Lookup Tables

Field-programmable gate arrays (FPGAs) enable efficient neural-network inference, but most deployment flows either accelerate multiply-accumulate operations or convert pretrained quantized models into lookup tables (LUTs). We present DiffLUT-Net, an FPGA-native network connected by six-input LUTs that are trained from scratch. We jointly learn the 64 truth-table entries of a LUT and the source to each of its six input ports using a differentiable LUT function relaxation and hardware source selection. After training, the truth tables and connections are discretized, unused logic can be pruned, and the network is exported directly as synthesizable Verilog. Across five benchmarks, DiffLUT-Net achieves favorable accuracy-resource trade-offs. These results demonstrate the effectiveness of jointly learning LUT functions and sparse connectivity for compact FPGA-native inference. The code is available at https://github.com/TUDa-HWAI/DiffLUT-Network.

Lookup table (LUT)-based image denoising methods have attracted increasing attention due to their high efficiency and hardware-friendly properties. However, existing RGB-LUT approaches require three identical LUTs to process RGB channels in parallel, resulting in large on-chip SRAM consumption. A simple alternative is to apply LUT processing only to the luminance (Y) channel in the YUV color space to reduce memory usage. However, this naive strategy leads to degraded restoration quality, since ignoring the chrominance (UV) channels introduces color distortion and residual artifacts. In this work, we propose Hybrid-LUT, a YUV-based asymmetric channel-processing framework that combines LUT and filtering in a unified design. Specifically, a multi-band LUT branch with pixel-level weight fusion is applied to the Y channel to recover fine textures, while lightweight filtering is used for the UV channels to maintain color consistency. This design reduces LUT storage by two-thirds compared with RGB-LUT methods while maintaining the same runtime throughput. Extensive experiments show that Hybrid-LUT achieves state-of-the-art (SOTA) performance across multiple benchmarks with only 421 KB of storage. In particular, our method surpasses existing LUT-based denoising approaches by at least 0.63 dB CPSNR on real-world datasets, demonstrating its effectiveness for image denoising on resource-constrained edge devices. The project is available at https://github.com/Ai-ZL/Hybrid-LUT .

Multimodal large language models have advanced visual understanding, yet perception-intensive reasoning remains challenging. Recent latent visual reasoning methods introduce hidden-space computation before answering, but they often rely on costly intermediate supervision, such as bounding boxes, sketches, or interleaved rationales. These strategies focus on how latent states should be shaped, but do not explicitly assess whether the latent is useful for the final answer. We propose LUT, a latent reasoning framework trained with only standard VQA pairs. LUT centers training on Latent Utility at two levels. At the trajectory level, we propose Utility-Aware Latent Distillation SFT, which explores answer-relevant latent trajectories, selects qualified trajectories by their information gain, and distills more reliable and learnable supervision through curriculum learning. At the step level, we propose Latent Attribution Policy Optimization, which uses answer-to-latent attribution to differentially optimize latent steps during reinforcement learning. Experiments on perception-intensive visual reasoning benchmarks show that LUT outperforms previous latent reasoning methods and remains competitive with latent-text interleaved methods with lower annotation cost.

Memoizing an expensive function of a sorted score vector is a data-structure problem before it is a numerical one: at a billion gridpoints, a hash map or a search tree spends most of its space on keys the grid already determines. We describe an implemented memo table that stores none. An entry's address is computed in closed form from the sorted argument itself, so $N$ values occupy $N$ slots, the argument is recoverable from the index, and the table can be memory-mapped and served from a file larger than RAM. Against a chained hash map it uses $5.7\times$ less memory at $37$M entries and $10.2\times$ less at $1.9$B, answers queries up to $2.9\times$ faster and builds up to $250\times$ faster; on 64 threads its construction needs no coordination; a sharded hash gains only $1.17\times$. Against an open-addressing table with inline keys it is $4$--$7\times$ smaller and $100\times$ faster to build but $1.5\times$ slower to query, a deficit we trace to the $O(d)$ index arithmetic. At $22$ GB on a $16$ GB desktop it serves each query in one disk access, where no key-storing container can be built; and its order-preserving addressing keeps a perturbation workload on the same pages that a hashed layout scatters. The closed form exists because the key set is the multiset combinations, whose index is the combinatorial number system. Memoizing Plackett--Luce normalization runs $25$--$55\times$ faster than Newton's method; memoizing $α$-entmax thresholds does not pay. The contrast says when this structure is worthwhile.

Underwater image enhancement is challenged by spatially non-uniform, wavelength-dependent attenuation. Propagation distance and wavelength govern this degradation, while YCbCr separates luminance from chrominance for restoration. We propose DY-LUT, a depth-aware YCbCr lookup-table framework for real-time enhancement. A dual-branch encoder predicts image-level fusion weights and a joint pair of pixel-wise degradation indices from image and depth features. These quantities condition learnable 4D LUTs, followed by lightweight local refinement. DY-LUT preserves traditional LUT efficiency while enabling depth-conditioned, spatially adaptive restoration. With externally supplied depth, its 3.56M-parameter enhancement network achieves competitive quality on UIEB-90 and LSUI and runs $9$--$304\times$ faster than representative high-capacity baselines. Adaptive inference further maintains real-time performance ($\sim7$ ms) for 4K UIQAD images. DY-LUT also benefits downstream detection and feature matching. Ablations show that YCbCr is a more effective basis than RGB for depth-conditioned lookup, while the jointly learned indices further improve adaptive querying. These results provide a physically grounded route to efficient UIE on practical platforms.

We introduce a novel method for both partial and full optimization of the connections in deep differentiable logic gate networks (LGNs) and lookup table networks (LUTNs). Our training method utilizes a probability distribution over a set of connections per gate/lookup table (LUT) input pin, selecting the connection with highest merit, all whilst the optimal gate types or LUT-entries are learned in parallel. We show that the connection-optimized LGNs outperform standard fixed-connection LGNs on the Yin-Yang, MNIST Handwritten Digits and Fashion-MNIST benchmarks, while requiring only a fraction of the number of logic gates. We achieve 98.92% on the MNIST dataset with two layers of 8000 gates. With only one layer of 8000 gates, we obtain 98.45%, showing that our method requires almost 50 times fewer gates compared to fixed-connection LGNs. Training stability up to ten layers has been ensured by employing a high learning rate, straight-through estimators and trimming constant-output gate types. Additionally, we present a LUT neuron description that enables stable training with backpropagation, tested up to 6-layer deep networks. The model requires four times fewer trainable parameters and still achieves a higher accuracy compared to the fixed-connection LGN training algorithm. Our connection-training algorithm also works well for the LUTNs, achieving an accuracy of 98.88% for two layers of 2000 6-input LUTs.

With the advancement of imaging technology, ultra-high-definition images have become increasingly essential in modern visual applications. However, existing multi-focus image fusion remains largely confined to low-resolution images and faces three major barriers in UHD scenarios, namely data availability, model adaptability, and deployment feasibility, which severely hinder its practical application. To shatter these barriers, first, we propose the UHD-MFF dataset, the first large-scale ultra-high-resolution multi-focus fusion dataset. Second, we propose a scale-specialized lookup-table framework tailored for ultra-high-resolution images, termed as UMF-LUT. It consists of Coarse-Region Lookup Table (C-LUT) and Detail-Edge Lookup Table (D-LUT). Specifically, C-LUT performs joint queries of multiple gradient cues and semantic cues at low-resolution scales to enable region-level decision-making. Also, D-LUT operates at high-resolution scales, leveraging efficient Laplacian cues to provide complementary edge-level decision information. Such a design makes the model particularly well-suited for ultra-high-resolution multi-focus image fusion. Finally, it offers strong deployability with minimal computational overhead, enabling real-time 4K multi-focus fusion and showing promising potential for smartphone. Extensive experiments demonstrate that it outperforms SOTA methods in both visual fidelity and quantitative metrics. It effectively advances the development of multi-focus image fusion toward ultra-high-resolution imaging scenarios. The code is available at https://github.com/zyb5/UHD-MFF.

Agentic tool-calling language models depend on large registries of callable APIs, functions, and local actions. Placing full tool specifications directly in the prompt incurs a cost that scales linearly with the size of the tool registry, rapidly consuming the context budget. As the registry grows, this leads to higher latency and degrades selection accuracy, particularly due to interference from irrelevant tools. We overcome these limitations by introducing NTILC, a neural tool selection and invocation framework that replaces in-context registry look-up with learned latent retrieval. NTILC maps both user intent and tool specifications into a shared embedding space, enabling tool selection via external retrieval rather than in-context lookup. The language model is conditioned only on the selected tool schema, allowing for precise, constrained argument generation. Central to our approach is a signature-aware composite objective, which augments semantic similarity with constraints derived from tool signatures (e.g., argument schema, type compatibility, and return types). By combining Circle Loss with a Functional Margin Loss, the model enforces separation between tools that are semantically similar but incompatible under their execution signatures. We evaluate NTILC on public tool-selection and function-calling datasets and report context token usage, retrieval accuracy, and selection latency metrics. Across these settings, NTILC reduces context window consumption by over 95% and inference latency by up to 74% compared to long-context ICT baselines.

We present Flash Cubical, a highly efficient computation of cubical persistence on a V-filtration for 2D and 3D images over $\mathbb{F}_2$. The implementation is built around three core ideas. First, cubical complexes satisfy properties that allow for the computation of persistence of the highest dimension via union-find and duality. Second, pruning of certain edges allows for a fast and efficient implementation of union-find. Third, the use of a lookup table, which exploits the regularity of cubical complexes to pre-compute local information. This avoids the need to compute local information at run time. To the best of our knowledge, this is the most efficient implementation of cubical persistence with a V-filtration, both in terms of time and memory costs. Although the paper focuses on persistence for V-filtration cubical complexes, the underlying ideas generalise naturally to T-filtrations on cubical complexes and suggest promising directions for other complexes.

3D Lookup Tables (3D LUTs) are widely used for color mapping, but their grid-based representation requires discretizing the RGB space, leading to a capacity-memory trade-off that becomes prohibitive when storing large numbers of LUTs. Recent approaches adopt implicit neural representations to improve scalability, yet their black-box nature limits interpretability and hinders intuitive, localized editing. In this paper, we propose Gaussian LUT (GLUT), a continuous and explicit color representation that models color transformations using a set of learnable 3D Gaussian primitives. By avoiding fixed-resolution grids, GLUT achieves flexible representational capacity while maintaining a compact memory footprint. Its explicit, spatially localized formulation further enables both accurate modeling and interpretability. Building on this representation, we introduce a compact conditional generator (CGLUT) that predicts GLUT parameters for multiple LUT instances, encoding diverse color styles in a single framework to enable smooth and controllable LUT style blending. Moreover, GLUT supports efficient, user-friendly editing by allowing localized adjustments to specific color regions without global retraining. Experimental results demonstrate that our approach outperforms prior neural LUT representations in both accuracy and efficiency, while offering improved interpretability and interactive control.

Scaled Outer Product (SOP) is a post-training quantization methodology for large language model weights, designed to deliver near-lossless fidelity at 4.5--6 bits per weight on hardware with per-layer LUT decode. The methodology combines per-layer search of fixed and dynamic codebook pairs selected by a per-block selection bit, signed per-block scales, activation-weighted cosine selection, and multiple-choice knapsack promotion of sensitive layers with outlier and sparse-residual correction. Fixed codebooks include NF4, BOF4, Split87, and SH4; per-layer optimized codebooks (DD4) are hosted in LUT SRAM. A new hardware-efficient LUT output format (HIF) is proposed to improve performance, energy, and cost. Across six open model families, the recommended FP6 operating point (E2M3sUE4M4, 6.5 bpw) achieves lower weight reconstruction error than the conventional per-layer-POT FP8 baseline (E4M3, 8.0 bpw) at 1.5 bpw lower storage cost, demonstrating that block-scaled small atoms with carefully chosen scale precision can replace conventionally-deployed FP8. Full evaluation across the 4.5--6 bpw range, including layer promotion and sparse residual correction, is reported in a companion paper.

Read-only memory (ROM) provides deterministic access to predefined data mappings. Extending ROM concepts to the optical domain enables high-bandwidth, low-latency, and parallel memory access, but realizing compact and reconfigurable optical ROM remains challenging due to loss, wavelength control, and integration constraints. This work presents a high-speed, reconfigurable photonic ROM architecture implemented using integrated microring resonators (MRRs). The ROM encodes predefined input-output mappings directly in the spectral response of the photonic devices, enabling deterministic lookup-based operation without dynamic computation during readout. To improve scalability and reduce cumulative insertion loss, the architecture employs compact banked sub-arrays that are selectively addressed through an optical decoding mechanism. Reconfigurability is achieved using transistor-based optical selectors, allowing different ROM banks to be activated without physical light rerouting or interferometric structures. The proposed photonic ROM is designed and evaluated using device-level simulations based on the GlobalFoundries 45SPCLO silicon photonics platform. Simulation results demonstrate reliable operation at data rates up to 12.5 GHz, with stable light-to-current transfer characteristics obtained through integrated photodiode readout. The optical ROM can be used to implement nonlinear activation functions utilised in photonic accelerator architectures, including sigmoid, tanh, ReLU, and exponential mappings.

The Airborne Collision Avoidance System Xu (ACAS-Xu) relies on large certified Look-Up Tables (LUTs) that encode the exact decision logic used in operation. Neural-network-based approximations have been proposed to reduce memory requirements, but they inherently introduce approximation errors and complicate formal verification. This paper presents a symbolic compression approach based on Binary Decision Diagrams (BDDs) that preserves the exact semantics of the ACAS-Xu LUTs. The resulting representation is canonical, deterministic, and fully equivalent to the original tables, enabling sound and exact reasoning over the complete decision logic. By expressing both the system behavior and domain-specific operational properties within a common Boolean framework, verification reduces to efficient BDD operations and emptiness checks, with precise counterexamples generated when properties are violated. We demonstrate that the proposed BDD-based representation significantly reduces memory usage, achieves predictable and low-latency execution, and can be deployed on embedded platforms. These results highlight BDDs as a compelling alternative for exact, verifiable, and embedded deployment of ACAS-Xu decision logic.

Achieving a desirable balance between rendering quality and real-time performance is a long-standing challenge in modern game and rendering engines, particularly on resource-constrained mobile devices such as laptops, tablets, and smartphones. Existing approaches to automatic rendering parameter optimization either depend on exhaustive per-scene pre-computation that spans several days, suffer from the prohibitive inference overhead of neural networks that prevents per-frame adaptation, or lack generalizability across heterogeneous hardware and diverse scenes. In this paper, we propose \textbf{LUT-Opt}, a lightweight, general-purpose framework for adaptive per-frame rendering parameter optimization. Our method decomposes the joint optimization of rendering time and image quality into a tractable two-stage pipeline. In the offline stage, we train a pair of XGBoost regressors to predict rendering time and image quality from rendering parameters, hardware state, and scene complexity descriptors. The trained ensemble models are then distilled into compact lookup tables (LUTs) through systematic discretization and a two-phase linear search that first constrains rendering time and subsequently maximizes structural similarity (SSIM). During runtime, the pre-computed LUT is queried every frame in sub-millisecond time, enabling truly adaptive parameter selection with negligible computational overhead. We validate LUT-Opt on two representative rendering techniques -- subsurface scattering (SSS) and hybrid-pipeline ambient occlusion (AO) -- implemented within Unreal Engine 5. Extensive experiments across multiple scenes and GPU configurations demonstrate that LUT-Opt reduces subsurface scattering rendering time by approximately 40% and ambient occlusion rendering time by roughly 70%, while incurring only about 2% increase in image quality error, with per-frame inference latency below 0.1\ ms.

Lookup-table (LUT) based neural networks can deliver ultra-low latency and excellent hardware efficiency on FPGAs by mapping arithmetic operations directly onto the logic primitives. However, state-of-the-art LUT-aware training (LAT) approaches remain difficult to use in practice: they are often orders of magnitude slower to train than conventional networks, require non-trivial manual tuning for hardware efficiency, and lack an end-to-end workflow. This work presents HGQ-LUT, integrated in https://github.com/calad0i/HGQ2, a new LAT approach that achieves state-of-the-art hardware efficiency while accelerating training by over 100 times on modern GPUs. HGQ-LUT introduces LUT-Dense and LUT-Conv layers that are implemented with regular, accelerator-efficient tensor operations during training, which are then compiled into logic LUTs for hardware. By combining these layers with fine-grained, element-wise heterogeneous quantization (including zero-bit pruning) and a LUT-aware resource surrogate, HGQ-LUT enables the automatic exploration of accuracy-resource trade-offs without manual bit-width tuning. We further integrate HGQ-LUT into open-source toolchains, enabling unified design, compilation, and bit-exact verification of hybrid architectures that mix LUT-based with conventional arithmetic blocks. These features make LAT-based DNNs practical for real-world deployment, such as at the CERN Large Hadron Collider's experiments.

Large token-indexed lookup tables provide a compute-decoupled scaling path, but their practical gains are often limited by poor parameter efficiency and rapid memory growth. We attribute these limitations to Zipfian under-training of the long tail, heterogeneous demand across layers, and "slot collapse" that produces redundant embeddings. To address this, we propose X-GRAM, a frequency-aware dynamic token-injection framework. X-GRAM employs hybrid hashing and alias mixing to compress the tail while preserving head capacity, and refines retrieved vectors via normalized SwiGLU ShortConv to extract diverse local n-gram features. These signals are integrated into attention value streams and inter-layer residuals using depth-aware gating, effectively aligning static memory with dynamic context. This design introduces a memory-centric scaling axis that decouples model capacity from FLOPs. Extensive evaluations at the 0.73B and 1.15B scales show that X-GRAM improves average accuracy by as much as 4.4 points over the vanilla backbone and 3.2 points over strong retrieval baselines, while using substantially smaller tables in the 50% configuration. Overall, by decoupling capacity from compute through efficient memory management, X-GRAM offers a scalable and practical paradigm for future memory-augmented architectures. Code aviliable in https://github.com/Longyichen/X-gram.

While Mixture-of-Experts (MoE) scales capacity via conditional computation, Transformers lack a native primitive for knowledge lookup, forcing them to inefficiently simulate retrieval through computation. To address this, we introduce conditional memory as a complementary sparsity axis, instantiated via Engram, a module that modernizes classic $N$-gram embedding for O(1) lookup. By formulating the Sparsity Allocation problem, we uncover a U-shaped scaling law that optimizes the trade-off between neural computation (MoE) and static memory (Engram). Guided by this law, we scale Engram to 27B parameters, achieving superior performance over a strictly iso-parameter and iso-FLOPs MoE baseline. Most notably, while the memory module is expected to aid knowledge retrieval (e.g., MMLU +3.4; CMMLU +4.0), we observe even larger gains in general reasoning (e.g., BBH +5.0; ARC-Challenge +3.7) and code/math domains~(HumanEval +3.0; MATH +2.4). Mechanistic analyses reveal that Engram relieves the backbone's early layers from static reconstruction, effectively deepening the network for complex reasoning. Furthermore, by delegating local dependencies to lookups, it frees up attention capacity for global context, substantially boosting long-context retrieval (e.g., Multi-Query NIAH: 84.2 to 97.0). Finally, Engram establishes infrastructure-aware efficiency: its deterministic addressing enables runtime prefetching from host memory, incurring negligible overhead. We envision conditional memory as an indispensable modeling primitive for next-generation sparse models.

3D visual grounding aims to locate objects based on natural language descriptions in 3D scenes. Existing supervised methods are limited by generalization and recent zero-shot methods typically rely on a predefined Object Lookup Table (OLT) to query Visual Language Models (VLMs) for reasoning about object locations via a single step grounding, which limits the applications in scenarios with undefined targets and complex queries. To address these problems, we present OpenGround, a novel zero-shot framework for open-world 3D visual grounding that remains compatible with recent zero-shot methods. OpenGround integrates Task-Chain Planning to decompose a query into a plan of context-to-target sub-goals for progressive grounding, and Context-Guided Perception to perceive novel objects online under context guidance from the task chain. We also propose a new dataset named OpenTarget, which contains over 7000 object-description pairs to mimic open-world evaluation. Extensive experiments demonstrate that OpenGround achieves competitive performance on Nr3D, state-of-the-art on ScanRefer, and delivers a substantial 17.6% improvement on OpenTarget. Project Page at https://why-102.github.io/openground.io/.