Determinantal Point Processes (DPPs)

• Determinantal Point Processes (DPPs) are probabilistic models that use determinants of kernel submatrices to ensure diversity and negative correlation among selected subsets.
• They are widely applied in machine learning tasks such as extractive summarization, recommendation systems, and randomized numerical linear algebra due to their efficient sampling and inference algorithms.
• Recent developments extend DPPs to fixed-size, continuous, structured, and nonsymmetric settings, enhancing their scalability and applicability in complex real-world scenarios.

A determinantal point process (DPP) is a probabilistic model over subsets of a base set, with probabilities governed by determinants of submatrices of a kernel matrix. Originally arising in random matrix theory and quantum mechanics as models of repulsion (e.g., eigenvalue distributions of Hermitian ensembles or noncolliding fermion systems), DPPs have become foundational in diverse—often combinatorial—machine learning and statistical contexts that require subset selection with explicit diversity or negative correlation. The defining property of a DPP is that for any finite ground set, the probability of observing a particular configuration (subset) is proportional to the determinant of the kernel submatrix restricted to that subset, favoring selection of sets whose elements are dissimilar according to the geometry induced by the kernel.

1. Mathematical Foundations and Core Definitions

For a finite ground set $\mathcal{Y} = \{1,\dots,N\}$, a DPP defines a random subset $Y \subseteq \mathcal{Y}$ via a kernel matrix $K\in \mathbb{R}^{N\times N}$ (or $L$ in the $L$-ensemble representation). The DPP is fully specified by requiring for any $S \subseteq \mathcal{Y}$: $\mathbb{P}(S \subseteq Y) = \det(K_S)$ where $K_S$ denotes the principal submatrix indexed by $S$.

The most common forms are:

• Marginal (correlation) kernel representation: $K$ is symmetric, $0 \preceq K \preceq I$.
• $L$-ensemble representation: For $L\succ 0$, the probability mass function for $Y$ is

$\mathbb{P}(Y) = \frac{\det(L_Y)}{\det(I + L)}$

where $L_Y$ is the submatrix of $L$ indexed by $Y$.

Fixed-size DPPs (k-DPPs) condition the process on $|Y|=k$, leading to: $$\mathbb{P}_k(Y) \propto \det(L_Y) \quad \text{if } |Y| = k$$ Normalization involves elementary symmetric polynomials of the eigenvalues.

Extended L-ensembles form a unifying generalization capturing all DPPs (including projection and partial-projection cases), where the likelihood is: $$\mathbb{P}(X) \propto (-1)^p \det\begin{pmatrix} L_X & V_X\ V_X^\top & 0 \end{pmatrix}$$ for a pair $(L, V)$, with $V$ spanning the deterministic “projection” part.

DPPs extend naturally to continuous domains and to structured settings via operator or dual representations.

2. Diversity, Negative Correlation, and Submodularity

The determinant in the DPP probabilities enforces diversity through its geometric interpretation: for feature vectors $v_i$ assigned to each item, $\det([v_i^\top v_j]_{i,j\in Y})$ measures the squared volume of the parallelepiped spanned by $\{v_i\}_{i\in Y}$. As such, the DPP discourages redundant (similar) elements, favoring spread-out, “repulsive” selections. Negative correlations between items are explicit: the marginal probability that two items co-occur is less than or equal to the product of their individual marginals.

The log-determinant is known to be submodular but non-monotone, granting DPP-based objectives strong theoretical guarantees (especially for greedy maximization) and connections to combinatorial optimization.

ML practitioners exploit the decomposition

$L_{ij} = q_i q_j S_{ij}$

to separate item quality ($q_i$) from pairwise similarity/dissimilarity ($S_{ij}$), enabling explicit control and learning of the relevance-diversity tradeoff.

3. Inference, Learning, and Scalability

Inference

• Marginals: $\mathbb{P}(i\in Y) = K_{ii}$, computable via $K = L(I+L)^{-1}$.
• Exact sampling: Standard algorithms use eigendecomposition and a two-phase procedure: select a random subset of eigenvectors, then sample items greedily via orthogonalization (Gautier et al., 2018).
• MAP inference: Selecting the most probable subset (often under budget constraints) is NP-hard, but the submodular structure allows efficient greedy approximations with provable guarantees (Kulesza et al., 2012).

Learning

• Maximum likelihood: For $L$-ensembles with parametric forms (e.g., $q_i = \exp(\theta^\top f_i)$), the log-likelihood

$$\ell(\theta) = \sum_{i\in Y} \theta^\top f_i + \log \det(S_Y) - \log \det(I+L)$$

is concave in $\theta$, enabling convex optimization (Kulesza et al., 2012).

• Efficient gradients: Owing to DPP tractability, gradients of the log-likelihood involve efficiently computable item marginals and expectations over subsets.
• Kernel parameterization
  • Log-linear quality models (Kulesza et al., 2012)
  • Multiple kernel learning for flexible similarity structure (Gong et al., 2014)
  • Deep neural architectures capturing nonlinear interactions (Gartrell et al., 2018)
• Variational and MCMC inference: Nonspectral methods (Bardenet et al., 2015) and low-rank factorizations (Dupuy et al., 2016, Gartrell et al., 2016) provide scalable learning for large item sets and continuous ground spaces.
• Structured/Combinatorial DPPs: Factorized and dual approaches extend DPPs to exponentially large or structured ground sets (SDPPs), with inference via second-order message passing and dual eigenspaces (Kulesza et al., 2012).

Scalability

Modern DPPs support large-scale applications via: | Methodology | Complexity | Scalability Context | |------------------------------------|------------------------|----------------------------------------------| | Low-rank/mixture models | $O(MK^2)$ | Large catalogs, recommendation, basket data | | Kronecker structure (KronDPPs) | $O(N^{3/2})$ | Massive ground sets ($N\gg10^3$) | | Sublinear learning (Dupuy et al., 2016) | $O(V^3)$ ($V\ll N$) | Continuous/exponential-size item spaces | | Nonspectral variational bounds | $O(m^3)$ ($m\ll n$) | Variational/MCMC, finite/continuous |

4. Extensions: Fixed Size, Continuous DPPs, Structured and Nonsymmetric Kernels

• k-DPPs: Sampling with fixed cardinality is common in applications (e.g., document or basket summarization of fixed length). While standard DPPs (Bernoulli eigenvalue selection) have variable size, the k-DPP conditions on $|Y|=k$ (Kulesza et al., 2012, Barthelmé et al., 2018). Inclusion probabilities in k-DPPs are more complex, involving ratios of elementary symmetric polynomials; asymptotic and saddlepoint approximations make k-DPPs tractable and demonstrate the convergence of k-DPPs to standard DPPs in large-scale limits.
• Continuous (and Large/Exponential) Domains: DPPs defined by kernel operators are leveraged for spatial statistics and continuous diversity. Low-rank approximations (Nyström, RFF), dual representations, and Gibbs samplers address the computational intractability of integral operator-based kernels (Affandi et al., 2013, Dupuy et al., 2016).
• Structured DPPs (SDPPs): Factorized models, dual representations, and second-order semiring message-passing extend DPPs to ground sets of combinatorial or structured objects (e.g., paths, thread sequences, forested graphs) (Kulesza et al., 2012).
• Nonsymmetric and Block DPPs: Recent work relaxes symmetry, yielding kernels $K$ not necessarily $K=K^\top$, encompassing both repulsive and attractive (“positive” and “negative” cross-correlation) interactions. The necessary and sufficient condition for a DPP kernel (possibly nonsymmetric) is that for all $p\in\{0,1\}^n$,

$\det(D(p)(I-K) + D(1_n - p)K) \geq 0$

and for $p\in(0,1)^n$, strict positivity is required (Arnaud, 5 Jun 2024, Gartrell et al., 2019). This yields new families of attractive couplings for modeling marked spatial processes, where marginal repulsion within types coexists with global attraction across types.

5. Applications in Machine Learning, Statistics, and Beyond

DPPs underpin principled subset selection in settings where diversity and coverage are crucial:

• Extractive summarization: DPPs select subsets of sentences maximizing relevance and minimizing redundancy (Kulesza et al., 2012, Kulesza et al., 2012).
• Image and video summarization: Frame or patch selection combines coverage and diversity, benefiting from flexible kernel parameterization and large-margin objectives (Gong et al., 2014).
• Recommendation systems: DPPs serve as priors for shopping basket composition, leveraging low-rank, mixture, or deep kernel parameterizations for scalable inference and superior predictive performance (Gartrell et al., 2016, Gartrell et al., 2018).
• Structured prediction: SDPPs provide tractable diversity-aware predictions for structured tasks such as pose estimation, timeline construction, and graph threading (Kulesza et al., 2012).
• Randomized numerical linear algebra: DPP sampling yields optimally diverse matrix row/column selection—critical for least squares regression, Nyström approximations, and low-rank modeling—with strong guarantees on estimator bias and variance, and connections to leverage score sampling (Dereziński et al., 2020).
• Object detection: DPP-based filtering offers a principled, parameter-free alternative to non-maximum suppression (NMS), enhancing detection precision in crowded scenes (Some et al., 2020).
• Statistical modeling of spatial and marked point patterns: Nonsymmetric and coupled DPP frameworks model complex repulsion/attraction phenomena beyond the reach of traditional processes (Arnaud, 5 Jun 2024).

Software toolkits such as DPPy encapsulate exact and approximate DPP algorithms for both finite and continuous spaces (Gautier et al., 2018).

6. Current Directions, Open Problems, and Theoretical Advances

Recent work expands the theoretical and algorithmic landscape of DPPs:

• Flat Limit and Universality: For kernel functions tending to a constant (the “flat limit”), limiting DPPs are characterized by the smoothness of the kernel and the determinant structure reduces to Vandermonde determinants or partial-projection processes, yielding parameter-free “default” DPPs (Barthelmé et al., 2020, Barthelmé et al., 2021).
• Extended L-ensembles: Provide a unified formalism subsuming classical, fixed-size, projection, and partial-projection DPPs, naturally supporting kernels from a broader class (conditionally positive-definite functions), and enabling new sampling and normalization techniques based on saddle-point matrices and the extended Cauchy–Binet theorem (Tremblay et al., 2021, Barthelmé et al., 2020).
• Algorithmic and Inferential Challenges: Open questions persist regarding efficient computation of higher-order sums (e.g., $\sum_Y [\det(L_Y)]^p$ for $p > 1$), entropy functionals, scalability to extreme ground set sizes, and interpretability of learned kernels in the context of neural parameterizations and mixture models (Kulesza et al., 2012, Gartrell et al., 2018, Gartrell et al., 2016).
• Extensions to Markov and Temporal Models: Markov DPPs extend negative correlation and diversity constraints to sequential decision processes, yielding diversity both within and across time slices in streaming or interactive selection (Affandi et al., 2012).

7. Relation to Alternative Models and Broader Significance

DPPs offer computational and statistical advantages over traditional models for negative correlation (e.g., repulsive Markov random fields), particularly due to the tractable algebraic structure (sampling, marginals, normalization) enabled by determinants and linear algebra. DPP-induced negative correlations are “transitive” and governed globally by the positive semidefinite constraint, contrasting with the local specification in MRFs.

By providing a mathematically rigorous and computationally efficient approach to modeling subset selection with explicit diversity and coverage, DPPs serve as a bridge between algebraic, probabilistic, and geometric perspectives. They have become a foundational component in machine learning, spatial statistics, computational physics, and randomized algorithms, with active research extending their scope to structured, continuous, temporally dependent, and nonsymmetric domains.