NeurIPS 2026 Papers

A central limitation of modern machine learning systems is the lack of formal certificates guaranteeing their behavior beyond empirical, sample-based evaluation. Sum-of-squares (SOS) methods provide such guarantees by certifying polynomial inequalities over continuous domains via semidefinite programming (SDP). While prior work has introduced SDP layers as differentiable components in neural networks, their scalability is limited by two key bottlenecks, that we are fixing in this paper: (i) failure to preserve sparsity from the forward SOS program in the backward pass, leading to inefficient training, and (ii) loss of certificate structure during differentiation, preventing its use in learning. We propose a differentiable sparse SOS layer whose backward pass mirrors the sparse structure of the forward solve, reusing local blocks, overlaps, and sparse constraints. The method avoids building dense certificates for scalar losses and computes gradients using the same sparse structure when certificate details are required. Experiments show that this enables practical SOS-based layers within learning systems, reducing backward computation time by one to three orders of magnitude while preserving certification quality. We evaluate on sparse polynomial certification, pendulum control, combinatorial relaxations, robotics projection, oscillator chains, and pose graph synchronization. Results demonstrate that sparse certificate objectives can be effectively integrated into learning pipelines when the backward pass respects the verifier's structure.

Token-based world models enable fine-grained latent planning, but iterative search inside the predictor is dominated by the number of spatial tokens processed during rollout. We introduce COSTGRAD, a training-free, goal-conditioned selector that ranks spatial tokens by the gradient norm of the planning cost with respect to each input token. By deriving importance from the downstream control objective, COSTGRAD targets tokens that matter for planning rather than merely for prediction. On AdaLN-conditioned predictors at $50\%$ sparsity, COSTGRAD matches or exceeds full-token planning on three of four continuous-control benchmarks, while giving a measured $2.6\times$ wall-clock speedup per environment planning step. The sparsity dividend can be combined with reduced CEM search, yielding a $\sim5\times$ total speedup while still exceeding the full-token baseline. We also identify an architecture-dependent failure mode: in a matched AdaLN-vs-concat comparison, concat maintains comparable full-token performance but pure COSTGRAD loses its advantage over random selection. This failure tracks action-pathway drift under selected token removal: AdaLN largely preserves how actions affect kept tokens, whereas concat perturbs the action-conditioned computation. Mixing in random anchors partially recovers COSTGRAD's advantage on some matched-concat checkpoints, suggesting selector--architecture compatibility as a design axis for sparse world-model planning.

Koopman autoencoders (KAEs) seek a higher-dimensional latent representation in which nonlinear dynamics evolve linearly. However, many interesting systems have multiple basins of attraction, and both theoretical and empirical work has shown these multibasin systems cannot generally admit a single finite-dimensional global Koopman embedding under standard assumptions. We posit that encoders with a sparsity-inducing objective encouraging few active latent coefficients will provide latent supports as an inspectable basin-modeling principle for Koopman autoencoders. We use these encoders producing sparse latents in training Sparse Koopman Autoencoders (SKAEs) without basin labels or other regime annotations, and treat the learned latent supports as model-produced regime variables after training. Across a range of procedurally generated multibasin systems and chaotic flows, we show that SKAEs have superior forecasting performance compared to dense-latent KAEs. We also perform a mechanistic study that shows latent supports produced by SKAEs are both essential for the quality of the representation and useful for identifying basins on held-out basin interior states, whereas dense-latent KAEs collapse to an uninformative single family. These results identify sparse latents and their corresponding supports as label-free, interpretable regime variables for Koopman learning in nonlinear systems with multiple local dynamical laws.

Language models are increasingly used in text generation, decision support, and automated interaction, where their behavior must be controlled in a localized, reversible, and selective way. Existing methods either update shared parameters or constrain generation through external interfaces, leaving open whether frozen models contain compact internal control. We introduce sparse internal control, a framework for steering target behavior by applying inference-time interventions to a small set of internal model nodes. The framework formulates control through local target controllability, where candidate site–direction pairs are characterized by their behavioral effects under target, preservation, and energy constraints. This yields two complementary selectors: Driver-OMP, which selects nodes that sparsely reconstruct a desired behavioral displacement, and Coverage-Driver, which favors nodes whose effects cover prompts stably. We further propose a six-axis control-evaluation protocol measuring reachability, signed reversal, dose response, feedback controllability, off-target preservation, and held-out reuse. Across IOI, MMLU MCQA, and sentiment steering on models from GPT-2 small to Qwen3-4B, sparse driver sets act as executable actuators: they steer target behavior, respond monotonically to dose, support closed-loop control, and preserve unrelated next-token behavior. On refusal steering, the same protocol exposes a boundary case that passes all evaluated axes except strict reachability. Held-out reuse separates prompt-specific controls requiring refitted strengths from population-level controls that transfer as frozen interventions. These results show that model control can move beyond output-side steering: mechanistic internal variables can be selected, certified, and reused as localized control handles for frozen language models.

LLM chain-of-thought reasoning improves with additional compute, but standard 2 inference scaling [Wei et al., 2022, Wang et al., 2022, Snell et al., 2024, Brown 3 et al., 2024] treats all steps equally. We ask: which step in a reasoning chain benefits 4 most from extra computation? We make three layered contributions, each scoped 5 explicitly. (I) Measurement. We define expansion utility Uk as the marginal 6 gain in correctness probability from resampling from step k onward, measured via 7 a step-oracle sweep. Across nine model families (8B–123B), seven benchmarks 8 (math, science, code), and 6,500 chains, every one of the 41 healthy (model, bench) 9 cells shows positive oracle gap at 10% step budget (95% Wilson CI [0.914, 1.000]; 10 mean +22.0 pp, range +1 to +48 pp). (II) Structure. Per-step utility is sparse 11 (97.6% of cells: Gini ≥ 0.80); Gini and mean oracle gap are tightly anti-correlated (Pearson r = −0.913, p = 9.5 × 10−17 12 ). A resample-stability simulation and 13 a segmentation-rule sensitivity check (Appendix J) bound the headline against 14 noise and rule-choice confounders. (III) Routing proof-of-concept. A cross15 model router trained with a within-example listwise loss [Cao et al., 2007] and a 16 learned depth prior, combined at inference with a bench-conditional depth mask 17 and a confidence gate, recovers 5-seed-averaged gated GR@10% = +0.149 (95% 18 bootstrap CI [+0.085, +0.217]) of the cross-model oracle gap at matched step19 budget, with no architectural change to the underlying model. We frame this as 20 proof the framework is actionable cross-model, not as deployment-ready scaling; 21 matched-token comparison to self-consistency / best-of-N requires a separate 22 generation campaign and is left to follow-up (§7). Methodological observation. 23 Within-example pointwise top-k BCE under cross-model pooling produces a high24 AUC discriminator (AUC@10 = 0.70) that selects the wrong steps (gated GR 25 = −0.15); the listwise loss is shift-invariant to per-(model, bench) utility scale and 26 removes this inversion.

Linear attention models allow a fixed state size and a fixed amount of compute per token. However, due to their limited state size, linear attention models fall behind in long-context recall compared to softmax-attention-based transformer architectures. Increasing the state size of linear attention improves recall performance but at the cost of higher FLOPs. In this work, we introduce Sparse Delta Memory (SDM), an architecture that scales the hidden state of gated linear RNNs to orders of magnitude higher capacity using a sparse addressing scheme. SDM extends the Gated DeltaNet architecture by replacing the dense key-value outer product with sparse reads and writes to a large explicit memory. We show that, under an isoFLOP constraint and with an identical number of parameters, a higher state memory capacity significantly improves performance on in-context learning and long-context retrieval tasks. Moreover, by learning the initial state of the SDM memory and therefore using it as a parametric memory, we show that the model further improves on a wide range of common-knowledge and reasoning tasks.

Generative protein models are increasingly used for functional sequence design, yet the process by which biological information is organized during generation remains poorly understood. Diffusion protein language models provide a tractable setting for this question, as sequence generation proceeds through an explicit denoising trajectory from highly corrupted states to complete proteins. This trajectory enables a temporal view of interpretability: beyond identifying which biological signals are represented, it allows us to examine when these signals emerge and when specific residues become committed. Here, we analyze DPLM representations using sparse autoencoders trained across layers and noise levels. The resulting features capture biologically meaningful signals at both residue and protein scales, align with functional annotations, and preserve downstream biological information under reconstruction. Following these features along the denoising trajectory reveals a consistent functional ordering: catalytic-enriched features pre-activate at still-masked catalytic positions before residue identity is resolved, and catalytic residues are subsequently recovered earlier by the iterative denoiser. This prioritization remains significant after controlling for prediction difficulty, sequence context, amino-acid identity, structural environment, and evolutionary conservation, and is not reproduced by a random-feature null. Across ProteinGym deep-mutational-scanning assays, residues recovered earlier during denoising are also more mutation-sensitive. These results reveal a temporal organization of biological information in diffusion-based protein generation, in which functionally important residues are not only represented, but preferentially committed during sequence formation.

Sparse attention improves LLM inference efficiency by selecting a subset of key–value (KV) entries, but at the cost of potential accuracy degradation. In particular, omitting critical KV entries can induce substantial errors in model outputs. Existing methods typically operate under fixed or adaptive token budgets and provide empirical robustness or partial theoretical guarantees, yet they do not ensure zero false negatives across decoding steps, particularly since the set of relevant tokens is both query- and step-dependent. Our empirical observations confirm that missing even one critical key can lead to sharp error spikes, especially in long-output reasoning tasks where the set of important tokens varies throughout decoding. This observation motivates the need for indexing methods that dynamically adapt to these variations across decoding steps while guaranteeing a full recall of the relevant keys above a certain threshold. We address this challenge by reformulating sparse attention as the computational geometry problem of halfspace range searching. However, existing range searching data structures are not suitable for modern LLM inference due to their computational and implementation overheads. To overcome this, we introduce Louver, a novel index structure tailored for efficient KV cache retrieval. Louver (i) guarantees zero false negatives with respect to a specified threshold in both theory and practice, (ii) is lightweight to integrate into existing LLM inference pipelines, and (iii) incorporates hardware-aware optimizations for both CPU and GPU executions. Our experiments demonstrate that Louver outperforms prior sparse attention methods in both accuracy and runtime, and is faster than highly optimized dense implementations such as FlashAttention. These results highlight that recall guarantees are a critical and overlooked dimension of sparse attention, and open a new direction for building theoretically grounded, efficient KV cache indices.

Recent work has revealed a link between self-attention mechanisms in transformers and test-time kernel regression via the Nadaraya-Watson estimator, with standard softmax attention corresponding to a Gaussian kernel. However, a kernel-theoretic understanding of sparse attention mechanisms is currently missing. In this paper, we establish a formal correspondence between sparse attention and compact (bounded support) kernels. We show that normalized ReLU and sparsemax attention arise from Epanechnikov kernel regression under fixed and adaptive normalizations, respectively. More generally, we demonstrate that widely used kernels in nonparametric density estimation---including Epanechnikov, biweight, and triweight---correspond to $\alpha$-entmax attention with $\alpha = 1 + \frac{1}{n}$ for $n \in \mathbb{N}$, while the softmax/Gaussian relationship emerges in the limit $n \to \infty$. This unified perspective explains how sparsity naturally emerges from kernel design and provides principled alternatives to heuristic top-$k$ attention and other associative memory mechanisms. Experiments with a kernel-regression-based variant of transformers---Memory Mosaics---show that kernel-based sparse attention achieves competitive performance on language modeling, in-context learning, and length generalization tasks, offering a principled framework for designing attention mechanisms.

Domain generalised semantic segmentation (DGSS) has recently benefited from combining multiple foundation models, such as vision-language and vision foundation models, to leverage their diverse representations for generalisation. Existing multi-model approaches share a common design choice: cross-model fusion is restricted to depth-aligned layers, implicitly assuming that mutually beneficial information resides at the same depth across models. However, foundation models pretrained under different objectives may develop distinct representation hierarchies, making the optimality of depth-aligned fusion questionable. Moreover, we observe that the foundation models have different depth-wise domain sensitivity. Motivated by these observations, we propose a Sparse All-Layer Connector (SALC) that enables each layer to access information from preceding layers of another model. As the number of accessible layers grows with depth, dense aggregation may hurt generalisation. SALC therefore learns to adaptively select a sparse subset of informative layers. We further introduce a candidate dropout regularisation that strengthens sparsity and encourages SALC to explore diverse layers, leading to more robust selection. Across four foundation model combinations and three DGSS evaluation settings, the proposed design demonstrates stronger generalisation than conventional depth-aligned fusion. Code will be public upon acceptance.

In recent years, open-source efforts like Senorita-2M have propelled video editing toward natural language instruction. However, current publicly available datasets predominantly focus on local editing or style transfer, which largely preserve the original scene structure and are easier to scale. In contrast, *Background Replacement*, a task central to creative applications such as film production and advertising, requires synthesizing entirely new, temporally consistent scenes while maintaining accurate foreground-background interactions, making large-scale data generation significantly more challenging. Consequently, this complex task remains largely underexplored due to a scarcity of high-quality training data. This gap is evident in poorly performing state-of-the-art models, *e.g.,* Kiwi-Edit, because the primary open-source dataset that contains this task, *i.e.,* OpenVE-3M, frequently produces static, unnatural backgrounds. In this paper, we trace this quality degradation to *a lack of precise background guidance* during data synthesis. Accordingly, we design a scalable pipeline that generates foreground and background guidance in a decoupled manner with strict quality filtering. Building on this pipeline, we introduce *Sparkle*, a dataset of ~140K video pairs spanning five common background-change themes, alongside *Sparkle-Bench*, the largest evaluation benchmark tailored for background replacement to date. Experiments demonstrate that our dataset and the model trained on it achieve substantially better performance than all existing baselines on both OpenVE-Bench and Sparkle-Bench. Our proposed dataset, benchmark, and model are fully open-sourced at https://showlab.github.io/Sparkle/.

Sparse attention reduces compute and memory bandwidth for long-context LLM inference. However, two key challenges remain: (1) KV cache capacity still grows with sequence length, and offloading to CPU memory introduces a PCIe transfer bottleneck; (2) the sparse selection step itself retains $O(T^2)$ complexity and can dominate attention cost at long contexts. We propose SparDA, a decoupled sparse attention architecture that introduces a fourth per-layer projection, the Forecast, alongside Query, Key, and Value. The Forecast predicts the KV blocks needed by the next layer, enabling lookahead selection that overlaps CPU-to-GPU prefetch with current-layer execution. Because Forecast is decoupled from the attention query, our GQA implementation uses one Forecast head per GQA group, reducing selection overhead versus the original multi-head selector. SparDA adds $<$0.5\% parameters and trains only the Forecast projections by matching the original selector's attention distribution. On two sparse-pretrained 8B models, SparDA matches or slightly improves accuracy and delivers up to 1.25$\times$ prefill speedup and 1.7$\times$ decode speedup over the sparse-attention offload baseline. By enabling larger feasible batch sizes on a single GPU, SparDA further reaches up to 5.3$\times$ higher decode throughput than the non-offload sparse baseline.

Object detection in high-resolution wide (HRW) shots, where a single image can span gigapixel resolutions and kilometre-scale fields of view, suffers from extreme foreground sparsity, dramatically varying foreground density from sparse to crowded scenes, and object clusters whose spatial extent ranges from dozens to thousands of pixels within the same dataset. State-of-the-art sparse vision transformers for gigapixel detection select a fixed top-k fraction of windows by hand-crafted variance scores, which over-promote textured but object-free background such as foliage and patterned facades, cannot adapt the keep ratio to the wide range of crowd density, and confine each kept window to a 7x7 receptive field that is far smaller than the spatial extent of many real-world object clusters. We propose SpanFormer, a sparse vision transformer that broadens the computational span at three complementary granularities: (i) Prototype Routing replaces variance with learnable foreground/background prototypes regularised by a balance constraint that prevents prototype collapse, widening the semantic span of the scoring criterion; (ii) Dynamic Top-k predicts a per-image keep ratio with a lightweight selector, making the compute span elastic so that crowded scenes receive proportionally more compute while near-empty scenes are skipped; (iii) Cluster Memory Attention performs union-find clustering over the kept windows by joint cosine similarity and Chebyshev spatial radius and augments each window's K and V with cross-window memory drawn from its cluster, extending the attention span beyond the local window with all projection parameters shared with the local branch and therefore no added projection cost. On the PANDA gigapixel benchmark, SpanFormer improves AP50 from 78.0% to 80.3% over the SparseFormer baseline while reducing backbone FLOPs by over 75%.

Inorganic crystals are periodic, highly symmetric arrangements of atoms in three-dimensional space. Their structures are constrained by the symmetry operations of a crystallographic \emph{space group} and restricted to lie in specific affine subspaces known as \emph{Wyckoff positions}. The frequency an atom appears in the crystal and its rough positioning are determined by its Wyckoff positions. Most generative models that predict atomic coordinates overlook these symmetry constraints, leading to unrealistically high populations of proposed crystals exhibiting limited symmetry. We introduce Space Group Conditional Flow Matching (SGFM), a novel generative framework that samples significantly closer to the target population of highly symmetric, stable crystals. We achieve this by conditioning the entire generation process on a given space group and set of Wyckoff positions; specifically, we define a conditionally symmetric noise distribution and a group-conditioned, equivariant, parametric vector field that restricts the motion of atoms to their initial Wyckoff position. Our form of group-conditioned equivariance is achieved using an efficient reformulation of \emph{group averaging} tailored for symmetric crystals. It reduces the computational overhead of symmetrization to a negligible level. SGFM achieves state-of-the-art performance in \textit{De Novo Generation} on the MP-$20$ benchmark.

Despite remarkable progress in text-to-image generation, current Diffusion Transformers (DiTs) inherently struggle with learning intricate visual content, such as accurate multilingual scene text rendering and novel visual concepts. This limitation stems from the inherent contradiction between vague semantic guidance provided by text and the objects of accurate reconstruction during diffusion training. To address this bottleneck, we propose SpaG-DiT, a novel spatial grounding training framework. SpaG-DiT directly injects explicit spatial priors into the DiTs before the attention module via Contextual Diagonal Position Encoding (CDPE), which dynamically modulates spatial coordinates without destroying the inherent sequential structure of the linguistic input. Furthermore, we introduce VLM Grounding Guidance (VLM-GG) as an automated pipeline to extract implicit spatial priors from pre-trained Vision-Language Models, eliminating human annotation costs. To evaluate the grounding ability, we establish two new datasets: SpaG-DiT-MultiLingual for non-Latin/Chinese text rendering and SpaG-DiT-Creature for new concept learning. Extensive experiments demonstrate that SpaG-DiT achieves superior training efficiency and significantly outperforms vanilla self-finetuning on the above two tasks and the widely adopted GenEval benchmark. These results validate the plug-and-play nature of SpaG-DiT and its potential to enhance DiT training processes. Code, model, and data will be released.

Vision Transformers are increasingly applied to data defined on the sphere, particularly in physics, meteorology, and related scientific domains. Position embeddings provide the spatial information that attention itself does not encode, making their design central to the expressive power of Transformers. However, most existing position embedding schemes are designed for Cartesian grids and therefore do not naturally handle longitude periodicity, polar singularities, or geodesic relations on spherical domains. This mismatch limits the ability of standard attention to model spherical geometry without specialized architectural modifications. We propose Spherical Reflection Position Embedding (SpRePE), a drop-in position embedding scheme for Vision Transformers on spherical data. SpRePE encodes each absolute spherical position by applying Householder reflections to query and key representations. Although each token is encoded only from its own absolute coordinates, the resulting attention inner products induce an explicit spherical relative-position term with a clear geometric interpretation. This formulation injects sphere-aware relative geometric information into standard attention without constructing pairwise attention-bias matrices, introducing task-specific modules, or modifying the backbone architecture. It also avoids the quadratic overhead of relative position biases and preserves the same asymptotic overhead as RoPE. We evaluate SpRePE on spherical image classification, panoramic depth estimation, and global weather forecasting. Across these tasks, SpRePE achieves competitive performance compared with strong position embedding baselines, with particularly clear gains in settings where spherical geometry is important.

Large language models (LLMs) increasingly answer queries by citing web sources, but existing evaluations emphasize the final answer rather than evidence quality. We introduce SourceBench, a benchmark for measuring the quality of cited web sources across 100 real-world queries spanning informational, factual, argumentative, social, and shopping intents. SourceBench uses an eight-metric framework covering content quality (content relevance, factual accuracy, objectivity) and page-level signals (e.g., freshness, authority/accountability, clarity), and includes a human-labeled dataset with a calibrated LLM-based evaluator that matches expert judgments closely. We evaluate eight LLMs, Google Search, and three AI search tools over 3996 cited sources using SourceBench and conduct further experiments to understand the evaluation results. Overall, our work reveals four key new insights that can guide future research in the direction of GenAI and web search.

Longitudinal radiology report generation increasingly conditions on prior images and reports. This history is clinically valuable, but it creates a source-validity failure mode: a generator can carry prior-only findings into the current report. Weintroduce source-causal robustness, a stress-test and control framework that holds the current chest X-ray fixed while intervening on historical context. On a full eligible MIMIC-CXR longitudinal study (18,716 pairs; 93,580 generations across five source conditions), historical context consistently trades fewer missed findings for more unsupported positives. Matched wrong-history, the key negative control, increases proxy-defined false positives by +0.366 per report while reducing false negatives by −0.129; random wrong-history yields an even larger false positive stress signal (+0.465 FP/report). We propose the Source-Causal Finding Router (SCFR), a source-validity controller trained with within-evaluation subject exclusion. Across three held-out evaluations, SCFR repairs matched wrong-history false-positive inflation by −0.343, −0.550, and −0.457 per report, passes Holm corrected primary testing at the bootstrap resolution floor (Holm-adjusted p ≤ 0.0012), and satisfies fixed current-only parity margins. Cross-generator stress panels with RadInfer and Libra, together with ReXPref-Prior, RadGraph2, Chest ImaGenome, ablations, and case review, support the mechanism. These results establish a protocol-defined state-of-the-art source-causal robustness/control result for longitudinal radiology report generation under matched wrong-history stress tests.

Open-weight Vision-Language Models (VLMs) score 85--94\% on existing spatial benchmarks (BLINK Spatial, What's Up CLEVR) yet collapse to chance on simple geometric questions when object, scene, and texture priors are stripped away. We introduce and open-source Sophon, a procedurally generated diagnostic of paired colored point-cloud stimuli across three coordinate systems (1D linear, 2D Cartesian, 2D polar) and four question types (binary, categorical, comparative, quantitative). Across four open architectures spanning every major projector design (LLaVA-NeXT, LLaVA-OneVision, Qwen2.5-VL, Idefics3), accuracy clusters at 47--52\% while frontier models (Opus 4.6, Gemini 2.5 Pro, GPT-5.4) clear 77--83\% and humans reach 84--94\%. We trace the open-weight failure end-to-end. The vision tower works: encoder probes recover Sophon spatial direction at 80--97\%. The language tower's reasoning circuit works: a text-only control restores $+40$ pp, and probes preserve the spatial signal through every LLM decoder layer at 80--86\%. Per-layer attention attribution localizes visual integration to mid-stack layers. Yet at the readout, the model elevates the correct answer-set tokens but selects among them at chance. Fine-tuning experiments rule out the unembedding as the mechanism. By moving the unembedding layer through the residual stream, we observe the emergence of a small but discriminative contrast at late-stack residual layers in fine-tuned models, which in practice converts the random selection into a confident one more likely to be correct. However, ablation experiments locate the mechanism upstream: training only the upstream layers with the late stack frozen recovers the full $+29$ pp gain, while training only the late stack captures less than a third. The late-stack contrast is the downstream readout of upstream changes. No layer in either base or fine-tuned models shows geometric alignment between the spatial signal and the unembedding's vocabulary axes, suggesting that 7B-class VLMs have the weights to reason spatially but lack the residual-stream pathways to convert that reasoning into answers.