Model Complexity

A popular selling point of diffusion large language models (dLLMs) is their capacity for parallelism: the ability to generate sequences of text far more efficiently than autoregressive models, which require one forward pass per token. Yet among the many competing paradigms for dLLMs, from masked to uniform to Gaussian diffusion, principled understanding of how these different proposals compare in parallelism remains limited. In this work, we initiate a fine-grained comparison of the capacity for parallelism among these three leading approaches and prove the following: - Uniform and Gaussian diffusion can sample in a number of forward passes which scales with the dual total correlation of the underlying distribution, a measure of intrinsic complexity which can be much smaller than the context length. Previously, it was only known how to achieve this using masked diffusion. - For a certain family of random empirical measures, we show that $\widetildeΘ(\sqrt{d})$ forward passes are necessary and sufficient to sample using uniform or Gaussian diffusion, yet there exist approximate score oracles for which $\widetildeΩ(d)$ forward passes are needed for masked diffusion. This establishes the first provable separation in parallelism between the three prevailing dLLM paradigms. Contrary to popular intuition that masked diffusions are harder to parallelize because they must commit to token values, the latter separation instead comes from the fact that the critical windows in masked diffusion sampling are asymptotically narrower than those in uniform and Gaussian diffusion sampling.

In previous papers, we began the study of sequence prediction algorithms adapted to stringological word complexity measures. In particular, we defined a complexity measure called Arithmetic Repetition Complexity (ARC) which admits a polynomial-time prediction algorithm with a mistake bound quasilinear in the complexity. Here, we show a weaker complexity measure related to ARC that admits an especially efficient prediction algorithm: an algorithm that runs in quasilinear time and polylog space for appropriate highly-structured sequences. The complexity measure is defined via a restricted class of "zipline programs" (a variant of straight-line programs), which we call layered. We thus get a less expressive measure with a more efficient algorithm (compared to our results for ARC), demonstrating a possible tradeoff.

Complexity measured from generated code is failure-dependent: a difficult prompt can yield a short failing program and be assigned low output complexity. We introduce a six-dimension prompt-side structural-complexity index scored before generation and kept separate from correctness. We select 5,000 Python prompts across six bands of a preliminary single-rater rubric. Four out-of-panel LLM raters rescore the locked prompts, giving 19,997 score rows; composite inter-rater reliability is ICC = 0.872 on the 4,998 prompts with all four ratings. We evaluate 21 models per prompt, yielding 105,000 generations. In the unadjusted mean-pooled analysis, pass rate has a nonmonotone breakpoint at composite 13.75, with 79.9% at or below and 87.6% above. This is not a universal failure cutoff. Task-type fixed effects shift the breakpoint to 10.75 and cut the regime gap from 7.6 to 2.1 points. A construction-frame control shifts it to 8.50 with a raw gap of -3.5 points, and neither frame alone reproduces the pooled +7.6-point change. Model-specific fits include 16 upward and five downward changes. A 365-prompt audit-clean extension matches the original five-model estimates at bins 15 and 16 but adds only 14 prompts above bin 16. Among zero-pass generations with computable Lizard complexity, 28.5% pair a prompt composite above 8 with output complexity at most 10. Human agreement is moderate and rater-dependent on a disagreement-enriched calibration set; paraphrase and cross-language rescoring preserve score ordering. Overidentification tests reject the joint restrictions on the six dimensions, so we treat the composite as an index and make no causal interpretation of the 2SLS estimates. The contribution is a pre-generation measurement framework and a bounded observational analysis of reliability regimes.

We introduce a new single-loop algorithmic framework for smooth nonconvex--concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of $O(L^2D_Y\barΔ_0\varepsilon^{-3})$, where $L$ is the gradient Lipschitz constant, $D_Y$ bounds the diameter of the dual feasible set, and $\barΔ_0$ is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to $O(L^2D_YΔ_φ\varepsilon^{-3})$, up to an additive lower-order cost, where $Δ_φ:=φ(x_0)-\inf_xφ(x)$. We further establish a lower bound of $Ω(L^2D_YΔ_φ\varepsilon^{-3})$ for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves $\mathcal{O}\!(L^{3/2}D_Y^{1/2}Δ_φ\varepsilon^{-5/2})$ gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves $O\!(\sqrtκ\,LΔ_φ\varepsilon^{-2})$ leading complexity for both stationarity criteria, where $κ=L/μ$ is the dual condition number, up to an additive initialization cost. The $\varepsilon^{-2}$ accuracy dependence is optimal under fixed regularity and initialization bounds.

An ensemble can improve photovoltaic forecasts while adding components that contribute little or increase computation. We assess these effects through matched comparisons and ablations of a fixed heterogeneous predictor bank. Hourly experiments use GEFCom2014 and three additional public datasets, with chronological partitions and three seeds. Under retrospective ERA5 assistance, static fusion reduces scaled mean absolute error against matched boosting by 1.11%, 4.41%, and 1.63% on PVDAQ, OPSD, and Ausgrid; only OPSD remains supported after multiple-comparison correction. Weather gating offers no consistent incremental benefit. Exploratory member removals show group-level dependence alongside individual redundancy. A separate, previously inspected fifteen-minute case replaces one neural member with a tree predictor: normalized error falls by 1.72%, but measured inference is slower. These findings support component-wise evaluation with explicit limits on weather availability and test-set reuse.

We investigate the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in any fixed Euclidean dimension. The potential is $\mu$-strongly convex and $L$-smooth, with an unknown mode in the ball of radius $\mu^{-1/2}$ about the origin. We have access to unbiased stochastic oracles with the variance at most $\sigma^2$. For every $\sigma^2\ge0$ and total variation (TV) accuracy $0<\varepsilon\le1/10$, we prove that the tight complexity of sampling a distribution within $\epsilon$-TV distance from the target distribution is $$N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)+ \frac{\sigma^2}{\mu\epsilon}\right),$$ where $\kappa:=\frac L\mu$ is the condition number. Note that this complexity bound is simultaneously tight for the condition number $\kappa$ and accuracy $\epsilon$. Besides, our tight complexity bound is adaptive to noiseless setting $\sigma=0$, which is $N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)\right)$.

No Free Lunch theorem implies that any performance gains achieved by a classifier on a particular image distribution are necessarily offset by a loss of performance over the set of all possible problems; thus, no single model is universally optimal. Selecting the most suitable classifier for image datasets is a critical yet challenging task due to the intrinsic complexity and diversity of images. This paper proposes a meta-learning framework that leverages a comprehensive set of meta-features capturing dataset complexity to predict classifier performance without exhaustive training. By extracting and selecting features using methods such as autoencoders, pre-trained networks, and dimensionality reduction techniques, we train regression models to efficiently estimate classifier accuracies. Additionally, clustering techniques are employed to group classifiers with similar performance patterns, simplifying the recommendation process. The datasets used span a wide range of concepts, including nature, animals, numbers, motorcycles, medical images, and human bodies, to ensure broad generalization. Evaluated on 56 diverse image datasets, our approach achieves an average ranking prediction accuracy exceeding 86%, demonstrating its effectiveness in guiding model selection. This scalable and interpretable framework provides a practical solution to improve classification performance while reducing computational costs.

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

Assessing suicide risk from social media text is a small-data, high-stakes setting requiring not only severity prediction but also supporting evidence and clinically relevant risk and protective factors. Yet common NLP techniques, including model scaling, synthetic data, loss reweighting, ensembling, and threshold tuning, are often applied without testing whether their gains hold up under severe class imbalance, coupled outputs, and limited author-level data. We study 1,635 clinician-annotated posts and audit 31 pre-specified techniques from 7 methodological families through roughly 300 controlled experiments on author-disjoint partitions. We found no prior audit of this playbook in this regime. The findings guide a task-grounded system for three outputs: 4-level suicide risk, evidence spans, and 24 clinical risk and protective factors. Only 5 of 31 comparisons produced reliable gains. We reformulate factor prediction as entailment between each post and its codebook definitions, using an architecturally diverse ensemble with class-balanced training and score rescaling. Risk predictions condition a 7-model evidence tagger ensemble; evidence restricts symbolic risk rules; and a difficult risk class is routed separately. The factor predictor remains independent because risk evidence provides no additional factor signal. We also correct a mismatch between validation scores used for threshold fitting and test-time ensemble scores through deployment-consistent calibration, yielding the largest improvement to the factor system. The final system achieves 0.8203 for risk, 0.7953 for evidence, and 0.7045 macro-F1 for factors, with a 0.7781 composite, ranking third among 53 teams. We call the underlying principle task-conditioned technique selection: retain techniques only when task-specific knowledge, structure, or empirical evidence justifies them.

What can a single layer of self-attention compute? We study head complexity: the minimum number of attention heads required to compute a function in a one-layer attention-only model. We establish an exact hierarchy under this measure: $k$ heads compute $k$-bit parity but cannot compute $(k+1)$-bit parity. The lower bound is unconditional in the two resources a transformer might otherwise exploit; it holds at unbounded embedding dimension and unbounded numerical precision. The proof rests on an alternating-sum obstruction: after clearing the softmax denominators, every monomial in the resulting decision polynomial omits at least one of the $k+1$ input bits, forcing its correlation with parity to vanish. The same obstruction yields lower bounds for related tasks, including the well-studied multi-hop induction-head task. We also establish compactness bounds for embedding dimension and numerical precision. Specifically, a compactness theorem shows that any function computable at all can be computed with embedding dimension and precision bounded by the discrete data of the task, namely, head count, alphabet size, and length. Thus, potentially unbounded dimension or precision provably cannot substitute for heads. Finally, we derive nearly matching universal bounds for general binary functions: $2^n$ heads suffice to compute every $n$-bit binary function, with one head per monomial in its multilinear expansion, while a counting argument shows almost all such functions require $Ω(2^n/n^2)$ heads. This lower bound matches the upper bound to within a $\operatorname{poly}(n)$ factor, even when dimension and precision are unbounded. Together, these results characterize head requirements for Boolean computation in this model.

Data-driven remaining useful life (RUL) prediction requires complete degradation trajectories for training, yet such run-to-failure data are scarce and expensive. Practitioners currently lack principled guidance on how many failure examples suffice for a given model and accuracy target. This paper develops a sample complexity framework for RUL prediction comprising seven main results organised around three themes. First, we establish fundamental learning rates: a distribution-free generalization bound shows that the uniform deviation of the mean squared error decreases as $O(B^{2}\sqrt{p/n})$, where $p$ is the model complexity and $n$ the number of trajectories, and a minimax lower bound proves that the $Θ(p/n)$ rate is unimprovable.} \rev{Second, we quantify how domain knowledge accelerates learning: incorporating degradation physics reduces data requirements by up to two orders of magnitude for deep networks, a Bernstein-type analysis achieves the minimax-optimal $O(p/n)$ rate under high signal-to-noise conditions, and closed-form penalties reveal when an incorrectly assumed physics model hurts rather than helps. Third, we characterise the impact of data quality: fleet variability induces an irreducible bias$-$variance tradeoff, while right-censored observations suffer an efficiency loss that depends critically on the degradation class.} Closed-form expressions are provided for exponential, power-law, and stretched-exponential degradation. \rev{Cross-domain validation against published turbofan, battery, and bearing benchmarks confirms the theoretical predictions within a factor of 2$-$3 on average. The results yield practical guidelines for planning data collection, selecting model complexity, and evaluating physics model assumptions in prognostics applications.

We study information bottlenecks in modern deep-learning architectures -- RNNs, softmax transformers, linear-attention transformers and state-space models -- through the lens of the indexing primitive. In this primitive, the input consists of $n$ bits and one integer $i$ from $1$ to $n$ called the index, and the output equals the value of the $i$-th bit. We introduce causal complexity for masked architectures. We show that architectures with low causal complexity cannot solve the indexing primitive in any constant number of layers when the index appears at the end of the input. In particular, this limitation applies to low-parameter RNNs, SSMs and masked linear-attention transformers. In contrast, small softmax transformers can solve it in one layer, while non-masked linear-attention transformers can solve it in 2, which separates them from their masked counterparts. In turn, when the index appears at the beginning, we show that small RNNs are capable of solving this task in 1 layer, while all the other architectures require 2. All our impossibility results are unconditional and apply even to models that employ infinite-precision real arithmetic. Moreover, experiments for up to $n=64$ qualitatively align with our theory: configurations with low-parameter theoretical solutions learn the indexing task easily, while configurations that do not admit such theoretical solutions struggle to learn as the sequence length grows.

Knowledge-Intensive Visual Question Answering (KI-VQA) benchmarks evaluate Vision-Language Models (VLMs) as multimodal knowledge assistants by requiring external information beyond a provided image to answer questions. KI-VQA involves multiple sub-problems -referring expression understanding, visual grounding, object recognition, knowledge retrieval, and reasoning-yet existing benchmarks typically report only end-task accuracy, obscuring where failures arise. To analyze the full KI-VQA pipeline, we introduce CRAG-MM-Diagnostics, a diagnostic benchmark with stage-wise data annotations that isolate 1) language-based visual grounding, 2) object identification, and 3) knowledge retrieval and reasoning. We evaluate fully parametric and retrieval-augmented VLMs, providing fine-grained analyses using newly collected metadata, such as target ROIs, entity names, and visual complexity scores. Our results point to knowledge retrieval and reasoning as the primary bottleneck, but also highlight issues in the other parts of the KI-VQA pipeline, such as the fact that VLMs struggle with target object identification or that image retrievers struggle to integrate textual cues. These findings expose fundamental limitations in current KI-VQA systems and motivate stage-aware evaluation. We, lastly, leverage these findings to propose a grounded bimodal RAG pipeline that integrates a visual grounding module to crop targets before image retrieval, boosting GPT-5 and Qwen's respective accuracies by 13.3 and 8.5 percentage points.

Conformal prediction certifies that a classifier's prediction sets cover the truth, and that certificate is marginal. Many recognition benchmarks build distribution shift into evaluation, placing disjoint conditions in the training and test splits. Under that shift the certificate stays reassuring while per class coverage fails silently: on a real cross subject skeleton benchmark marginal coverage holds near ninety percent while the worst class is covered about seventy percent and ten of sixty classes fall below eighty percent. This class specific undercoverage stays hidden behind a single reassuring marginal number. Once the shift acts jointly on covariates and labels, the target class conditional score law is unidentified, so no label free method is at once per class valid and efficient uniformly over target laws consistent with the observed source joint distribution and target covariate marginal. The per class labels needed to recover every class threshold to a given tolerance grow as the inverse square of that tolerance and the logarithm of the class count, with matching bounds for classwise threshold procedures. Pseudo labels do not shortcut it: the best prediction powered estimator gains at most a small constant factor where coverage collapses. Across three real shifts and an image corruption benchmark, source label calibration recovers much of the gap while marginal coverage holds, and stops once it breaks.

Predicting comprehension from eye movements could support adaptive reading interfaces. We present LEXIC, a compact recurrent model that predicts response correctness from fixation sequences, word frequency, and character length. It has 41.6K parameters and requires no language-model inference. Mean area under the receiver operating characteristic curve (AUROC) reaches 0.529 for Unseen Text and 0.554 for Unseen Reader on OneStop. Matched comparisons with AhnCNN show gains from both encoder redesign and lexical augmentation in these settings. Separate training and evaluation on SB-SAT also yield higher mean AUROC than AhnCNN. LEXIC occupies only 166 KB of model weights and requires 1.4 ms per trial on a single CPU core, supporting lightweight on-device inference.

At Amazon Prime Video, we face the critical operational challenge of managing code deployments during live events and rapid feature releases without causing service outages. Current change control approaches use blanket deployment freezes that block all changes regardless of risk, creating significant developer toil. While prior research has explored risky change predictors, these rely on developer-specific metadata or extensive historical data, raising privacy concerns and limiting applicability to new projects. We introduce a framework centered on diff-aware features, characteristics derived directly from code modifications. Our key contribution is the systematic identification of which quantitative metrics (code-level and change-level metrics) and qualitative indicators (coding style violations, change type classification) are necessary for risk prediction. We employ LLMs as multi-language feature extractors, demonstrating their effectiveness for code analysis beyond generation tasks and eliminating the need for language-specific tooling. We evaluated our framework on two datasets: Prime Video's production environment and the public ApacheJIT dataset. Our best-performing model achieves an average recall of 0.83 and F1 score of 0.81 across both datasets for detecting risky code changes. Notably, ablation analysis reveals that change-level volume metrics (e.g., lines added/deleted) are noisy predictors, while structural code complexity provides a substantially stronger risk signal. These results demonstrate that thoughtful feature curation enables effective change risk assessment across different programming languages and organizational contexts while avoiding privacy concerns.

Parameterized quantum circuits (PQCs) are increasingly used as policies and value functions in quantum reinforcement learning, yet it remains unclear when and why quantum policies generalize. We give a PAC-Bayesian account in which generalization is governed not by the raw number of circuit parameters, but by the effective dimension of the Fisher geometry induced by the circuit. This quantity is inflated by entanglement, making entangling connectivity an independent axis of complexity.In controlled experiments that fix the number of trainable rotations and vary only entanglement, we find that circuits with larger Fisher effective dimension exhibit larger train-test gaps, while parameter count is a weak predictor. The resulting bound acts primarily as a ranking certificate: it correctly orders circuits with identical parameter count, which parameter-counting bounds cannot do. We validate this mechanism across supervised classification, quantum contextual bandits, and value-function generalization, where entangled circuits consistently generalize worse than non-entangled circuits of equal parameter count, with gaps shrinking as sample size increases.Our strongest evidence comes from low-variance decision models, including single-observable classifiers, value heads, and one-step policies. In end-to-end multi-step policy learning, entanglement effects remain statistically significant but high return variance leaves the full ordering only partially resolved. Partial-correlation analysis shows that Fisher effective dimension screens off entangling pattern, and controls for training accuracy, readout, and optimizer rule out major optimization confounders. The effect also persists on an IBM Heron quantum processor under real noise. Overall, our results reframe quantum policy design around an entanglement--generalization trade-off rather than expressivity alone.

Mutual information (MI) estimation is a central problem in machine learning and statistics; however, existing benchmarks typically evaluate estimators on simplified, low-dimensional distributions, leaving their performance on complex, realistic data largely unexplored. We address this gap with a comprehensive benchmarking framework grounded in a unified copula-theoretic perspective that subsumes existing benchmarks as special cases. Within this framework, we propose two complementary families of tests: a copula-first family that systematically varies ground-truth MI, dimensionality, and marginal complexity using synthetic and flow-based transformations; and a marginals-first family that couples real-world image data with controlled dependency structures, extending the classic same-class-pairing paradigm. We use this suite to extensively evaluate three classes of estimators: non-parametric, discriminative, and generative. Contrary to prevailing assumptions, our results indicate that there is no universal winner: each category can systematically outperform all other estimators under specific setups. By analyzing these cases, we identify fundamental estimation barriers and propose new tests that more effectively stress these specific limitations. We share the open source code at https://github.com/VanessB/mutinfo.

With the increase in model parameters and training data, the instruction following and generalization capabilities of Large VisionLanguage Models (LVLMs) have been significantly improved. Based on the Mixture of Experts (MoE) architecture, LVLMs expand their parameter capacity while maintaining the inference cost. However, traditional MoE methods employ a Top-k static routing strategy, which fails to account for variations in the input and adaptively select the number of experts, resulting in suboptimal resource utilization. In this paper, we propose viewing token routing as an information encoding task, framing dynamic routing as a Minimum Description Length (MDL) problem in encoding By validating the connection between MDL and gating entropy in the MoE scenario, we introduce Gating Entropy-based Uncertainty-aware Adaptive Routing (GeMoE) for MoE. Unlike traditional static or heuristic-based dynamic routing methods, GeMoE explicitly models the trade-off between model complexity and performance. By using gating entropy to assess the complexity of tokens, GeMoE adaptively determines the number of experts each token should engage. On a wide range of backbones and benchmarks, our method achieves 99.5% average performance retention compared to the original static routing, while improving average expert activation sparsity by 36.5%.

Multimodal Large Language Models (MLLMs) demonstrate strong performance on standard visual question answering benchmarks, yet their scalability under controlled structural complexity remains poorly understood. We introduce TriViewBench, a controlled three-view visual reasoning benchmark constructed from synthetic 3D scenes with explicitly parameterized object count and occlusion. The benchmark contains 1,923 scenes and over 14K Question-Answer (QA) pairs organized into four complexity levels and three reasoning categories: Local Decision, Object Counting, and Global Recovery. We evaluate 18 open- and closed-source MLLMs under a unified prompting protocol. All 18 models exhibit an identical capability hierarchy without exception (Local Decision > Object Counting > Global Recovery), and performance degrades monotonically with complexity: Local Decision tasks decline modestly (12.11% relative drop), while Object Counting degrades substantially (59.14%) and Global Recovery collapses severely (80.02%). Error analysis on Object Counting reveals two mechanistically independent failure modes: single-view tasks are dominated by undercounting due to occlusion blindness, whereas the multi-view task reverses to overcounting due to cross-view identity confusion. Chain-of-Thought (CoT) prompting yields near-zero overall benefit ($Δ= -0.16\%$) and its effect on Global Recovery is strongly capability-gated, suggesting that the bottleneck lies in cross-view spatial representation rather than reasoning strategy. These findings reveal fundamental scalability limitations in current MLLMs and position TriViewBench as a controlled diagnostic framework for analyzing structural reasoning failures.

Fully homomorphic encryption (FHE) allows computations to be performed directly on encrypted data without decryption, offering strong privacy guarantees for sensitive data analysis. This capability is important for privacy-sensitive applications like secure cloud computing, finance, and healthcare. The complexity of FHE schemes, however, has hindered their practical adoption. To make FHE accessible to a broader range of developers, a new generation of specialized frameworks has emerged to translate high-level FHE programs into complex FHE operations, introducing a new programming paradigm. However, the inherent complexity of FHE frameworks makes them prone to incorrect implementation logic. Unlike mere crashes, logic bugs in these frameworks can silently corrupt encrypted computation, potentially leading to severe financial losses and security vulnerabilities in FHE-enhanced applications. In this work, we introduce HERTA, the first automated testing tool tailored for FHE frameworks. HERTA leverages metamorphic testing to uncover deep-seated implementation bugs and vulnerabilities across the multi-layered FHE software stack. To that end, we design a set of novel metamorphic relations (MRs) derived specifically from FHE semantics. These MRs stress the most challenging aspects of the pipeline, enabling automated correctness testing without the need for a manual ground truth. Our evaluation of HERTA on 3 leading industry frameworks discovered 21 previously unknown bugs, several of which have already been confirmed and fixed by developers. Furthermore, our hazard analysis reveals the critical security impact these bugs pose to the integrity and availability of FHE-based services.

Piecewise-affine neural networks (e.g., with ReLU or LeakyReLU activations) implement continuous piecewise-affine maps, and the number of affine regions provides a natural proxy for expressive capacity. However, the gap between theoretical region capacity and the affine regions realized after training remains insufficiently understood. We study this gap from two complementary perspectives. First, we give a rigorous, architecture-dependent theorem for affine line-segment probes: for multilayer perceptrons with piecewise-affine activations, the number of affine pieces realized along an affine line-segment probe is upper bounded by an explicit product of layer-wise width terms (and activation breakpoint factors). This yields a neuron-threshold lower bound for representing target functions with prescribed one-dimensional piece complexity, formalizing the minimal region budget required for complex signals. Second, we exactly enumerate affine regions realized within bounded 2D and higher-dimensional domains under controlled task complexity. Under fixed architectures and training protocols, increasing input--label complexity yields trained solutions with markedly fewer realized regions in the evaluation domain, even though worst-case architectural capacity is unchanged; we call this reduced region usage expressivity saturation. Moreover, in the most challenging regimes, 2D visualizations show that region-usage collapse often coincides with degraded decision boundaries. Finally, we visualize the training dynamics of affine-region partitions and decision boundaries, revealing a consistent refinement process during optimization.

Compositionality is believed to be the foundation for generalization, enabling models to reuse meaningful primitives in novel combinations. Yet, models trained with standard gradient-based optimization rarely, and often only weakly, exhibit compositional internal structure, and it remains unclear how or why such compositionality forms. In this work, we show that compositionality emerges in a narrow connectivity-depth sweet spot. Along the connectivity axis, compositionality only appears in some specifically sparse networks, heavily depends on which connections remain rather than on weights' sparsity alone. Along the depth axis, compositionality emerges within a narrow, target-dependent regime, peaking at specific depths, while both shallower and deeper networks fail. When either the depth or connectivity condition is violated, gradient descent silently converges to fractured solutions rather than compositional ones. To discover and exploit this emergence, we introduce (i) similarity-based pruning (SP) to recover compositional connectivity and (ii) a heuristic depth predictor to estimate where compositionality is most likely to appear. Finally, we support these empirical findings with a theoretical framework based on compositional sparsity, volume-ratio arguments, and feature-interference bounds, explaining why compositional solutions are reachable only in a narrow depth-connectivity regime.

Prompt-based LLMs are increasingly used for stance detection, but harder examples are not always repaired by clearer instructions, reasoning prompts, retrieval, or debate. We introduce SICI (Stance Inference Complexity Index), a seven-dimensional diagnostic measure of the semantic-pragmatic burden imposed by a target--text pair. Across SemEval-2016 and VAST, SICI predicts LLM accuracy better than surface proxies and shows substantial cross-scorer reliability ($α=0.771$). More importantly, LLM errors change regime as SICI increases: low-complexity examples invite over-attribution, especially Against predictions; intermediate examples form an unstable boundary; and high-complexity examples rapidly concentrate on None. This phase-transition-like structure persists across GPT-3.5, GPT-4o-mini, DeepSeek-V3, and GPT-4o, although stronger models move the boundaries. A 15-method intervention study further shows that prompting, retrieval, and debate often shift models along the attribution--abstention axis rather than removing the high-complexity bottleneck.

We introduce the ladderpath index as a measure of language complexity grounded in algorithmic information theory. It counts the minimum steps needed to reconstruct a sequence through hierarchical reuse of repeated substructures, capturing an exactly computable but constrained form of algorithmic compressibility related to, but distinct from, Kolmogorov complexity. We apply the ladderpath approach to 21 parallel corpora from the Parallel Universal Dependencies dataset. The ladderpath index is approximately invariant across the languages, and varies much less than the corpus length. This is more pronounced when all corpora are mapped to a unified binary representation, providing evidence for the equi-complexity hypothesis from a representation-independent perspective. We also observe trade-offs between character inventory size and corpus length, and between vocabulary-level and corpus-level reconstruction complexity, supporting the trade-off hypothesis that total complexity is conserved and redistributed across linguistic levels. The reusable substructures identified by the ladderpath approach, without any linguistic input, overlap with words and morphological components attested in the natural vocabulary. The hierarchical reuse captured by the ladderpath approach parallels the chunking mechanisms proposed in cognitive science, where the human cognitive system compresses linguistic input into nested, reusable units under shared memory and processing constraints. This connection between cognitive chunking and the ladderpath approach provides a new interpretation for the equi-complexity and trade-off hypotheses, grounding both in the shared cognitive architecture that underlies language processing across human languages.

We study the task of agnostically learning general (as opposed to homogeneous) ReLUs under the Gaussian distribution with respect to the squared loss. In the passive learning setting, recent work gave a computationally efficient algorithm that uses $poly(d,1/ε)$ labeled examples and outputs a hypothesis with error $O(opt)+ε$, where $opt$ is the squared loss of the best fit ReLU. Here we focus on the interactive setting, where the learner has some form of query access to the labels of unlabeled examples. Our main result is the first computationally efficient learner that uses $d polylog(1/ε)+\tilde{O}(\min\{1/p, 1/ε\})$ black-box label queries, where $p$ is the bias of the target function, and achieves error $O(opt)+ε$. We complement our algorithmic result by showing that its query complexity bound is qualitatively near-optimal, even ignoring computational constraints. Finally, we establish that query access is essentially necessary to improve on the label complexity of passive learning. Specifically, for pool-based active learning, any active learner requires $\tildeΩ(d/ε)$ labels, unless it draws a super-polynomial number of unlabeled examples.

Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attention for its ability to infer governing laws directly from experimental or simulated data, especially when the underlying physics is unclear. However, the field has expanded rapidly along diverse methodological directions, particularly with the emergence of AI-based approaches, and still lacks a clear organizing perspective. In this Review, we propose a problem-oriented perspective on data-driven differential equation discovery. We first introduce a two-dimensional phase diagram of equation discoverability, where discovery problems are organized according to structural complexity and coefficient complexity. This phase diagram shows how the field has moved from the discovery of sparse equations with simple coefficients toward more complex governing laws with richer structures and more flexible parameterizations. It also clarifies why different methodological families succeed or fail in different problem settings. We then present the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. By identifying the core problems of equation discovery that persist across algorithmic variations, REO shifts the discussion from individual algorithms to the fundamental principles that determine discoverability. We connect these perspectives to applications across physics and adjacent sciences, and argue that the next challenge is not merely recovering equations, but using them to revise existing theories, distil mechanisms and form new scientific concepts.

Modern deep neural networks usually have large parameter scales and nonlinear hierarchical structures, and they have achieved strong performance in computer vision. However, the source of their generalization performance remains difficult to explain using traditional statistical learning theory. Among the factors that may affect visual generalization, data scale, model complexity, and input modalities are fundamental and controllable variables. This study empirically analyzes how these three factors influence model generalization performance. Specifically, in a preliminary experiment, we construct a one-dimensional nonlinear function and vary the number of training samples and the polynomial degree to observe the effects of data scale and model complexity on model performance. In the main experiments, we compare model performance on CIFAR-10 and CIFAR-100 under different training data scales, model architectures, and input modalities. The experimental results show that increasing the training data scale consistently improves generalization performance, whereas changes in model complexity do not provide stable gains. In addition, removing color information degrades model performance, while explicit prior features such as gradients, edges, and wavelets have inconsistent effects across different model architectures. Overall, this study provides an empirical analysis of the relationships among data scale, model complexity, input modalities, and visual generalization performance. Code and experimental logs are available at: https://github.com/YidiZhouluo/DeepLearning-Empirical-Studies/tree/main/Exp_01.

A new framework for the estimation of the complexity posed by video-question pairs to video-LLMs, Video Attribute-Based Complexity (VideoABC), is proposed. Video complexity is defined as the probability of failure of a video-LLM for a given video-question pair. VideoABC is a non-parametric complexity measure, using a reference video dataset and a pre-defined vocabulary of video attributes informative of complexity, \eg the scene complexity or the speed of the video event informative of the question. In a training phase, reference videos are projected into the space of these attributes, which is then quantized. The expected ABC of each quantization cell is then computed. Given a new video and its projection into the attribute space, complexity is estimated by the expected ABC of the associated quantization cell. To enable the use of VideoABC with small reference video datasets, two quantizers are combined: a k-means quantizer that enables accurate complexity estimates for samples in the distribution of the reference dataset and a universal lattice quantizer that guarantees generalization to out-of-distribution samples. A synthetic video generation procedure, inspired by target-distractor manipulations of psychophysics studies, is proposed to populate the cells of the lattice quantizer during training, enabling the computation of their expected ABCs. Experimental results show that VideoABCis effective even with very low-dimensional attribute representations, substantially outperforming approaches like `video-LLM as judge' with much less complexity. Finally, the explainable nature of the VideoABC score, in terms of well-defined attributes, is shown to provide insights on how the attribute composition of benchmarks affects their complexity.

Card payment fraud detection is usually framed as a supervised classification problem. Although this approach has generated practical progress, improvement has remained incremental despite major advances in model architecture. We argue that this is not mainly a failure of function approximation or optimization, but a consequence of structural information impairments inherent to the payment ecosystem. We formalize card authorization as a sequential decision problem with delayed, censored, corrupted, and counterfactually missing feedback. We derive a minimax regret lower bound showing that these impairments enter multiplicatively in the denominator of the achievable learning rate. The bound implies that improving issuer reporting quality or reducing censorship can yield larger reductions in the regret floor than increasing model complexity. We also show that heterogeneity across issuers worsens learnability beyond what average impairment rates suggest. The paper contributes a theory of why fraud detection in payment networks is fundamentally harder than in standard online learning settings, identifies ecosystem information quality as the key bottleneck, and provides a theoretical basis for prioritizing investments in reporting infrastructure, dispute process quality, and selective exploration. The paper is theory-first and does not rely on proprietary transaction data.

An accurate assessment of a model's complexity is crucial for topics such as interpretation, generalization, and model selection. However, most existing complexity measures either rely on heuristic assumptions or are computationally prohibitive. In this paper, we present a mathematically rigorous yet easy-to-compute measure of model complexity that is based on the similarities between the model gradients across inputs. It is thus well-defined for any parametric model, but also for kernel-based non-parametric models. We prove that our measure of complexity generalizes model-specific complexity measures such as polynomial degree (for polynomial regression), kernel length scale (for Matérn kernels), number of neighbors (for k-nearest neighbors), number of splits (for decision trees), and number of trees (for random forests). We also use our measure to obtain new insights into the double descent phenomenon for random Fourier features, random forests, neural networks, and gradient boosting.

Objective: The primary goal of this study was to systematically examine the impact of commonly used imbalance handling methods (IHMs) on predictive performance in biomedical binary classification, considering the interplay between model complexity and diverse data modalities. Material and Methods: We evaluated five representative IHMs: random undersampling (RUS), random oversampling (ROS), SMOTE, re-weighting (RW), and direct F1-score optimization (DMO), against a raw training (RAW) baseline. The evaluation encompassed three public biomedical datasets: MIMIC-III (tabular), ADE-Corpus-V2 (text), and MURA (image), spanning three common biomedical data modalities. To assess varying model complexity, we employed a range of architectures, from classical logistic regression and random forest to deep neural networks, including multilayer perceptron (MLP), BiLSTM, BERT, DenseNet, and DINOv2. Results: For simpler models such as logistic regression on tabular data, IHMs yielded no significant advantage over the RAW baseline, aligning with prior findings. However, clear benefits were observed for more complex models and unstructured data: (a) ROS and RW consistently enhanced the performance of powerful models; (b) direct F1-score optimization demonstrated utility primarily for unstructured text and image data; and (c) RUS and SMOTE consistently degraded performance and are therefore not recommended. Conclusion: The effectiveness of IHMs depends on both model complexity and data modality. Performance gains are most pronounced when leveraging appropriate IHMs, such as ROS, RW, and DMO, on high-complexity models.

Graph neural networks (GNNs) have emerged as a fundamental tool for learning from graph-structured data, achieving strong performance across a wide range of applications. However, understanding their generalization capabilities remains challenging due to the complex structural dependencies inherent in such data. Existing generalization analyses largely follow the classical machine learning paradigm, focusing primarily on model complexity while overlooking the fundamental role of graph structure. Therefore, in this work, we systematically investigate this role by asking: does the graph structure actually influence generalization, and if so, by how much? To answer the first question and validate our intuition, we theoretically prove that incorporating more edges into the prediction process transforms the input representations to be overly accommodating to the output model, thereby inducing overfitting. To address the second question, we formulate a structural complexity measure based on the number of effective edges and derive a Rademacher complexity-based generalization bound. In doing so, we demonstrate that GNN generalization depends explicitly on structural complexity, alongside traditional parameter-dependent factors. Motivated by these theoretical findings, we propose a structural entropy regularization method. This approach controls structural complexity by regulating effective edges to balance underfitting and overfitting, ultimately improving the generalization performance of GNNs.

Strategic classification studies learning settings in which individuals can modify their features, at a cost, in order to influence the classifier's decision. A central question is how the sample complexity of the induced (strategic) hypothesis class depends on the complexities of the underlying hypothesis class and the cost structure governing feasible manipulations. Prior work has shown that in several natural settings, such as linear classifiers with norm costs, the induced complexity can be controlled. We begin by showing that such guarantees fail in general - even in simple cases: there exist hypothesis classes of VC dimension $1$ on the real line such that, even under the simplest interval neighborhoods, the induced class has infinite VC dimension. Thus, strategic behavior can turn an easy learning problem into a non-learnable one. To overcome this, we introduce structure via a geometric definability assumption: both the hypothesis class and the cost-induced neighborhood relation can be defined by first-order formulas over $\mathbb{R}_{\mathtt{exp}}$. Intuitively, this means that hypotheses and costs can be described using arithmetic operations, exponentiation, logarithms, and comparisons. This captures a broad range of natural classes and cost functions, including $\ell_p$ distances, Wasserstein distance, and information-theoretic divergences. Under this assumption, we prove that learnability is preserved, with sample complexity controlled by the complexity of the defining formulas.

Progression, the task of updating a knowledge base to reflect action effects, generally requires second-order logic. Identifying first-order special cases, by restricting either the knowledge base or action effects, has long been a central topic in reasoning about actions. It is known that local-effect, normal, and acyclic actions, three increasingly expressive classes, admit first-order progression. However, a systematic analysis of the size of such progressions, crucial for practical applications, has been missing. In this paper, using the framework of Situation Calculus, we show that under reasonable assumptions, first-order progression for these action classes grows only polynomially. Moreover, we show that when the KB belongs to decidable fragments such as two-variable first-order logic or universal theories with constants, the progression remains within the same fragment, ensuring decidability and practical applicability.

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.

We introduce TableVista, a comprehensive benchmark for evaluating foundation models in multimodal table reasoning under visual and structural complexity. TableVista consists of 3,000 high-quality table reasoning problems, where each instance is expanded into 10 distinct visual variants through our multi-style rendering and transformation pipeline. This process encompasses diverse scenario styles, robustness perturbations, and vision-only configurations, culminating in 30,000 multimodal samples for a multi-dimensional evaluation. We conduct an extensive evaluation of 29 state-of-the-art open-source and proprietary foundation models on TableVista. Through comprehensive quantitative and qualitative analysis, we find that while evaluated models remain largely stable across diverse rendering styles, they exhibit pronounced performance degradation on complex structural layouts and vision-only settings, revealing that current models struggle to maintain reasoning consistency when structural complexity combines with visually integrated presentations. These findings highlight critical gaps in current multimodal capabilities, providing insights for advancing more robust and reliable table understanding models.

Using multimodal foundation models to analyze table images is a high-value yet challenging application in consumer and enterprise scenarios. Despite its importance, current evaluations rely largely on structured-text tables or clean rendered images, leaving the visual complexity of in-the-wild table images underexplored. Such images feature varied layouts and diverse domains that demand sophisticated structural perception and numerical reasoning. To bridge this gap, we introduce WildTableBench, the first question-answering benchmark for naturally occurring table images from real-world settings. WildTableBench comprises 402 high-information-density table images collected from online forums and websites across diverse domains, together with 928 manually annotated and verified questions spanning 17 subtypes across five categories. We evaluate 21 frontier proprietary and open-source multimodal foundation models on this benchmark. Only one model exceeds 50% accuracy, while all remaining models range from 4.1% to 49.9%. We further conduct diagnostic analyses to characterize model failures and reveal persistent weaknesses in structural perception and reasoning. These results and analyses provide useful insights into current model capabilities and establish WildTableBench as a valuable diagnostic benchmark for table image understanding. Dataset: https://huggingface.co/datasets/jzhuang/WildTableBench Code: https://github.com/hjzhe/WildTableBench Leaderboard: https://hjzhe.github.io/WildTableBench

Topological neural networks have emerged as effective tools for modeling higher-order relational structures beyond pairwise graphs, including hypergraphs, simplicial complexes, and cell complexes. However, existing Weisfeiler-Leman type expressivity analyses are typically developed on different structural domains and rely on domain-specific neighborhood systems, making their expressive powers difficult to compare within a common formalism. In this paper, we introduce the Combinatorial Complex Weisfeiler-Leman (CCWL) framework, a unified expressive power refinement defined on combinatorial complexes. By exploiting the ability of combinatorial complexes to represent both set-type relations and part-whole hierarchies, CCWL performs topological color refinement through four structural neighborhoods: boundary, co-boundary, lower adjacency, and upper adjacency. We show that, under specified lifting maps, CCWL can simulate several domain-specific WL-type refinements, thereby providing a common theoretical baseline for analyzing topological message passing. We further study the neighborhood sufficiency problem and prove that, under explicit coverage conditions, a reduced refinement using only lower- and upper-adjacent bridge information preserves the distinguishing power of the full four-neighborhood CCWL refinement. Guided by this theoretical result, we instantiate the reduced refinement as the Combinatorial Complex Isomorphism Network (CCIN). Experiments on synthetic and real-world benchmarks demonstrate that CCIN achieves competitive performance against representative graph and topological neural network baselines. Ablation studies and resource-efficiency analyses further support the effectiveness of the proposed lower/upper-neighborhood design.

As the complexity of System-on-Chip (SoC) designs grows, the shift-left paradigm necessitates the rapid development of high-fidelity reference models (typically written in SystemC) for early architecture exploration and verification. While Large Language Models (LLMs) show promise in code generation, their application to hardware modeling faces unique challenges: (1) Rigid, static workflows fail to adapt to varying design complexity, causing inefficiency; (2) Context window overflow in multi-turn interactions leads to catastrophic forgetting of critical specifications; and (3) the Coupled Validation Failure problem--where generated Testbenches (TBs) incorrectly validate flawed models due to correlated hallucinations--severely undermines reliability. To address these limitations, we introduce RefEvo, a dynamic multi-agent framework designed for agile and reliable reference modeling. RefEvo features three key innovations: (1) A Dynamic Design Planner that autonomously decomposes design specifications and constructs tailored execution workflows based on semantic complexity; (2) A Co-Evolutionary Verification Mechanism, which employs a Dialectical Arbiter to simultaneously rectify the model and verification logic against the specification (Spec) oracle, effectively mitigating false positives; and (3) A Spec Anchoring Strategy for lossless context compression. Evaluated on a diverse benchmark of 20 hardware modules, RefEvo achieves a 95% pass rate, outperforming static baselines by a large margin. Furthermore, our context optimization reduces token consumption by an average of 71.04%, achieving absolute savings of over 70,000 tokens per session for complex designs while maintaining 100% specification recall.

Understanding social polarization requires integrating insights from psychology, sociology, and complex systems science. Agent-based modeling provides a natural framework to combine perspectives from different fields and explore how individual cognition shapes collective outcomes. This study introduces a novel agent-based model that integrates two cognitive and social mechanisms: the desire to be unique within a group (optimal distinctiveness theory) and the tendency to simplify complex information (cognitive compression). In the model, virtual agents interact in pairs and decide whether to adopt each other's opinions by balancing two opposing drives: maximizing opinion diversity within their local social group while simplifying the overall opinion landscape, with both evaluated using Shannon entropy. We show that the combination of these mechanisms can reproduce real-world patterns, such as the emergence of distinct heterogeneous opinion clusters. Moreover, unlike many existing models where opinions become fixed once opinion groups form, individuals in our model continue to adjust their opinions after clusters emerge, leading to ongoing variation within and between opinion groups. Computational experiments reveal that polarization emerges when local group sizes are moderate (consistent with Dunbar's number), while smaller groups cause fragmentation and larger ones hinder distinct cluster formation. Higher cognitive compression increases unpredictability, while lower compression produces more consistent group structures. These results demonstrate how simple psychological rules can generate complex, realistic social behavior and advance understanding of polarization in human societies.

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

\emph{Integrated communication and computation} (IC$^2$) has emerged as a new paradigm for enabling efficient edge inference in sixth-generation (6G) networks. However, the design of IC$^2$ technologies is hindered by the lack of a tractable theoretical framework for characterizing \emph{end-to-end} (E2E) inference performance. The metric is highly complicated as it needs to account for both channel distortion and artificial intelligence (AI) model architecture and computational complexity. In this work, we address this challenge by developing a tractable analytical model for E2E inference accuracy and leveraging it to design a \emph{channel-adaptive AI} algorithm that maximizes inference throughput, referred to as the edge processing rate (EPR), under latency and accuracy constraints. Specifically, we consider an edge inference system in which a server deploys a backbone model with early exit, which enables flexible computational complexity, to perform inference on data features transmitted by a mobile device. The proposed accuracy model characterizes high-dimensional feature distributions in the angular domain using a Mixture of von Mises (MvM) distribution. This leads to a desired closed-form expression for inference accuracy as a function of quantization bit-width and model traversal depth, which represents channel distortion and computational complexity, respectively. Building upon this accuracy model, we formulate and solve the EPR maximization problem under joint latency and accuracy constraints, leading to a channel-adaptive AI algorithm that achieves full IC$^2$ integration. The proposed algorithm jointly adapts transmit-side feature compression and receive-side model complexity according to channel conditions to maximize overall efficiency and inference throughput. Experimental results demonstrate its superior performance as compared with fixed-complexity counterparts.

Instruction-based image editing enables natural-language control over visual modifications, yet existing models falter under Instruction-Visual Complexity (IV-Complexity), where intricate instructions meet cluttered or ambiguous scenes. We introduce RePlan (Region-aligned Planning), a plan-then-execute framework that couples a vision-language planner with a diffusion editor. The planner decomposes instructions via step-by-step reasoning and explicitly grounds them to target regions; the editor then applies changes using a training-free attention-region injection mechanism, enabling precise, parallel multi-region edits without iterative inpainting. To strengthen planning, we apply GRPO-based reinforcement learning using 1K instruction-only examples, yielding substantial gains in reasoning fidelity and format reliability. We further present IV-Edit, a benchmark focused on fine-grained grounding and knowledge-intensive edits. Across IV-Complex settings, RePlan consistently outperforms strong baselines trained on far larger datasets, improving regional precision and overall consistency. Our project page: https://replan-iv-edit.github.io