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Preference Shapes Relevance: Cross-component Hierarchical Semantic Alignment for Personalized Generative Retrieval

Generative Retrieval (GR) has emerged as a promising paradigm by mapping queries directly to Semantic IDs (SIDs) with powerful representation capabilities for candidate items. However, existing SIDs derived solely from item content create a semantic gap, failing to align dynamic query intents with static item representations. Furthermore, current generative paradigms rarely model user behavior sequences and are always bottlenecked by the high inference latency of beam-search autoregressive decoding. To address these challenges, we propose $\textbf{C}$ross-component $\textbf{H}$ierarchical semantic $\textbf{A}$lignment for $\textbf{P}$ersonalized generative retrieval ($\textbf{CHAP}$), a novel personalized GR framework from a hierarchical perspective. First, we design a Hierarchical Semantic Alignment module to align query's latent space with item's quantization path and synchronize multi-granular semantics. Second, we construct a personalized GR framework that models user behavior by synergizing discrete SIDs for structural guidance and continuous representations for fine-grained semantic refinement. Notably, we introduce a Residual Cascading Generation mechanism to restrict the costly multi-step Transformer Decoder to a single-pass inference, boosting inference throughput while mitigating information loss. Extensive experiments on three public datasets, one proprietary industrial dataset, and online A/B tests demonstrate CHAP's superiority, validating the effectiveness and practical value of our approach. The code is publicly available at https://github.com/zzzgm/CHAP.

cs.IR

Efficient GPU Retrieval for Semantic Search

Semantic Search on LinkedIn must retrieve relevant profiles from a corpus of hundreds of millions in response to natural-language queries such as "a fintech founder in Berlin who worked in payments." The deployed relevance policy is bottleneck-oriented: every active non-negotiable facet must be satisfied, and a pre-existing LLM Graded Relevance (GR) judge operationalizes this through a fixed min/median aggregation over facet grades. Cosine similarity instead averages evidence, letting a strong match on one facet mask failure on another, capping the recall of the first-stage (L0) retriever. We present a policy-aligned retrieval framework: embeddings are partitioned into eight category-supervised segments whose scores follow the same min/median rule at serving time; for multi-vector retrieval, this segment score is computed independently per tagged document slot and maximized across slots. A lightweight single-slot Stage-1 scorer generates high-recall candidates, while scale-invariant relative-norm gating keeps category activation consistent across training, evaluation, and serving. On 21K held-out queries, this representation improves offline relevance over a matched-capacity baseline, with gains broadly distributed across facet combinations. We serve this framework with a two-stage GPU architecture: an FP8 coarse ranker scores the full corpus, increasing per-shard capacity by 71% and Stage-1 matmul throughput by 36%, then an FP16 stage exactly re-ranks an oversampled candidate set, recovering 99.6-99.8% of full-FP16 recall at over 500 QPS per shard replica. In a member-randomized A/B test, exploratory-query Precision@10 under the unchanged GR judge rises from 63.7% to 79.0% and navigational Precision@1 from 65.5% to 74.7%, with a blinded human evaluation independently confirming the Precision@10 gain.

cs.AI

Beyond Semantic IDs: Encoding Business-Value Ranking into Document Identifiers for Generative Retrieval

Generative Retrieval (GR) formulates retrieval as a sequence-to-sequence generation task, assigning each document a document identifier (DocID) and retrieving it through autoregressive decoding, making DocID design a critical factor in retrieval quality. However, existing schemes based on discrete representation learning suffer from inherent collision issues and create a mismatch between the DocID's encoding objective and the system's business optimization target. To address these limitations, we propose \textbf{Cluster-Ranked Identifier (CRID)}, which decouples DocID into \textit{semantic clustering} and \textit{business-value ranking}, yielding collision-free identifiers that support incremental updates via intra-cluster reranking. We further introduce an analytical framework that decomposes retrieval gains into \textit{personalized preference} and \textit{statistical prior} generalization, revealing how semantic cluster size governs the balance between the two components. Experiments on a Taobao e-commerce corpus of over 300M items show that CRID surpasses the strongest embedding-based retrieval baseline on top-K Hitrate, and delivers +1.06\% GMV in full-traffic deployment.

cs.IR

Semantic Trimming and Auxiliary Multi-step Prediction for Generative Recommendation

Generative Recommendation (GR) has recently transitioned from atomic item-indexing to Semantic ID (SID)-based frameworks to capture intrinsic item relationships and enhance generalization. However, the adoption of high-granularity SIDs leads to two critical challenges: prohibitive training overhead due to sequence expansion and unstable performance reliability characterized by non-monotonic accuracy fluctuations. We identify that these disparate issues are fundamentally rooted in the Semantic Dilution Effect, where redundant tokens waste massive computation and dilute the already sparse learning signals in recommendation. To counteract this, we propose STAMP (Semantic Trimming and Auxiliary Multi-step Prediction), a framework utilizing a dual-end optimization strategy. We argue that effective SID learning requires simultaneously addressing low input information density and sparse output supervision. On the input side, Semantic Adaptive Pruning (SAP) dynamically filters redundancy during the forward pass, converting noise-laden sequences into compact, information-rich representations. On the output side, Multi-step Auxiliary Prediction (MAP) employs a multi-token objective to densify feedback, strengthening long-range dependency capture and ensuring robust learning signals despite compressed inputs. Unifying input purification and signal amplification, STAMP enhances both training efficiency and representation capability. Experiments on public Amazon and large-scale industrial datasets show STAMP achieves 1.23--1.38$\times$ speedup and 17.2\%--54.7\% VRAM reduction while maintaining or improving performance across multiple architectures.

cs.IR

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG

Geometric mean and Lebesgue-type decomposition of completely positive maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

math.OA

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA

Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.

math.PR

Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

math.PR

Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

math-ph

Edge codes constructed from unicyclic graphs

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

math.CO

Difference equations of average entropies

Exact cumulants of entanglement entropies of random state ensembles have traditionally been studied within the random matrix framework. In this work, we propose an alternative approach based on the intrinsic connection to integrable systems. The central idea is to embed entropic quantities into tau functions satisfying Toda-type lattice equations, which in turn yield linear difference equations for their averages. Directly solving the difference equations recovers exact entropy formulas in the literature. The integrable systems approach bypasses the case-by-case, ensemble-dependent derivations required by random matrix methods. The approach also suggests a possible route towards unified and more efficient higher-order cumulant calculations by exploring integrable hierarchies.

math-ph

A Complete Characterization of Tensorizable $f$-divergences

Csiszar's formulation of the $f$-divergence introduced a vast family of functionals for quantifying dissimilarity between probability distributions. However, many applications in statistics and information theory rely only on a few $f$-divergences, such as the Kullback-Leibler divergence, the $χ^2$-divergence, and the squared Hellinger distance. These divergences are especially useful because they admit simple compositional formulas under product measures, a property sometimes referred to as tensorization. In this work, we refine a formalism of tensorization previously introduced in the literature. Then, we show that any possible tensorization formula has a multi-affine form characterized by a single parameter, and identify all tensorizable $f$-divergences under our adopted notion of tensorization.

cs.IT

Spatial symmetry invariance of solution of Kolmogorov flow

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

physics.flu-dyn

A simple derivation of the Kalman filter

In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.

math.OC

Efficient Polynomial-Time Decoding of Simplicial Anticodes with Near-Optimal Performance

In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.

cs.IT