Daily Papers

Discrete autoregressive modeling, notably through next-token prediction, has become a cornerstone of modern language models. Extending this paradigm to vision allows visual and language models to converge on a unified framework, thereby fostering shared methodology and infrastructure. This work investigates a central question: how far can a finite discrete vocabulary go in representing the richness of the visual world? We present dNaViT, a Discrete Native-resolution Vision Transformer that produces discrete visual primitives analogous to language tokens. Built upon a frozen pretrained encoder, dNaViT applies lightweight residual modulation with a dual-branch objective to inject reconstruction-aware cues into semantic features. By leveraging grouped product quantization, it scales discrete capacity without inflating spatial sequence lengths, while a balanced training recipe maintains both semantic and pixel-level information. Consequently, dNaViT functions as a versatile discrete visual tokenizer, enabling unified encoding across downstream tasks alongside high-fidelity image decoding. Extensive evaluations show that dNaViT matches or exceeds task-specific baselines across 12 visual understanding benchmarks, 4 generation benchmarks, and an image editing benchmark. These findings demonstrate that a compact discrete vocabulary can serve as a universal visual currency and unlock the full potential of fully unified, native multimodal methodology.

Discrete diffusion models are gaining traction in the visual generative area for their efficiency and compatibility. However, the pioneered attempts still fall behind the continuous counterparts, which we attribute to the noise (absorbing state) design and sampling heuristics. In this study, we propose the rehashing noise framework for discrete diffusion transformer, termed ReDDiT, to extend absorbing states and improve expressive capacity of discrete diffusion models. ReDDiT enriches the potential paths that latent variables can traverse during training with randomized multi-index corruption. The derived rehash sampler, which reverses the randomized absorbing paths, guarantees the diversity and low discrepancy of the generation process. These reformulations lead to more consistent and competitive generation quality, mitigating the need for heavily tuned randomness. Experiments show that ReDDiT significantly outperforms the baseline (reducing gFID from 6.18 to 1.61) and is on par with the continuous counterparts with higher efficiency.

The representation disparity between visual generation and understanding imposes a critical gap in integrating these capabilities into a single framework. To bridge this gap, we introduce UniTok, a discrete visual tokenizer that encodes fine-grained details for generation while also capturing high-level semantics for understanding. Despite recent studies have shown that these objectives could induce loss conflicts in training, we reveal that the underlying bottleneck stems from limited representational capacity of discrete tokens. We address this by introducing multi-codebook quantization, which divides vector quantization with several independent sub-codebooks to expand the latent feature space, while avoiding training instability caused by overlarge codebooks. Our method significantly raises the upper limit of unified discrete tokenizers to match or even surpass domain-specific continuous tokenizers. For instance, UniTok achieves a remarkable rFID of 0.38 (versus 0.87 for SD-VAE) and a zero-shot accuracy of 78.6% (versus 76.2% for CLIP) on ImageNet. Our code is available at https://github.com/FoundationVision/UniTok.

Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates may violate global conservation over long rollouts. For the important subclass of mass-conservation-type equations, the problem is compounded by inherent physical bounds (e.g., nonnegativity or concentrations in [0,1]) whose violation further destabilizes predictions. We introduce FluxNet, which learns cumulative transport amounts representing the total conserved quantity redistributed between each cell and a configurable neighborhood over the full surrogate interval. A conservative update guarantees exact discrete conservation by construction; modular capacity-constrained transport heads (L, U, and D) enforce lower bounds, upper bounds, or near-zero dual-bound violations through architectural design. Unlike flux-rate surrogates that require temporal integration and thus inherit CFL constraints, FluxNet involves no such integration; configurable transport neighborhoods enable large-timestep prediction at full spatial resolution. Ghost cells extend the framework to non-periodic boundaries. Experiments on four benchmarks (1D convection--diffusion, 2D shallow water, 1D traffic flow, 2D Cahn--Hilliard) demonstrate exact conservation, structural bound preservation, architecture modularity, and superior stability over flux-rate surrogates at large temporal strides. The code is publicly available at: https://github.com/Lan-zs/FluxNet.

Auto-regressive models have achieved impressive results in 2D image generation by modeling joint distributions in grid space. In this paper, we extend auto-regressive models to 3D domains, and seek a stronger ability of 3D shape generation by improving auto-regressive models at capacity and scalability simultaneously. Firstly, we leverage an ensemble of publicly available 3D datasets to facilitate the training of large-scale models. It consists of a comprehensive collection of approximately 900,000 objects, with multiple properties of meshes, points, voxels, rendered images, and text captions. This diverse labeled dataset, termed Objaverse-Mix, empowers our model to learn from a wide range of object variations. However, directly applying 3D auto-regression encounters critical challenges of high computational demands on volumetric grids and ambiguous auto-regressive order along grid dimensions, resulting in inferior quality of 3D shapes. To this end, we then present a novel framework Argus3D in terms of capacity. Concretely, our approach introduces discrete representation learning based on a latent vector instead of volumetric grids, which not only reduces computational costs but also preserves essential geometric details by learning the joint distributions in a more tractable order. The capacity of conditional generation can thus be realized by simply concatenating various conditioning inputs to the latent vector, such as point clouds, categories, images, and texts. In addition, thanks to the simplicity of our model architecture, we naturally scale up our approach to a larger model with an impressive 3.6 billion parameters, further enhancing the quality of versatile 3D generation. Extensive experiments on four generation tasks demonstrate that Argus3D can synthesize diverse and faithful shapes across multiple categories, achieving remarkable performance.

Sparse mixture of experts (SMoE) is an effective solution for scaling up model capacity without increasing the computational costs. A crucial component of SMoE is the router, responsible for directing the input to relevant experts; however, it also presents a major weakness, leading to routing inconsistencies and representation collapse issues. Instead of fixing the router like previous works, we propose an alternative that assigns experts to input via indirection, which employs the discrete representation of input that points to the expert. The discrete representations are learnt via vector quantization, resulting in a new architecture dubbed Vector-Quantized Mixture of Experts (VQMoE). We provide theoretical support and empirical evidence demonstrating the VQMoE's ability to overcome the challenges present in traditional routers. Through extensive evaluations on both large language models and vision tasks for pre-training and fine-tuning, we show that VQMoE achieves a 28% improvement in robustness compared to other SMoE routing methods, while maintaining strong performance in fine-tuning tasks.

Recent advances in quantum information science enabled the development of quantum communication network prototypes and created an opportunity to study full-stack quantum network architectures. This work develops SeQUeNCe, a comprehensive, customizable quantum network simulator. Our simulator consists of five modules: Hardware models, Entanglement Management protocols, Resource Management, Network Management, and Application. This framework is suitable for simulation of quantum network prototypes that capture the breadth of current and future hardware technologies and protocols. We implement a comprehensive suite of network protocols and demonstrate the use of SeQUeNCe by simulating a photonic quantum network with nine routers equipped with quantum memories. The simulation capabilities are illustrated in three use cases. We show the dependence of quantum network throughput on several key hardware parameters and study the impact of classical control message latency. We also investigate quantum memory usage efficiency in routers and demonstrate that redistributing memory according to anticipated load increases network capacity by 69.1% and throughput by 6.8%. We design SeQUeNCe to enable comparisons of alternative quantum network technologies, experiment planning, and validation and to aid with new protocol design. We are releasing SeQUeNCe as an open source tool and aim to generate community interest in extending it.

Multimodal large language models (MLLMs) extend the success of language models to visual understanding, and recent efforts have sought to build unified MLLMs that support both understanding and generation. However, constructing such models remains challenging: hybrid approaches combine continuous embeddings with diffusion or flow-based objectives, producing high-quality images but breaking the autoregressive paradigm, while pure autoregressive approaches unify text and image prediction over discrete visual tokens but often face trade-offs between semantic alignment and pixel-level fidelity. In this work, we present Bridge, a pure autoregressive unified MLLM that augments pre-trained visual understanding models with generative ability through a Mixture-of-Transformers architecture, enabling both image understanding and generation within a single next-token prediction framework. To further improve visual generation fidelity, we propose a semantic-to-pixel discrete representation that integrates compact semantic tokens with fine-grained pixel tokens, achieving strong language alignment and precise description of visual details with only a 7.9% increase in sequence length. Extensive experiments across diverse multimodal benchmarks demonstrate that Bridge achieves competitive or superior results in both understanding and generation benchmarks, while requiring less training data and reduced training time compared to prior unified MLLMs.

We rigorously state the connection between the EHZ-capacity of convex Lagrangian products Ktimes TsubsetR^ntimesR^n and the minimal length of closed (K,T)-Minkowski billiard trajectories. This connection was made explicit for the first time by Artstein-Avidan and Ostrover under the assumption of smoothness and strict convexity of both K and T. We prove this connection in its full generality, i.e., without requiring any conditions on the convex bodies K and T. This prepares the computation of the EHZ-capacity of convex Lagrangian products of two convex polytopes by using discrete computational methods.

Autoregressive visual generation models typically rely on tokenizers to compress images into tokens that can be predicted sequentially. A fundamental dilemma exists in token representation: discrete tokens enable straightforward modeling with standard cross-entropy loss, but suffer from information loss and tokenizer training instability; continuous tokens better preserve visual details, but require complex distribution modeling, complicating the generation pipeline. In this paper, we propose TokenBridge, which bridges this gap by maintaining the strong representation capacity of continuous tokens while preserving the modeling simplicity of discrete tokens. To achieve this, we decouple discretization from the tokenizer training process through post-training quantization that directly obtains discrete tokens from continuous representations. Specifically, we introduce a dimension-wise quantization strategy that independently discretizes each feature dimension, paired with a lightweight autoregressive prediction mechanism that efficiently model the resulting large token space. Extensive experiments show that our approach achieves reconstruction and generation quality on par with continuous methods while using standard categorical prediction. This work demonstrates that bridging discrete and continuous paradigms can effectively harness the strengths of both approaches, providing a promising direction for high-quality visual generation with simple autoregressive modeling. Project page: https://yuqingwang1029.github.io/TokenBridge.

Diffusion models have gained prominence in generating high-quality sequences of text. Nevertheless, current approaches predominantly represent discrete text within a continuous diffusion space, which incurs substantial computational overhead during training and results in slower sampling speeds. In this paper, we introduce a soft absorbing state that facilitates the diffusion model in learning to reconstruct discrete mutations based on the underlying Gaussian space, thereby enhancing its capacity to recover conditional signals. During the sampling phase, we employ state-of-the-art ODE solvers within the continuous space to expedite the sampling process. Comprehensive experimental evaluations reveal that our proposed method effectively accelerates the training convergence by 4x and generates samples of similar quality 800x faster, rendering it significantly closer to practical application. The code is released at \url{https://github.com/Shark-NLP/DiffuSeq

The Universal Approximation Theorem posits that neural networks can theoretically possess unlimited approximation capacity with a suitable activation function and a freely chosen or trained set of parameters. However, a more practical scenario arises when these neural parameters, especially the nonlinear weights and biases, are bounded. This leads us to question: Does the approximation capacity of a neural network remain universal, or does it have a limit when the parameters are practically bounded? And if it has a limit, how can it be measured? Our theoretical study indicates that while universal approximation is theoretically feasible, in practical numerical scenarios, Deep Neural Networks (DNNs) with any analytic activation functions (such as Tanh and Sigmoid) can only be approximated by a finite-dimensional vector space under a bounded nonlinear parameter space (NP space), whether in a continuous or discrete sense. Based on this study, we introduce the concepts of ε outer measure and Numerical Span Dimension (NSdim) to quantify the approximation capacity limit of a family of networks both theoretically and practically. Furthermore, drawing on our new theoretical study and adopting a fresh perspective, we strive to understand the relationship between back-propagation neural networks and random parameter networks (such as the Extreme Learning Machine (ELM)) with both finite and infinite width. We also aim to provide fresh insights into regularization, the trade-off between width and depth, parameter space, width redundancy, condensation, and other related important issues.

Memory systems can store vastly different amounts of information despite similar hardware constraints. Here, we show that superior spatial memory emerges from a discrete stiffening of hippocampal population geometry-a transition from disorganized to crystalline collective coding. Comparing food-caching chickadees to non-caching zebra finches, we found that the caching hippocampus maintains a topologically rigid, "crystalline" geometry with significantly higher geometric stability (Shesha 0.245 v 0.166) and nearly two-fold greater temporal coherence (Shesha 0.393 v 0.209), while the non-caching hippocampus resembles a disorganized "mist." This stability is actively constructed by synergistic circuit dynamics: excitatory neurons form the spatial scaffold while inhibitory populations contribute orthogonal decorrelation, a circuit motif in which excitatory and inhibitory populations occupy largely non-overlapping representational subspaces. A double dissociation with Valiant's Stable Memory Allocator, a model predicting that dedicated neuron ensembles underlie each memory, confirms this advantage reflects continuous topological organization rather than discrete neuron allocation: caching networks exhibit near-zero split-half allocation reliability despite their geometric superiority. Computational modeling across 10k configurations reveals topological rigidity as the mathematical prerequisite for scale: crystalline codes sustain high-fidelity readout beyond M=1k locations while mist codes fail below M=10, a >100-fold capacity advantage. This capacity requires a 169fold representational redundancy: a "geometric tax" stabilizing the manifold against biological noise. These results establish geometric stability as a candidate organizing principle of biological memory: evolution achieves high-capacity memory not by proliferating neurons, but by engineering the geometry of the neural code itself.

The DNA storage channel is considered, in which the M Deoxyribonucleic acid (DNA) molecules comprising each codeword are stored without order, sampled N times with replacement, and then sequenced over a discrete memoryless channel. For a constant coverage depth M/N and molecule length scaling Θ(log M), lower (achievability) and upper (converse) bounds on the capacity of the channel, as well as a lower (achievability) bound on the reliability function of the channel are provided. Both the lower and upper bounds on the capacity generalize a bound which was previously known to hold only for the binary symmetric sequencing channel, and only under certain restrictions on the molecule length scaling and the crossover probability parameters. When specified to binary symmetric sequencing channel, these restrictions are completely removed for the lower bound and are significantly relaxed for the upper bound in the high-noise regime. The lower bound on the reliability function is achieved under a universal decoder, and reveals that the dominant error event is that of outage -- the event in which the capacity of the channel induced by the DNA molecule sampling operation does not support the target rate.

Speech watermarking techniques can proactively mitigate the potential harmful consequences of instant voice cloning techniques. These techniques involve the insertion of signals into speech that are imperceptible to humans but can be detected by algorithms. Previous approaches typically embed watermark messages into continuous space. However, intuitively, embedding watermark information into robust discrete latent space can significantly improve the robustness of watermarking systems. In this paper, we propose DiscreteWM, a novel speech watermarking framework that injects watermarks into the discrete intermediate representations of speech. Specifically, we map speech into discrete latent space with a vector-quantized autoencoder and inject watermarks by changing the modular arithmetic relation of discrete IDs. To ensure the imperceptibility of watermarks, we also propose a manipulator model to select the candidate tokens for watermark embedding. Experimental results demonstrate that our framework achieves state-of-the-art performance in robustness and imperceptibility, simultaneously. Moreover, our flexible frame-wise approach can serve as an efficient solution for both voice cloning detection and information hiding. Additionally, DiscreteWM can encode 1 to 150 bits of watermark information within a 1-second speech clip, indicating its encoding capacity. Audio samples are available at https://DiscreteWM.github.io/discrete_wm.

Discrete flow matching generates text by iteratively transforming noise tokens into coherent language, but may require hundreds of forward passes. Distillation uses the multi-step trajectory to train a student to reproduce the process in a few steps. When the student underperforms, the usual explanation is insufficient capacity. We argue the opposite: the trajectory is the bottleneck, not the student. Each training trajectory is built through a chain of blind stochastic jumps with no evaluation of sequence quality; a single bad decision at an early midpoint propagates through subsequent steps, yet the student must imitate the result. Trajectory-Shaped Discrete Flow Matching (TS-DFM) replaces these blind jumps with guided navigation: a lightweight energy compass evaluates candidate continuations at each midpoint, selecting the most coherent. All shaping is training-only; inference cost is unchanged. On 170M-parameter language modeling, the shaped student at 8 steps achieves 32% lower perplexity than the 1,024-step teacher while being 128x faster, with gains consistent across source distributions and three evaluators of increasing scale. TS-DFM achieves the best perplexity of any discrete-generation baseline we compare against, including methods trained on 6x more data or using 5x larger models.

We address Human Activity Recognition (HAR) utilizing Wi-Fi Channel State Information (CSI) under the joint requirements of causal interpretability, symbolic controllability, and direct operation on high-dimensional raw signals. Deep neural models achieve strong predictive performance on CSI-based HAR (CHAR), yet rely on continuous latent representations that are opaque and difficult to modify; purely symbolic approaches, in contrast, cannot process raw CSI streams. We propose a fully automatic and strictly decoupled pipeline in which CSI magnitude windows are compressed by a categorical variational autoencoder with Gumbel-Softmax latent variables under a capacity-controlled objective, yielding a compact discrete representation. The encoder is then frozen and used as a deterministic mapping to one-hot latent trajectories. Causal discovery is performed on these trajectories to estimate class-conditional temporal dependency graphs. Statistically supported lagged dependencies are translated into Linear Temporal Logic (LTL) rules, producing a fully symbolic and deterministic classifier based solely on rule evaluation and aggregation, without any learned discriminative head. Because rules are defined over discrete latent variables, antenna-specific rule sets can in principle be combined at the symbolic level, enabling structured multi-antenna fusion without retraining the encoder. Results from CHAR Latent Temporal Rule Extraction (CHARL-TRE) indicate competitive performance while preserving explicit temporal and causal structure, showing that deterministic symbolic classification grounded in unsupervised discrete latent representations constitutes a viable alternative to end-to-end black-box models for wireless HAR.

Despite remarkable progress in autoregressive language models, alternative generative paradigms beyond left-to-right generation are still being actively explored. Discrete diffusion models, with the capacity for parallel generation, have recently emerged as a promising alternative. Unfortunately, these models still underperform the autoregressive counterparts, with the performance gap increasing when reducing the number of sampling steps. Our analysis reveals that this degradation is a consequence of an imperfect approximation used by diffusion models. In this work, we propose Energy-based Diffusion Language Model (EDLM), an energy-based model operating at the full sequence level for each diffusion step, introduced to improve the underlying approximation used by diffusion models. More specifically, we introduce an EBM in a residual form, and show that its parameters can be obtained by leveraging a pretrained autoregressive model or by finetuning a bidirectional transformer via noise contrastive estimation. We also propose an efficient generation algorithm via parallel important sampling. Comprehensive experiments on language modeling benchmarks show that our model can consistently outperform state-of-the-art diffusion models by a significant margin, and approaches autoregressive models' perplexity. We further show that, without any generation performance drop, our framework offers a 1.3times sampling speedup over existing diffusion models.

This paper introduces a novel architecture for trajectory-conditioned forecasting of future 3D scene occupancy. In contrast to methods that rely on variational autoencoders (VAEs) to generate discrete occupancy tokens, which inherently limit representational capacity, our approach predicts multi-frame future occupancy in an end-to-end manner directly from raw image features. Inspired by the success of attention-based transformer architectures in foundational vision and language models such as GPT and VGGT, we employ a sparse occupancy representation that bypasses the intermediate bird's eye view (BEV) projection and its explicit geometric priors. This design allows the transformer to capture spatiotemporal dependencies more effectively. By avoiding both the finite-capacity constraint of discrete tokenization and the structural limitations of BEV representations, our method achieves state-of-the-art performance on the nuScenes benchmark for 1-3 second occupancy forecasting, outperforming existing approaches by a significant margin. Furthermore, it demonstrates robust scene dynamics understanding, consistently delivering high accuracy under arbitrary future trajectory conditioning.

Autoregressive large language models (LLMs) scale well by expressing diverse tasks as sequences of discrete natural-language tokens and training with next-token prediction, which unifies comprehension and generation under self-supervision. Extending this paradigm to multimodal data requires a shared, discrete representation across modalities. However, most vision-language models (VLMs) still rely on a hybrid interface: discrete text tokens paired with continuous Vision Transformer (ViT) features. Because supervision is largely text-driven, these models are often biased toward understanding and cannot fully leverage large-scale self-supervised learning on non-text data. Recent work has explored discrete visual tokenization to enable fully autoregressive multimodal modeling, showing promising progress toward unified understanding and generation. Yet existing discrete vision tokens frequently lose information due to limited code capacity, resulting in noticeably weaker understanding than continuous-feature VLMs. We present Kelix, a fully discrete autoregressive unified model that closes the understanding gap between discrete and continuous visual representations.

Time series prototype learning is fundamentally challenged by observational ambiguity. Discrete architectures fail to resolve this, as they lack the capacity to decouple stochastic noise from continuous dynamics. Furthermore, rigid closed-set assumptions fail to capture unseen diversity. To address these limitations, we propose a hierarchical ordinary differential equation clustering network, which utilizes neural ordinary differential equation to model latent state evolution as a continuous integral curve. This formulation enforces temporal continuity to effectively disentangle smooth feature trends from stochastic noise, while our adaptive hierarchical mechanism autonomously determines the appropriate number of prototypes without rigid prior constraints. Validated on the early link failure detection task with irregularly sampled time series, the proposed method effectively extracts underlying physical prototypes, thereby enabling robust failure detection. Our code is available at https://github.com/NJ-LNN/Hierarchical-ODE.

Unified multimodal generation covers discrete, continuous, and hybrid paradigms. Discrete approaches with single-codebook visual tokenizers align naturally with text modeling but have limited representational capacity. Continuous and hybrid approaches retain richer visual representations, but they model text and images in heterogeneous spaces with different training objectives, potentially introducing optimization conflicts between modalities with misaligned representation spaces. These choices expose a central question: how to efficiently model high-capacity visual representations alongside textual content in a shared optimization space within a single backbone. We present Intern Lumina U2, to our knowledge the first diffusion language model to represent multi-codebook visual tokens in a fully discrete space. Each visual position is represented by a group of codes from complementary codebooks, increasing its representational capacity while preserving compatibility with discrete text modeling. At the input, the embeddings are concatenated at each position; at the output, a lightweight multi-codebook autoregressive head predicts the codes within each position sequentially, while the diffusion backbone processes all positions in parallel. Built on the unified 4D representation of AToken, Intern Lumina U2 supports image, video, and 3D inputs within a single backbone and achieves leading performance among unified models in visual understanding, image generation, and image editing. We further build a full-stack training, inference, and evaluation system on both Ascend and NVIDIA platforms to verify the cross-platform effectiveness of our model. We also improve the inference efficiency through a stability-aware sampling strategy, which combines confidence-plateau-guided unmasking approach with drift-triggered cache refreshing mechanisms, accelerating image generation by 4.45x.

We introduce a novel generative model, the Discrete Distribution Networks (DDN), that approximates data distribution using hierarchical discrete distributions. We posit that since the features within a network inherently capture distributional information, enabling the network to generate multiple samples simultaneously, rather than a single output, may offer an effective way to represent distributions. Therefore, DDN fits the target distribution, including continuous ones, by generating multiple discrete sample points. To capture finer details of the target data, DDN selects the output that is closest to the Ground Truth (GT) from the coarse results generated in the first layer. This selected output is then fed back into the network as a condition for the second layer, thereby generating new outputs more similar to the GT. As the number of DDN layers increases, the representational space of the outputs expands exponentially, and the generated samples become increasingly similar to the GT. This hierarchical output pattern of discrete distributions endows DDN with unique properties: more general zero-shot conditional generation and 1D latent representation. We demonstrate the efficacy of DDN and its intriguing properties through experiments on CIFAR-10 and FFHQ. The code is available at https://discrete-distribution-networks.github.io/

Vision-Language Models have excelled at textual reasoning, but they often struggle with fine-grained spatial understanding and continuous action planning, failing to simulate the dynamics required for complex visual reasoning. In this work, we formulate visual reasoning by means of video generation models, positing that generated frames can act as intermediate reasoning steps between initial states and solutions. We evaluate their capacity in two distinct regimes: Maze Navigation for sequential discrete planning with low visual change and Tangram Puzzle for continuous manipulation with high visual change. Our experiments reveal three critical insights: (1) Robust Zero-Shot Generalization: In both tasks, the model demonstrates strong performance on unseen data distributions without specific finetuning. (2) Visual Context: The model effectively uses visual context as explicit control, such as agent icons and tangram shapes, enabling it to maintain high visual consistency and adapt its planning capability robustly to unseen patterns. (3) Visual Test-Time Scaling: We observe a test-time scaling law in sequential planning; increasing the generated video length (visual inference budget) empowers better zero-shot generalization to spatially and temporally complex paths. These findings suggest that video generation is not merely a media tool, but a scalable, generalizable paradigm for visual reasoning.

The ability to render scenes at adjustable fidelity from a single model, known as level of detail (LoD), is crucial for practical deployment of 3D Gaussian Splatting (3DGS). Existing discrete LoD methods expose only a limited set of operating points, while concurrent continuous LoD approaches enable smoother scaling but often suffer noticeable quality degradation at full capacity, making LoD a costly design decision. We introduce Matryoshka Gaussian Splatting (MGS), a training framework that enables continuous LoD for standard 3DGS pipelines without sacrificing full-capacity rendering quality. MGS learns a single ordered set of Gaussians such that rendering any prefix, the first k splats, produces a coherent reconstruction whose fidelity improves smoothly with increasing budget. Our key idea is stochastic budget training: each iteration samples a random splat budget and optimises both the corresponding prefix and the full set. This strategy requires only two forward passes and introduces no architectural modifications. Experiments across four benchmarks and six baselines show that MGS matches the full-capacity performance of its backbone while enabling a continuous speed-quality trade-off from a single model. Extensive ablations on ordering strategies, training objectives, and model capacity further validate the designs.

Orbital angular momentum (OAM) multiplexing provides an infinite-dimensional discrete Hilbert space ideally suited for high-capacity free-space quantum key distribution (QKD). Nevertheless, pure scalar spatial modes carrying topological charge (|ell| ge 1) undergo severe decoherence when transmitted through terrestrial atmospheric turbulence. Turbulent refractive-index eddies split high-order vortex singularities, induce catastrophic intermodal crosstalk across adjacent topological channels, and rapidly drive the quantum bit error rate (QBER) well above the unconditional 11% security threshold associated with individual cloning attacks. Here, we demonstrate that hybrid polarization-OAM entangled states, known as vector vortex beams (VVBs), provide intrinsic, hardware-free immunity against turbulent perturbations. Because the optical anisotropy of terrestrial air is exceedingly small (Δn < 10^{-9}), refractive-index fluctuations couple symmetrically to orthogonal circular polarization modes as an identical common-mode scalar phase screen that cancels in the relative polarization-phase degree of freedom. By numerically propagating modal fields through modified power-spectral phase screens over turbulence strengths ranging from D/r_0 = 0 to 3.0, we show that the VVB encoding protocol suppresses the asymptotic QBER from 42.0% to 4.8%, yielding an error-suppression factor of approximately 11.6 without requiring active adaptive optics or deformable mirrors.

Large language models increasingly rely on sampling as a driver of their own improvement, making the fidelity of their learned distributions more critical than ever. Yet, not all distributions are equally easy to learn. In this work, we identify a curse of ambiguity: in large language models, and more broadly in all neural networks that produce discrete probability distributions, the more ambiguous a next-token distribution is, the harder it is to learn accurately. Through an extensive theoretical analysis, we trace this curse to architectural and learning roots. More ambiguous distributions require more capacity to be stored, larger embeddings to be represented, more steps to be fitted, and amplify token-sampling noise. We validate these findings on synthetic tasks with controlled ground truth and observe the same signatures in language models trained on real data. Our results provide a new perspective on the statistical capabilities of large language models and a practical framework for when to trust their output distribution.

Large language models increasingly serve as persistent knowledge workers, with in-context memory - facts stored in the prompt - as the default strategy. We benchmark in-context memory against Knowledge Objects (KOs), discrete hash-addressed tuples with O(1) retrieval. Within the context window, Claude Sonnet 4.5 achieves 100% exact-match accuracy from 10 to 7,000 facts (97.5% of its 200K window). However, production deployment reveals three failure modes: capacity limits (prompts overflow at 8,000 facts), compaction loss (summarization destroys 60% of facts), and goal drift (cascading compaction erodes 54% of project constraints while the model continues with full confidence). KOs achieve 100% accuracy across all conditions at 252x lower cost. On multi-hop reasoning, KOs reach 78.9% versus 31.6% for in-context. Cross-model replication across four frontier models confirms compaction loss is architectural, not model-specific. We additionally show that embedding retrieval fails on adversarial facts (20% precision at 1) and that neural memory (Titans) stores facts but fails to retrieve them on demand. We introduce density-adaptive retrieval as a switching mechanism and release the benchmark suite.

Latent Action Models (LAMs) have emerged as an effective paradigm for handling heterogeneous datasets during Vision-Language-Action (VLA) model pretraining, offering a unified action space across embodiments. However, existing LAMs often rely on discrete quantization encode and decode pipelines, which can lead to trivial frame reconstruction behavior, limited representational capacity, and a lack of physically meaningful structure. We introduce RotVLA, a VLA framework built on a continuous rotational latent action representation. Latent actions are modeled as elements of SO(n), providing continuity, compositionality, and structured geometry aligned with real-world action dynamics. A triplet frame learning framework further enforces meaningful temporal dynamics while avoiding degeneration. RotVLA consists of a VLM backbone and a flow-matching action head, pretrained on large-scale cross-embodiment robotic datasets and human videos with latent-action supervision. For downstream robot control, the flow-matching head is extended into a unified action expert that jointly denoises latent and robot actions. Here, latent actions serve as a latent planner, providing high-level guidance that conditions action generation. With only 1.7B parameters and 1700+ hours of pretraining data, RotVLA achieves 98.2% on LIBERO and 89.6% / 88.5% on RoboTwin2.0 under clean and randomized settings, respectively. It also demonstrates strong real-world performance on manipulation tasks, consistently outperforming existing VLA models.

Recently slot filling has witnessed great development thanks to deep learning and the availability of large-scale annotated data. However, it poses a critical challenge to handle a novel domain whose samples are never seen during training. The recognition performance might be greatly degraded due to severe domain shifts. Most prior works deal with this problem in a two-pass pipeline manner based on metric learning. In practice, these dominant pipeline models may be limited in computational efficiency and generalization capacity because of non-parallel inference and context-free discrete label embeddings. To this end, we re-examine the typical metric-based methods, and propose a new adaptive end-to-end metric learning scheme for the challenging zero-shot slot filling. Considering simplicity, efficiency and generalizability, we present a cascade-style joint learning framework coupled with context-aware soft label representations and slot-level contrastive representation learning to mitigate the data and label shift problems effectively. Extensive experiments on public benchmarks demonstrate the superiority of the proposed approach over a series of competitive baselines.

Quantization-aware training (QAT) is essential for extremely low-bit large language models (LLMs). Current QAT methods are mainly based on scalar quantization (SQ), which enables efficient optimization but suffers from severe performance degradation at 2-bit precision. On the other hand, vector quantization (VQ) provides substantially higher representational capacity, but its discrete codebook lookup prevents end-to-end training. We propose LC-QAT, a 2-bit weight-only VQ-QAT framework that represents quantized weights via a learned affine mapping over discrete vectors, which yields a high-quality PTQ initialization and enables fully differentiable end-to-end optimization without explicit codebook lookup in the training forward pass. This strong post-training initialization makes LC-QAT highly data-efficient. Experiments across diverse LLMs demonstrate that LC-QAT consistently outperforms state-of-the-art QAT methods while using only 0.1%--10% of the training data. Our results establish LC-QAT as a practical and scalable solution for extreme low-bit model deployment. Codes are publicly available at https://github.com/AI9Stars/UniSVQ.

Humans rely on their visual and tactile senses to develop a comprehensive 3D understanding of their physical environment. Recently, there has been a growing interest in exploring and manipulating objects using data-driven approaches that utilise high-resolution vision-based tactile sensors. However, 3D shape reconstruction using tactile sensing has lagged behind visual shape reconstruction because of limitations in existing techniques, including the inability to generalise over unseen shapes, the absence of real-world testing, and limited expressive capacity imposed by discrete representations. To address these challenges, we propose TouchSDF, a Deep Learning approach for tactile 3D shape reconstruction that leverages the rich information provided by a vision-based tactile sensor and the expressivity of the implicit neural representation DeepSDF. Our technique consists of two components: (1) a Convolutional Neural Network that maps tactile images into local meshes representing the surface at the touch location, and (2) an implicit neural function that predicts a signed distance function to extract the desired 3D shape. This combination allows TouchSDF to reconstruct smooth and continuous 3D shapes from tactile inputs in simulation and real-world settings, opening up research avenues for robust 3D-aware representations and improved multimodal perception in robotics. Code and supplementary material are available at: https://touchsdf.github.io/

Scaling the capacity of language models has consistently proven to be a reliable approach for improving performance and unlocking new capabilities. Capacity can be primarily defined by two dimensions: the number of model parameters and the compute per example. While scaling typically involves increasing both, the precise interplay between these factors and their combined contribution to overall capacity remains not fully understood. We explore this relationship in the context of sparse Mixture-of-Experts (MoEs), which allow scaling the number of parameters without proportionally increasing the FLOPs per example. We investigate how varying the sparsity level, i.e., the fraction of inactive parameters, impacts model's performance during pretraining and downstream few-shot evaluation. We find that under different constraints (e.g., parameter size and total training compute), there is an optimal level of sparsity that improves both training efficiency and model performance. These results provide a better understanding of the impact of sparsity in scaling laws for MoEs and complement existing works in this area, offering insights for designing more efficient architectures.

We study the problem of predicting numeric labels that are constrained to the integers or to a subrange of the integers. For example, the number of up-votes on social media posts, or the number of bicycles available at a public rental station. While it is possible to model these as continuous values, and to apply traditional regression, this approach changes the underlying distribution on the labels from discrete to continuous. Discrete distributions have certain benefits, which leads us to the question whether such integer labels can be modeled directly by a discrete distribution, whose parameters are predicted from the features of a given instance. Moreover, we focus on the use case of output distributions of neural networks, which adds the requirement that the parameters of the distribution be continuous so that backpropagation and gradient descent may be used to learn the weights of the network. We investigate several options for such distributions, some existing and some novel, and test them on a range of tasks, including tabular learning, sequential prediction and image generation. We find that overall the best performance comes from two distributions: Bitwise, which represents the target integer in bits and places a Bernoulli distribution on each, and a discrete analogue of the Laplace distribution, which uses a distribution with exponentially decaying tails around a continuous mean.

Despite Flow Matching and diffusion models having emerged as powerful generative paradigms for continuous variables such as images and videos, their application to high-dimensional discrete data, such as language, is still limited. In this work, we present Discrete Flow Matching, a novel discrete flow paradigm designed specifically for generating discrete data. Discrete Flow Matching offers several key contributions: (i) it works with a general family of probability paths interpolating between source and target distributions; (ii) it allows for a generic formula for sampling from these probability paths using learned posteriors such as the probability denoiser (x-prediction) and noise-prediction (epsilon-prediction); (iii) practically, focusing on specific probability paths defined with different schedulers considerably improves generative perplexity compared to previous discrete diffusion and flow models; and (iv) by scaling Discrete Flow Matching models up to 1.7B parameters, we reach 6.7% Pass@1 and 13.4% Pass@10 on HumanEval and 6.7% Pass@1 and 20.6% Pass@10 on 1-shot MBPP coding benchmarks. Our approach is capable of generating high-quality discrete data in a non-autoregressive fashion, significantly closing the gap between autoregressive models and discrete flow models.

Model dimension (d_{model}) is a fundamental hyperparameter in transformer language models, yet its role in setting the geometric limits of feature representation remains under-explored. Grounded in the Linear Representation and Superposition Hypotheses - which propose that models encode features as near-orthogonal directions in latent space - we develop a framework for estimating how many such directions a model can support. We first establish the embedding matrix as a measurable proxy for near-orthogonality constraints across the latent space: the boundary between meaningful token relationships and incidental similarity in the pairwise cosine similarity distribution gives a concrete estimate of the model's accepted deviation varepsilon from perfect orthogonality. Applying this metric across dozens of open-source models reveals two classes: models with high varepsilon whose embeddings lack near-orthogonal structure, and models with low varepsilon that maintain it. We then show that the standard Johnson-Lindenstrauss lemma greatly underestimates the packing efficiency of trained representations, and derive an adjusted capacity formula in which the number of near-orthogonal directions depends on the ratio of vectors to dimensions (k/d) rather than the raw count - a single modification that cuts prediction error by two orders of magnitude with no extra parameters. Combining these results, we define representational capacity as an upper bound on the number of distinguishable directions available for features and embeddings in a model's latent space. Capacity is exponentially sensitive to varepsilon, and larger models favor tighter orthogonality constraints over maximizing raw capacity - a pattern compatible with several explanations (a stability-capacity trade-off, a ceiling on usable concepts, or confounds with model scale) that we leave to future work.

A new information theoretic condition is presented for reconstructing a discrete random variable X based on the knowledge of a set of discrete functions of X. The reconstruction condition is derived from Shannon's 1953 lattice theory with two entropic metrics of Shannon and Rajski. Because such a theoretical material is relatively unknown and appears quite dispersed in different references, we first provide a synthetic description (with complete proofs) of its concepts, such as total, common and complementary informations. Definitions and properties of the two entropic metrics are also fully detailed and shown compatible with the lattice structure. A new geometric interpretation of such a lattice structure is then investigated that leads to a necessary (and sometimes sufficient) condition for reconstructing the discrete random variable X given a set { X_1,ldots,X_{n} } of elements in the lattice generated by X. Finally, this condition is illustrated in five specific examples of perfect reconstruction problems: reconstruction of a symmetric random variable from the knowledge of its sign and absolute value, reconstruction of a word from a set of linear combinations, reconstruction of an integer from its prime signature (fundamental theorem of arithmetic) and from its remainders modulo a set of coprime integers (Chinese remainder theorem), and reconstruction of the sorting permutation of a list from a minimal set of pairwise comparisons.

The Entropic Optimal Transport (EOT) problem and its dynamic counterpart, the Schrödinger bridge (SB) problem, play an important role in modern machine learning, linking generative modeling with optimal transport theory. While recent advances in discrete diffusion and flow models have sparked growing interest in applying SB methods to discrete domains, there is still no reliable way to evaluate how well these methods actually solve the underlying problem. We address this challenge by introducing a benchmark for SB on discrete spaces. Our construction yields pairs of probability distributions with analytically known SB solutions, enabling rigorous evaluation. As a byproduct of building this benchmark, we obtain two new SB algorithms, DLightSB and DLightSB-M, and additionally extend prior related work to construct the α-CSBM algorithm. We demonstrate the utility of our benchmark by evaluating both existing and new solvers in high-dimensional discrete settings. This work provides the first step toward proper evaluation of SB methods on discrete spaces, paving the way for more reproducible future studies.

We introduce the Discrete Min-Max Violation (DMMV) as a general optimization problem which seeks an assignment of discrete values to variables that minimizes the largest constraint violation. This context-free mathematical formulation is applicable to a wide range of use cases that have worst-case performance requirements. After defining the DMMV problem mathematically, we explore its properties to establish a foundational understanding. To tackle DMMV instance sizes of practical relevance, we develop a GPU-accelerated heuristic that takes advantage of the mathematical properties of DMMV for speeding up the solution process. We demonstrate the versatile applicability of our heuristic by solving three optimization problems as use cases: (1) post-training quantization of language models, (2) discrete tomography, and (3) Finite Impulse Response (FIR) filter design. In quantization without outlier separation, our heuristic achieves 14% improvement on average over existing methods. In discrete tomography, it reduces reconstruction error by 16% under uniform noise and accelerates computations by a factor of 6 on GPU. For FIR filter design, it nearly achieves 50% ripple reduction compared to using the commercial integer optimization solver, Gurobi. Our comparative results point to the benefits of studying DMMV as a context-free optimization problem and the advantages that our proposed heuristic offers on three distinct problems. Our GPU-accelerated heuristic will be made open-source to further stimulate research on DMMV and its other applications. The code is available at https://anonymous.4open.science/r/AMVM-5F3E/

Generative models have advanced significantly in sampling material systems with continuous variables, such as atomistic structures. However, their application to discrete variables, like atom types or spin states, remains underexplored. In this work, we introduce a Boltzmann generator built on discrete flow matching, specifically tailored for systems with discrete phase-space coordinates (e.g., the Ising model or crystalline compounds). This approach enables a single model to sample free energy surfaces over a wide temperature range with minimal training overhead. In addition, the model generation is scalable to larger lattice sizes than those in the training set. We demonstrate the effectiveness of our approach on the 2D Ising model, showing efficient and reliable free energy sampling. This framework provides a scalable and computationally efficient solution for discrete coordinate systems and can be extended to sample the alchemical degrees of freedom in crystalline compounds.

Discrete adversarial attacks are symbolic perturbations to a language input that preserve the output label but lead to a prediction error. While such attacks have been extensively explored for the purpose of evaluating model robustness, their utility for improving robustness has been limited to offline augmentation only. Concretely, given a trained model, attacks are used to generate perturbed (adversarial) examples, and the model is re-trained exactly once. In this work, we address this gap and leverage discrete attacks for online augmentation, where adversarial examples are generated at every training step, adapting to the changing nature of the model. We propose (i) a new discrete attack, based on best-first search, and (ii) random sampling attacks that unlike prior work are not based on expensive search-based procedures. Surprisingly, we find that random sampling leads to impressive gains in robustness, outperforming the commonly-used offline augmentation, while leading to a speedup at training time of ~10x. Furthermore, online augmentation with search-based attacks justifies the higher training cost, significantly improving robustness on three datasets. Last, we show that our new attack substantially improves robustness compared to prior methods.

An important feature of many real world facility location problems are capacity limits on the facilities. We show here how capacity constraints make it harder to design strategy proof mechanisms for facility location, but counter-intuitively can improve the guarantees on how well we can approximate the optimal solution.

Ternary Large Language Models (LLM) store every weight as one of three symbols {-1,0,+1}, so the cost of a ternary model is conventionally referenced to the information-theoretic log_2 3 approx 1.585 bits per weight. The prevailing deployment format packs five ternary weights into one byte (five-trit packing), and due to the power-of-two group sizes used in practice this rounds up to 1.625 bits per weight. This effective storage bit-width treats the three symbols {-1,0,+1} as equiprobable. We measure the actual symbol distribution of 29 ternary LLM models and find that zeros account for up to 51.5% of all weights. Motivated by this finding, we introduce BITCOS, a simple distribution-adaptive layout comprised of a dense presence bitmap plus a compacted sign vector, and costs 2 - z bits per weight element given a zero density z in the model's weights. BITCOS stores weights more compactly than the five-trit packing in 26 of the 29 tested models, and reaches 1.485 bits per weight on the sparsest of them. BITCOS is amenable to efficient unpacking on modern processors and GPUs, and we present optimized unpacking sequences for AVX-512, AVX2 and Intel Xe2 GPUs. Measured against production state-of-the-art ternary matrix-vector multiplication kernels, at the zero densities real-world ternary models exhibit, the realized gain with our proposed layout is up to 1.28times. Finally, we illustrate end-to-end LLM inference results on 5 different platforms (client and server CPUs, integrated and discrete Xe2 GPUs) where decode throughput improves by up to 1.18times on CPUs and 1.27times on GPUs.

Scaling laws have shaped recent advances in machine learning by enabling predictable scaling of model performance based on model size, computation, and data volume. Concurrently, the rise in computational cost for AI has motivated model compression techniques, notably quantization and sparsification, which have emerged to mitigate the steep computational demands associated with large-scale training and inference. This paper investigates the interplay between scaling laws and compression formats, exploring whether a unified scaling framework can accurately predict model performance when training occurs over various compressed representations, such as sparse, scalar-quantized, sparse-quantized or even vector-quantized formats. Our key contributions include validating a general scaling law formulation and showing that it is applicable both individually but also composably across compression types. Based on this, our main finding is demonstrating both theoretically and empirically that there exists a simple "capacity" metric -- based on the representation's ability to fit random Gaussian data -- which can robustly predict parameter efficiency across multiple compressed representations. On the practical side, we extend our formulation to directly compare the accuracy potential of different compressed formats, and to derive better algorithms for training over sparse-quantized formats.

Learning compact discrete representations of data is a key task on its own or for facilitating subsequent processing of data. In this paper we present a model that produces Discrete InfoMax Codes (DIMCO); we learn a probabilistic encoder that yields k-way d-dimensional codes associated with input data. Our model's learning objective is to maximize the mutual information between codes and labels with a regularization, which enforces entries of a codeword to be as independent as possible. We show that the infomax principle also justifies previous loss functions (e.g., cross-entropy) as its special cases. Our analysis also shows that using shorter codes, as DIMCO does, reduces overfitting in the context of few-shot classification. Through experiments in various domains, we observe this implicit meta-regularization effect of DIMCO. Furthermore, we show that the codes learned by DIMCO are efficient in terms of both memory and retrieval time compared to previous methods.

Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership i in S and estimating set intersection sizes |X cap Y| for two sets of symbols X and Y, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

Information-theoretic quantities play a crucial role in understanding non-linear relationships between random variables and are widely used across scientific disciplines. However, estimating these quantities remains an open problem, particularly in the case of high-dimensional discrete distributions. Current approaches typically rely on embedding discrete data into a continuous space and applying neural estimators originally designed for continuous distributions, a process that may not fully capture the discrete nature of the underlying data. We consider Continuous-Time Markov Chains (CTMCs), stochastic processes on discrete state-spaces which have gained popularity due to their generative modeling applications. In this work, we introduce INFO-SEDD, a novel method for estimating information-theoretic quantities of discrete data, including mutual information and entropy. Our approach requires the training of a single parametric model, offering significant computational and memory advantages. Additionally, it seamlessly integrates with pretrained networks, allowing for efficient reuse of pretrained generative models. To evaluate our approach, we construct a challenging synthetic benchmark. Our experiments demonstrate that INFO-SEDD is robust and outperforms neural competitors that rely on embedding techniques. Moreover, we validate our method on a real-world task: estimating the entropy of an Ising model. Overall, INFO-SEDD outperforms competing methods and shows scalability to high-dimensional scenarios, paving the way for new applications where estimating MI between discrete distribution is the focus. The promising results in this complex, high-dimensional scenario highlight INFO-SEDD as a powerful new estimator in the toolkit for information-theoretical analysis.

In this paper, we present a new technique to obtain upper bounds on undirected unicast network information capacity. Using this technique, we characterize an upper bound, called partition bound, on the symmetric rate of information flow in undirected unicast networks and give an algorithm to compute it. Two classes of networks are presented for which the bound is tight and the capacity is achievable by routing thus confirming the undirected unicast conjecture for these classes of networks. We also show that the bound can be loose in general and present an approach to tighten it.

We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in mathbb R^d, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in mathbb R^d. We study a computationally efficient estimator initially proposed by Pooladian and Niles-Weed (2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate n^{-1/2}, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.

This paper proposes a fundamentally new paradigm for image generation through set-based tokenization and distribution modeling. Unlike conventional methods that serialize images into fixed-position latent codes with a uniform compression ratio, we introduce an unordered token set representation to dynamically allocate coding capacity based on regional semantic complexity. This TokenSet enhances global context aggregation and improves robustness against local perturbations. To address the critical challenge of modeling discrete sets, we devise a dual transformation mechanism that bijectively converts sets into fixed-length integer sequences with summation constraints. Further, we propose Fixed-Sum Discrete Diffusion--the first framework to simultaneously handle discrete values, fixed sequence length, and summation invariance--enabling effective set distribution modeling. Experiments demonstrate our method's superiority in semantic-aware representation and generation quality. Our innovations, spanning novel representation and modeling strategies, advance visual generation beyond traditional sequential token paradigms. Our code and models are publicly available at https://github.com/Gengzigang/TokenSet.

We propose a new method for estimating how much a model ``knows'' about a datapoint and use it to measure the capacity of modern language models. Prior studies of language model memorization have struggled to disentangle memorization from generalization. We formally separate memorization into two components: unintended memorization, the information a model contains about a specific dataset, and generalization, the information a model contains about the true data-generation process. When we completely eliminate generalization, we can compute the total memorization, which provides an estimate of model capacity: our measurements estimate that GPT-style models have a capacity of approximately 3.6 bits per parameter. We train language models on datasets of increasing size and observe that models memorize until their capacity fills, at which point ``grokking'' begins, and unintended memorization decreases as models begin to generalize. We train hundreds of transformer language models ranging from 500K to 1.5B parameters and produce a series of scaling laws relating model capacity and data size to membership inference.

Autoregressive language models (ARMs) deliver strong likelihoods, but are inherently serial: they generate one token per forward pass, which limits throughput and inflates latency for long sequences. Diffusion Language Models (DLMs) parallelize across positions and thus appear promising for language generation, yet standard discrete diffusion typically needs hundreds to thousands of model evaluations to reach high quality, trading serial depth for iterative breadth. We introduce FS-DFM, Few-Step Discrete Flow-Matching. A discrete flow-matching model designed for speed without sacrificing quality. The core idea is simple: make the number of sampling steps an explicit parameter and train the model to be consistent across step budgets, so one big move lands where many small moves would. We pair this with a reliable update rule that moves probability in the right direction without overshooting, and with strong teacher guidance distilled from long-run trajectories. Together, these choices make few-step sampling stable, accurate, and easy to control. On language modeling benchmarks, FS-DFM with 8 sampling steps achieves perplexity parity with a 1,024-step discrete-flow baseline for generating 1,024 tokens using a similar-size model, delivering up to 128 times faster sampling and corresponding latency/throughput gains.

Deep neural networks perform well on classification tasks where data streams are i.i.d. and labeled data is abundant. Challenges emerge with non-stationary training data streams such as continual learning. One powerful approach that has addressed this challenge involves pre-training of large encoders on volumes of readily available data, followed by task-specific tuning. Given a new task, however, updating the weights of these encoders is challenging as a large number of weights needs to be fine-tuned, and as a result, they forget information about the previous tasks. In the present work, we propose a model architecture to address this issue, building upon a discrete bottleneck containing pairs of separate and learnable key-value codes. Our paradigm will be to encode; process the representation via a discrete bottleneck; and decode. Here, the input is fed to the pre-trained encoder, the output of the encoder is used to select the nearest keys, and the corresponding values are fed to the decoder to solve the current task. The model can only fetch and re-use a sparse number of these key-value pairs during inference, enabling localized and context-dependent model updates. We theoretically investigate the ability of the discrete key-value bottleneck to minimize the effect of learning under distribution shifts and show that it reduces the complexity of the hypothesis class. We empirically verify the proposed method under challenging class-incremental learning scenarios and show that the proposed model - without any task boundaries - reduces catastrophic forgetting across a wide variety of pre-trained models, outperforming relevant baselines on this task.

Representing speech and audio signals in discrete units has become a compelling alternative to traditional high-dimensional feature vectors. Numerous studies have highlighted the efficacy of discrete units in various applications such as speech compression and restoration, speech recognition, and speech generation. To foster exploration in this domain, we introduce the Interspeech 2024 Challenge, which focuses on new speech processing benchmarks using discrete units. It encompasses three pivotal tasks, namely multilingual automatic speech recognition, text-to-speech, and singing voice synthesis, and aims to assess the potential applicability of discrete units in these tasks. This paper outlines the challenge designs and baseline descriptions. We also collate baseline and selected submission systems, along with preliminary findings, offering valuable contributions to future research in this evolving field.

Discrete speech representations have garnered recent attention for their efficacy in training transformer-based models for various speech-related tasks such as automatic speech recognition (ASR), translation, speaker verification, and joint speech-text foundational models. In this work, we present a comprehensive analysis on building ASR systems with discrete codes. We investigate different methods for codec training such as quantization schemes and time-domain vs spectral feature encodings. We further explore ASR training techniques aimed at enhancing performance, training efficiency, and noise robustness. Drawing upon our findings, we introduce a codec ASR pipeline that outperforms Encodec at similar bit-rate. Remarkably, it also surpasses the state-of-the-art results achieved by strong self-supervised models on the 143 languages ML-SUPERB benchmark despite being smaller in size and pretrained on significantly less data.

An important concept in the fifth generation of mobile networks is multitenancy, which allows diverse operators sharing the same wireless infrastructure. To support this feature in conjunction with the challenging performance requirements of future networks, more automated and faster planning of the required radio capacity is needed. Likewise, installing small cells is an effective resource to provide greater performance and capacity to both indoor and outdoor places. This paper proposes a new framework for automated cell planning in multitenant small cell networks. In particular, taking advantage of the available network data, a set of detailed planning specifications over time and space domains are generated in order to meet the contracted capacity by each tenant. Then, the network infrastructure and configuration are updated according to an algorithm that considers different actions such as adding/removing channels and adding or relocating small cells. The simulation results show the effectiveness of various methods to derive the planning specifications depending on the correlation between the tenant's and network's traffic demands.

Large Language Models (LLMs) have emerged as a milestone in artificial intelligence, and their performance can improve as the model size increases. However, this scaling brings great challenges to training and inference efficiency, particularly for deploying LLMs in resource-constrained environments, and the scaling trend is becoming increasingly unsustainable. This paper introduces the concept of ``capacity density'' as a new metric to evaluate the quality of the LLMs across different scales and describes the trend of LLMs in terms of both effectiveness and efficiency. To calculate the capacity density of a given target LLM, we first introduce a set of reference models and develop a scaling law to predict the downstream performance of these reference models based on their parameter sizes. We then define the effective parameter size of the target LLM as the parameter size required by a reference model to achieve equivalent performance, and formalize the capacity density as the ratio of the effective parameter size to the actual parameter size of the target LLM. Capacity density provides a unified framework for assessing both model effectiveness and efficiency. Our further analysis of recent open-source base LLMs reveals an empirical law (the densing law)that the capacity density of LLMs grows exponentially over time. More specifically, using some widely used benchmarks for evaluation, the capacity density of LLMs doubles approximately every three months. The law provides new perspectives to guide future LLM development, emphasizing the importance of improving capacity density to achieve optimal results with minimal computational overhead.

For any fixed dimension d geq 3 we construct a Nikodym set in F_q^d of cardinality q^d - (d-2{log 2} +1+o(1)) q^{d-1} log q in the limit q to infty, when q is an odd prime power. This improves upon the naive random construction, which gives a set of cardinality q^d - (d-1+o(1)) q^{d-1} log q, and is new in the regime where F_q has unbounded characteristic and q not a perfect square. While the final proofs are completely human generated, the initial ideas of the construction were inspired by output from the tools AlphaEvolve and DeepThink. We also present a simple construction of Nikodym sets in F_q^2 for q a perfect square that is a special case of known unital-based constructions, and matches the existing bounds of q^2 - q^{3/2} + O(q log q), assuming that q is not the square of a prime p equiv 3 4.

Post-training quantization (PTQ) is widely regarded as one of the most efficient compression methods practically, benefitting from its data privacy and low computation costs. We argue that an overlooked problem of oscillation is in the PTQ methods. In this paper, we take the initiative to explore and present a theoretical proof to explain why such a problem is essential in PTQ. And then, we try to solve this problem by introducing a principled and generalized framework theoretically. In particular, we first formulate the oscillation in PTQ and prove the problem is caused by the difference in module capacity. To this end, we define the module capacity (ModCap) under data-dependent and data-free scenarios, where the differentials between adjacent modules are used to measure the degree of oscillation. The problem is then solved by selecting top-k differentials, in which the corresponding modules are jointly optimized and quantized. Extensive experiments demonstrate that our method successfully reduces the performance drop and is generalized to different neural networks and PTQ methods. For example, with 2/4 bit ResNet-50 quantization, our method surpasses the previous state-of-the-art method by 1.9%. It becomes more significant on small model quantization, e.g. surpasses BRECQ method by 6.61% on MobileNetV2*0.5.

Large Language Models (LLMs) are typically trained on data mixtures: most data come from web scrapes, while a small portion is curated from high-quality sources with dense domain-specific knowledge. In this paper, we show that when training LLMs on such data mixtures, knowledge acquisition from knowledge-dense datasets, unlike training exclusively on knowledge-dense data (arXiv:2404.05405), does not always follow a smooth scaling law but can exhibit phase transitions with respect to the mixing ratio and model size. Through controlled experiments on a synthetic biography dataset mixed with web-scraped data, we demonstrate that: (1) as we increase the model size to a critical value, the model suddenly transitions from memorizing very few to most of the biographies; (2) below a critical mixing ratio, the model memorizes almost nothing even with extensive training, but beyond this threshold, it rapidly memorizes more biographies. We attribute these phase transitions to a capacity allocation phenomenon: a model with bounded capacity must act like a knapsack problem solver to minimize the overall test loss, and the optimal allocation across datasets can change discontinuously as the model size or mixing ratio varies. We formalize this intuition in an information-theoretic framework and reveal that these phase transitions are predictable, with the critical mixing ratio following a power-law relationship with the model size. Our findings highlight a concrete case where a good mixing recipe for large models may not be optimal for small models, and vice versa.

We study the problem of counting the number of distinct elements in a dataset subject to the constraint of differential privacy. We consider the challenging setting of person-level DP (a.k.a. user-level DP) where each person may contribute an unbounded number of items and hence the sensitivity is unbounded. Our approach is to compute a bounded-sensitivity version of this query, which reduces to solving a max-flow problem. The sensitivity bound is optimized to balance the noise we must add to privatize the answer against the error of the approximation of the bounded-sensitivity query to the true number of unique elements.

A range of recent works addresses the problem of compression of sequence of tokens into a shorter sequence of real-valued vectors to be used as inputs instead of token embeddings or key-value cache. These approaches allow to reduce the amount of compute in existing language models. Despite relying on powerful models as encoders, the maximum attainable lossless compression ratio is typically not higher than x10. This fact is highly intriguing because, in theory, the maximum information capacity of large real-valued vectors is far beyond the presented rates even for 16-bit precision and a modest vector size. In this work, we explore the limits of compression by replacing the encoder with a per-sample optimization procedure. We show that vectors with compression ratios up to x1500 exist, which highlights two orders of magnitude gap between existing and practically attainable solutions. Furthermore, we empirically show that the compression limits are determined not by the length of the input but by the amount of uncertainty to be reduced, namely, the cross-entropy loss on this sequence without any conditioning. The obtained limits highlight the substantial gap between the theoretical capacity of input embeddings and their practical utilization, suggesting significant room for optimization in model design.

Semiparametric discrete choice models are widely used in a variety of practical applications. While these models are point identified in the presence of continuous covariates, they can become partially identified when covariates are discrete. In this paper we find that classical estimators, including the maximum score estimator, (Manski (1975)), loose their attractive statistical properties without point identification. First of all, they are not sharp with the estimator converging to an outer region of the identified set, (Komarova (2013)), and in many discrete designs it weakly converges to a random set. Second, they are not robust, with their distribution limit discontinuously changing with respect to the parameters of the model. We propose a novel class of estimators based on the concept of a quantile of a random set, which we show to be both sharp and robust. We demonstrate that our approach extends from cross-sectional settings to classical static and dynamic discrete panel data models.

A fixed network of McCulloch-Pitts threshold units with weights in {-1,0,1} can hold other threshold networks in its state and run them: the state is a ring of banks of records, each record a unit of a stored network, and each step evaluates one record. We use such a network to carry out von Neumann's universal construction and self-reproduction exactly. As a cellular automaton keeps its rule, the fixed network keeps its weights, and what reproduces is a stored network. A constructor of 143 records reads the description of a network from its tape, builds that network in the next bank, copies the description onto the next tape and hands control to what it built; started on its own description, it rebuilds itself, weights included, in every generation. The scheme scales to a universal computer. A SUBLEQ computer of 17,598 ternary units, stored as 36,080 records and running a program of 27 instructions, builds any network that fits a bank and reproduces itself in the same way, at every word width from eight bits on; run directly, it emits the serialization of its own weights, memory and tape. Integer pre-activations give every orbit a margin of 1/2, and replicating each unit r times multiplies it by r. That suffices against noise of any size on the pre-activations, but against von Neumann's output flips only below a threshold inversely proportional to the fan-in. These results are proved in Rocq.

Deep Neural Networks are highly susceptible to shortcut learning, frequently memorizing low-dimensional spurious correlations instead of underlying causal mechanisms. This phenomenon not only degrades out-of-distribution robustness but also induces severe demographic biases in sensitive applications. In this paper, we propose a geometric a priori methodology to mitigate shortcut learning. By deploying a zero-hidden-layer (N=1) Topological Auditor, we mathematically isolate features that monopolize the gradient without human intervention. We empirically demonstrate a Capacity Phase Transition: once linear shortcuts are pruned, networks are forced to utilize higher geometric capacity (N geq 16) to curve the decision boundary and learn ethical representations. Our approach outperforms L1 Regularization -- which collapses into demographic bias -- and operates at a fraction of the computational cost of post-hoc methods like Just Train Twice (JTT), successfully reducing counterfactual gender vulnerability from 21.18\% to 7.66\%.

In this article, we propose a new counter-based implementation of John von Neumann's middle-square random number generator (RNG). Several rounds of squaring are applied to a counter to produce a random output. We discovered that four rounds are sufficient to provide satisfactory data. Two versions of the RNG are presented, a 4-round version with 32-bit output and a 5-round version with 64-bit output. Both pass stringent tests of randomness and may be the fastest counter-based generators.

Discrete diffusion has recently emerged as a promising paradigm in discrete data modeling. However, existing methods typically rely on a fixed rate transition matrix during training, which not only limits the expressiveness of latent representations, a fundamental strength of variational methods, but also constrains the overall design space. To address these limitations, we propose Discrete Markov Bridge, a novel framework specifically designed for discrete representation learning. Our approach is built upon two key components: Matrix Learning and Score Learning. We conduct a rigorous theoretical analysis, establishing formal performance guarantees for Matrix Learning and proving the convergence of the overall framework. Furthermore, we analyze the space complexity of our method, addressing practical constraints identified in prior studies. Extensive empirical evaluations validate the effectiveness of the proposed Discrete Markov Bridge, which achieves an Evidence Lower Bound (ELBO) of 1.38 on the Text8 dataset, outperforming established baselines. Moreover, the proposed model demonstrates competitive performance on the CIFAR-10 dataset, achieving results comparable to those obtained by image-specific generation approaches.

Discrete visual tokens represent vision in a discrete form analogous to language, offering a native interface for multimodal modeling under a shared next-token prediction objective. However, quantization error is widely assumed to cause unavoidable semantic loss, fundamentally limiting the semantic fidelity of discrete visual tokens and positioning continuous visual representations as an intrinsic upper bound. Using multimodal understanding as a proxy for semantic fidelity, we revisit this assumption and find quantization error unreliable, as discrete representations with substantially larger errors can achieve better understanding. Instead, the effective bit budget of discrete indices emerges as a key determinant of downstream performance. However, existing quantizers eventually saturate as additional codebooks contribute progressively less semantic information. As one approach to alleviating this saturation, we introduce Factorized Vector Quantization, which quantizes complementary, fixed-dimensional views of the visual feature with symmetric parallel codebooks. It continues to improve at larger bit budgets, ultimately matching continuous visual representations in multimodal understanding. Trained on industrial-scale multimodal data, our model achieves state-of-the-art performance among discrete-representation models, outperforming UniAR and Emu3-Chat by 22.8% and 59.2%, respectively, on a nine-metric average. To our knowledge, it is also the first discrete-representation model competitive with strong continuous-representation models such as Qwen3-VL, including on fine-grained OCR tasks. We further identify the key principles and a practical recipe enabling this competitiveness. These results and insights support the potential of discrete visual representations to underpin large-scale native multimodal foundation models. Code and models are available at https://github.com/hustvl/FVQ

Discrete diffusion models iteratively denoise a noisy sequence of categorical data such as text, enabling any-order generation with self-correction and in-filling capabilities. This makes them promising candidates for reasoning, planning, and constrained-generation tasks like coding and math. Yet masked and uniform diffusion erase a token's identity in a single corruption event, so their noisy states cannot explicitly represent partially resolved uncertainty in which several alternatives remain plausible. We close this gap with Binary Discrete Diffusion, a family of continuous-time discrete-diffusion processes whose noisy state is a set of candidate tokens per position, represented as a binary membership vector. Binary diffusion changes candidate sets through additions and removals, allowing information to be lost and recovered gradually while the diffusion state itself represents partial uncertainty. On Sudoku-Extreme, Boxoban, and graph coloring, the symmetric formulation of binary diffusion consistently outperforms strong, well-tuned masked and uniform baselines under matched architectures and controlled training, sampling, and tuning budgets, with its largest gains on the hardest instances, while remaining competitive on language modeling (Text8, TinyStories, OpenWebText). These results support candidate sets as an effective intermediate representation for iterative reasoning and show that the corruption kernel can materially change the computation learned by the reverse model.

Large language model compression has made substantial progress through pruning, quantization, and low-rank decomposition, yet a fundamental limitation persists across all existing methods: compression budgets are allocated without any representation of what individual model components functionally encode. We term this the capability-blind compression problem and argue it is a root cause of two well-documented failures -- the insensitivity of perplexity-based evaluation to reasoning capability loss, and the abrupt phase transitions in model performance recently characterized by Ma et al. (2026). We propose Capability-Guided Compression (CGC), a framework that addresses this by using Sparse Autoencoder (SAE)-derived capability density maps to allocate differential compression budgets across transformer components. Capability density is a formally defined scalar measure combining the feature breadth, activation entropy, and cross-input consistency of a component's SAE feature activation distribution. We prove theoretically that components with higher capability density exhibit lower structural redundancy and reach their individual phase transition points at lower compression ratios, providing the first pre-compression mechanism for component-level phase transition prediction. Experiments on GPT-2 Medium confirm that capability density is statistically independent of Wanda importance scores (Spearman rho = -0.054, n = 384 heads), establishing it as a genuinely novel compression signal orthogonal to all existing importance metrics. We report a negative result on PPL-based compression comparison and provide a principled diagnosis identifying GPT-2 Medium as an insufficient test bed for the full CGC hypothesis. The theoretical framework, density formalism, and orthogonality finding constitute a foundation for capability-aware compression research.

Visual generation with discrete tokens has gained significant attention as it enables a unified token prediction paradigm shared with language models, promising seamless multimodal architectures. However, current discrete generation methods remain limited to low-dimensional latent tokens (typically 8-32 dims), sacrificing the semantic richness essential for understanding. While high-dimensional pretrained representations (768-1024 dims) could bridge this gap, their discrete generation poses fundamental challenges. In this paper, we present Cubic Discrete Diffusion (CubiD), the first discrete generation model for high-dimensional representations. CubiD performs fine-grained masking throughout the high-dimensional discrete representation -- any dimension at any position can be masked and predicted from partial observations. This enables the model to learn rich correlations both within and across spatial positions, with the number of generation steps fixed at T regardless of feature dimensionality, where T ll hwd. On ImageNet-256, CubiD achieves state-of-the-art discrete generation with strong scaling behavior from 900M to 3.7B parameters. Crucially, we validate that these discretized tokens preserve original representation capabilities, demonstrating that the same discrete tokens can effectively serve both understanding and generation tasks. We hope this work will inspire future research toward unified multimodal architectures. Code is available at: https://github.com/YuqingWang1029/CubiD.

This paper challenges the dominance of continuous pipelines in visual generation. We systematically investigate the performance gap between discrete and continuous methods. Contrary to the belief that discrete tokenizers are intrinsically inferior, we demonstrate that the disparity arises primarily from the total number of bits allocated in the latent space (i.e., the compression ratio). We show that scaling up the codebook size effectively bridges this gap, allowing discrete tokenizers to match or surpass their continuous counterparts. However, existing discrete generation methods struggle to capitalize on this insight, suffering from performance degradation or prohibitive training costs with scaled codebook. To address this, we propose masked Bit AutoRegressive modeling (BAR), a scalable framework that supports arbitrary codebook sizes. By equipping an autoregressive transformer with a masked bit modeling head, BAR predicts discrete tokens through progressively generating their constituent bits. BAR achieves a new state-of-the-art gFID of 0.99 on ImageNet-256, outperforming leading methods across both continuous and discrete paradigms, while significantly reducing sampling costs and converging faster than prior continuous approaches. Project page is available at https://bar-gen.github.io/

Deep neural network pruning and quantization techniques have demonstrated it is possible to achieve high levels of compression with surprisingly little degradation to test set accuracy. However, this measure of performance conceals significant differences in how different classes and images are impacted by model compression techniques. We find that models with radically different numbers of weights have comparable top-line performance metrics but diverge considerably in behavior on a narrow subset of the dataset. This small subset of data points, which we term Pruning Identified Exemplars (PIEs) are systematically more impacted by the introduction of sparsity. Compression disproportionately impacts model performance on the underrepresented long-tail of the data distribution. PIEs over-index on atypical or noisy images that are far more challenging for both humans and algorithms to classify. Our work provides intuition into the role of capacity in deep neural networks and the trade-offs incurred by compression. An understanding of this disparate impact is critical given the widespread deployment of compressed models in the wild.

Discrete diffusion models (DDMs) have shown powerful generation ability for discrete data modalities like text and molecules. However, their practical application is hindered by inefficient sampling, requiring a large number of sampling steps. Accelerating DDMs by using larger step sizes typically introduces significant problems in generation quality, as it amplifies the impact of both the compounding decoding error due to factorized predictions and discretization error from numerical approximations, leading to a significant decrease in sampling quality. To address these challenges, we propose learnable sampler distillation (LSD), a novel approach to train fast and high-fidelity samplers for DDMs. LSD employs a distillation approach where a student sampler with a few steps learns to align its intermediate score trajectory with that of a high-quality teacher sampler with numerous steps. This alignment is achieved by optimizing learnable sampler coefficients that adaptively adjust sampling dynamics. Additionally, we further propose LSD+, which also learns time schedules that allocate steps non-uniformly. Experiments across text generation, image generation, and synthetic tasks demonstrate that our proposed approaches outperform existing samplers for DDMs, achieving substantially higher sampling quality with significantly fewer sampling steps. Our code is available at https://github.com/feiyangfu/LSD{https://github.com/feiyangfu/LSD}.

Transformers often fail to learn generalizable algorithms, instead relying on brittle heuristics. Using graph connectivity as a testbed, we explain this phenomenon both theoretically and empirically. We consider a simplified Transformer architecture, the disentangled Transformer, and prove that an L-layer model has capacity to solve for graphs with diameters up to exactly 3^L, implementing an algorithm equivalent to computing powers of the adjacency matrix. We analyze the training-dynamics, and show that the learned strategy hinges on whether most training instances are within this model capacity. Within-capacity graphs (diameter leq 3^L) drive the learning of a correct algorithmic solution while beyond-capacity graphs drive the learning of a simple heuristic based on node degrees. Finally, we empirically demonstrate that restricting training data within a model's capacity leads to both standard and disentangled transformers learning the exact algorithm rather than the degree-based heuristic.

Probabilistic circuits (PCs) are a tractable representation of probability distributions allowing for exact and efficient computation of likelihoods and marginals. There has been significant recent progress on improving the scale and expressiveness of PCs. However, PC training performance plateaus as model size increases. We discover that most capacity in existing large PC structures is wasted: fully-connected parameter layers are only sparsely used. We propose two operations: pruning and growing, that exploit the sparsity of PC structures. Specifically, the pruning operation removes unimportant sub-networks of the PC for model compression and comes with theoretical guarantees. The growing operation increases model capacity by increasing the size of the latent space. By alternatingly applying pruning and growing, we increase the capacity that is meaningfully used, allowing us to significantly scale up PC learning. Empirically, our learner achieves state-of-the-art likelihoods on MNIST-family image datasets and on Penn Tree Bank language data compared to other PC learners and less tractable deep generative models such as flow-based models and variational autoencoders (VAEs).

It is currently difficult to distill discrete diffusion models. In contrast, continuous diffusion literature has many distillation approaches methods that can reduce sampling steps to a handful. Our method, Discrete Moment Matching Distillation (D-MMD), leverages ideas that have been highly successful in the continuous domain. Whereas previous discrete distillation methods collapse, D-MMD maintains high quality and diversity (given sufficient sampling steps). This is demonstrated on both text and image datasets. Moreover, the newly distilled generators can outperform their teachers.

In practical federated learning scenarios, the participating devices may have different bitwidths for computation and memory storage by design. However, despite the progress made in device-heterogeneous federated learning scenarios, the heterogeneity in the bitwidth specifications in the hardware has been mostly overlooked. We introduce a pragmatic FL scenario with bitwidth heterogeneity across the participating devices, dubbed as Bitwidth Heterogeneous Federated Learning (BHFL). BHFL brings in a new challenge, that the aggregation of model parameters with different bitwidths could result in severe performance degeneration, especially for high-bitwidth models. To tackle this problem, we propose ProWD framework, which has a trainable weight dequantizer at the central server that progressively reconstructs the low-bitwidth weights into higher bitwidth weights, and finally into full-precision weights. ProWD further selectively aggregates the model parameters to maximize the compatibility across bit-heterogeneous weights. We validate ProWD against relevant FL baselines on the benchmark datasets, using clients with varying bitwidths. Our ProWD largely outperforms the baseline FL algorithms as well as naive approaches (e.g. grouped averaging) under the proposed BHFL scenario.

When training large-scale models, the performance typically scales with the number of parameters and the dataset size according to a slow power law. A fundamental theoretical and practical question is whether comparable performance can be achieved with significantly smaller models and substantially less data. In this work, we provide a positive and constructive answer. We prove that a generic permutation-invariant function of d objects can be asymptotically compressed into a function of polylog d objects with vanishing error, which is proved to be the optimal compression rate. This theorem yields two key implications: (Ia) a large neural network can be compressed to polylogarithmic width while preserving its learning dynamics; (Ib) a large dataset can be compressed to polylogarithmic size while leaving the loss landscape of the corresponding model unchanged. Implication (Ia) directly establishes a proof of the dynamical lottery ticket hypothesis, which states that any ordinary network can be strongly compressed such that the learning dynamics and result remain unchanged. (Ib) shows that a neural scaling law of the form Lsim d^{-α} can be boosted to an arbitrarily fast power law decay, and ultimately to exp(-α' sqrt[m]{d}).

Discrete diffusion models have achieved success in tasks like image generation and masked language modeling but face limitations in controlled content editing. We introduce DICE (Discrete Inversion for Controllable Editing), the first approach to enable precise inversion for discrete diffusion models, including multinomial diffusion and masked generative models. By recording noise sequences and masking patterns during the reverse diffusion process, DICE enables accurate reconstruction and flexible editing of discrete data without the need for predefined masks or attention manipulation. We demonstrate the effectiveness of DICE across both image and text domains, evaluating it on models such as VQ-Diffusion, Paella, and RoBERTa. Our results show that DICE preserves high data fidelity while enhancing editing capabilities, offering new opportunities for fine-grained content manipulation in discrete spaces. For project webpage, see https://hexiaoxiao-cs.github.io/DICE/.

The recent trend in deep neural networks (DNNs) research is to make the networks more compact. The motivation behind designing compact DNNs is to improve energy efficiency since by virtue of having lower memory footprint, compact DNNs have lower number of off-chip accesses which improves energy efficiency. However, we show that making DNNs compact has indirect and subtle implications which are not well-understood. Reducing the number of parameters in DNNs increases the number of activations which, in turn, increases the memory footprint. We evaluate several recently-proposed compact DNNs on Tesla P100 GPU and show that their "activations to parameters ratio" ranges between 1.4 to 32.8. Further, the "memory-footprint to model size ratio" ranges between 15 to 443. This shows that a higher number of activations causes large memory footprint which increases on-chip/off-chip data movements. Furthermore, these parameter-reducing techniques reduce the arithmetic intensity which increases on-chip/off-chip memory bandwidth requirement. Due to these factors, the energy efficiency of compact DNNs may be significantly reduced which is against the original motivation for designing compact DNNs.

We introduce Generalized Discrete Diffusion from Snapshots (GDDS), a unified framework for discrete diffusion modeling that supports arbitrary noising processes over large discrete state spaces. Our formulation encompasses all existing discrete diffusion approaches, while allowing significantly greater flexibility in the choice of corruption dynamics. The forward noising process relies on uniformization and enables fast arbitrary corruption. For the reverse process, we derive a simple evidence lower bound (ELBO) based on snapshot latents, instead of the entire noising path, that allows efficient training of standard generative modeling architectures with clear probabilistic interpretation. Our experiments on large-vocabulary discrete generation tasks suggest that the proposed framework outperforms existing discrete diffusion methods in terms of training efficiency and generation quality, and beats autoregressive models for the first time at this scale. We provide the code along with a blog post on the project page : https://oussamazekri.fr/gdds{https://oussamazekri.fr/gdds}.

We conjecture that the universal logarithmic coefficient for capacity of entanglement c_{cap} is bounded below by the corresponding entropy coefficient c_{eff} at conformal interfaces. We prove the bound in free boson and fermion CFTs and in holographic models with interface branes, and establish it using defect perturbation theory. Numerical calculations at interacting interfaces between the tricritical Ising and Ising CFTs further support the bound. Despite this constraint, capacity provides independent interface data. We construct interfaces with identical entropy and energy transmission coefficients but different capacity coefficients, and show that capacity can even increase as entropy decreases.

Idempotence is the stability of image codec to re-compression. At the first glance, it is unrelated to perceptual image compression. However, we find that theoretically: 1) Conditional generative model-based perceptual codec satisfies idempotence; 2) Unconditional generative model with idempotence constraint is equivalent to conditional generative codec. Based on this newfound equivalence, we propose a new paradigm of perceptual image codec by inverting unconditional generative model with idempotence constraints. Our codec is theoretically equivalent to conditional generative codec, and it does not require training new models. Instead, it only requires a pre-trained mean-square-error codec and unconditional generative model. Empirically, we show that our proposed approach outperforms state-of-the-art methods such as HiFiC and ILLM, in terms of Fr\'echet Inception Distance (FID). The source code is provided in https://github.com/tongdaxu/Idempotence-and-Perceptual-Image-Compression.

A service system may use the same categories to assign standard allocations and to check whether each service is fulfilled. Finer categories can match individual needs more closely, but they divide the observations available for monitoring. We study this conflict for a fixed menu from which participants select by declaring a category, with fulfilment assessed from each category's aggregate outcomes during a fixed period. We relate allocation loss to variation in preferred allocations within categories and identify conditions under which aggregate observations preserve the verification performance of individual records. For nested refinements under stated utility and observation assumptions, an allocation-loss tolerance and a per-category detection target define a feasibility band. Categories must be fine enough to provide suitable allocations but sufficiently populated to support verification. A source-dependent lower bound on declaration entropy and a minimum contributor requirement give necessary information and population constraints. For an explicit finite population with quadratic utility and binary service outcomes, we prove the exact feasible range across all categorical designs and exhibit designs attaining the information lower bound at specified tolerances. The results provide conditions for choosing categories jointly for allocation and verification.

Flow matching has emerged as a scalable paradigm for training high-quality generative models, but sampling from the learned probability flow requires many network evaluations. Distillation can reduce this cost to one or a few evaluations; however, one-step generation often sacrifices quality, making few-step generation the practical operating regime. Existing few-step methods perform their iterative computation along the probability flow and therefore require a fixed, manually chosen timestep discretization. This discretization is often chosen heuristically and is expensive to tune; it may also be restrictive when refinement difficulty differs across samples or spatial locations. We introduce data-space iteration, a few-step generation framework that removes flow discretization altogether. Starting from noise, a shared generator directly refines its prediction in data space, with every iteration trained to produce the best sample permitted by its capacity. Our formulation integrates with distribution matching distillation (DMD) with minimal changes, enabling a controlled comparison between iteration methods under matched training settings. On class-conditional ImageNet 256x256, data-space iteration outperforms standard discretization baselines and matches or improves upon variants selected through schedule search, without requiring schedule-specific training. These results show that data-space iteration provides a simple and effective alternative to discretized flow-space iteration for fast generation.

Understanding neural networks is challenging in part because of the dense, continuous nature of their hidden states. We explore whether we can train neural networks to have hidden states that are sparse, discrete, and more interpretable by quantizing their continuous features into what we call codebook features. Codebook features are produced by finetuning neural networks with vector quantization bottlenecks at each layer, producing a network whose hidden features are the sum of a small number of discrete vector codes chosen from a larger codebook. Surprisingly, we find that neural networks can operate under this extreme bottleneck with only modest degradation in performance. This sparse, discrete bottleneck also provides an intuitive way of controlling neural network behavior: first, find codes that activate when the desired behavior is present, then activate those same codes during generation to elicit that behavior. We validate our approach by training codebook Transformers on several different datasets. First, we explore a finite state machine dataset with far more hidden states than neurons. In this setting, our approach overcomes the superposition problem by assigning states to distinct codes, and we find that we can make the neural network behave as if it is in a different state by activating the code for that state. Second, we train Transformer language models with up to 410M parameters on two natural language datasets. We identify codes in these models representing diverse, disentangled concepts (ranging from negative emotions to months of the year) and find that we can guide the model to generate different topics by activating the appropriate codes during inference. Overall, codebook features appear to be a promising unit of analysis and control for neural networks and interpretability. Our codebase and models are open-sourced at https://github.com/taufeeque9/codebook-features.

Learning deep discrete latent presentations offers a promise of better symbolic and summarized abstractions that are more useful to subsequent downstream tasks. Inspired by the seminal Vector Quantized Variational Auto-Encoder (VQ-VAE), most of work in learning deep discrete representations has mainly focused on improving the original VQ-VAE form and none of them has studied learning deep discrete representations from the generative viewpoint. In this work, we study learning deep discrete representations from the generative viewpoint. Specifically, we endow discrete distributions over sequences of codewords and learn a deterministic decoder that transports the distribution over the sequences of codewords to the data distribution via minimizing a WS distance between them. We develop further theories to connect it with the clustering viewpoint of WS distance, allowing us to have a better and more controllable clustering solution. Finally, we empirically evaluate our method on several well-known benchmarks, where it achieves better qualitative and quantitative performances than the other VQ-VAE variants in terms of the codebook utilization and image reconstruction/generation.

Modern Transformer design and compression both reduce to allocating capacity under a budget. The standard scalars for these decisions, #Params and #FLOPs, capture size and compute but not architectural structure: two architectures with identical parameter budgets but different depth-width, head, or FFN allocations receive identical scores yet behave differently. We propose Neural Spectral Capacity (NSC), a closed-form scalar grounded in the singular-value spectrum of each weight matrix. Under standard random initialization, the Marchenko-Pastur law renders NSC computable from the architectural specification alone, with no model instantiation, data, or gradients. Its layer-wise additive structure admits NSC-DP, an exact dynamic-programming solver returning the architecture globally maximizing NSC under resource constraints in seconds on a CPU -- a guarantee that black-box search over existing training-free proxies cannot provide. Empirically, NSC outperforms #Params, #FLOPs, and representative training-free proxies in ranking across seven Transformer and CNN families (on FlexiBERT, τ= 0.505 on pairs differing in #Params by less than 10%, where #Params collapses to 0.082); NSC-DP discovers a Transformer-XL architecture on WikiText-103 that beats the human-designed baseline in 2 seconds; and prunes LLaMA-7B to the best 5.7B model across eight commonsense reasoning tasks without any calibration data, about 5900x faster than the strongest training-free proxy baseline.

We discovered the underlying physics in Next-token Prediction (NTP). We identified the law of information conservation within NTP and proposed the First Law of Information Capacity (IC-1), demonstrating that the essence of intelligence emergence in auto-regressive models is fundamentally a process of information transfer. We also introduced Landauer's Principle into NTP, formulating the Second Law of Information Capacity (IC-2), which establishes the relationship between auto-regressive model training and energy consumption. Additionally, we presented several corollaries, which hold practical significance for production practices. Finally, we validated the compatibility and complementarity of our findings with existing theories.

Recently, considerable efforts have been directed towards compressing Large Language Models (LLMs), which showcase groundbreaking capabilities across diverse applications but entail significant deployment costs due to their large sizes. Meanwhile, much less attention has been given to mitigating the costs associated with deploying multiple LLMs of varying sizes despite its practical significance. Thus, this paper introduces any-precision LLM, extending the concept of any-precision DNN to LLMs. Addressing challenges in any-precision LLM, we propose a lightweight method for any-precision quantization of LLMs, leveraging a post-training quantization framework, and develop a specialized software engine for its efficient serving. As a result, our solution significantly reduces the high costs of deploying multiple, different-sized LLMs by overlaying LLMs quantized to varying bit-widths, such as 3, 4, ..., n bits, into a memory footprint comparable to a single n-bit LLM. All the supported LLMs with varying bit-widths demonstrate state-of-the-art model quality and inference throughput, proving itself to be a compelling option for deployment of multiple, different-sized LLMs. The source code will be publicly available soon.

We study sparse locally private discrete mechanisms generated by a nonnegative kernel and an input-dependent admissible output support. The support set is intentionally small relative to the ambient alphabet and acts as a mechanism-design parameter. This formulation covers metric-ball supports, sparse exponential mechanisms, sparse staircase mechanisms, sparse randomized response, sparse additive kernels such as Skellam perturbations, and Poisson-binomial-inspired bounded-support channels. We give a general exact characterization of pure and approximate local differential privacy for this sparse-kernel class. Pure local differential privacy forces all supports to coincide, so genuinely input-dependent sparse supports are incompatible with pure privacy. In the approximate regime, the privacy defect decomposes exactly into support leakage and overlap excess loss. We instantiate the formula for several discrete mechanism families. For sparse staircase mechanisms, a shell-ratio condition eliminates overlap excess loss. For graph-metric supports, overlap of metric balls is necessary for nontrivial privacy. For sparse randomized response, the support-leakage term has a closed form in terms of support mismatch. For generic radius-truncated additive kernels and, in particular, sparse Skellam mechanisms, we obtain exact finite-support formulas in terms of the Skellam CDF and an exact Bessel-ratio overlap condition. Numerical evaluations illustrate how support radius and kernel shape separately affect support leakage and overlap excess loss.

The Platonic Representation Hypothesis (PRH) holds that as models scale, representations of heterogeneous networks converge toward a shared model of reality. We propose its sequel and boundary, the Capability Convergence Hypothesis (CCH): under a fixed per-token inference budget, representational convergence does not entail capability convergence. Capability instead converges toward a class, the access-complete hybrid: any architecture holding both a compressive O(1)-state channel and a scalable verbatim-index channel. We anchor it on a witness task, the Newton's-apple problem in an infinite stream, and name three resource walls: a Shannon wall barring any o(Nb)-state architecture, a horizon wall barring any fixed window, and a circuit wall barring fixed-depth attention-only composition (conditional on TC0 != NC1). Under an explicit separability assumption a hybrid crosses all three by paying each wall's price, so capability is strictly super-additive under composition. We separate what we prove from what we conjecture: the access-completeness principle rests on information-theoretic lower bounds and pre-registered experiments, while the field-level convergence trend is an economics-motivated conjecture. We report the first pre-registered small-scale tests under criteria frozen before the data: the predicted scissors gap is measured (exact-retrieval error 0.994 vs. 0.000 once a 64-scalar state gains one global-attention layer), the state-tracking bifurcation lands at the registered boundary, and a conjunction witness shows an irreducibly two-channel solution; one prediction failed with its direction reversed and is reported as such. Representational convergence is given freely by scale; capability convergence must be purchased by access structure.

The Schr\"odinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. Most SB-related research focuses on continuous data space R^{D} and leaves open theoretical and algorithmic questions about applying SB methods to discrete data, e.g, on finite spaces S^{D}. Notable examples of such sets S are codebooks of vector-quantized (VQ) representations of modern autoencoders, tokens in texts, categories of atoms in molecules, etc. In this paper, we provide a theoretical and algorithmic foundation for solving SB in discrete spaces using the recently introduced Iterative Markovian Fitting (IMF) procedure. Specifically, we theoretically justify the convergence of discrete-time IMF (D-IMF) to SB in discrete spaces. This enables us to develop a practical computational algorithm for SB which we call Categorical Schr\"odinger Bridge Matching (CSBM). We show the performance of CSBM via a series of experiments with synthetic data and VQ representations of images.

Constraints are critical in text generation as LLM outputs are often unreliable when it comes to ensuring generated outputs adhere to user defined instruction or general safety guidelines. To address this gap, we present Constrained Discrete Diffusion (CDD), a novel method for enforcing constraints on natural language by integrating discrete diffusion models with differentiable optimization. Unlike conventional text generators, which often rely on post-hoc filtering or model retraining for controllable generation, we propose imposing constraints directly into the discrete diffusion sampling process. We illustrate how this technique can be applied to satisfy a variety of natural language constraints, including (i) toxicity mitigation by preventing harmful content from emerging, (ii) character and sequence level lexical constraints, and (iii) novel molecule sequence generation with specific property adherence. Experimental results show that our constraint-aware procedure achieves high fidelity in meeting these requirements while preserving fluency and semantic coherence, outperforming auto-regressive and existing discrete diffusion approaches.

Capability and efficiency are two key dimensions of reasoning in large language models (LLMs). Capability refers to the ability to solve a given problem correctly, whereas efficiency refers to the ability to do so with limited resources. When LLMs use Chain-of-Thought (CoT) reasoning to solve problems of controlled hardness, both the number of problems solved correctly and the number of tokens required to reach a correct answer depend on problem hardness and model size. However, how these factors jointly shape capability and efficiency remains poorly understood. Here, we use hierarchical Bayesian models to evaluate the capability and efficiency of LLMs from the DeepSeek-R1-Distill model family across four classes of arithmetic and algorithmic reasoning problems. At a fixed model size, the probability of correctly solving an instance decays approximately exponentially with instance size, our proxy for problem hardness. The decay scale grows sublinearly with model size, indicating that larger models are more capable, but that capability gains diminish with scale. Output length grows as a power law with instance size, which serves as a proxy for difficulty. However, the parameters of this power law do not vary systematically with model size, suggesting that larger models do not become more efficient. Together, these findings reveal potential limitations of naive scaling as a strategy for developing more capable AI systems: capability improves with diminishing returns, while efficiency shows little to no improvement.

The optimal bit-width for achieving the best trade-off between quantized model size and accuracy has been a subject of ongoing debate. While some advocate for 4-bit quantization, others propose that 1.58-bit offers superior results. However, the lack of a cohesive framework for different bits has left such conclusions relatively tenuous. We present ParetoQ, the first unified framework that facilitates rigorous comparisons across 1-bit, 1.58-bit, 2-bit, 3-bit, and 4-bit quantization settings. Our findings reveal a notable learning transition between 2 and 3 bits: For 3-bits and above, the fine-tuned models stay close to their original pre-trained distributions, whereas for learning 2-bit networks or below, the representations change drastically. By optimizing training schemes and refining quantization functions, ParetoQ surpasses all previous methods tailored to specific bit widths. Remarkably, our ParetoQ ternary 600M-parameter model even outperforms the previous SoTA ternary 3B-parameter model in accuracy, using only one-fifth of the parameters. Extensive experimentation shows that ternary, 2-bit, and 3-bit quantization maintains comparable performance in the size-accuracy trade-off and generally exceeds 4-bit and binary quantization. Considering hardware constraints, 2-bit quantization offers promising potential for memory reduction and speedup.

Generating texts with a large language model (LLM) consumes massive amounts of memory. Apart from the already-large model parameters, the key/value (KV) cache that holds information about previous tokens in a sequence can grow to be even larger than the model itself. This problem is exacerbated in one of the current LLM serving frameworks which reserves the maximum sequence length of memory for the KV cache to guarantee generating a complete sequence as they do not know the output sequence length. This restricts us to use a smaller batch size leading to lower GPU utilization and above all, lower throughput. We argue that designing a system with a priori knowledge of the output sequence can mitigate this problem. To this end, we propose S^{3}, which predicts the output sequence length, schedules generation queries based on the prediction to increase device resource utilization and throughput, and handle mispredictions. Our proposed method achieves 6.49times throughput over those systems that assume the worst case for the output sequence length.

The capacity of a neural network to absorb information is limited by its number of parameters. Conditional computation, where parts of the network are active on a per-example basis, has been proposed in theory as a way of dramatically increasing model capacity without a proportional increase in computation. In practice, however, there are significant algorithmic and performance challenges. In this work, we address these challenges and finally realize the promise of conditional computation, achieving greater than 1000x improvements in model capacity with only minor losses in computational efficiency on modern GPU clusters. We introduce a Sparsely-Gated Mixture-of-Experts layer (MoE), consisting of up to thousands of feed-forward sub-networks. A trainable gating network determines a sparse combination of these experts to use for each example. We apply the MoE to the tasks of language modeling and machine translation, where model capacity is critical for absorbing the vast quantities of knowledge available in the training corpora. We present model architectures in which a MoE with up to 137 billion parameters is applied convolutionally between stacked LSTM layers. On large language modeling and machine translation benchmarks, these models achieve significantly better results than state-of-the-art at lower computational cost.

As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.

In the conventional cellular system, devices are not allowed to communicate directly with each other in the licensed cellular bandwidth and all communications take place through the base stations. The users requirements has led the technology to become fast and faster. Multimedia rich data exchange, fast service, high quality voice calls, newer and more demanding applications, information at fingertips, everything requires technology and communication between devices. A constant need to increase network capacity for meeting the users growing demands has led to the growth of cellular communication networks from the first generation(1G) to the fifth generation(5G). There will be crores of connected devices in the coming future . A large number of connections are going to be heterogeneous, demanding lesser delays, higher data rates, superior throughput and enhanced system capacity. The available spectrum resources are limited and has to be flexibly used by mobile network operators to cope with the rising demands. An emerging facilitator of the upcoming high data rate demanding next-generation networks are device-to-device(D2D) communication. This paper has developed a model that establishes Device-to-Device (D2D) communication between two end-users without involving the eNB (evolved Node B). We have sharded the UEs and CUs based on the criteria of DISTANCE. To do so, we used the K-means clustering method.

We present Bit Diffusion: a simple and generic approach for generating discrete data with continuous state and continuous time diffusion models. The main idea behind our approach is to first represent the discrete data as binary bits, and then train a continuous diffusion model to model these bits as real numbers which we call analog bits. To generate samples, the model first generates the analog bits, which are then thresholded to obtain the bits that represent the discrete variables. We further propose two simple techniques, namely Self-Conditioning and Asymmetric Time Intervals, which lead to a significant improvement in sample quality. Despite its simplicity, the proposed approach can achieve strong performance in both discrete image generation and image captioning tasks. For discrete image generation, we significantly improve previous state-of-the-art on both CIFAR-10 (which has 3K discrete 8-bit tokens) and ImageNet-64x64 (which has 12K discrete 8-bit tokens), outperforming the best autoregressive model in both sample quality (measured by FID) and efficiency. For image captioning on MS-COCO dataset, our approach achieves competitive results compared to autoregressive models.

Combinatorial optimization (CO) is naturally discrete, making machine learning based on differentiable optimization inapplicable. Karalias & Loukas (2020) adapted the probabilistic method to incorporate CO into differentiable optimization. Their work ignited the research on unsupervised learning for CO, composed of two main components: probabilistic objectives and derandomization. However, each component confronts unique challenges. First, deriving objectives under various conditions (e.g., cardinality constraints and minimum) is nontrivial. Second, the derandomization process is underexplored, and the existing derandomization methods are either random sampling or naive rounding. In this work, we aim to tackle prevalent (i.e., commonly involved) conditions in unsupervised CO. First, we concretize the targets for objective construction and derandomization with theoretical justification. Then, for various conditions commonly involved in different CO problems, we derive nontrivial objectives and derandomization to meet the targets. Finally, we apply the derivations to various CO problems. Via extensive experiments on synthetic and real-world graphs, we validate the correctness of our derivations and show our empirical superiority w.r.t. both optimization quality and speed.

This work presents code to procedurally generate examples for the ARC training tasks. For each of the 400 tasks, an example generator following the transformation logic of the original examples was created. In effect, the assumed underlying distribution of examples for any given task was reverse engineered by implementing a means to sample from it. An attempt was made to cover an as large as reasonable space of possible examples for each task. That is, whenever the original examples of a given task may be limited in their diversity e.g. by having the dimensions of the grids, the set of symbols or number of objects constant or within tight bounds, even though the transformation does not require it, such constraints were lifted. Having access to not just a few examples per task, as the case for ARC, but instead very many, should enable a wide range of experiments that may be important stepping stones towards making leaps on the benchmark.

Diffusion models have emerged as a promising alternative to autoregressive models in modeling discrete categorical data. Yet diffusion models that directly work on discrete data space do not fully exploit the power of iterative refinement, as the signals are lost during the transition between discrete states. Existing continuous diffusion models for discrete data have limited performance compared to discrete approaches, and the unclear link between them restricts the development of diffusion models for discrete data. In this work, we propose a continuous diffusion model for language modeling that incorporates the geometry of the underlying categorical distribution. We establish a connection between the discrete diffusion and continuous flow on the statistical manifold, and building on the analogy, we introduce a simple design for the diffusion process that generalizes previous discrete diffusion models. We further propose a simulation-free training framework based on radial symmetry and a simple technique to address the high dimensionality of the manifold. Comprehensive experiments on language modeling benchmarks and other modalities show that our method outperforms existing discrete diffusion models and approaches the performance of autoregressive models. Codes available at https://github.com/harryjo97/RDLM{https://github.com/harryjo97/RDLM}.

Radio-frequency portion of the electromagnetic spectrum is a scarce resource. Cognitive radio technology has emerged as a promising solution to overcome the spectrum scarcity bottleneck. Through this technology, secondary users (SUs) sense the spectrum opportunities free from primary users (PUs), and opportunistically take advantage of these (temporarily) idle portions, known as spectrum holes. In this correspondence, we consider a variant of the cognitive radio resource allocation problem posed by Martinovic et al. in 2017. The distinguishing feature of this version of the problem is that each SU, due to its hardware limitations, imposes the requirement that the to-be-aggregated spectrum holes cannot be arbitrarily far from each other. We call this restriction as the Maximal Aggregation Range (MAR) constraint, and refer to this variant of the problem as the MAR-constrained hole assignment problem. The problem can be formalized as an NP-hard combinatorial optimization problem. We propose a novel binary integer linear programming (ILP) formulation to the problem. The number of constraints in this formulation is the number of spectrum holes plus the number of SUs. On the other hand, the number of binary decision variables in the formulation can be prohibitively large, as for each legitimate spectrum allocation to each SU, one variable is needed. We propose a branch-and-price (B&P) framework to tackle this challenge. This framework is in fact a branch-and-bound procedure in which at each node of the search tree, we utilize the so-called (delayed) column generation technique for solving the LP relaxation of the corresponding subproblem. As evidenced by the numerical results, the LP relaxation bounds are very tight. This allows for a very effective pruning of the search space. Compared to the previously suggested formulations, the proposed technique can require much less computational effort.

We study the capacity of the self-attention key-query channel: for a fixed budget, how many distinct token-token relations can a single layer reliably encode? We introduce Relational Graph Recognition, where the key-query channel encodes a directed graph and, given a context (a subset of the vertices), must recover the neighbors of each vertex in the context. We measure resources by the total key dimension D_K = h,d_k. In a tractable multi-head model, we prove matching information-theoretic lower bounds and upper bounds via explicit constructions showing that recovering a graph with m' relations in d_{model}-dimensional embeddings requires D_K to grow essentially as m'/d_{model} up to logarithmic factors, and we obtain corresponding guarantees for scaled-softmax attention. This analysis yields a new, capacity-based rationale for multi-head attention: even in permutation graphs, where all queries attend to a single target, splitting a fixed D_K budget into multiple heads increases capacity by reducing interference from embedding superposition. Controlled experiments mirror the theory, revealing sharp phase transitions at the predicted capacity, and the multi-head advantage persists when adding softmax normalization, value routing, and a full Transformer block trained with frozen GPT-2 embeddings.

Network binarization emerges as one of the most promising compression approaches offering extraordinary computation and memory savings by minimizing the bit-width. However, recent research has shown that applying existing binarization algorithms to diverse tasks, architectures, and hardware in realistic scenarios is still not straightforward. Common challenges of binarization, such as accuracy degradation and efficiency limitation, suggest that its attributes are not fully understood. To close this gap, we present BiBench, a rigorously designed benchmark with in-depth analysis for network binarization. We first carefully scrutinize the requirements of binarization in the actual production and define evaluation tracks and metrics for a comprehensive and fair investigation. Then, we evaluate and analyze a series of milestone binarization algorithms that function at the operator level and with extensive influence. Our benchmark reveals that 1) the binarized operator has a crucial impact on the performance and deployability of binarized networks; 2) the accuracy of binarization varies significantly across different learning tasks and neural architectures; 3) binarization has demonstrated promising efficiency potential on edge devices despite the limited hardware support. The results and analysis also lead to a promising paradigm for accurate and efficient binarization. We believe that BiBench will contribute to the broader adoption of binarization and serve as a foundation for future research. The code for our BiBench is released https://github.com/htqin/BiBench .

Recent work on KV cache quantization, culminating in TurboQuant, has approached the Shannon entropy limit for per-vector compression of transformer key-value caches. We observe that this limit applies to a strictly weaker problem than the one that actually matters: compressing the KV cache as a sequence. The tokens stored in a KV cache are not arbitrary floating-point data -- they are samples from the exact formal language the model was trained on, and the model is by construction a near-optimal predictor of that language. We introduce sequential KV compression, a two-layer architecture that exploits this structure. The first layer, probabilistic prefix deduplication, identifies semantically equivalent shared prefixes across sessions using the trie metric d_T(s, s') = -log_2 P_M(s ^ s') from Probabilistic Language Tries (PLTs). The second layer, predictive delta coding, stores only the residual of each new KV vector from the model's own prediction of it, achieving a per-token entropy bound of H(KV_{i+1} | KV_{<=i}) <= H(token_{i+1} | token_{<=i}). We prove that at typical language model perplexity -- approximately 10-20 for fluent English text -- this bound is 3.3-4.3 bits on average per token position, compared to TurboQuant's 3 bits per vector component (with typical attention heads having 64-128 components). The theoretical compression ratio over TurboQuant is approximately 914,000x at the Shannon limit. Even at 1000x above the entropy floor -- a deliberately pessimistic worst-case overhead, two orders of magnitude above the 2-5x typical of practical source coders -- the ratio remains approximately 914x over TurboQuant, with compression improving rather than degrading as context length grows. The two layers are orthogonal and compose with existing per-vector quantization methods including TurboQuant.

The growing interest in artificial intelligence has created workloads that require both sequential and random access. At the same time, NVMe-backed storage solutions have emerged, providing caching capability for large columnar datasets in cloud storage. Current columnar storage libraries fall short of effectively utilizing an NVMe device's capabilities, especially when it comes to random access. Historically, this has been assumed an implicit weakness in columnar storage formats, but this has not been sufficiently explored. In this paper, we examine the effectiveness of popular columnar formats such as Apache Arrow, Apache Parquet, and Lance in both random access and full scan tasks against NVMe storage. We argue that effective encoding of a column's structure, such as the repetition and validity information, is the key to unlocking the disk's performance. We show that Parquet, when configured correctly, can achieve over 60x better random access performance than default settings. We also show that this high random access performance requires making minor trade-offs in scan performance and RAM utilization. We then describe the Lance structural encoding scheme, which alternates between two different structural encodings based on data width, and achieves better random access performance without making trade-offs in scan performance or RAM utilization.

Large language models (LLMs) have revolutionized numerous applications, yet their deployment remains challenged by memory constraints on local devices. While scaling laws have enhanced LLM capabilities, the primary bottleneck has shifted from capability to availability, emphasizing the need for efficient memory management. Traditional compression methods, such as quantization, often require predefined compression ratios and separate compression processes for each setting, complicating deployment in variable memory environments. In this paper, we introduce BitStack, a novel, training-free weight compression approach that enables megabyte-level trade-offs between memory usage and model performance. By leveraging weight decomposition, BitStack can dynamically adjust the model size with minimal transmission between running memory and storage devices. Our approach iteratively decomposes weight matrices while considering the significance of each parameter, resulting in an approximately 1-bit per parameter residual block in each decomposition iteration. These blocks are sorted and stacked in storage as basic transmission units, with different quantities loaded based on current memory availability. Extensive experiments across a wide range of tasks demonstrate that, despite offering fine-grained size control, BitStack consistently matches or surpasses strong quantization baselines, particularly at extreme compression ratios. To the best of our knowledge, this is the first decomposition-based method that effectively bridges the gap to practical compression techniques like quantization. Code is available at https://github.com/xinghaow99/BitStack.

Given a set of K probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Minwise hashing is a fundamental and one of the most successful hashing algorithm in the literature. Recent advances based on the idea of densification~Proc:OneHashLSH_ICML14,Proc:Shrivastava_UAI14 have shown that it is possible to compute k minwise hashes, of a vector with d nonzeros, in mere (d + k) computations, a significant improvement over the classical O(dk). These advances have led to an algorithmic improvement in the query complexity of traditional indexing algorithms based on minwise hashing. Unfortunately, the variance of the current densification techniques is unnecessarily high, which leads to significantly poor accuracy compared to vanilla minwise hashing, especially when the data is sparse. In this paper, we provide a novel densification scheme which relies on carefully tailored 2-universal hashes. We show that the proposed scheme is variance-optimal, and without losing the runtime efficiency, it is significantly more accurate than existing densification techniques. As a result, we obtain a significantly efficient hashing scheme which has the same variance and collision probability as minwise hashing. Experimental evaluations on real sparse and high-dimensional datasets validate our claims. We believe that given the significant advantages, our method will replace minwise hashing implementations in practice.

Quantization is a fundamental technique to handle the growing sizes of embeddings generated by modern models. Existing quantization schemes are largely embedding agnostic and allocate bits uniformly across dimensions. However, recent models produce embeddings with significant geometric structure. In this work, we investigate whether a variable bit allocation scheme can improve quantization quality under a fixed memory budget. We propose a simple variable bit allocation framework that partitions an embedding into contiguous buckets and allocates storage non-uniformly across them. Using a greedy allocation strategy, we instantiate this framework for both Product Quantization (PQ) and Scalar Quantization (SQ). We perform a series of experiments on embeddings known to have the Matryoshka property (MRL), and consistently observe that non-uniform allocations outperform uniform baselines at identical storage budgets. The largest improvements occur in the low-bit regime, where uniform allocation is particularly inefficient for MRL embeddings. At the same compression rates, variable allocation improves recall by up to 8\% for PQ and up to 18\% for SQ. Our results suggest a new direction for structure-aware compression and indexing techniques for large-scale retrieval systems.

Randomly initialized dense networks contain subnetworks that achieve high accuracy without weight learning--strong lottery tickets (SLTs). Recently, Gadhikar et al. (2023) demonstrated that SLTs could also be found within a randomly pruned source network. This phenomenon can be exploited to further compress the small memory size required by SLTs. However, their method is limited to SLTs that are even sparser than the source, leading to worse accuracy due to unintentionally high sparsity. This paper proposes a method for reducing the SLT memory size without restricting the sparsity of the SLTs that can be found. A random subset of the initial weights is frozen by either permanently pruning them or locking them as a fixed part of the SLT, resulting in a smaller model size. Experimental results show that Edge-Popup (Ramanujan et al., 2020; Sreenivasan et al., 2022) finds SLTs with better accuracy-to-model size trade-off within frozen networks than within dense or randomly pruned source networks. In particular, freezing 70% of a ResNet on ImageNet provides 3.3 times compression compared to the SLT found within a dense counterpart, raises accuracy by up to 14.12 points compared to the SLT found within a randomly pruned counterpart, and offers a better accuracy-model size trade-off than both.

Recent years have witnessed the rapid advancements of large language models (LLMs) and their expanding applications, leading to soaring demands for computational resources. The widespread adoption of test-time scaling further aggravates the tension between model capability and resource consumption, highlighting the importance of inference efficiency. However, a unified metric that accurately reflects an LLM's efficiency across different model sizes and architectures remains absent. Motivated by the correlation between compression and intelligence, we introduce information capacity, a measure of model efficiency based on text compression performance relative to computational complexity. Larger models can predict the next token more accurately, achieving greater compression gains but at higher computational costs. Empirical evaluations on mainstream open-source models show that models of varying sizes within a series exhibit consistent information capacity. This metric enables a fair efficiency comparison across model series and accurate performance prediction within a model series. A distinctive feature of information capacity is that it incorporates tokenizer efficiency, which affects both input and output token counts but is often neglected in LLM evaluations. We assess the information capacity of 49 models on 5 heterogeneous datasets and observe consistent results on the influences of tokenizer efficiency, pretraining data, and the mixture-of-experts architecture.

Discrete flow matching provides a flexible framework for generative modeling on discrete structures. However, the standard factorized training objective exposes the model to targets of varying difficulty, mixing well-conditioned, predictable tokens with ambiguous, high-entropy ones. We empirically demonstrate that the uncertainty over the value of each token is closely related to the density of available context in its neighborhood. Motivated by this observation, we propose a simple modification to the underlying continuous-time Markov chain (CTMC) that incorporates local context information. Our context-weighted sampler improves generation quality with negligible computational overhead, while our scaled cross-entropy loss function reweights the training signal from different tokens and reduces generative perplexity by up to 63% on OpenWebText. Moreover, our approach matches a strong semi-autoregressive block diffusion baseline in quality while retaining the ability to perform generation in any order. These results highlight the role of local context as an important factor in discrete generative modeling and show that simple context-aware modifications can significantly improve both sampling and training efficiency.

Discrete diffusion models have emerged as a powerful generative modeling framework for discrete data with successful applications spanning from text generation to image synthesis. However, their deployment faces challenges due to the high dimensionality of the state space, necessitating the development of efficient inference algorithms. Current inference approaches mainly fall into two categories: exact simulation and approximate methods such as tau-leaping. While exact methods suffer from unpredictable inference time and redundant function evaluations, tau-leaping is limited by its first-order accuracy. In this work, we advance the latter category by tailoring the first extension of high-order numerical inference schemes to discrete diffusion models, enabling larger step sizes while reducing error. We rigorously analyze the proposed schemes and establish the second-order accuracy of the theta-trapezoidal method in KL divergence. Empirical evaluations on GPT-2 level text and ImageNet-level image generation tasks demonstrate that our method achieves superior sample quality compared to existing approaches under equivalent computational constraints.

Accurate battery capacity estimation is key to alleviating consumer concerns about battery performance and reliability of electric vehicles (EVs). However, practical data limitations imposed by stringent privacy regulations and labeled data shortages hamper the development of generalizable capacity estimation models that remain robust to real-world data distribution shifts. While self-supervised learning can leverage unlabeled data, existing techniques are not particularly designed to learn effectively from challenging field data -- let alone from privacy-friendly data, which are often less feature-rich and noisier. In this work, we propose a first-of-its-kind capacity estimation model based on self-supervised pre-training, developed on a large-scale dataset of privacy-friendly charging data snippets from real-world EV operations. Our pre-training framework, snippet similarity-weighted masked input reconstruction, is designed to learn rich, generalizable representations even from less feature-rich and fragmented privacy-friendly data. Our key innovation lies in harnessing contrastive learning to first capture high-level similarities among fragmented snippets that otherwise lack meaningful context. With our snippet-wise contrastive learning and subsequent similarity-weighted masked reconstruction, we are able to learn rich representations of both granular charging patterns within individual snippets and high-level associative relationships across different snippets. Bolstered by this rich representation learning, our model consistently outperforms state-of-the-art baselines, achieving 31.9% lower test error than the best-performing benchmark, even under challenging domain-shifted settings affected by both manufacturer and age-induced distribution shifts. Source code is available at https://github.com/en-research/GenEVBattery.

Discrete denoising diffusion models (DDMs) have recently emerged as a compelling alternative to autoregressive (AR) modeling for discrete data, offering parallel generation and iterative global refinement capabilities. Unlike continuous diffusion, where the state space is fixed, DDMs are fundamentally shaped by how the discrete state space is constructed: the tokenization scheme, the vocabulary topology, and domain-specific structural alphabets. This work introduces a unified conceptual framework that views discrete diffusion models through the construction of the underlying discrete state space. Within this framework, existing formulations, including transition-matrix, masking/absorbing-state, and score/ratio-based approaches, emerge as different instantiations of a common design space. The framework further exposes common design trade-offs across training objectives, inference algorithms, scaling behavior, systems optimization, and evaluation protocols, suggesting several promising directions for future research.