Personalized Determinantal Point Processes

• Personalized DPPs are user-conditioned probabilistic models that integrate item quality and global similarity to balance relevance and diversity in recommendation sets.
• They personalize recommendations by using user-specific quality scores—often derived from cosine similarities in two-tower models—and employ k-DPPs to sample fixed-length subsets.
• Various learning paradigms and privacy-preserving measures illustrate trade-offs in scalability, expressiveness, and online performance for production recommender systems.

Searching arXiv for recent and foundational papers on personalized DPPs and related recommendation formulations. Personalized Determinantal Point Processes (DPPs) are user-conditioned probabilistic models over subsets of items in which a recommendation set is assigned probability through the determinant of a user-specific kernel, thereby coupling relevance and diversity at the set level. In recommender-system practice, they are used either as explicit personalized subset models, as k-DPP ranking objectives over user-specific candidate sets, or as post-retrieval diversity filters layered on top of a base recommender. The literature spans conditional feature-based DPP learning (Kulesza et al., 2012), low-rank and mixture formulations for heterogeneous basket data (Gartrell et al., 2016), nonsymmetric extensions with both attraction and repulsion (Gartrell et al., 2019), private and privacy-aware analyses (Fitzsimons et al., 2024), personalized k-DPP ranking objectives (Liu et al., 2024), and production-scale deployment of personalized DPP sampling for cultural recommendations (Ibrahim et al., 12 Sep 2025).

1. Core mathematical structure

A discrete DPP is defined on a finite ground set $\mathcal{Y} = \{1,\dots,N\}$ or $Y=\{1,\dots,M\}$ by an $L$-ensemble kernel $L$, with subset probability

$P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$

In the standard formulation, $L$ is positive semidefinite; for a subset $Y$, the determinant $\det(L_Y)$ is large when the selected items are individually strong and mutually non-redundant. In the Gram decomposition used by Kulesza and Taskar, the kernel is written as

$L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$

with $q_i$ a quality term and $Y=\{1,\dots,M\}$0 a unit-norm feature vector, so that $Y=\{1,\dots,M\}$1 with $Y=\{1,\dots,M\}$2. The induced set probability factors into item quality and a determinant over similarities, giving the standard relevance-diversity interpretation (Kulesza et al., 2012).

The determinant has a geometric interpretation as volume. In the pass Culture deployment, the same idea is written as

$Y=\{1,\dots,M\}$3

where $Y=\{1,\dots,M\}$4 is a user-specific quality score and $Y=\{1,\dots,M\}$5 is the volume spanned by item embeddings. This decomposition makes explicit that DPPs reward sets that are simultaneously high-quality and spread in semantic space (Ibrahim et al., 12 Sep 2025).

For fixed-length recommendation lists, the relevant object is often the k-DPP, which conditions the DPP on $Y=\{1,\dots,M\}$6: $Y=\{1,\dots,M\}$7 This conditioning is central when recommendation is evaluated as Top-$Y=\{1,\dots,M\}$8 ranking, because the probability becomes a ranking over subsets of equal size rather than over all cardinalities (Liu et al., 2024).

A common misconception is that DPPs are merely “diversity boosters.” In fact, the kernel simultaneously encodes item prominence and item similarity. Another important limitation is structural: symmetric DPPs encode negative correlations only. In marginal-kernel form, $Y=\{1,\dots,M\}$9, so large off-diagonal terms suppress co-occurrence rather than encourage it (Kulesza et al., 2012).

2. Mechanisms of personalization

The dominant personalized construction keeps diversity global and makes quality user-dependent. In its simplest form,

$L$0

or equivalently

$L$1

where $L$2 reflects the relevance of item $L$3 for user $L$4, while $L$5 is a user-independent item-item similarity term. This is the formulation used in the pass Culture system: personalization enters entirely through $L$6, while item-item similarity is computed from semantic embeddings of item title and description (Ibrahim et al., 12 Sep 2025).

In that deployment, the personalized relevance signal comes from a two-tower retrieval model. The user tower produces $L$7, the item tower produces $L$8, and the quality term is

$L$9

normalized to $L$0. Semantic diversity is represented by all-MiniLM-L6-v2 sentence-transformer embeddings of dimension $L$1, reduced to $L$2 dimensions for latency. For each request, the system retrieves $L$3 candidates and samples $L$4 items with a k-DPP using

$L$5

This is a minimal personalized DPP: user-specific quality, global similarity, stochastic subset selection (Ibrahim et al., 12 Sep 2025).

A closely related formulation appears in personalized ranking with k-DPPs, where the user-specific kernel is written as

$L$6

Here $L$7 is any differentiable vector of user-specific item scores—produced by MF, GCN, NeuMF, or another base recommender—while $L$8 is a user-independent diversity kernel. This preserves personalization without learning a full $L$9 kernel per user (Liu et al., 2024).

The privacy analysis literature supplies another route to personalization. In the interaction-matrix construction, one starts from a binary user-item matrix $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$0 and defines $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$1, conceptually yielding item popularity on the diagonal and pairwise co-consumption off-diagonal. A personalized extension can then restrict the candidate ground set for user $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$2 or modulate item quality for that user while keeping the shared similarity structure fixed. This suggests that many collaborative-filtering signals can be embedded into personalized DPP kernels so long as the mapping from data to $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$3 remains controlled (Fitzsimons et al., 2024).

3. Learning paradigms

The earliest direct route to personalized DPPs is the conditional DPP. In the feature-based parameterization of Kulesza and Taskar,

$P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$4

with

$P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$5

Because the log-likelihood is concave in $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$6, learning reduces to convex optimization; the gradient depends on DPP marginals $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$7, and the paper uses L-BFGS with Gaussian-prior regularization. This conditional construction does not explicitly model user identity, but it suggests the personalized extension $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$8 obtained by placing user or user-item interaction features inside $P_L(Y) = \frac{\det(L_Y)}{\det(L+I)}.$9, with the same spectral inference machinery (Kulesza et al., 2012).

A second line of work addresses latent heterogeneity rather than explicit user conditioning. The Bayesian low-rank DPP mixture model represents the kernel as $L$0 and then replaces a single low-rank kernel with a mixture

$L$1

Each basket has a latent component assignment $L$2, mixture weights have a symmetric-Dirichlet prior, item-trait matrices are given Gaussian priors with Gamma precisions, and inference uses Gibbs sampling together with SGHMC. The model is not explicitly personalized by user ID, but it can be interpreted as a latent mixture of behavior types or shopping missions. The paper itself proposes extensions such as user-specific mixture weights $L$3 or user-conditioned combinations of shared component kernels, which is a direct path to shared-component personalized DPPs (Gartrell et al., 2016).

A third learning strategy is to optimize recommendation models with a k-DPP objective directly. The L$L$4P family forms a user-specific ground set of size $L$5, containing $L$6 positives and $L$7 sampled negatives, and maximizes the k-DPP probability of the positive $L$8-subset: $L$9 The positive-only objective is

$Y$0

and the positive-plus-negative variant adds $Y$1. Because the loss is differentiable in the parameters underlying $Y$2, it can be optimized by Adam and applied to MF, GCN, GCMC, or NeuMF. This makes the DPP not a post-processing layer but the primary set-level ranking criterion (Liu et al., 2024).

4. Sampling, inference, and deployment patterns

In exact sampling from an $Y$3-ensemble DPP, the kernel is eigendecomposed as $Y$4; eigenvectors are selected independently with probability $Y$5, and a sequential subset-construction step then samples items using squared projection norms. This spectral decomposition underlies both sampling and computation of marginals (Kulesza et al., 2012). In the pass Culture implementation, exact k-DPP sampling is performed with DPPy’s FiniteDPP.sample_exact_k_dpp, using a size-$Y$6 k-DPP on a candidate pool of $Y$7 items (Ibrahim et al., 12 Sep 2025).

A recurring systems pattern is to use DPPs as a post-retrieval diversity filter rather than as a full-catalog recommender. The pass Culture pipeline has three stages: Two-Tower Retrieval, DPP Diversity Filter, and Compliance filtering & popularity ranking. Semantic embeddings are precomputed offline; online processing fetches or computes the user embedding, retrieves $Y$8 candidates via vector search, constructs the personalized kernel over that candidate set, samples $Y$9 items, applies business rules, and ranks the remaining items by popularity (Ibrahim et al., 12 Sep 2025).

This architecture is motivated by complexity. Naive DPP sampling on a $\det(L_Y)$0 kernel would require $\det(L_Y)$1, but with a low-rank linear kernel $\det(L_Y)$2 and embedding dimension $\det(L_Y)$3, the pass Culture system uses $\det(L_Y)$4. For $\det(L_Y)$5 and $\det(L_Y)$6, the paper reports approximately $\det(L_Y)$7 operations per user, which it considers feasible in real time (Ibrahim et al., 12 Sep 2025).

The choice between stochastic and deterministic inference matters operationally. Greedy max-determinant selection yields a single highest-determinant set but always the same list for a fixed user; stochastic DPP sampling yields different sets on repeated calls, enabling exploration. This is particularly relevant in recommender systems where repeated refreshes and long-term preference elicitation are part of the product behavior (Ibrahim et al., 12 Sep 2025).

A plausible implication is that personalized DPPs are best viewed as set-aware rerankers or samplers attached to a strong candidate generator. That reading is explicit in the cultural-recommendation deployment, in the k-DPP ranking objective, and in the conditional-DPP formulation, even though the exact deployment strategy differs across papers (Ibrahim et al., 12 Sep 2025).

5. Differential privacy and naturally private personalized DPPs

The privacy literature treats DPP sampling as structurally close to the exponential mechanism. In the eigenvalue-sampling phase, the DPP selects eigenvector $\det(L_Y)$8 with probability

$\det(L_Y)$9

By choosing the score $L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$0, this has the same functional form as an exponential mechanism with binary response. The central claim is therefore that a standard DPP sampler is already performing a randomized, score-based selection whose privacy parameter can be tied to the sensitivity of $L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$1 under neighboring databases (Fitzsimons et al., 2024).

For privacy, strict positive definiteness is necessary: if any eigenvalue approaches zero, $L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$2 diverges to $L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$3 and sensitivity becomes unbounded. The remedy is jitter,

$L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$4

which guarantees $L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$5. Under an operator-norm kernel sensitivity bound

$L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$6

the paper derives the eigenvalue-score sensitivity

$L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$7

and hence the implicit privacy expenditure for the eigenvalue-sampling step,

$L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$8

The analysis is for pure $L_{ij} = q_i\,\phi_i^\top \phi_j\,q_j,$9-DP and is expressed at the sampler level rather than via explicit noise injection (Fitzsimons et al., 2024).

For personalized recommenders, the paper suggests controlling privacy through the personalized kernel construction itself. If user $q_i$0 receives a kernel

$q_i$1

then privacy hinges on bounding

$q_i$2

when one subject’s data changes. The recommended strategy is to keep $q_i$3 dependent only on public or DP-protected data, learn $q_i$4 with a DP procedure such as DP-SGD or objective perturbation, add jitter, and then apply the same spectral sensitivity analysis to each user’s DPP (Fitzsimons et al., 2024).

The full privacy guarantee remains incomplete. The paper gives only a partial treatment of the SubsetConstruction phase, with sensitivity depending on eigenvector perturbation $q_i$5, jitter $q_i$6, and eigengaps via Davis–Kahan, but no closed-form relation from $q_i$7 to $q_i$8. It explicitly suggests that DPPs may need to be combined with standard DP mechanisms such as the Sparse Vector Technique rather than used as standalone private recommenders (Fitzsimons et al., 2024).

6. Empirical behavior, expressive extensions, and open questions

The strongest direct evidence on personalized DPPs in production comes from cultural recommendation. On pass Culture, three systems were compared: Model A with no DPP, Model B with personalized $q_i$9, and Model C with $Y=\{1,\dots,M\}$00. Offline, for recommended set size $Y=\{1,\dots,M\}$01, relevance measured by mean cosine similarity was $Y=\{1,\dots,M\}$02 for A, $Y=\{1,\dots,M\}$03 for B $Y=\{1,\dots,M\}$04, and $Y=\{1,\dots,M\}$05 for C $Y=\{1,\dots,M\}$06; the volume ratio relative to A was $Y=\{1,\dots,M\}$07, $Y=\{1,\dots,M\}$08, and $Y=\{1,\dots,M\}$09; the business diversity metric was $Y=\{1,\dots,M\}$10, $Y=\{1,\dots,M\}$11 $Y=\{1,\dots,M\}$12, and $Y=\{1,\dots,M\}$13 $Y=\{1,\dots,M\}$14. Online, CTR was $Y=\{1,\dots,M\}$15 for A, $Y=\{1,\dots,M\}$16 for B $Y=\{1,\dots,M\}$17, and $Y=\{1,\dots,M\}$18 for C $Y=\{1,\dots,M\}$19; the volume ratio across distinct items recommended was $Y=\{1,\dots,M\}$20, $Y=\{1,\dots,M\}$21, and $Y=\{1,\dots,M\}$22; the business diversity metric was $Y=\{1,\dots,M\}$23, $Y=\{1,\dots,M\}$24 $Y=\{1,\dots,M\}$25, and $Y=\{1,\dots,M\}$26 $Y=\{1,\dots,M\}$27. The central empirical conclusion is that personalization in the quality term retains more clicks than pure-diversity DPPs, but online relevance loss is still substantial (Ibrahim et al., 12 Sep 2025).

Set-level k-DPP optimization also improves standard recommender models. On Amazon Beauty, MovieLens-1M, and Anime, L$Y=\{1,\dots,M\}$28P variants improve relevance metrics, category coverage, and the harmonic-mean F-score relative to BCE, BPR, SetRank, and Set2SetRank. The paper reports, for example, that in the GCN setting on Beauty, Nd@20 rises from about $Y=\{1,\dots,M\}$29 under BPR to about $Y=\{1,\dots,M\}$30 for the best L$Y=\{1,\dots,M\}$31P variant, approximately $Y=\{1,\dots,M\}$32. The gains are largest on the sparsest dataset, which suggests that explicit set modeling is especially useful when user feedback is limited (Liu et al., 2024).

Low-rank mixture models provide a different empirical lesson: heterogeneity matters. The Bayesian low-rank DPP mixture significantly outperforms a single low-rank DPP, a Bayesian low-rank non-mixture DPP, and a full-rank DPP on basket completion; on the MS Store dataset, precision@5 at $Y=\{1,\dots,M\}$33 improves by about $Y=\{1,\dots,M\}$34 relative to the best competing model. This indicates that a single global kernel may be too capacity-limited for populations with multiple co-occurrence regimes, even before explicit user personalization is introduced (Gartrell et al., 2016).

Symmetric kernels are not the only option. Nonsymmetric DPPs relax the symmetric PSD constraint to $Y=\{1,\dots,M\}$35-matrices and can represent both repulsion and attraction. With the decomposition

$Y=\{1,\dots,M\}$36

the symmetric part remains PSD while the skew-symmetric part allows $Y=\{1,\dots,M\}$37 and $Y=\{1,\dots,M\}$38 to have different signs, which yields positive covariance for some item pairs. On recommendation-style basket data, the nonsymmetric model outperforms the symmetric one: on Amazon Apparel, MPR rises from $Y=\{1,\dots,M\}$39 to $Y=\{1,\dots,M\}$40 and AUC from $Y=\{1,\dots,M\}$41 to $Y=\{1,\dots,M\}$42; on the Amazon 3-category dataset, MPR rises from $Y=\{1,\dots,M\}$43 to $Y=\{1,\dots,M\}$44 and AUC from $Y=\{1,\dots,M\}$45 to $Y=\{1,\dots,M\}$46. For personalized DPPs, this suggests that user-specific attraction as well as repulsion may be important whenever recommendation lists should contain complements as well as substitutes (Gartrell et al., 2019).

Several limitations recur across the literature. Current production personalized DPPs often personalize only the quality term and keep similarity global (Ibrahim et al., 12 Sep 2025). Symmetric DPPs cannot model positive correlations (Kulesza et al., 2012). Low-rank DPPs cannot generate subsets larger than the rank of $Y=\{1,\dots,M\}$47, so expressivity is tied to rank unless mixtures are used (Gartrell et al., 2016). Full privacy guarantees for the whole DPP sampler are incomplete (Fitzsimons et al., 2024). The trade-off between relevance and diversity remains highly domain-sensitive: in the cultural-recommendation setting, diversity gains were large, but CTR losses remained significant even after personalization (Ibrahim et al., 12 Sep 2025).

Future work in the literature points in several directions: explicit weighting of the quality term through an exponent $Y=\{1,\dots,M\}$48 in

$Y=\{1,\dots,M\}$49

user-specific similarity rather than user-independent similarity, more scalable DPP approximations for large candidate sets, longer-horizon evaluation beyond CTR, user-conditioned mixtures of shared component kernels, and privacy-aware training combined with standard DP mechanisms (Ibrahim et al., 12 Sep 2025). Taken together, these directions suggest that the central challenge is no longer whether DPPs can personalize recommendation, but which part of the kernel should be personalized, how aggressively it should be personalized, and how that personalization interacts with scalability, privacy, and online utility.