Determinantal Sampling Designs

• Determinantal sampling designs are a class of methods that use Hermitian kernels to encode inclusion probabilities via determinant formulas, ensuring precise control over sample selection.
• They exhibit negative dependence, systematically reducing joint inclusions to promote diversity and yield exact variance estimates for linear estimators.
• These designs support scalable implementations through spectral decomposition and efficient selection algorithms, making them ideal for multi-stage and geometry-aware sampling.

Searching arXiv for recent and foundational papers on determinantal sampling designs. {"query":"determinantal sampling designs arXiv (Loonis et al., 2015, Loons, 26 Aug 2025, Belhadji et al., 2023, Tran et al., 7 Jul 2026)", "max_results": 10} Searching arXiv for foundational and recent determinantal sampling design papers by title and topic. {"query":"title:\"Determinantal Sampling Designs\" arXiv", "max_results": 5} Determinantal sampling designs are sampling designs on a finite population $U$ whose law is generated by a Hermitian kernel $K$ through determinant formulas for inclusion events. In the formulation developed by Loonis and Mary, a determinantal sampling design (DSD) combines an explicit probabilistic law, exact first- and second-order inclusion probabilities, and a simple selection algorithm, thereby importing recent point-process structure into sampling theory (Loonis et al., 2015). Subsequent work has emphasized a related feature that is especially important in complex survey constructions: DSDs provide known and parametrisable inclusion probabilities at any order, which makes them suitable for optimization problems in indirect and multi-stage sampling (Loons, 26 Aug 2025).

1. Formal definition and probabilistic structure

Let $U$ be a population of size $N$, and let $K$ be a Hermitian $N\times N$ matrix indexed by $U$, with eigenvalues in $[0,1]$; equivalently, $K$ is a contracting Hermitian kernel. A sampling design $\mathcal P$ on $K$0 is determinantal if there exists such a kernel $K$1 for which, for every $K$2,

$K$3

where $K$4 is the submatrix indexed by $K$5, and $K$6 is the random sample. The notation $K$7 is standard in this framework (Loonis et al., 2015).

This definition places all inclusion events under a single algebraic object. The determinant of a principal minor governs the probability of simultaneous inclusion, so the full law of the design is encoded in $K$8. In this sense, DSDs are finite-population analogs of determinantal point processes, but their role in sampling theory is more specific: they furnish directly usable inclusion probabilities for estimators, variance formulas, and design optimization.

A central structural property is negative dependence. For disjoint $K$9, DSDs satisfy

$U$0

and the class is described as satisfying the strong Rayleigh property and negative association (Loonis et al., 2015). In sampling terms, this means that joint inclusion is systematically damped relative to independence, which is precisely the mechanism through which DSDs promote spread and reduce redundancy.

2. Inclusion probabilities, covariance, and sample size

The first- and second-order inclusion probabilities have closed forms: $U$1 and, for $U$2,

$U$3

Hence the off-diagonal covariance is

$U$4

while $U$5 (Loonis et al., 2015).

Quantity	Expression
First-order inclusion	$U$6
Second-order inclusion	$U$7
Off-diagonal covariance	$U$8
Inclusion event of any order	$U$9

These formulas immediately show that DSDs are not merely “diverse” in an informal sense; their pairwise dependence is exactly quantified by squared moduli of kernel entries. The negative sign in $N$0 is the finite-population expression of repulsion.

The sample-size law is equally explicit. If $N$1 are the eigenvalues of $N$2, then

$N$3

When $N$4 is a projection matrix, all eigenvalues are $N$5 or $N$6, so the design is fixed size. When $N$7 is diagonal, the design reduces to Poisson sampling (Loonis et al., 2015). A common misconception is therefore that determinantal designs are intrinsically fixed size; the theory is broader, and fixed-size designs correspond to the projection-kernel subcase.

Another misconception is that simple random sampling is generically determinantal. The finite-population theory does not support that claim: simple random sampling of size $N$8 is not generally determinantal, although for many $N$9 an equiangular tight frame yields a DSD matching simple random sampling first- and second-order inclusion probabilities (Loonis et al., 2015).

3. Construction principles and exact selection algorithms

One of the main constructive results is existence with prescribed first-order inclusion probabilities. Given $K$0 with $K$1 and $K$2, there exists a projection matrix $K$3 with diagonal $K$4, hence a fixed-size DSD of size $K$5; the existence argument uses the Schur-Horn theorem. The same work gives explicit formulas for a real symmetric projection matrix $K$6 with prescribed diagonal, constructed recursively using plane rotations. Equal-probability fixed-size designs are also constructed through equiangular tight frames and via a Toeplitz kernel built from roots of unity (Loonis et al., 2015).

For exact selection, the general design decomposes spectrally. If

$K$7

one first samples independent Bernoulli variables with parameters $K$8, then forms the projection kernel on the span of the selected eigenvectors, and finally samples from that fixed-size projection design (Loonis et al., 2015). This is the standard exact determinantal mechanism in the finite setting.

The computational bottleneck is the eigendecomposition and the orthogonalization stage. In the discrete DPP algorithmics literature, the standard three-phase procedure consists of eigendecomposition, Bernoulli eigenvector sampling with probabilities $K$9, and a Gram-Schmidt-type orthogonalisation procedure. The naive implementation of the third phase has average cost $N\times N$0, whereas an optimized algorithm reduces this to $N\times N$1, with a dual low-rank variant for memory-constrained settings (Tremblay et al., 2018). A different exact route replaces eigendecomposition by Cholesky decompositions and a two-step Bernoulli-process plus thinning strategy; it remains $N\times N$2 in complexity, but the constant is lower than for spectral methods and it can be competitive or faster in some regimes (Launay et al., 2018).

These algorithmic developments belong to the broader determinantal sampling literature rather than to the original survey-sampling formulation alone, but they are directly relevant whenever DSDs are implemented at large scale.

4. Linear estimation, variance identities, and optimal design

The principal statistical advantage of DSDs is that the variance of linear estimators can be written exactly from first- and second-order inclusion probabilities. For weights $N\times N$3, variable $N\times N$4, and $N\times N$5, the mean squared error is

$N\times N$6

Specializing to DSDs gives

$N\times N$7

For the Horvitz-Thompson estimator, with $N\times N$8,

$N\times N$9

where $U$0 denotes the entrywise product (Loonis et al., 2015).

The same framework yields finite-sample and asymptotic theorems. Mean-square convergence of the estimator of the mean is ensured, for example, by

$U$1

The theory also provides a central limit theorem under explicit sufficient conditions, and Bernstein-type deviation bounds; for fixed-size DSDs, one such inequality is

$U$2

with $U$3 (Loonis et al., 2015).

The optimization problem is to minimize a variance criterion over the projected spectrahedron

$U$4

For small $U$5, semidefinite programming is proposed; for larger $U$6, practical strategies include sorting the population according to auxiliary variables, building a projection with the prescribed diagonal, and applying greedy plane-rotation improvements (Loonis et al., 2015). The extreme small-sample case is especially transparent: when $U$7, the unique minimizer of the variance is a DSD with a rank $U$8 kernel.

A notable exact characterization concerns perfect estimation. Zero mean squared error occurs when the kernel $U$9 and the diagonal matrix $[0,1]$0 commute, which requires the DSD to be stratified with fixed allocations in each stratum (Loonis et al., 2015). This identifies an algebraic condition for a classical design-theoretic objective.

5. Constraints, indirect sampling, and high-order coordination

A major recent extension places DSDs inside two-stage indirect sampling. In that setting, the central advantage is again explicit control of inclusion probabilities: determinantal designs have known and parametrisable inclusion probabilities at any order, which makes it possible to implement the Generalized Weight Share Method with closed-form expressions for the optimal weight matrix and, under hypotheses $[0,1]$1 and $[0,1]$2, a formula for the optimal inclusion probabilities used in the second stage (Loons, 26 Aug 2025).

The unbiasedness constraint for the weight matrix $[0,1]$3 is

$[0,1]$4

and the variance of the associated estimator is expressed as a quadratic form in the stacked weight vector. The optimal solution is obtained through a Moore-Penrose pseudoinverse formula, while the second-stage optimal inclusion probabilities are

$[0,1]$5

When the intermediate and second-stage designs are determinantal, closed-form target first-order and joint inclusion probabilities are available, which in turn enable an alternative application of the Horvitz-Thompson estimator for totals in the target population (Loons, 26 Aug 2025).

Constraint handling in the algorithmic DPP literature shows both the power and the limits of the paradigm. For partition constraints, there is an exact polynomial-time algorithm when the number of partitions is constant, based on multivariate characteristic polynomials; the distribution enforces $[0,1]$6 while retaining determinant-based diversity (Kathuria et al., 2016). By contrast, for general matroid constraints, exact DPP sampling is subject to a complexity-theoretic barrier: for transversal matroids, the problem is $[0,1]$7-hard (Kathuria et al., 2016). A plausible implication is that the determinantal formalism scales naturally to structured constraints only when those constraints preserve enough algebraic tractability.

6. Scalable implementations and adjacent determinantal design paradigms

The broader determinantal sampling literature has turned DSD-style repulsion into a practical tool for large datasets, clustering, geometry-aware sampling, and function reconstruction. In determinantal consensus clustering, DPP sampling is used to select diverse centroids; because full eigendecomposition of the Gram matrix is prohibitive for large $[0,1]$8, two scalable alternatives were introduced: a nearest-neighbor Gaussian process sparsification that replaces the dense kernel by a sparse approximation with a similar eigenspectrum, and random small submatrix sampling with cost $[0,1]$9 instead of $K$0. On simulated Gaussian mixtures and on UCI smartphone activity, MNIST, and Fashion-MNIST, these methods outperform or match PAM and $K$1-means in ARI and show lower variability (Vicente et al., 2021).

Determinantal sampling has also been extended beyond finite Euclidean setups. Spectral DPPs built from Laplacian-type operators on compact Riemannian manifolds and weighted networks attain

$K$2

mean-squared-error rates that adapt automatically to intrinsic dimension, using kernels formed from eigenspaces of Laplace–Beltrami operators, graph Laplacians, or more general Markov diffusions (Tran et al., 7 Jul 2026). This suggests that the design principle embodied by DSDs—repulsion calibrated by a kernel—extends naturally to geometry-aware sampling outside classical finite-population survey settings.

In RKHS reconstruction, determinantal sampling based on the kernel eigenstructure yields mean-square $K$3 guarantees, fast convergence rates, and the instance optimality property for a smaller number of function evaluations than i.i.d. Christoffel sampling; projection DPPs and continuous volume sampling are the main constructions there (Belhadji et al., 2023). In scalable manifold learning, determinantal landmark selection on non-Euclidean spaces uses geodesic-distance kernels and an efficient approximation running in linear time, combined with local covariance matrices and Bhattacharyya-distance neighborhoods to improve sparse embeddings (Wachinger et al., 2015).

Taken together, these developments do not alter the core finite-population definition of a determinantal sampling design. They show, rather, that the same determinant-based mechanism supports a wide class of sampling constructions in which explicit inclusion structure, negative dependence, and kernel-driven geometry are the central organizing principles.