Curse of Dimensionality

Kolmogorov-Arnold networks (KANs) have a dimension-independent approximation exponent when the target admits a sufficiently smooth representation by univariate edge functions. We ask whether this favorable scaling belongs to the architecture or to the assumed target class. For arbitrary-depth standard spline-KANs, we prove a worst-case lower bound on isotropic Sobolev balls: below a dimension-independent accuracy threshold, the required number of spline coefficients grows at least exponentially with input dimension. Existing composition-matched MLP results, by contrast, show that favorable exponents on low-arity targets are not exclusive to KANs. We also revisit empirical comparisons. On four scaling-law targets, spline-KAN approximants constructed from known decompositions transfer exactly to MLPs with the cubic ReLU activation $σ(x)=(\max\{0,x\})^3$, although their full dense parameter counts are larger. In controlled ridge-profile scans, KANs have a strong advantage when each nonlinear component depends on one input coordinate. When a component couples multiple input coordinates, this advantage may narrow sharply. At tight budgets, interacting targets often favor MLPs. In the random-label test on MNIST, ReLU-MLPs fit the randomized training labels more efficiently than spline-only KANs. Together, these results locate favorable scaling in target structure and a compatible inductive bias, not a general KAN-specific escape from the curse of dimensionality.

As the input dimension $n$ grows, rule-based machine learning, such as Learning Classifier Systems (LCSs), faces a fundamental scalability bottleneck for function approximation: both rule count and parameter count grow exponentially with $n$. Traditional LCSs partition the $n$-dimensional input space directly, requiring $\mathcal{O}(m^n)$ rules for adequate coverage, where $m$ is the per-variable resolution. This article breaks from this paradigm by reorganizing rules dimension-wise, guided by the Kolmogorov-Arnold representation theorem: any continuous $n$-dimensional function can be expressed as a finite superposition of one-dimensional functions. The proposed Kolmogorov-Arnold Classifier System (KACS) decomposes the target function into one-dimensional subproblems and assigns a dedicated ruleset to each, reducing the worst-case rule count from $\mathcal{O}(m^n)$ to $\mathcal{O}(mn^2)$ and replacing $n$-dimensional local models with one-dimensional models requiring only two parameters per rule, independent of $n$. We also provide the first constructive proof that an LCS, namely KACS, is a universal approximator for continuous functions on compact domains. Evaluated against a direct $n$-dimensional input space partitioning approach under otherwise identical conditions, KACS achieves competitive accuracy in many settings while using only 2% to 40% of the parameters. Our implementation is available at https://github.com/YNU-NakataLab/KACS.

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η\varepsilon^{-η}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $ξ>0$, the explicit bound $C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}$ on the number of parameters.

Large language models (LLMs) have become the default tool for a remarkable range of tasks, yet they have had conspicuously little success at one of the most common machine learning workloads: predictive analytics over tabular data. This gap is the founding premise of the fast-growing field of tabular foundation models, but the question of why generic LLMs fail has remained open. We study a frontier LLM in its purest inference regime - a single generation pass over a prompt containing the full training and test data, with no tools, no agentic scaffolding, and no fine-tuning - and systematically evaluate five hypotheses for the failure: (a) an inability to handle noisy or non-linearly-separable data; (b) the linearised CSV format obscuring column structure; (c) the tokenisation of numeric values; (d) the number of test points classified per query; and (e) the dimensionality of the input. Controlled experiments falsify (a)-(d). Dimensionality, in contrast, is decisive: sweeping random linear projections of thirty-one benchmark datasets, the LLM is the only method among nine whose accuracy decreases as dimensionality grows, while every classical baseline stays flat or improves. A behavioural comparison against 252 configured classical models finds that in two dimensions the LLM predicts like a local, distance-based method (up to 91.6% grid agreement), but in higher dimensions no classical model - even when augmented with tuned, dimension-dependent noise - reproduces its predictions. We do not claim to have identified the internal mechanism; our results show, more modestly, that the LLM's capability dissolves with dimension in a way no noise-corrupted classical learner mimics - which explains why LLMs, so capable elsewhere, keep losing to fifty-year-old baselines on tables, while leaving the mechanism of the prediction as an open question.

Traditional approximation theory measures convergence rates in terms of the number of parameters or degrees of freedom. However, practical computation operates under finite precision: parameters must be encoded using a finite number of bits. Therefore, approximation efficiency should be evaluated in terms of computational bit complexity, which is intrinsically connected to the metric entropy of the underlying function class. In this work, we develop a unified approximation framework based on binary encoding and metric entropy. We analyze classical methods (including polynomial approximation, sparse grids, and finite elements) as well as shallow and deep neural networks, and compare their approximation rates for function classes with comparable metric entropy. We observe that, when evaluated in terms of bits, most classical methods are in general suboptimal relative to the intrinsic limits dictated by metric entropy, while neural network methods may exhibit different behaviors. We show that when complexity is measured in bits rather than parameters, no method fundamentally exceeds the approximation order achieved by classical approaches. Our results also indicate that many seeming advantages of neural networks, including dimension-independent rates and superconvergence phenomena, stem from differences in function class complexity rather than intrinsic architectural superiority. In this sense, the traditional curse of dimensionality can be misleading; the fundamental limitation is instead a curse of bit complexity, governed by metric entropy.

This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators. While standard ReLU networks achieve minimax-optimal rates in the $L^2(P)$ norm for various nonparametric regression tasks, we establish a theoretical lower bound demonstrating that least-squares ReLU estimators can suffer from the curse of dimensionality in their uniform convergence behavior. Motivated by the need for reliable uniform guarantees in downstream tasks requiring worst-case reliability, we address this limitation by analyzing smoothly activated DNNs (smooth DNNs), encompassing both feedforward and residual structures. We establish novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds for the approximators of these models. Leveraging these results, we derive non-asymptotic uniform convergence rates for smooth DNN estimators across multiple statistical contexts, including Huber, least-squares, quantile, and logistic regression. We prove that smooth DNNs can mitigate the {curse of dimensionality} in uniform convergence by adaptively exploiting the low-dimensional hierarchical composition structure of the target function. Supported by both simulation studies and a real-world application, our results position smooth DNNs as a theoretically grounded and practically viable alternative to ReLU networks for statistical learning tasks requiring uniform guarantees.

Generative models are often conditioned on a small set of examples via cross-attention. Under the Gaussian optimal-transport path, we show that the exact velocity field induced by a finite support set is a Nadaraya--Watson kernel smoother whose bandwidth decreases with flow time, from broad averaging at early steps to nearest-neighbor at late steps. A single Gaussian-kernel attention head exactly computes this field, connecting cross-attention conditioning to classical kernel theory. The theory predicts three failure regimes: nearest-neighbor collapse of the kernel at high dimension, mismatch between the isotropic kernel and the data geometry, and insufficient support for nonparametric estimation. Experiments on Gaussian mixtures, spherical shells, and DINOv2 ImageNet features confirm that learned conditioning improves in precisely these regimes, and that IP-Adapter's cross-attention implements approximate NW smoothing in practice.

Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited interpretability. This work investigates how sparsity can help address these challenges in functional learning, a central ingredient in operator learning. We propose a framework that employs convolutional architectures to extract sparse features from a finite number of samples, together with deep fully connected networks to effectively approximate nonlinear functionals. Using universal discretization methods, we show that sparse approximators enable stable recovery from discrete samples. In addition, both the deterministic and the random sampling schemes are sufficient for our analysis. These findings lead to improved approximation rates and reduced sample sizes in various function spaces, including those with fast frequency decay and mixed smoothness. They also provide new theoretical insights into how sparsity can alleviate the curse of dimensionality in functional learning.

Diffusion models are state-of-the-art tools for various generative tasks. Yet training these models involves estimating high-dimensional score functions, a task that in principle suffers from the curse of dimensionality. It is therefore important to understand how low-dimensional structure in the target distribution can be exploited in these models. Here we consider locality structure, which describes certain sparse conditional dependencies among the target random variables. Given some locality structure, the score function is effectively low-dimensional, so that it can be estimated by a localized neural network with significantly reduced sample complexity. This observation motivates the localized diffusion model, where a localized score matching loss is used to train the score function within a localized hypothesis space. We prove that such localization enables diffusion models to circumvent the curse of dimensionality with dimension-independent error bounds, at the price of additional localization error. Under realistic sample size scaling, we then show both theoretically and numerically that a moderate localization radius can balance the statistical and localization errors, yielding better overall performance. Locality structure also facilitates parallel training, making localized diffusion models potentially more efficient for large-scale applications.