arXiv subjects

Jonathan W. Siegel

Publications and source records attributed to Jonathan W. Siegel.

At least 19 recordsLinked to original sources

arXiv2026

Constrained Kolmogorov widths

The main theme of approximation theory is to understand how well a general function $f$ can be approximated by a simpler function $g$ such as a polynomial or spline. In many applications, one wants $g$ to retain known properties of $f$ such as its inherent smoothness or a geometrical property such as monotonicity or convexity. Additional requirements on $g$ of this type are known as constraints. In this paper, we do a systematic study of constrained approximation to understand how the imposition of such constraints limits the efficiency of the approximation. We study constrained approximation in the setting of linear approximation where $g$ is to be taken from a finite dimensional linear space $V$ of a fixed dimension $n$. Kolmogorov widths describe how well one can approximate when using such linear spaces $V$. The first part of this paper introduces and studies several types of constrained widths, including the constrained Kolmogorov widths, and gives comparisons between them. The second part of the paper is restricted to classical settings where the constraint imposes a smoothness requirement on $g$. In this case, our results prove that the additional constraint can typically be imposed with no loss in the efficiency of the approximation.

math.NA↗

arXiv2026

Sampling and reconstruction of convex functions

We discuss optimal recovery for classes of multivariate convex functions from given point samples, as well as the sampling numbers of these classes, corresponding to optimal sample choices. Upper and lower bounds for either variant are established when the reconstruction error is measured in $L_p$ for $1\leq p\leq \infty$. These bounds match, sometimes up to logarithmic factors, and therefore characterize the respective optimal rate of decay. For classical smoothness classes such as Sobolev, Hölder or Besov spaces, it is well known that the optimal decay rate of sampling numbers can be achieved by sampling on uniform tensor product grids and using linear methods of reconstruction, such as piecewise polynomial interpolation. One of the main findings in this paper is that for classes of convex functions, these procedures generally produce suboptimal rates, except when $p=1$ and $p=\infty$, and are outperformed by nonlinear reconstruction methods that do not employ tensor product grids.

math.NA↗

arXiv2026

Embedding Dimension Lower Bounds for Universality of Deep Sets and Janossy Pooling

In many practical applications it is important to build symmetries into neural network architectures. Consider the important case of permutation symmetry on point clouds consisting of $n$ points in $d$ dimensions. In this case the network learns a function on a set of $n$ points in $\mathbb{R}^d$, and a natural paradigm for constructing invariant networks is Janossy pooling, which generalizes the popular Deep Sets architecture. We study the universality of this approach, in particular the important question of how large the embedding dimension must be to guarantee universality of this architecture. Specifically, using a novel technique, we prove new lower bounds on the required size of this embedding dimension. For Deep Sets, this gives the correct minimal dimension up to a constant factor for all $d > 1$. For $k$-ary Janossy pooling, we prove the first non-trivial lower bound on the required embedding dimension when $k > 1$.

cs.LG↗

arXiv2026

Quantitative Approximation Rates for Group Equivariant Learning

The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of $α$-Hölder functions $f: [0,1]^N \to \mathbb{R}$. The goal of this paper is to provide similar quantitative approximation results in the context of group equivariant learning, where the learned $α$-Hölder function is known to obey certain group symmetries. While there has been much interest in the literature in understanding the universal approximation properties of equivariant models, very few quantitative approximation results are known for equivariant models. In this paper, we bridge this gap by deriving quantitative approximation rates for several prominent group-equivariant and invariant architectures. The architectures that we consider include: the permutation-invariant Deep Sets architecture; the permutation-equivariant Sumformer and Transformer architectures; joint invariance to permutations and rigid motions using invariant networks based on frame averaging; and general bi-Lipschitz invariant models. Overall, we show that equally-sized ReLU MLPs and equivariant architectures are equally expressive over equivariant functions. Thus, hard-coding equivariance does not result in a loss of expressivity or approximation power in these models.

cs.LG↗

arXiv2026

In-Context Multi-Operator Learning with DeepOSets

An important application of neural networks to scientific computing has been the learning of non-linear operators. In this framework, a neural network is trained to fit a non-linear map between two infinite dimensional spaces, for example, the solution operator of ordinary and partial differential equations. Recently, inspired by the discovery of in-context learning for large language models, an even more ambitious paradigm has been explored, called multi-operator learning. In this approach, a neural network is trained to learn many different operators at the same time. In order to evaluate one of the learned operators, the network is passed example inputs and outputs to disambiguate the desired operator. In this work, we provide a precise mathematical formulation of the multi-operator learning problem. In addition, we modify a simple efficient architecture, called DeepOSets, for multi-operator learning and prove its universality for multi-operator learning. Finally, we provide a comprehensive set of experiments that demonstrate the ability of DeepOSets to learn multiple operators corresponding to different initial-value and boundary-value differential equations and use in-context examples to predict accurately the solutions corresponding to queries and differential equations not seen during training. The main advantage of DeepOSets is its architectural simplicity, which allows the derivation of theoretical guarantees and training times that are in the order of minutes, in contrast to similar transformer-based alternatives that are empirically justified and require hours of training.

cs.LG↗

arXiv2025

Approximation Rates for Shallow ReLU$^k$ Neural Networks on Sobolev Spaces via the Radon Transform

Let $Ω\subset \mathbb{R}^d$ be a bounded domain. We consider the problem of how efficiently shallow neural networks with the ReLU$^k$ activation function can approximate functions from Sobolev spaces $W^s(L_p(Ω))$ with error measured in the $L_q(Ω)$-norm. Utilizing the Radon transform and recent results from discrepancy theory, we provide a simple proof of nearly optimal approximation rates in a variety of cases, including when $q\leq p$, $p\geq 2$, and $s \leq k + (d+1)/2$. The rates we derive are optimal up to logarithmic factors, and significantly generalize existing results. An interesting consequence is that the adaptivity of shallow ReLU$^k$ neural networks enables them to obtain optimal approximation rates for smoothness up to order $s = k + (d+1)/2$, even though they represent piecewise polynomials of fixed degree $k$.

stat.ML↗

arXiv2025

Acceleration via silver step-size on Riemannian manifolds with applications to Wasserstein space

There is extensive literature on accelerating first-order optimization methods in a Euclidean setting. Under which conditions such acceleration is feasible in Riemannian optimization problems is an active area of research. Motivated by the recent success of dynamic stepsize methods in the Euclidean setting, we undertake a study of such algorithms in the Riemannian setting. We provide the new class of algorithms determined by the choice of vector transport that allows the dynamic stepsize acceleration on Riemannian manifolds for the function classes associated with the corresponding vector transport. As a core application, we show our algorithm recovers the standard Wasserstein gradient descent on 2-Wasserstein space, and as a result provides the first provable accelerated gradient method in Wasserstein space. In addition, we validate the numerical strength of the algorithm for standard benchmark tasks on the space of symmetric positive definite matrices.

math.OC↗

arXiv2025

Nearly optimal bounds on the Fourier sampling numbers of Besov spaces

Let $\mathbb{T}^d$ denote the $d$-dimensional torus. We consider the problem of optimally recovering a target function $f^*:\mathbb{T}^d\rightarrow \mathbb{C}$ from samples of its Fourier coefficients. We make classical smoothness assumptions on $f^*$, specifically that $f^*$ lies in a Besov space $B^s_\infty(L_q)$ with $s > 0$ and $1\leq q\leq \infty$, and measure recovery error in the $L_p$-norm with $1\leq p\leq \infty$. Abstractly, the optimal recovery error is characterized by a `restricted' version of the Gelfand widths, which we call the Fourier sampling numbers. Up to logarithmic factors, we determine the correct asymptotics of the Fourier sampling numbers in the regime $s/d > 1 - 1/p$. We also give a description of nearly optimal Fourier measurements and recovery algorithms in each of these cases. In the other direction, we prove a novel lower bound showing that there is an asymptotic gap between the Fourier sampling numbers and the Gelfand widths when $q = 1$ and $p_0 < p\leq 2$ with $p_0 \approx 1.535$. Finally, we discuss the practical implications of our results, which imply a sharper recovery of edges, and provide numerical results demonstrating this phenomenon.

math.FA↗

arXiv2025

Sharp Lower Bounds on Interpolation by Deep ReLU Neural Networks at Irregularly Spaced Data

We study the interpolation power of deep ReLU neural networks. Specifically, we consider the question of how efficiently, in terms of the number of parameters, deep ReLU networks can interpolate values at $N$ datapoints in the unit ball which are separated by a distance $δ$. We show that $Ω(N)$ parameters are required in the regime where $δ$ is exponentially small in $N$, which gives the sharp result in this regime since $O(N)$ parameters are always sufficient. This also shows that the bit-extraction technique used to prove lower bounds on the VC dimension cannot be applied to irregularly spaced datapoints. Finally, as an application we give a lower bound on the approximation rates that deep ReLU neural networks can achieve for Sobolev spaces at the embedding endpoint.

cs.LG↗

arXiv2025

Optimal Recovery Meets Minimax Estimation

A fundamental problem in statistics and machine learning is to estimate a function $f$ from possibly noisy observations of its point samples. The goal is to design a numerical algorithm to construct an approximation $\hat f$ to $f$ in a prescribed norm that asymptotically achieves the best possible error (as a function of the number $m$ of observations and the variance $σ^2$ of the noise). This problem has received considerable attention in both nonparametric statistics (noisy observations) and optimal recovery (noiseless observations). Quantitative bounds require assumptions on $f$, known as model class assumptions. Classical results assume that $f$ is in the unit ball of a Besov space. In nonparametric statistics, the best possible performance of an algorithm for finding $\hat f$ is known as the minimax rate and has been studied in this setting under the assumption that the noise is Gaussian. In optimal recovery, the best possible performance of an algorithm is known as the optimal recovery rate and has also been determined in this setting. While one would expect that the minimax rate recovers the optimal recovery rate when the noise level $σ$ tends to zero, it turns out that the current results on minimax rates do not carefully determine the dependence on $σ$ and the limit cannot be taken. This paper handles this issue and determines the noise-level-aware (NLA) minimax rates for Besov classes when error is measured in an $L_q$-norm with matching upper and lower bounds. The end result is a reconciliation between minimax rates and optimal recovery rates. The NLA minimax rate continuously depends on the noise level and recovers the optimal recovery rate when $σ$ tends to zero.

math.ST↗

arXiv2025

Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks

We study the following two related problems. The first is to determine to what error an arbitrary zonoid in $\mathbb{R}^{d+1}$ can be approximated in the Hausdorff distance by a sum of $n$ line segments. The second is to determine optimal approximation rates in the uniform norm for shallow ReLU$^k$ neural networks on their variation spaces. The first of these problems has been solved for $d\neq 2,3$, but when $d=2,3$ a logarithmic gap between the best upper and lower bounds remains. We close this gap, which completes the solution in all dimensions. For the second problem, our techniques significantly improve upon existing approximation rates when $k\geq 1$, and enable uniform approximation of both the target function and its derivatives.

stat.ML↗

arXiv2025

On the expressiveness and spectral bias of KANs

Kolmogorov-Arnold Networks (KAN) \cite{liu2024kan} were very recently proposed as a potential alternative to the prevalent architectural backbone of many deep learning models, the multi-layer perceptron (MLP). KANs have seen success in various tasks of AI for science, with their empirical efficiency and accuracy demostrated in function regression, PDE solving, and many more scientific problems. In this article, we revisit the comparison of KANs and MLPs, with emphasis on a theoretical perspective. On the one hand, we compare the representation and approximation capabilities of KANs and MLPs. We establish that MLPs can be represented using KANs of a comparable size. This shows that the approximation and representation capabilities of KANs are at least as good as MLPs. Conversely, we show that KANs can be represented using MLPs, but that in this representation the number of parameters increases by a factor of the KAN grid size. This suggests that KANs with a large grid size may be more efficient than MLPs at approximating certain functions. On the other hand, from the perspective of learning and optimization, we study the spectral bias of KANs compared with MLPs. We demonstrate that KANs are less biased toward low frequencies than MLPs. We highlight that the multi-level learning feature specific to KANs, i.e. grid extension of splines, improves the learning process for high-frequency components. Detailed comparisons with different choices of depth, width, and grid sizes of KANs are made, shedding some light on how to choose the hyperparameters in practice.

cs.LG↗

arXiv2025

Convergence and error control of consistent PINNs for elliptic PDEs

We provide an a priori analysis of collocation methods for solving elliptic boundary value problems. They begin with information in the form of point values of the data and utilize only this information to numerically approximate the solution u of the PDE. For such a method to provide an approximation with guaranteed error bounds, additional assumptions on the data, called model class assumptions, are needed. We determine the best error of approximating u in the energy norm, in terms of the total number of point samples, under all Besov class model assumptions for the right hand side and boundary data. We then turn to the study of numerical procedures and analyze whether a proposed numerical procedure achieves the optimal recovery error. We analyze numerical methods which generate the numerical approximation to $u$ by minimizing specified data driven loss functions over a set $Σ$ which is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a data driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis a new loss function is proposed. We also address the recent methods of Physics Informed Neural Networks. We prove that minimization of the new loss over restricted neural network spaces $Σ$ provides an optimal recovery of the solution $u$, provided that the optimization problem can be numerically executed and $Σ$ has sufficient approximation capabilities. We also analyze variants of the new loss function which are more practical for implementation. Finally, numerical examples illustrating the benefits of the proposed loss functions are given.

math.NA↗

arXiv2024

Nesterov acceleration despite very noisy gradients

We present a generalization of Nesterov's accelerated gradient descent algorithm. Our algorithm (AGNES) provably achieves acceleration for smooth convex and strongly convex minimization tasks with noisy gradient estimates if the noise intensity is proportional to the magnitude of the gradient at every point. Nesterov's method converges at an accelerated rate if the constant of proportionality is below 1, while AGNES accommodates any signal-to-noise ratio. The noise model is motivated by applications in overparametrized machine learning. AGNES requires only two parameters in convex and three in strongly convex minimization tasks, improving on existing methods. We further provide clear geometric interpretations and heuristics for the choice of parameters.

stat.ML↗

arXiv2024

Weighted variation spaces and approximation by shallow ReLU networks

We investigate the approximation of functions $f$ on a bounded domain $Ω\subset \mathbb{R}^d$ by the outputs of single-hidden-layer ReLU neural networks of width $n$. This form of nonlinear $n$-term dictionary approximation has been intensely studied since it is the simplest case of neural network approximation (NNA). There are several celebrated approximation results for this form of NNA that introduce novel model classes of functions on $Ω$ whose approximation rates do not grow unbounded with the input dimension. These novel classes include Barron classes, and classes based on sparsity or variation such as the Radon-domain BV classes. The present paper is concerned with the definition of these novel model classes on domains $Ω$. The current definition of these model classes does not depend on the domain $Ω$. A new and more proper definition of model classes on domains is given by introducing the concept of weighted variation spaces. These new model classes are intrinsic to the domain itself. The importance of these new model classes is that they are strictly larger than the classical (domain-independent) classes. Yet, it is shown that they maintain the same NNA rates.

stat.ML↗

arXiv2024

Sharp Convergence Rates for Matching Pursuit

We study the fundamental limits of matching pursuit, or the pure greedy algorithm, for approximating a target function $ f $ by a linear combination $f_n$ of $n$ elements from a dictionary. When the target function is contained in the variation space corresponding to the dictionary, many impressive works over the past few decades have obtained upper and lower bounds on the error $\|f-f_n\|$ of matching pursuit, but they do not match. The main contribution of this paper is to close this gap and obtain a sharp characterization of the decay rate, $n^{-α}$, of matching pursuit. Specifically, we construct a worst case dictionary which shows that the existing best upper bound cannot be significantly improved. It turns out that, unlike other greedy algorithm variants which converge at the optimal rate $ n^{-1/2}$, the convergence rate $n^{-α}$ is suboptimal. Here, $α\approx 0.182$ is determined by the solution to a certain non-linear equation.

stat.ML↗

arXiv2024

Sharp Lower Bounds on the Manifold Widths of Sobolev and Besov Spaces

We consider the problem of determining the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$ or $1 \leq p \leq 2$. We close this gap and obtain sharp lower bounds for all $1 \leq p,q \leq \infty$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $\ell^M_q$-balls in the $\ell_p$-norm when $p\leq q$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.

math.NA↗

arXiv2024

Equivariant Frames and the Impossibility of Continuous Canonicalization

Canonicalization provides an architecture-agnostic method for enforcing equivariance, with generalizations such as frame-averaging recently gaining prominence as a lightweight and flexible alternative to equivariant architectures. Recent works have found an empirical benefit to using probabilistic frames instead, which learn weighted distributions over group elements. In this work, we provide strong theoretical justification for this phenomenon: for commonly-used groups, there is no efficiently computable choice of frame that preserves continuity of the function being averaged. In other words, unweighted frame-averaging can turn a smooth, non-symmetric function into a discontinuous, symmetric function. To address this fundamental robustness problem, we formally define and construct \emph{weighted} frames, which provably preserve continuity, and demonstrate their utility by constructing efficient and continuous weighted frames for the actions of $SO(2)$, $SO(3)$, and $S_n$ on point clouds.

cs.LG↗