Papers

As generated images become increasingly realistic, reliable forgery detection is essential for maintaining trust in visual information. However, existing methods primarily rely on task-specific supervision to adapt vision foundation model representations, without fully exploiting internal forensic knowledge to guide detection. To address this limitation, we propose Reserve-Guided Elicitation (RGE), a framework that treats sparse, origin-sensitive internal components in pretrained models as a forensic reserve and translates their localization into structural constraints for lightweight adaptation. Specifically, we first use the Forensic Lens (F-lens) to decompose activations across layers and token groups into independent components and globally screen them by their response differences between real and generated images, identifying reserve sites and directions. Next, we map the selected directions back to hidden-state space to construct fixed reserve subspaces and insert Forensic Reserve Adapters (FRA) only at the identified sites. Finally, with the backbone parameters, previously fitted reference classifier, and subspace bases fixed, we train only the FRA coefficient maps to generate input-dependent residual updates constrained to the corresponding subspaces, strengthening existing forensic responses. Using only 500 labeled training images and a trainable parameter budget below 0.2% of the backbone, RGE achieves competitive performance across three detection benchmarks without target-benchmark adaptation. Furthermore, RGE consistently improves over the corresponding frozen detectors across eight encoders spanning self-supervised and vision-language pretraining, eliciting a latent forensic capacity broadly shared across pretrained vision models.

The Akutsu-Deguchi-Ohtsuki (ADO) invariants of links are a non-semisimple version of the colored Jones polynomials. We prove that the top coefficients of the ADO invariants of links in $S^3$ are multiplicative under Murasugi-sum along connected minimal genus Seifert surfaces. More generally, for any simple complex Lie algebra $\mathfrak{g}$ of type ADE and any $r\geq 3$, we prove that the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant is multiplicative under Murasugi-sum. As a corollary, we get that all the $\mathfrak{g}$-ADO invariants of fibred links are $q$-monic, generalizing a previous result of the authors for $\mathfrak{sl}_2$. Moreover, whenever $\mathfrak{u}_q(\mathfrak{g})$ is ribbon, we find an expression for the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant in terms of Hopf algebra integrals of universal $\mathfrak{u}_q(\mathfrak{g})$-invariants of bottom tangles.

Quadruped robots are increasingly expected to carry objects while moving through human environments. But what happens when a person interacts directly with the payload rather than with the robot? If the payload is unrestrained, the robot must distinguish intentional external interactions from ordinary payload motion, while still keeping the load balanced and maintaining stable locomotion. How can a quadruped infer and compliantly respond to such interactions using only onboard measurements? In this work, we develop a force-aware locomotion framework that treats payload interactions as commands that shape the motion of the combined robot-payload system. Our approach separates the learning of force-aware locomotion and force estimation on an unrestrained payload. We combine a compliant load-carrying policy with a causal force estimator, trained through estimator-in-the-loop data aggregation and finetuning, to predict interactions from onboard robot measurements. Our simulations and real-world experiments show that the resulting controller can maintain stable payload-carrying locomotion, yield compliantly to external interactions, and use the inferred force to support human-guided changes in the robot's trajectory.

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

Broadly studied origami crease patterns such as the Miura-ori are capable of rigid folding that maintains their spatial periodicity. However, periodic crease patterns generically require additional creases and fold into quasi-cylindrical shapes. Here, using group theory, we show that the folding properties of such sheets are largely determined by their spatial symmetries, known as layer groups. Within each symmetry class, the allowed linear shape-periodic modes of sheets are constrained and classified by the irreducible representation they belong to. Such classification further gives a sufficient condition, containing a symmetry constraint and a topological constraint, for the existence of a nonlinear strain mode. In this way, familiar special cases are placed within a unified framework that connects symmetry, network topology, and deployability. Finally, we turn our classification results into a predictive design principle that enables the rapid design of novel sheets with prescribed properties.

We study the problem of recovering low-dimensional latent geometry from a random geometric graph generated by noisy, high-dimensional data. Specifically, we analyze the performance of a spectral embedding algorithm on the Signal+Noise Graph Model, in which vertices are associated to points perturbed by Gaussian noise, and edges are included for pairs whose inner product exceeds a specified alignment threshold. In the high-dimensional regime where the number $n$ of points and the ambient dimension $d$ both tend to infinity, we show that under a spectral gap condition, the top eigenvectors and eigenvalues of the graph's adjacency matrix can be used to approximately recover the point cloud up to an orthogonal transformation. We illustrate our results on point clouds sampled from nested spheres and high-dimensional sinusoid curves.

Recently, Girsch-Kurinczuk have obtained new formulas for local newforms in depth-zero and minimax integral-depth cuspidal representations of $p$-adic $\mathrm{GL}(n)$. One of their main results is an averaging formula which provides an integral representation of the Whittaker newform in terms of the Paškūnas-Stevens Whittaker function. In the present paper, we first prove a reverse averaging formula by establishing an integral representation of the Paškūnas-Stevens Whittaker function in terms of the Whittaker newform. Together, these formulas give an explicit two-way passage between these two distinguished vectors. Finally, we use our averaging formula and earlier work of Kurinczuk-Matringe to give new explicit test vectors in terms of Whittaker newforms for such cuspidal $\mathrm{GL}(n)\times\mathrm{GL}(n)$ pairs, whenever their Rankin-Selberg $L$-factor is not identically equal to $1$.

We calculate the conservative deflection of a photon by a massive spinless source, such as a black hole, at third post-Minkowskian order using scattering amplitudes. We work in the heavy-source and long-range limits and adopt the geometric-optics approximation. The required five-point photon-graviton tree amplitude is constructed by on-shell recursion and sewn to three scalar-graviton vertices across a five-fold cut. Polynomial division organizes the cut information into sixteen contributions, including the residues with fewer propagators. We perform the source energy integrations and retain the near-on-shell photon regions in the remaining spatial integrals. Within the static region in which all exchanged momenta are soft, we relate a photon pole prescription to the subtraction of the cubic tree iteration from the conservative kernel. The sum of the sixteen contributions determines the classical logarithmic term at two loops. Its transverse Fourier transform gives the interaction radial action and yields the third-order deflection angle. The result agrees with the weak-deflection expansion of Schwarzschild null geodesics and demonstrates how nonlinear classical light bending is encoded in loop amplitudes around flat spacetime.

We investigate a continuous-time financial market where the asset price exhibits weak (sublinear) mean reversion and has a nonzero drift. Complementing earlier work on strong (superlinear) mean reversion, we show that, for an investor maximizing expected exponential utility, the certainty equivalent grows as $O(T^{2\beta+1})$ where $0<\beta<1$ is the strength of mean reversion. An explicit asymptotically optimal strategy is also given.

Recently Large Language Models (LLMs) and LLM-based agents increasingly need to incorporate knowledge acquired after pretraining, e.g., domain facts, user preferences, documents, and interaction experience. In-context learning (ICL) and ICL-based agent harness remain flexible, but they consume context capacity and incur repeated discretized encoding cost that grows with context length. In-parameter memory offers a complementary substrate: reusable memory information is represented in model parameters, adapters, or other parameter-like objects that are composed into the forward pass at inference time. This survey focuses on methods that augment LLMs with such parametric memory at deployment: a memory-bearing parameter object is plugged into the forward pass during inference, whether it is acquired before or during deployment. We organize the landscape with two orthogonal axes: Parameter Placement, which includes Embedding, Attention, FFN layers, or Hybrid when two or more layers are used; and Parameter Acquisition Time, which distinguishes methods whose memory object is acquired during deployment (online) from those acquired before it (offline). We clarify boundaries, conduct comparisons, and discuss open directions in interference, safety, co-design with ICL, and recursive self-improvement.

We denote by $f_k(\Gamma)$ the largest integer with the property that every subset of a finite abelian group $\Gamma$ of size at least $|\Gamma| - f_k(\Gamma)$ is a $k$-fold sumset. Extending a recent result of Alon and Pham, we prove that $$ f_k(\Gamma) \leq \widetilde{O} \left(n^{(2k-1)/(4k-3)}\right) $$ holds for all finite abelian groups $\Gamma$ and integers $k \geq 2$, where $n = |\Gamma|$. Additionally, we also show that the lower bound $f_k(\Gamma) \geq \widetilde{\Omega} \left(n^{1/k}\right)$ holds if $\Gamma$ has no nontrivial element of order dividing $k$. Our upper bound improves a previous result of Balogh, Liu, and Sharifzadeh, and recovers the bound of Alon and Pham in the case $k = 2$. The proof relies on a new upper bound for the independence number of random Cayley sum hypergraphs, which may be of independent interest.

We study the critical points of a quadratic functional depending on the gradient of the Bach tensor on Riemannian four-manifolds, which generalize the Bach-flat condition. We show that, on every closed four-manifold, there exists a weak Bach-parallel metric, i.e. a critical point for this functional with respect to conformal variations: in particular, we prove that there exist infinitely many conformal classes which contain a unique minimizer for the functional, up to constant positive rescaling. Next, we analyze the global minima of the functional, i.e. metrics with parallel Bach tensor, relating these metrics to well-known variational problems. Using a version of de Rham's splitting theorem on complete four-manifolds, we provide a classification result for products of surfaces, exploiting the theory of conformal gradient solitons; we also construct a new explicit example of a Bach-flat metric which is neither locally conformally flat nor conformally Einstein and we characterize HCMU metrics on complete surfaces. Finally, we prove an equivalence between the Bach-parallel condition on 4D cylinders and the existence of critical metrics for a well-known quadratic curvature functional in dimension three: in this direction, we also prove a characterization of flat three-manifolds, under some curvature and finite energy assumptions.

Long-horizon world-model planning typically relies on autoregressive rollouts, where predicted states are repeatedly fed back into the model. This preserves temporal structure but creates a horizon-length sequential path and exposes later predictions to recursive decoded-state feedback. We introduce Parallel Predictive World Models (PPWM), which predict a finite-horizon trajectory in parallel while retaining causal interaction among future representations. Each horizon is conditioned on its causal action prefix, and future representations interact before decoding, separating temporal causality from state-by-state output recursion. We formalize this distinction by viewing autoregressive rollout as a causal trajectory map and identifying the decoded-state feedback pathway removed by PPWM. Across four visual-control tasks, PPWM achieves the lowest long-horizon prediction error and the highest Cross-Entropy Method (CEM) simulator success among the evaluated predictive interfaces. Meanwhile, PPWM achieves more than a 3$\times$ average CEM planning speedup over the autoregressive LeWM baseline. These results suggest that accurate and efficient long-horizon world-model planning does not require state-by-state autoregression, but can instead be achieved through parallel causal trajectory prediction.

Diffusion models generate data through a continuum of denoising problems, and are widely observed to reveal coarse structure before fine detail. Yet, this intuition is mostly empirical and qualitative. We introduce feature information dynamics, an information-theoretic framework for localizing when a feature is generated during diffusion. Using the I-MMSE identity, we connect the rate of feature mutual information change to a gap between optimal unconditional and feature-conditional denoising losses, yielding practical estimators for feature information density. We further develop a chained decomposition that separates shared from incremental information in a feature hierarchy. We use this framework first to quantitatively confirm spectral autoregression in pixel diffusion, and then to extend the analysis beyond frequency: under a class $\to$ mask $\to$ Canny conditioning chain, the per-feature information densities differ across pixel, SDVAE, VAVAE, and RAE, exposing fundamental differences between these representations and suggesting that ordered generation could be beneficial for training diffusion models. Our code is available at https://github.com/AI4Science-WestlakeU/feature-information-dynamics.

We establish a counterpart, for individual finite open covers, of the global entropy-mean-dimension comparison of Lindenstrauss and Weiss. More precisely, the mean dimension of a finite open cover is quantitatively bounded in terms of its topological entropy and cardinality. In particular, a finite open cover with zero entropy has zero mean dimension. The same comparison holds for actions of countable amenable groups and in the sofic setting. It follows that every mean dimension pair is an entropy pair and, in particular, that uniformly positive mean dimension (UPMD) implies uniformly positive entropy (UPE). The cover and pair implications answer Questions 2.27 and 4.9 of García-Ramos and Gutman, respectively. The proof associates a polytope to a minimal subcover and combines the Lindenstrauss-Weiss compression argument with Maurey's empirical method to estimate the volumes of its coordinate projections.

We propose a method for choosing the shared memory parameter $\beta_1=\beta_2=\beta$ in Adam from a short pilot training. The selected $\beta$ remains fixed during the subsequent full training. A local model of Adam's normalized direction balances sampling variability against the delay introduced by averaging past gradients. This balance gives a cubic memory rule, whose two coefficients are estimated from gradient probes at a few pilot checkpoints. The estimator uses the numerator and denominator jointly, preserving their covariance. With a 200-update pilot and sixteen probe gradients at each of four checkpoints, a seed-matched retrospective evaluation on eleven vision and language workloads reduces mean relative validation gap by 40.7% and worst-quarter mean gap by 44.3% against the grid representative of shared $\beta=0.95$. The mean gap is also 32.3% lower than that of the best constant $\beta$ chosen across all eleven workloads.

Using differential Galois theory—specifically, Magid's theory of the complete Picard-Vessiot closure—we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\ Amer.\ Math.\ Soc.\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.

System administrators of Internet-scale services need to resolve failure incidents to maintain reliability of such services. Ideally, we want a troubleshooting system to be: (1) expressive to known and unknown incidents with high accuracy; (2) cost efficient at scale; (3) explainable to provide actionable insights operators can act on; and (4) entail low effort from the operators. Unfortunately, most existing systems, including emerging LLM-assisted agentic workflows and structured frameworks for authoring diverse RCA algorithms fall short of achieving all four requirements. We present E4, a novel agentic system for troubleshooting for Internet-scale services. E4 embodies the paradigm of constrained creativity that combines the best of LLM-assisted automation and exploration with the explainability and efficiency of a structured approach. Instead of allowing an LLM agent to write arbitrary code or generate arbitrary responses, we provide the agent a restricted DSL to generate its response via simple loop-free data flow programs. This DSL, equipped with high level operators for troubleshooting, makes E4's output accurate, verifiable and explainable. On a mix of synthetic and real-world workloads, E4 achieves up to 62% better accuracy compared to state-of-the-art solutions, while providing more explainable responses at up to 12x reduced cost.

Recent game design agents have made substantial progress in generating playable games. However, program correctness does not ensure an enjoyable experience for players. We present Recursive Game Creator, an experience-oriented harness to advance agentic game development from rough game prototypes into entertaining games. Recursive Game Creator organizes recursive development around four components: Designer, Builder, Player, and Reviewer. The Designer translates user instructions and Reviewer's feedback into detailed plans. The Builder turns these plans into candidate games. The coding-native Player creates and executes reusable policies through programmatic interfaces to efficiently collect diverse gameplay trajectories, mitigating evaluation bias caused by slow GUI-based collection. The Reviewer uses carefully designed trajectory-based metrics to induce player preferences, integrating with visual evidence and explicit textual preferences to evaluate games against game-specific criteria. Finally, the Reviewer accepts the better version and provides improvement reviews for the next round, closing the recursive loop. Our method achieves state-of-the-art overall performance of 77.89 on GameCraft-Bench. On GameASG-Bench, it achieves a strict task success rate of 53.2%, a 34.1% improvement over the same-model baseline, and the highest mean runtime-check pass rate at 93.4% among compared methods. A user study shows longer playtime and higher ratings. Project page is at: https://imbaldy.github.io/recursive-game-creator/.

High-beam-count LiDAR sensors are costly, yet many perception pipelines require dense angular sampling. Using a pretrained Stable Diffusion model as the backbone, we fine-tune a LiDAR-conditioned depth model with pseudo-depth targets from a 2D foundation model. During training, the LiDAR conditioning is randomly decimated at different beam budgets. We then investigate how much of a LiDAR scan can be recovered from heavily decimated input and characterize performance across the input beam budget. We evaluate against physically held-out real beams on nuScenes and report recovery separately from fit accuracy. Our model yields its largest advantage in very sparse regimes, achieving a $\delta_{1.25}$ accuracy of $66.8$% from $4$-beam input where scattered interpolation reaches only $45.1$%. A class-stratified error breakdown further reveals that planar surfaces recover first while objects introducing depth discontinuities degrade earliest. Together, these results quantify the recovery/resolution trade-off for foundation-model-guided LiDAR enhancement.

Recent years have seen an increasing need for MR tools for game development and visual game analytics. However, the number of visual game analytics tools for MR games remains scarce due to the unique complexities in integrating real components from both the physical and the digital worlds in MR games. Existing visual analytics tools for such games are not suitable for space-agnostic generalization. To tackle this problem, we introduce a space-agnostic visual game analytics tool that incorporates adaptive spatial reconstruction and real-time object detection built for the Microsoft HoloLens 2 MR headset. We believe that the outcomes of our work will increase insights into player environment and pave a new direction for visual game analytics for MR game development.

A robot that offloads its control loop to the network carries the receiver with it, so link quality is decided by where it goes. Simulating this requires both a physics engine and a site-specific propagation model at once; to our knowledge no simulator natively unifies both, with existing couplings of the two limited to offline analyses. We demonstrate a real-time co-simulation framework coupling NVIDIA Isaac Sim and NVIDIA Sionna over ROS 2 that closes the perception-action-communication (PAC) loop between them. Sionna ray-traces the base-station-to-robot channel over the exact geometry Isaac Sim simulates on, rather than modeling it stochastically, and feeds channel states back into the control loop in real time. Ray-traced on GPU, the coverage map is refreshed in ~16ms (~60 Hz), fast enough for real-time control. To showcase the framework's utility, we implement a wireless-aware navigation application in an OpenStreetMap(OSM)-derived SUTD campus twin with two Nova Carter robots: the closed-loop planner eliminates communication outage at only +7.4% traversal time over the shortest-path baseline (which spends 7.9 s of its 81.2 s run disconnected).

Binary interaction is common amongst massive stars and can significantly alter their pre-supernova structure and composition. This has important consequences for the nucleosynthetic yields and observable properties of the resulting supernovae. However, binary-evolved progenitors remain largely unexplored in modern neutrino-driven core-collapse supernova simulations. In this work, we present one- and two-dimensional core-collapse supernova simulations of single- and binary-star progenitors. We investigate the impact of binary interaction on the explosion and examine the effect of in-situ networks on the dynamics and resulting nucleosynthesis relative to post-processing approaches. Our progenitors highlight the impact of binary interaction on the pre-supernova core structure of the star and its significance for explodability. We find that binary progenitors are predicted to be generally more explodable than single stars with the same initial mass, indicating that the supernova outcome cannot be well correlated with the initial mass. However, a systematic comparison of the explosion outcomes of binary- and single-star progenitors from self-consistent simulations is still required to confirm this trend. Energy generation from nuclear reactions, as well as the abundances of key isotopes such as ${}^{56}$Ni, can be well approximated by a small $\alpha$-chain network. However, there are notable discrepancies in the production of Ti, Cr, and Fe isotopes which remain sensitive to the network size and thermodynamic history.

A Banach space $X$ is $L$-embedded in its bidual if there is a projection $P\colon X^{**}\to X$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in X^{**}$. We show that, as long as $X$ has dimension at least $2$, its Lipschitz-free space, denoted by $\mathcal{F}(X)$, is not $L$-embedded in its bidual. Our methods have applications to the problem of when the $L$-embeddability of $\mathcal{F}(M)$ passes to $\mathcal{F}(A)$ for a metric space $M$ and $A\subseteq M$. This is the case when $A$ is compact or when $A$ is geodesically closed.

Extending the well-known Riemannian positive mass theorem to low-regularity metrics is an ongoing effort, starting from the early 2000s, that has recently seen renewed interest. Contrary to many other approaches we do not wish to assume smoothness at infinity nor to introduce a new definition of mass tailored to our approach, but rather go back to the coordinate isoperimetric mass originally proposed by Huisken. We establish new asymptotic expressions for the coordinate isoperimetric mass of continuous asymptotically flat metrics, which constitute the main ingredient in proving a finiteness criterion, coordinate invariance and equality to the generalized ADM mass of Lee and LeFloch under reasonable assumptions. In particular, for $W^{1,p}_{-\tau}$ asymptotically flat metrics, with $p>n$, $\tau>(n-2)/2$, and nonnegative distributional scalar curvature, the coordinate isoperimetric mass is indeed coordinate invariant, and we establish a positive mass theorem in this regularity without any topological assumptions aside from asymptotic flatness. Our proof relies on a density theorem similar to that of Lee, Lesourd, and Unger, but under a weaker regularity and decay assumption, which may be of independent interest.

We study entropy production for stationary diffusions on separable Hilbert spaces with possibly degenerate, state-dependent trace-class covariance. We first define the reversible–irreversible drift decomposition to provide a candidate drift for the reversed dynamics. Then by working directly with the invariant measure, we establish a lower bound in terms of the extended stationary Cameron–Martin energy of the irreversible drift, without requiring a diffusion representation of the stationary reversal. The bound implies infinite entropy production when the energy is infinite and we establish sufficient conditions for equality in the finite-energy regime. We also develop complementary criteria for identifying the stationary reversal as a Hilbert-space diffusion with constant or continuous state-dependent diffusion coefficients, respectively. Two nonlinear infinite-rank examples are provided to show that finite entropy production does not require the irreversible drift to lie in the covariance range, whereas entropy production may be infinite even with globally Lipschitz coefficients and injective covariance.

Grain-boundary (GB) segregation in austenitic Fe-Cr-Ni alloys has been studied mainly in irradiated or sensitised material, rarely in the nanocrystalline state. Hybrid molecular dynamics/Monte Carlo (MD/MC) simulations are used to segregate three independent nanocrystalline Fe-18Cr-12Ni polycrystals, which are compared with annealed and random controls on the same microstructures. Ni is depleted at the GB plane and Cr is enriched in the adjacent atomic shell; both directions are established at the production sampling budget and strengthen when one of the three microstructures is sampled further. The Cr enrichment is resolved only against the far-field grain interior; the common partition into face-centred-cubic (fcc) and non-fcc atoms places the enriched shell in its own reference. Under a constant uniaxial stress of 4.6 GPa at 300 K, the segregated polycrystals take about 16% longer than random ones to reach 50% strain; measured against annealed controls, boundary chemistry contributes about two-thirds of this gain and boundary relaxation the rest. The difference arises from a lower strain rate early in plastic flow, with no resolvable change in dislocation content. At a fixed strain rate the segregated state has a 6-7% higher peak stress than its annealed control at 300, 750 and 950 K, and at matched ratios of applied to flow stress the gain is the same at all three temperatures, decreasing only as that ratio rises. Early plastic flow is thus the stage at which GB chemistry acts, and simulated GB composition in nanocrystals depends strongly on the reference region.

We consider a modular associative neural network made of $L$ Hopfield models (HMs), coupled so that intra-module interactions are Hebbian and inter-module interactions are anti-Hebbian; this competitive coupling is known to endow the network with pattern-disentanglement capabilities. The integral representation of this system coincides with an assembly of $L$ restricted Boltzmann machines (RBMs) whose hidden layers are coupled, thereby extending the HM-RBM duality to the modular setting. We then train this modular RBM, via one-step contrastive divergence, to perform a classification task in which a query encoded on the visible layers is mapped onto an $L$-tuple of labels read off the hidden layers. When the query is composed of $L$ patterns that are mutually orthogonal on average, we prove that the RBM weights obtained as empirical means over the training dataset, as suggested by the HM-RBM equivalence, constitute a fixed point of the learning dynamics, and we derive an explicit, non-asymptotic bound on the residual drift. We then turn to a harder classification task in which each module is queried with a mixture of $L$ patterns and we show numerically that the same setting for the RBM weights still provides an effective set-up, letting the network jointly classify and disentangle the mixture.

We study online profit maximization in multilateral trade, a setting that generalizes both dynamic pricing and bilateral trade. At each round, an intermediary posts prices to buyers and offers payments to sellers; trade occurs only if every agent accepts. We consider valuations drawn from an unknown joint distribution, allowing arbitrary correlation across agents, under the assumption that each agent's marginal valuation distribution has unknown support of size at most $K$. For constant dimension $d$, we design an algorithm with regret $\widetilde O(\sqrt{K^dT})$. This extends optimal support-dependent guarantees for one-dimensional dynamic pricing to the multidimensional setting of multilateral trade. At the core of our approach is a normalized zooming procedure that adapts to the local probability of trade, avoiding the excessive refinement of nearly flat regions that can arise with standard zooming.

Understanding the limits of list-decodability of Reed-Solomon codes has been one of the most important open problems in algebraic coding theory. Recently, Brakensiek, Chen, Putterman, Zhang, and Zheng, in a remarkable breakthrough, showed that Reed-Solomon (RS) codes over fields of large characteristic are algorithmically list-decodable all the way up to capacity. Building on this result, Jeronimo subsequently extended these techniques to solve the proximity-gaps question for RS codes. In this article, we give a unified and transparent exposition of these results.