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arXiv2026

A General Plotkin-type Bound on Function-Correcting Codes with Wyner-Graham Distance

Function correcting codes (FCCs) are designed to protect a specified function evaluation of messages at a higher level than the level of protection for messages, against errors while reducing the redundancy required for reliable communication. FCCs have thus far been studied for channels matched to various distances, including the Hamming and Lee distances. Every function partitions the message space into preimage sets corresponding to its distinct function values. Existing Plotkin-type bounds on the optimal redundancy of FCCs under the studied distances, applicable to arbitrary functions on the message space, depend on the pairwise distances among all the message vectors. This makes these bounds difficult to compute. We derive a general Plotkin-type bound on the optimal redundancy of FCCs under Wyner Graham distances, which include the Hamming and Lee distances as special cases. Our bound depends only on the cardinalities of the preimage sets and the sum of pairwise distances only among vectors within each preimage set. This approach significantly reduces the computations required to evaluate the existing Plotkin-type bounds and yields simplified bounds that are easier to compute for specific functions. We obtain simplified Plotkin-type bound for linear functions under the Wyner-Graham distance. Furthermore, the existing simplified bounds for linear functions under the Hamming and Lee distances are recovered as special cases of the proposed bound. We also obtain simplified bounds for several important classes of functions, including the Hamming weight function, the Hamming weight distribution function, the monomial functions under the Hamming distance, and the modular sum function, the Lee weight function, and the Lee weight distribution function under the Lee distance.

cs.IT↗

arXiv2026

Unary Functions, Automorphisms, and Unlabeled First-Order Model Counting

Every fixed first-order sentence $φ$ determines an enumerative sequence $n\mapsto\mathrm{FOMC}(φ,n)$, counting its models on the labeled domain $[n]$. We study the complexity of these sequences when logical specifications may use genuine unary function symbols and hence nested terms $x,f(x),f^2(x),\ldots$. We first prove that, for every fixed sentence $φ\in\mathrm{C}^1_{=}[f]$, with one unary function and an arbitrary finite relational vocabulary, $\mathrm{FOMC}(φ,n)$ is computable in time polynomial in $n$. By contrast, permitting either a second variable or a second unary function already yields hardness. Without counting quantifiers, there is a fixed sentence in $\mathrm{FO}^2_{=}[f]$ whose model-counting function is $\#\mathrm{P}_1$-complete. With one variable and two unary functions, there is a fixed constant-free universal sentence in $\mathrm{FO}^1_{=}[f,g]$, using only unary predicates besides $f$ and $g$, whose model-counting function is again $\#\mathrm{P}_1$-complete. We also relate labeled and unlabeled enumeration exactly. For every relational sentence $φ$, we construct an extension $φ_{\mathrm{aut}}$ in which a unary function records an automorphism and $\mathrm{FOMC}(φ_{\mathrm{aut}},n)=n!\cdot\mathrm{UFOMC}(φ,n)$, where $\mathrm{UFOMC}(φ,n)$ denotes the number of $n$-element models of $φ$ up to isomorphism. Thus automorphism marking gives a one-query exact reduction from unlabeled to labeled model counting at the same domain size. Over relational vocabularies of maximum arity at most $k$, where $k\geq2$, eliminating the auxiliary function yields single-query reductions from unlabeled $\mathrm{FO}^k_{=}$ and $\mathrm{C}^k$ model counting to labeled $\mathrm{FO}^{k+1}_{=}$ and $\mathrm{C}^{k+1}$ model counting, respectively.

cs.LO↗

arXiv2026

Almost Linear 3-Spanners of Temporal Cliques

Temporal graphs model dynamic networks by assigning positive integer time labels to the edges, while information propagates along temporal paths, whose edge labels are traversed in nondecreasing order. A temporal $α$-spanner of a temporal graph with $n$ vertices is a temporal subgraph that approximates the minimum-hop temporal distance between every pair of vertices within a factor of $α$. While general temporal graphs may not admit sparse temporal $α$-spanners for any value of $α$, temporal cliques are known to admit temporal $(2k-1)$-spanners of size $\widetilde{\mathcal{O}}(kn^{1+1/k})$ for every positive integer $k$. We present a simple recursive algorithm that computes, for every temporal clique on $n$ vertices, a temporal $3$-spanner of size $n^{1+2/\sqrt{\ln n}}=n^{1+o(1)}$, thereby improving the previous best upper bound of $\widetilde{\mathcal{O}}(n^{3/2})$. We also show that a modified version of our algorithm computes temporal $3$-spanners of size $\mathcal{O}(nL)$ when the lifetime is bounded by $L$, i.e., all time labels are in $\{1,\ldots,L\}$, thus improving the previous bound of $\mathcal{O}(2^Ln\log n)$. Both results are particularly striking in light of the known lower bound of $Ω(n^2)$ on the size of temporal $2$-spanners, which already holds for temporal cliques of lifetime $L\geq 3$. Both algorithms rely on a new simple recursive decomposition that certifies temporal connectivity for a large collection of source-target pairs using only $\mathcal{O}(n)$ carefully selected edges and recursively processes only the remaining pairs. Besides yielding substantially improved upper bounds, this approach is significantly simpler than previous constructions.

cs.DS↗

arXiv2026

An almost quadratic bound for the minimal excluded minors for a surface

As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs embeddable in a given surface can be characterized by a finite set of minimal excluded minors. However, the proof is purely existential and therefore provides no explicit information about these excluded minors. In 1993, Seymour established the first general upper bound on the order of such minimal excluded minors. Recently, Houdaigoui and Kawarabayashi improved this result by deriving a quasi-polynomial upper bound. Despite this advance, the gap between this bound and the known linear lower bound $Ω(g)$ (where $g$ denotes the genus) remains substantial. In particular, they conjectured that a polynomial upper bound should hold. In this paper, we confirm this conjecture by showing that the order of the minimal excluded minors for a surface of genus $g$ is $g^{2+o(1)}$. This result significantly narrows the gap between the known lower and upper bounds, bringing the asymptotic behavior much closer to the conjectured optimum. Our approach relies on a new structural property of minimal excluded minors. Let $G$ be a minimal excluded minor for a surface of Euler genus $g$. Houdaigoui and Kawarabayashi showed that $G$ contains $O(\log g)$ pairwise disjoint cycles that are contractible and nested in some embedding of $G$. We strengthen this result by proving a separator-based variant: for any contractible subgraph $H \subseteq G$ with a separator of size $s$ (with $H$ contained entirely in one side), the subgraph $H$ contains $O(\log s)$ disjoint cycles that are contractible and nested in some embedding of $G$. This allows us to replace a genus-dependent bound with a separator-dependent one, which is the main new ingredient in deriving our polynomial bound.

math.CO↗

arXiv2026

Distributed $(Δ+1)$-Coloring in Graphs of Bounded Neighborhood Independence

The distributed coloring problem is arguably one of the key problems studied in the area of distributed graph algorithms. The most standard variant of the problem asks for a proper vertex coloring of a graph with $Δ+ 1$ colors, where $Δ$ is the maximum degree of the graph. Despite an immense amount of work on distributed coloring problems in the distributed setting, determining the deterministic complexity of $(Δ+ 1)$-coloring in the standard message passing model remains one of the most important open questions of the area. In this paper, we aim to improve our understanding of the deterministic complexity of $(Δ+ 1)$-coloring as a function of $Δ$ in a special family of graphs for which significantly faster algorithms are already known. The neighborhood independence $θ$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. In general, in graphs of neighborhood independence $θ= O(1)$ (e.g., line graphs), it is known that $(Δ+ 1)$-coloring can be solved in $2^{O(\sqrt{\log Δ})} + O(\log^* n)$ rounds. In the present paper, we significantly improve this result, and we show that in graphs of bounded neighborhood independence, a $(Δ+ 1)$-coloring can be computed in $(\log Δ)^{O(\log \log Δ/ \log \log \log Δ)} + O(\log^* n)$ rounds and thus in quasipolylogarithmic time in $Δ$. Additionally, we show that when $θ= o(Δ^{1/8})$, one can color the graph in $o(\sqrtΔ) + O(\log^* n)$ rounds, which is faster than the current best known coloring algorithms do on general graphs. We also show that the known approach that leads to a polylogarithmic in $Δ$ algorithm for $(2Δ- 1)$-edge coloring already fails for edge colorings of hypergraphs of rank at least 3.

cs.DC↗

arXiv2026

Ordered Ruzsa-Szemeredi Numbers at Matching Size Two

Bondy and Szwarcfiter defined $\mathrm{ex}^*(n,F)$ as the largest number of edges in an $n$-vertex graph whose edge set partitions into induced copies of $F$; for $F=2K_2$ the deficiency $\binom{n}{2}-\mathrm{ex}^*(n,2K_2)$ is $Θ(n^{3/2})$. We study the ordered relaxation at fixed matching size, in which each part need only be induced in the union of itself with the parts that follow it; write $\mathrm{ORS}_n(r)$ for the largest number of parts, so that $r\,\mathrm{ORS}_n(r)$ is the ordered analogue of $\mathrm{ex}^*(n,rK_2)$. Our main tool is a characterisation valid for every $r$: an ordered decomposition into induced $r$-matchings is a sequence of steps that start from $K_n$ and repeatedly delete a perfect matching from $2r$ vertices currently spanning a clique. Reading a decomposition backwards turns a condition about the ordering into a reachability question that an exhaustive search can settle. For $r=2$ we determine $\mathrm{ORS}_n(2)$ exactly at orders five through nineteen, where it takes the values $1,3,5,8,11,14,19,23,28,34,40,47,54,62,70$, and we confine $\mathrm{ORS}_{20}(2)$ to $\{78,79\}$. The counting bound $\lfloor n(n-4)/4\rfloor$ is attained at orders five through nine and at eleven, and missed by exactly one part at every other order below twenty, so order eleven is an isolated exception, not a parity effect. Across this range the ordered deficiency equals $\frac32n+O(1)$, and along powers of two a dyadic construction keeps it below $O(n\log n)$; whether it is linear for all $n$ is our main open question. The structural results are formalised in Lean 4, and the searches are certified by fail-closed sweeps and an independent checker.

math.CO↗

arXiv2026

Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs

In the uniform sparsest cut problem we are asked to find a vertex set that cuts few edges relative to the number of vertex pairs it separates. The Goemans-Linial SDP coupled with the Arora-Rao-Vazirani rounding gives an $\mathcal{O}(\sqrt{\log n})$ approximation on arbitrary graphs on $n$ vertices. We study this relaxation on finite Abelian Cayley graphs. First we show that when the second normalized Laplacian eigenvalue of $G= \mathrm{Cayley}(Γ, S)$ is realized by a Fourier character with image size at most four then $λ_2(G)=\mathrm{SDP}_{\mathrm{GL}}(G)=ψ(G)$. Geometrically, a character maps the vertices onto a regular polygon where the squared chord distance satisfies the triangle inequalities exactly when the polygon has at most four vertices. Grouping equal character fibers gives a cyclic quotient where the optimal cut can be found exactly and so the relaxation is exact on finite Abelian Cayley graphs on groups of exponent at most four. Second, we replace each generator $s$ of $S$ by a uniformly random element of its cyclic subgroup (including identity). If $r_s$ is the order of $s$, we let $α(r_s)$ to be the average number of $\pm s$ steps needed to simulate such a move, and let $ρ(S)=\max_{s\in S}α(r_s)$ be its worst case. Full cyclic averaging eliminates character phases and choosing a nontrivial character $χ^*$ minimizing the auxiliary eigenvalue and taking $K=\mathrm{ker}χ^*$ gives \[ ψ(G)\leqψ_G(K)\leq\frac{q^*}{q^*-1} \cdotρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G)\leq 2ρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G), \] where $q^*=|χ^*(Γ)|$. If all generator orders are at most $R$, this is an $R/2$ approximation. Finally, we construct an infinite family of finite Abelian Cayley graphs with Goemans-Linial integrality gap exactly $16/15$.

math.CO↗

arXiv2026

Characterizations and Complexity of Minimum Forward and Integer Cycle Bases

The cycle space of a directed graph is generated by a cycle basis, where, in general, cycles are allowed to have both forward and backward arcs. In a forward cycle, all arcs must follow the given direction. Several open questions remain regarding the complexity of the minimum cycle basis problem, in particular the minimum-weight integral cycle basis problem, and the minimum-weight weakly and strictly fundamental forward cycle basis problems. In this paper, we address these open questions. First, we study the existence, structure, and computational complexity of minimum-weight forward cycle bases. We give a complete structural characterization of digraphs that admit weakly fundamental (and hence integral) forward cycle bases. We further provide a characterization when a strongly connected digraph admits a forward fundamental cycle basis, proving that such a basis exists if and only if the set of directed cycles has cardinality equal to the cycle rank; in this case, the basis is unique. Lastly, we show that while minimum-weight forward fundamental cycle bases can be found in polynomial time whenever they exist, the minimum-weight forward weakly fundamental cycle basis problem is APX-hard via an L-reduction from the minimum-weight weakly fundamental cycle basis problem on digraphs with metric weights. Second, we introduce opt-in graphs, i.e., the family of graphs for which minimum cycle bases are integral for any weight function. We show that this family is minor-closed and hence, by the Robertson-Seymour theorem, is characterized by a finite set of forbidden minors, so that the opt-in recognition problem is solvable in polynomial time. Lastly, we present an algorithm to check whether a graph is opt-in, and if not, to identify which of its minors belong to the set of forbidden minors. Applying this algorithm, we show that the complete graph $K_n$ is opt-in if and only if $n \leq 7$.

math.OC↗

arXiv2026

Joint Alignment and Distillation for Video Generation via Sample-Guided Distribution Matching

Aligning video generative models to human preferences heavily relies on Reinforcement Learning (RL), which suffers from extensive computational overhead. Existing workflows typically treat RL and distillation as disconnected stages: applying RL before distillation incurs prohibitive computational costs, whereas applying RL after distillation frequently leads to model collapse. To overcome these limitations, we propose a unified, single-stage optimization framework grounded in Distribution Matching (DM). In the standard DM framework, distillation updates the model via a gradient direction that minimizes the gap between the real and fake models, guiding generations toward clarity and high fidelity. Building upon this, we introduce DM-Align, which derives a complementary gradient direction to guide the model toward human-preferred samples. Inspired by DPO and GRPO, our method leverages the distributional gap -- formulated from either preference pairs or intra-group exploration -- to directly construct this preference-guided gradient. By synergizing these two gradient directions, our approach eliminates the need for multi-step reward evaluation and complex ODE-SDE conversions inherent in traditional RL. Comprehensive experiments across multiple foundational video models demonstrate that this sample-guided framework robustly enhances both distillation quality and preference alignment, consistently outperforming both standalone variants and sequential two-stage pipelines.

cs.CV↗

arXiv2026

The Cayley Completion of a Graph

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $γ^{+}$ (additions only) and the Cayley edit distance $γ_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $γ^{+}(G)\ge nΔ^{*}/(2m)-1$, where $Δ^{*}$ is the least $d\geΔ$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $γ^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $γ^{+}$, while $γ_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $γ^{+}(P_n)=γ^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $γ_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

cs.DM↗

arXiv2026

Layer-Based Width for PAFP

The Path Avoiding Forbidden Pairs problem (PAFP) asks whether, in a directed graph $G$ with terminals $s,t$ and a set $\mathcal{F}$ of forbidden vertex pairs, there is an $s$-$t$ path that contains at most one endpoint from each forbidden pair. We initiate the study of PAFP through a layer-based width measure. Our first focus is the union digraph $G\cup\mathcal{F}$, obtained by adding to $G$ one arc per forbidden pair, oriented according to a fixed reachability-compatible order. Let the BFS layer $L_d$ be all vertices at directed shortest-path distance $d$ from $s$, where the BFS-width from $s$ is $\max_d |L_d|$. We show if $G\cup\mathcal{F}$ has BFS-width $b$ from $s$ and only $β$ arcs going from a later BFS layer to an earlier one, then PAFP is FPT parameterized by $b+β$. The backward-arc hypothesis is essential: we show PAFP remains NP-complete when the union digraph is a DAG with BFS-width 2. We also show if the input DAG has BFS-width at most $2$ and only $k$ backward input arcs, then PAFP can be decided in $2^k |I|^{O(1)}$ time, with unrestricted forbidden pairs. This width-$2$ result is tight: inspection of a classical reduction shows NP-completeness on input DAGs of BFS-width $3$ with no backward input arcs. Moreover, we study exact-length layers in the input graph, where the $d$-th layer consists of the vertices reachable from $s$ by a directed path of length exactly $d$. For DAGs of exact-length width at most $2$, we show PAFP is polynomial-time decidable by a 2-SAT encoding of fixed-length paths. This bound is tight: the same classical reduction yields NP-completeness on DAGs of exact-length width $3$. Unlike previously known polynomial-time regimes for PAFP, which restrict the forbidden-pair set in order to obtain tractability, our two input-graph tractability results allow unrestricted forbidden pairs and input graphs with exponentially many $s$-$t$ paths.

cs.DS↗

arXiv2026

Strong Conflict-Free Vertex-Connection via Twin Cover: Kernelization and Chromatic Bounds

A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a coloring is the strong conflict-free vertex-connection number $\operatorname{svcfc}(G)$. We study this problem under the parameter twin cover. Let $X$ be a twin cover of $G$ of size $t$, and let $k$ be the target number of colors. In our first result, given $(G,k)$ together with a twin cover $X$, we reduce in polynomial time to an equivalent annotated instance on at most $\max\{2,t+(t+1)k2^{t+k-1}\}$ vertices. Hence the annotated version of Strong CFVC Number, in which a twin cover is supplied as part of the input, is fixed-parameter tractable parameterized by $t+k$. Using this bound, we then obtain a kernel parameterized by $\operatorname{tc}(G)+k$; in particular, for every fixed $k$, the problem is fixed-parameter tractable parameterized by the twin-cover number alone. In our second result, we prove every connected graph $G$ with twin cover $X$ of size $t$ satisfies $χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$. More generally, if $Y\subseteq X$ intersects every shortest path of length at least $3$, then $\operatorname{svcfc}(G)\le χ(G)+|Y|$. We also derive an exact expression for the chromatic number on graphs of bounded twin-cover number: for every proper coloring $φ$ of $G[X]$, the minimum number of colors needed to extend $φ$ to all of $G$ is $K_φ=\max_{S\subseteq X}(|φ(S)|+m(S))$, and hence $χ(G)=\min_{φ\text{ proper on }G[X]} K_φ$. Our results provide the first evidence that twin cover is a useful parameter for strong conflict-free vertex-connection and show that, once a twin cover is fixed, the remaining difficulty is concentrated in a bounded additive gap above the chromatic number.

cs.DM↗

arXiv2026

Persistent Memory Through Triple-Loop Consolidation Under Stochastic Unit Turnover

Dissipative cognitive architectures maintain computation through continuous energy expenditure, where units that exhaust their energy are stochastically replaced with fresh random state. This creates a fundamental challenge: how can persistent, context-specific memory survive when all learnable state is periodically destroyed? Existing memory mechanisms -- including elastic weight consolidation, synaptic intelligence, and surprise-driven gating -- rely on gradient computation and are inapplicable to systems that do not perform it. We introduce Deep Memory (DM), a backpropagation-free persistent memory mechanism operating through a triple-loop consolidation cycle: (1) recording of expert-specific content centroids, (2) seeding of replaced units with stored representations, and (3) stabilization through continuous re-entry. Discrete expert routing via Mixture-of-Experts (MoE) gating is required, in the regimes tested, to prevent the centroid convergence that would render stored memories identical. We derive a Foster-Lyapunov drift bound for the full triple loop, showing that seeding rescales the turnover noise floor. Across $1{,}007$ simulation runs over thirteen blocks: (i) removing stable context-expert binding removes specialization ($\mathrm{MI}=1.10$ vs. $0.001$; $n=91$); (ii) DM achieves $R=0.984$ vs. $0.385$ without memory ($n=16$); (iii) continuous seeding reconstructs representations after interference ($R_\mathrm{recon}=0.978$; one-shot fails; $n=30$); (iv) the mechanism operates within a characterized $(K,p)$ envelope ($n=350$); (v) recording $\times$ seeding is the minimal critical dyad ($n=40$); (vi) associative and reservoir baselines (Hopfield, ESN) are compared under matched turnover ($n=370$). DM is thus a falsifiable, bounded mechanism for persistent memory in backpropagation-free cognitive systems, with functional parallels to hippocampal consolidation.

cs.NE↗

arXiv2026

Decoupling Disaggregated Memory Optimizations from Indexing: A Compiler-Runtime Approach

Disaggregated memory (DM) decouples compute and memory into independently scalable pools, connected over a slower interconnect rather than a local bus. This decoupling is exactly what makes DM attractive--but it also means that every index must now reason explicitly about remote-memory access and its associated optimizations. State-of-the-art index designs respond to this by embedding remote-memory logic and optimizations directly into their core data structures and concurrency control mechanisms. Consequently, an optimization tuned for one index cannot be lifted and reused in another, and even the same index cannot be ported to a different DM architecture without a fresh round of redesign. This escalating, per-index, per-platform engineering burden is unsustainable as hardware and index requirements evolve. In this paper, we present Nox, a compiler-runtime framework that breaks this coupling by taking an unmodified, concurrent index as input and automatically generating its disaggregated-memory counterpart, without touching the original index logic. A compiler layer rewrites the index's LLVM IR to expose allocation, address, and pointer-dependency information that a centralized runtime uses to drive caching and address translation. Empirically, Nox-generated B+-trees, hash tables, and skip lists scale robustly on real RDMA and CXL hardware across every workload tested. They can also outperform some specialized, hand-crafted indexes and match others, especially on workloads that are closer to real-world ones. These results show that today's fastest disaggregated-memory optimizations need not stay locked inside monolithic, hand-crafted code--a compiler-runtime stack can generalize them while preserving the scalability of proven index implementations, without sacrificing it for portability.

cs.DB↗

arXiv2026

RAPTOR: Role-Aware Private Training for Mixture-of-Experts

Differentially private (DP) fine-tuning methods treat sparse Mixture-of-Experts (MoE) models as a single dense block, ignoring that shared layers see all data while experts only see routed records. We identify and formally characterize three resulting failure modes: global clipping suppresses expert gradients, batch-level normalization dilutes sparse expert updates, and fixed privacy noise degrades signal-to-noise ratio on low-load experts. We introduce RAPTOR - a Role-Aware Private Training framework, which alternates shared and expert optimization and targets each failure directly, using expert-specific clipping and noise together with a public expected-owner denominator and a count-independent update schedule that avoids conditioning on private, realized expert counts. We prove the resulting mechanism satisfies $(\varepsilon,δ)$-DP: because each record is assigned to exactly one owner expert, per-expert mechanisms within a layer compose in parallel, so updating all $E$ experts costs no more, in privacy terms, than updating one, with shared and expert streams composing sequentially across training. We further derive a bias-variance decomposition of the public-denominator estimator showing its bias grows predictably with routing imbalance, yielding a privacy-free rule for selecting which layer to protect from routing entropy measured on a small public corpus. Experiments on Switch Transformer and OLMoE fine-tuning across GLUE tasks, and on DeepSeek-VL2-Tiny, show consistent gains over standard DP baselines across several privacy levels ($\varepsilon$), with the largest margins typically at the tightest budgets. Code and models are publicly available: https://github.com/leduckhai/RAPTOR

cs.LG↗

arXiv2026

Climate Physics Dynamic Matching

Deep generative models such as flow matching and diffusion models have shown potential for learning complex dynamical systems, but typically act as black boxes that neglect underlying physical structure, while physics-based models governed by partial differential equations are often incomplete due to missing source terms, or uncertain parametrisations. We present Climate Physics Dynamic Matching (ClimPhyDM), a variational simulation-free dynamics informed framework for weather forecasting that combines an advection-type physics prior with data-driven components in a variational framework. % to capture the stochasticity and multi-modality of unresolved atmospheric dynamics. On the ERA5 benchmark at hourly (42-hour) and monthly (5-month) resolutions, ClimPhyDM outperforms ClimODE, and GB-DM, keeping the lower error at extended horizon, indicating improved temporal stability and resistance to error accumulation, while its simulation-free paradigm also enables training on a single modest 12 GB consumer GPU.

stat.AP↗

arXiv2026

MGDiff: Multi-Interest Sequence Recommendation with Masking GNN-Guided Diffusion

We propose a novel Multi-Interest Sequence Recommendation Framework with \underline{M}asking \underline{G}NN-Guided \underline{Diff}usion Model (MGDiff), designed to generate accurate, bias-free user interest information during the diffusion process. First, we propose a semantics-enhanced Dual-layer Semantic Guidance (DSG) framework, which decomposes guidance into two synergistic stages: extracting latent item semantics and decoupling multidimensional user intent. We design a Weight-adaptive Masking Graph Neural Network reconstructs missing links to uncover deep item relationships beyond superficial co-occurrence, while a Dynamic Multi-Expert Network projects user preferences into distinct semantic subspaces to suppress irrelevant interference. This hierarchical design yields structured guidance that significantly improves the generation accuracy of diffusion models. Second, We propose a Popularity-Aware Guidance (PAG) mechanism that performs spatial geometric adjustments on the outputs of diffusion models: by using item popularity as a differentiable adjustment signal to recalibrate similarity metrics, we enable DMs to generate diverse recommendations free from popularity bias. Finally, we compare MGDiff with multiple baseline models across four widely used datasets, demonstrating its superior performance and validating its effectiveness.

cs.IR↗

arXiv2026

Exact Recovery Thresholds for Weighted Data Selection in Vector-Valued Linear Regression

We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. In vector-valued linear regression with square loss $\ell_{(x,y)}(W)=|Wx-y|_2^2$, where $x\in\mathbb{R}^d$, $y\in\mathbb{R}^m$ and the learner is the empirical risk minimizer of minimal Frobenius norm, we prove that the minimal budget of weighted examples that recovers the full-data loss on every finite dataset is exactly $n^*(d,m)=(m+1)d$. We further determine two more values of the weighted selection profile $F_w(d,m,n)$: at the near-threshold budget, $F_w(d,m,(m+1)d-1)=1+\frac{1}{dm^2}$, and at the spanning budget, $F_w(d,m,d)=d+1$ for every $m$, while $F_w(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_w(2,2,3)\in[13/8,15/8]$ and $F_w(2,2,4)\in[5/4,3/2]$, reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms, and establish strong structural evidence for the conjecture. The upper-bound techniques (a fixed-basis conic compression lemma, a determinant-facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems) are of independent interest. As a byproduct we correct an erroneous claim circulating in a recent unrefereed preprint, exhibiting an explicit dataset with $m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. All results are new only for $m\ge 2$; the scalar case $m=1$ is due to Hanneke et al.

cs.LG↗