Delta_{q}\equiv 0

. The remaining selector rows have

\Delta/\tau\geq 9.2

, so their slack is at most

1.5\times 10^{-5}

and they sit on the upper bound to the displayed precision.

Coreset Selection

Coreset selection chooses compact subsets that preserve a dataset’s learning, evaluation, or optimization value while reducing cost.

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Spectral DPPs via NEPv: A Scalable Continuous Relaxation of Determinantal MAP for Diversity-Aware Data Selection

Richard Yi Da Xu

Hong Kong Baptist University · TadReamk Limited

Determinantal Point Process Training Data Selection Matrix Optimization Spectral Methods

Open paper AskPDF arXiv

Selecting a small, diverse, high-quality subset from a massive pool of candidates is a recurring primitive in modern machine learning -- data curation and coreset selection for training and fine-tuning large models, active-learning batch acquisition, prompt and exemplar selection for in-context learning, retrieval diversification, and experimental design. Determinantal Point Processes (\DPP s) give a principled, well-calibrated notion of diversity for this task, but their MAP objective -- pick a size-

k

subset

S

maximizing

\logdet(L_S)

-- is NP-hard, and the standard greedy and sampling algorithms scale superlinearly in the ground-set size

n

. This cost is prohibitive precisely in the data-centric regime where diversity matters most, where

n

ranges over millions to billions of candidate examples, features, or embeddings. We recast \DPP-MAP as a continuous optimization problem over the Stiefel manifold, and show that its first-order optimality conditions form a Nonlinear Eigenvalue Problem with eigenvector dependency (\NEPv) of a previously unstudied form. This \NEPv\ admits a self-consistent field (\SCF) iteration with a spectral-gap-based local contraction guarantee, giving a principled iterative solver where the diversity objective drives an eigenvector-dependent operator. The resulting algorithm, \OurMethod, requires only matrix-vector products with the kernel and runs in time

O\!\big((ndk+nk^2)\,t\big)

for a small number of iterations

t

, scaling near-linearly in

n

and integrating directly with low-rank and feature-map kernels common in ML. This paper focuses on the relaxation, solver, and scaling analysis; full real-data benchmarking is left to a planned empirical study.

Abstract

Coreset Selection

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