Entropy Coding

The KV cache has become a major bottleneck in deploying LLMs, as its memory footprint grows linearly with sequence length and batch size, imposing substantial pressure on both memory capacity and bandwidth. Among various KV cache compression techniques, quantization is particularly attractive due to its effectiveness and ease of deployment. However, most existing methods rely on fixed-width quantization, where a $b$ bit representation is inherently limited to $2^b$ quantization levels. As the bit width decreases, the number of available levels shrinks exponentially, leading to severe information loss and rapid performance degradation. We further observe that fixed-width quantization fails to exploit the highly non-uniform distribution of KV cache. After rotation and normalization, KV values approximately follow a normal distribution, with most values concentrated near the center and only a small fraction appearing in the tails. Nevertheless, fixed-width coding allocates the same number of bits to frequent and rare symbols. Entropy coding naturally exploits such non-uniformity by assigning shorter codewords to frequent symbols and longer ones to rare symbols, substantially reducing the average number of bits required for representation. However, its variable-length output is not suited to highly parallel attention kernels, where efficient dequantization and computation rely on regular memory layouts and fixed-stride accesses. To bridge this gap, we propose \textbf{D-Quant}, a flexible KV cache quantization framework that introduces a \textbf{drift} mechanism to convert entropy-coded representations of each token into fixed-size bitstreams, enabling regular memory access and parallel dequantization within attention kernels.

Speculative sampling accelerates diffusion model generation by drafting inexpensive candidate states and correcting them under a coupling that preserves the target distribution exactly, reducing the number of expensive target evaluations. Existing diffusion samplers, notably those based on reflection maximal coupling, are topologically constrained: their lookahead drafts form a chain graph, a single linear sequence, which inherently limits the acceptance rate per target evaluation. We connect speculative sampling in diffusion models to relative entropy coding (REC). This perspective shows the lookahead need not be linear and motivates our central contribution, draft trees, which enrich the candidates considered per round and lower the target function evaluations. We further adopt greedy rejection sampling, an REC algorithm, as the draft-target coupling, improving acceptance while guaranteeing exact target samples. Experiments across diverse target and draft models demonstrate up to 8.3% acceleration over the reflection coupling baseline in practical settings.

Universal multimodal embedding (UME) increasingly demands encoder's capacity for handling a broad range of tasks and modalities with increased complexity. Prior scaling methods either increase the representation size, retrieval effort, or scales the encoder into a heavy multimodal LLM. Recent works, such as Think-Then-Embed (TTE), explore scaling via reasoning tokens. However, embedding models are hard to scale up: increasing parameters directly tradeoffs for the large training batch size that contrastive learning needs, and retrieval has to be served under tight latency. Moreover, UME tasks are diverse in complexity, where scaling up embedders can bring significant redundant computation. In this work, we propose MOEMB, which instead scales UME along the expert axis through mixture-of-experts (MoE), growing encoder capacity while preserving single-vector, non-autoregressive encoding. Through a systematic study of the design space and training recipes for MoE-based UME, MoEMB sets a new state of the art on both MMEB-V2 and MRMR among models trained on public MMEB-family data: with only 3B active parameters, MoEMB surpasses TTE-based methods with >4x active parameters, using significantly less computes. To further improve the scalability and efficiency, we conduct the first comprehensive study of adaptive computation for MoE-based embedding, spanning diverse strategies across training-based and inference-only methods. Together, these results support expert scaling as an effective and efficient direction for UME, with adaptive computation further improving efficiency for MLLM-based embedding models towards large-scale retrieval and recommendation systems.

Search and database engines still store text as UTF-8, a format built for humans. But the systems that increasingly read and write that text (embedders, rerankers, and language-model agents) work with token IDs, not characters, so every access pays to translate between the two. As agents become the primary readers and writers of stored text, we argue for token-native storage: keep the text as the model's own byte-pair-encoding (BPE) token IDs. Packing r50k IDs as uint16 already beats UTF-8 by 2.25x on English with no compression, and an entropy coder on top reaches 3.30x. Across six tokenizers and three corpora (English, code, Hindi), compressing token IDs matches or beats every byte codec, even a corpus-trained zstd dictionary. Two findings sharpen the case. BPE numbers tokens by merge order instead of frequency, and re-ranking by frequency lets a plain integer codec (streamvbyte) recover most of the entropy coder's ratio while decoding ~7x faster, a near-free change to how AI labs publish vocabularies. And because a model reads token IDs, not text, a token-native store hands over the IDs directly instead of re-tokenizing on every read. The only requirement is that reader and writer share a tokenizer, and different model families often use different ones today, so we argue for standardization: a published, shared vocabulary, the way ASCII and UTF-8 standardized text.

A watermark in a generative model's output is usually asked only whether a text is machine-made. The same mark can do more: attribute it to the user who produced it, extract a hidden payload, or localize the part that survives editing. These form a forensic ladder, and we ask what each rung costs in the sample length $n$. One object organizes the answers. Let $S$ be the secret the mark carries (a user's identity or payload), and let the information profile $ν(t)=I(S;X_t\mid X_{<t})$ record how much the $t$-th token reveals about $S$ given the earlier ones. Its total mass pays for attribution and extraction; how that mass is spread pays for localization; and detection alone is paid for not by information but by presence, the distance from the marked to the unmarked distribution. The literature's two quality models, a mark subtle on every token and one that stamps a few tokens loudly, are two incomparable ways of capping this profile. Our main theorem settles the ladder's entropy column. For statistically distortion-free schemes, attributing a text to one of $N$ users costs $Θ(\log N/h)$ tokens over every stationary-ergodic source of entropy rate $h$, sharp to a $(1+o(1))$ factor: to our knowledge the first tight entropy-rate law for multi-user attribution (via exact alignment). The natural collision-counting analysis overcharges without bound; only a decoder thresholding each candidate by its own realized surprisal attains the rate while almost never implicating an innocent user. A matching converse makes the law two-sided, and extraction of an $\ell$-bit payload costs $Θ(\ell/h)$. Two gaps are real, not modeling artifacts: a $Θ(\log N)$-token window in which a text is provably machine-made yet unattributable, and a footprint-resolution uncertainty principle. Experiments on GPT-2, Pythia-410M, and Qwen2.5 recover the predicted constants.

In standard federated learning systems, the parameter server broadcasts the global model to the participating devices in every iteration. Motivated by the temporal correlation between consecutive global models, differential coding can be applied to global model dissemination to reduce the information magnitude, thereby enabling communication with fewer quantization bits. However, due to wireless link failures, devices may occasionally miss differential updates and consequently fail to reconstruct the global model. As a result, they either continue local training based on an outdated model or remain idle until the next full-model broadcast becomes available. To address this challenge, we propose a mixed-timescale differential coding (MTDC) scheme that performs differential coding at two different levels by adjusting the reference model. With MTDC, a device can reconstruct the latest global model between two full-model broadcasts even if it misses a differential update. We provide a convergence analysis that motivates the design of an age-aware variant of MTDC, along with a device scheduling policy to further improve communication efficiency. Simulation results demonstrate that the proposed MTDC schemes achieve superior learning performance compared to baseline methods under similar communication resource budgets in the presence of downlink transmission failures.

Enhancing perceptual dimensions while miniaturizing imaging systems presents significant challenges for high-dimensional visual sensing. Conventionally, the acquisition of the 5D (x,y,u,v,λ) spectral light field (5D-SLF) data cube relies on bulky and expensive camera arrays, which are impractical for widespread application. Existing single-detector systems are fundamentally limited by a trade-off between the resolutions of different dimensions owing to insufficient coding capabilities. Here we introduce an Aperture-aware Dispersion Light-field Imaging Spectrometer (ADLIS), that targets a synergy between compactness and resolution through aperture-multiplexed modulation, leveraging the inherent spectral-filtering properties of birefringent material. Using only a manufacturing-friendly and cost-effective phase plate made of birefringent quartz crystal, the aperture of the proposed ADLIS enables compact angular-spectral encoding that is highly sensitive to both the incident angle and spectrum of incoming light. In contrast to the viewpoint-separation approach of microlens arrays, ADLIS employs aperture encoding to superimpose all viewpoints onto each sensor pixel. This shifts the design paradigm from spatial division to encoding integration, aiming to achieve full-resolution light field recovery. Thus, we develop the Aperture-aware Dispersion Light-field Imaging (ADLI) framework, which optimizes the aperture design and 5D-SLF reconstruction in an end-to-end (E2E) manner. Trained by simulation data and validated through real-world experiments, our system achieves robust high-performance 5D-SLF imaging while maintaining full spatial resolution.

Learned video codecs based on continuous latent representations struggle to operate reliably below 0.1 bits per pixel~(bpp): without a differentiable rate signal, Lagrangian optimisation cannot effectively trade reconstruction quality for bitrate at extreme compression ratios. We demonstrate that discrete latent representations sidestep this limitation entirely. In a vector-quantized~(VQ) codec, the codebook size~$K$ imposes a hard information ceiling of $\log_2 K$ bits per symbol; a learned autoregressive prior then exploits the non-uniform distribution of code usage -- which we show follows a power law -- to push actual bitrates well below this ceiling, without any rate-penalty tuning. Building on the MS-VQ-VAE architecture introduced in~\cite{kotthapalli2026msvqvae}, we sweep $K \in \{128, 256, 512, 1024\}$ under a uniform training protocol to trace four operating points on the rate-distortion~(RD) curve. We identify and resolve a critical training instability: gradient-based VQ collapses catastrophically at $K \leq 512$, whereas EMA-stabilised codebook updates with dead-code restart maintain full utilisation across all configurations. On 500 UCF101 test clips ($64\!\times\!64$, 32frames), our models operate at 0.043-0.064bpp -- 3.3-5$\times$ below H.264's practical floor and $5$-$7.6\times$ below H.265's floor at this resolution. Every MS-VQ-VAE configuration outperforms H.265 CRF,36 on perceptual quality (LPIPS) despite using $5$-$7.6\times$ fewer bits. At $K{=}1024$, the model surpasses H.265 CRF,36 on LPIPS by a margin of 0.072 absolute while using $5.1\times$ fewer bits. Codebook analysis confirms power-law index distributions and 70-85% entropy efficiency, establishing the pipeline as a principled learned entropy coder.

Vision-language models (VLMs) often underperform on evidence intensive tasks because decisive visual evidence are small, localized, and easy to overlook, leading to failures in evidence readout even when high-level reasoning is intact. Prior inference-time visual interventions can improve grounding without retraining, but they are largely open-loop and lack a mechanism to verify whether highlighted evidence is actually used. We study answer-span prediction entropy as a model-internal feedback signal and show that naive entropy minimization is ambiguous, since low entropy may arise from evidence-grounded confidence or shortcut collapse. To resolve this ambiguity, we introduce low-entropy anchors and an entropy-shaping objective that reduces answer uncertainty while preserving baseline high-confidence tokens. We instantiate this principle in SPOT-E, a plug-and-play test-time method that produces question-conditioned spotlights, optimized per instance via light-weight tuning based on Group Relative Policy Optimization (GRPO). Across all benchmarks and different VLM families, SPOT-E yields consistent gains and improved robustness under visual corruptions. Code is publicly available at: https://github.com/YinBo0927/SPOT-E

Many classically studied function classes are known to be approximated optimally by superpositional methods, i.e. with approximants constructed as the linear combination of elements in some dictionary. Here optimality means that the uniform approximation error viewed as a function of the number of parameters used has polynomial decay of the highest order achievable by any parametrized method whose parameters can be encoded as a bit string of length proportional, up to logarithmic factors, to the number of parameters. While compositional methods like neural networks are structurally different, their approximation rates can be made comparable by imposing constraints that ensure such a proportional bit string encoding. In this work we study function classes exhibiting structural properties that limit superpositional approximation rates to be strictly lower than compositional approximation rates. In particular, we construct explicit examples for which there is an arbitrarily large gap.

Lossy text compression reduces data size while preserving core meaning, making it well-suited for summarization, automated analysis, and digital archives. Despite the dominance of transformer-based models in language modeling, integrating context vectors and entropy coding into Sequence-to-Sequence (Seq2Seq) generation remains underexplored. A key challenge lies in identifying the most informative context vectors from encoder output and incorporating entropy coding to enhance storage efficiency while maintaining high-quality outputs, even under noisy text. We introduce TextEconomizer, an encoder-decoder framework paired with a transformer neural network that reduces variable-sized inputs by 50% to 80% without prior knowledge of dataset dimensions. Our model achieves competitive compression ratios via entropy coding while delivering near-perfect text quality, assessed by BLEU, ROUGE, METEOR, and semantic similarity scores. TextEconomizer operates with approximately 153x fewer parameters than comparable models, achieving a 5.39x compression ratio without sacrificing semantic quality. We also evaluate an LSTM-based autoencoder achieving a state-of-the-art 67x compression ratio with 196x fewer parameters, and LLaMAFormer, a modified transformer with 263x fewer parameters than ICAE while maintaining competitive text quality. TextEconomizer significantly surpasses existing transformer-based models in balancing memory efficiency and high-fidelity outputs, marking a breakthrough in lossy compression with optimal space utilization.

Lossless pixel-level image transmission is a fundamental regime beyond semantic communications, because exact recovery requires both accurate symbol probability modeling and reliable delivery over noisy channels. This paper proposes DDM-SSCC, a discrete-diffusion-model-based separate source-channel coding framework for lossless image transmission. Different from raster-order autoregressive coding, the proposed source codec adapts a diffusion language model to pixel-token restoration and performs synchronized reverse arithmetic coding under bidirectional attention, allowing multiple masked tokens to be coded within one reverse denoising step. This progressive restoration process also yields a more favorable source representation for noisy transmission, since newly restored tokens can serve as bidirectional context in subsequent denoising steps. To bridge the gap between generation-oriented masked denoising and lossless arithmetic coding, we further introduce a Halton-guided denoising order, a mask-ratio-aware cosine schedule, and a lightweight temperature calibration module. These designs respectively improve spatial coverage, adapt the denoising pace to context reliability, and calibrate the probability tables used by arithmetic coding. Experiments on CIFAR10, DIV2K-LR-X4, and Kodak over additive white Gaussian noise and Rayleigh fading channels show that DDM-SSCC achieves better exact-recovery performance than representative lossless and semantic communication baselines, while ablation studies verify the effectiveness of the proposed denoising order, schedule, and calibration modules.

State-of-the-art learned image compression (LIC) schemes are increasingly based on hybrid CNN-transformer architectures. To further improve rate-distortion performance, we introduce channel-wise wavelet transforms into both the transformer and entropy-coding components. First, we propose a channel-wise wavelet-domain transformer attention (ChWDTA) mechanism. ChWDTA keeps the efficient windowed spatial self-attention used in modern LIC backbones, but computes the Q/K/V projections on channel-wise wavelet-transformed features before mapping the attention output back with the inverse transform. The resulting Channel-wise Wavelet-Domain Transformer Block (ChWDTB) therefore preserves the spatial tokenization pattern of windowed attention while sparsifying the channel covariance seen by the attention projections. Second, in the entropy-coding stage, we introduce a channel-wise wavelet packet (ChWP) decomposition that produces four equal-sized subbands, which better fit channel-wise slice-based autoregressive entropy modeling. When each channel-wise subband is divided into two slices, we use eight slices for entropy coding. With this configuration, the proposed scheme obtains BD-rate reductions of -17.82%, -19.15%, and -22.56% on the Kodak, CLIC Professional Validation, and Tecnick test sets, respectively. Even when each channel-wise subband is coded as a single slice, the scheme still retains most of the coding gains with lower complexity. The results confirm the advantage of introducing wavelet transform in CNN-transformer-based LIC schemes.

While feed-forward 3D Gaussian splatting reconstructs renderable Gaussian primitives from sparse context views without per-scene optimization, existing pipelines do not provide a compact scene representation for storage or transmission. A natural solution is to apply existing 3DGS compression methods to the generated Gaussian primitives. However, this approach operates on the final irregular 3D representation and is decoupled from the internal feature-to-Gaussian generation process, which limits compression efficiency. To address this, we introduce CodecSplat, an ultra-compact latent coding framework for feed-forward 3D Gaussian splatting. CodecSplat first encodes an intermediate 2D Gaussian-generation feature into an entropy-coded scene bitstream. At the decoder, the latent feature is reconstructed and used to predict depth and Gaussian parameters, which are then mapped to 3D Gaussian primitives. Note that, by integrating compression into the feed-forward Gaussian generation pipeline, CodecSplat avoids inefficient compression over irregular 3D Gaussian primitives and allows the codec to exploit the structured intermediate feature representation. We instantiate CodecSplat on a feed-forward Gaussian splatting backbone with depth-guided multi-view feature refinement and a hierarchical learned feature codec. On DL3DV and RealEstate10K datasets, CodecSplat achieves 23.56-26.36 dB and 24.76-27.05 dB PSNR with only 20.00-107.77 KiB and 3.37-12.51 KiB per scene, respectively. This is roughly one order of magnitude smaller than compressing feed-forward generated Gaussian primitives, while preserving controllable rate-distortion behavior.

Entropy coding is widely used in typical learned image compression (LIC) that converts latents into a compact bitstream. However, entropy coding is typically sequential and becomes the coding latency bottleneck. To overcome it, we present Entropy-Coding Free Learned Image Compression (EF-LIC), a multi-rate framework that generates compact representation by removing statistical and correlation redundancy with low coding latency. First, we introduce unconstrained vector quantization and prove that its index distribution approaches the maximum-entropy bound, yielding minimal statistical redundancy. Second, we propose a context-conditioned autoregressive transform that directly reparameterizes the latents to reduce inter-dependency. Theoretical analysis shows that EF-LIC can remove correlation redundancy as effectively as typical LIC with entropy coding, leading to comparable compression performance. Experiments show EF-LIC achieves up to 67.86% bitrate reduction over MS-ILLM on Kodak with LPIPS. Ablation studies further show EF-LIC matches the compression performance of its entropy-coding based variant while achieving over $3\times$ faster encoding and $5\times$ faster decoding.

Generative recommendation reformulates recommendation as next-token prediction over discrete semantic identifiers (IDs). A fundamental yet unexplored design choice is that existing methods employ fixed-length tokenization for all items, implicitly assuming uniform encoding capacity regardless of item characteristics. Through systematic experiments across four datasets, we discover the Popularity-Length Paradox: popular items achieve optimal performance with short IDs, while tail items require substantially longer codes to capture discriminative semantics. This reveals a critical mismatch where popular items benefit from abundant collaborative signals and require minimal semantic detail, whereas tail items must rely on fine-grained content features due to sparse interaction data. To address this, we propose VarLenRec, a framework for learning variable-length tokenization. We develop Popularity-Weighted Information Budget Allocation (PIBA), an information-theoretic framework proving that optimal ID length should scale as a negative power of popularity. Directly implementing variable-length allocation faces two technical challenges: standard Euclidean residual quantization lacks geometric capacity to support diverse code lengths without distortion, and discrete length decisions are non-differentiable. We address these through Hyperbolic Residual Quantization, which leverages the exponential volume growth of the Poincaré ball to naturally stratify encoding capacity, and a Soft Length Controller, which enables differentiable length prediction via continuous layer retention probabilities regularized by PIBA-derived priors. Extensive experiments demonstrate that VarLenRec achieves significant improvements over state-of-the-art methods in recommendation accuracy and training/inference efficiency, revealing the importance of adaptive encoding capacity in generative recommendation.

With LLM watermarking already being deployed commercially, practical applications increasingly require multibit watermarks that encode more complex payloads, such as user IDs or timestamps, into the generated text. In this work, we propose a fundamentally new approach for multibit watermarking: introducing binomial encoding to directly encode every bit of the payload at every token position. We complement our approach with a stateful encoder that during generation dynamically redirects encoding pressure toward underencoded bits. Our evaluation against 8 baselines on up to 64-bit payloads shows that our scheme achieves superior message accuracy and robustness, with the gap to baseline methods widening in more relevant settings (i.e., large payloads and low-distortion regimes). At the same time, we challenge prior works' evaluation metrics, highlighting their lack of practical insights, and introduce per-bit confidence scoring as a practically relevant metric for evaluating multibit LLM watermarks.

Learning, prediction, and compression are intimately connected: a model that accurately predicts the next symbol in a sequence can be coupled with a source coder to compress that sequence near its information-theoretic limit. When tokenized characters arriving at a fixed reading pace are encoded into variable-length codewords and streamed over a fixed-rate channel, a queue forms whose per-token delay depends on the mean and variance of the bit lengths and on the coder's algorithmic latency. This paper investigates the compression--delay tradeoff that arises when a causal language model serves as the sequential predictor within a predict-then-code architecture for real-time text transmission. Several coding schemes are compared: Shannon (ideal), Huffman, arithmetic coding, rANS at various block sizes, and gzip. The analysis separates algorithmic delay, inherent to the coder, from computational delay, which shrinks as hardware improves. Huffman is the practical choice for over-provisioned channels, with zero algorithmic delay and modest compression overhead. Arithmetic coding achieves near-optimal compression at the cost of decodability delay. Findings are validated across two scales: GPT-2 (124M) and Llama~3.2 (3B), a twenty-five-fold parameter range. This scaling yields an approximately 38% reduction in bits per character, effectively over-provisioning the channel and thereby changing which coder is optimal.

We introduce Flashback, a reversible string decomposition that repeatedly peels the maximal leading and trailing character runs from a sentinel-wrapped input, recording each pair as one bilateral token. Decomposition and reconstruction both run in O(n) time and space. Our central result is a run-pairing theorem: Flashback is equivalent to pairing the first run of the string with the last, the second with the second-to-last, and so on. This gives an exact token count of 1+[r/2] for a string with r maximal runs, and matches a lower bound that holds for any admissible bilateral run-peeling scheme. From the run-pairing theorem the main structural properties follow as corollaries: the irreducible peeling kernel uses at most two symbols; palindromes are precisely the strings whose run-length encoding is symmetric with an odd number of runs; the image of the decomposition admits an explicit finite-state characterisation; and changing one run length rewrites exactly one content token.