Abstract The classical universal approximation (UA) theorem for neural networks establishes mild conditions under which a feedforward neural network can approximate a continuous function f with arbitrary accuracy. A recent result shows that neural networks also enjoy a more general interval universal approximation (IUA) theorem, in the sense that the abstract interpretation semantics of the network using the interval domain can approximate the direct image map of f (i.e., the result of applying f to a set of inputs) with arbitrary accuracy. These theorems, however, rest on the unrealistic assumption that the neural network computes over infinitely precise real numbers, whereas their software implementations in practice compute over finite-precision floating-point numbers. An open question is whether the IUA theorem still holds in the floating-point setting. This paper introduces the first IUA theorem for floating-point neural networks that proves their remarkable ability to perfectly capture the direct image map of any rounded target function f , showing no limits exist on their expressiveness. Our IUA theorem in the floating-point setting exhibits material differences from the real-valued setting, which reflects the fundamental distinctions between these two computational models. This theorem also implies surprising corollaries, which include (i) the existence of provably robust floating-point neural networks; and (ii) the computational completeness of the class of straight-line programs that use only floating-point additions and multiplications for the class of all floating-point programs that halt.

Existing works on the expressive power of neural networks typically assume real-valued parameters and exact mathematical operations during the evaluation of networks. However, neural networks run on actual computers can take parameters only from a small subset of the reals and perform inexact mathematical operations with round-off errors and overflows. In this work, we study the expressive power of floating-point neural networks, i.e., networks with floating-point parameters and operations. We first observe that for floating-point neural networks to represent all functions from floating-point vectors to floating-point vectors, it is necessary to distinguish different inputs: the first layer of a network should be able to generate different outputs for different inputs. We also prove that such distinguishability is sufficient, along with mild conditions on activation functions. Our result shows that with practical activation functions, floating-point neural networks can represent floating-point functions from a wide domain to all finite or infinite floats. For example, the domain is all finite floats for Sigmoid and tanh, and it is all finite floats of magnitude less than 1/8 times the largest float for ReLU, ELU, SeLU, GELU, Swish, Mish, and sin.

The exact minimum width that allows for universal approximation of unbounded-depth networks is known only for ReLU and its variants. In this work, we study the minimum width of networks using general activation functions. Specifically, we focus on squashable functions that can approximate the identity function and binary step function by alternatively composing with affine transformations. We show that for networks using a squashable activation function to universally approximate $L^p$ functions from $[0,1]^{d_x}$ to $\mathbb R^{d_y}$, the minimum width is $\max\\{d_x,d_y,2\\}$ unless $d_x=d_y=1$; the same bound holds for $d_x=d_y=1$ if the activation function is monotone. We then provide sufficient conditions for squashability and show that all non-affine analytic functions and a class of piecewise functions are squashable, i.e., our minimum width result holds for those general classes of activation functions.

Recently, there has been a growing focus on determining the minimum width requirements for achieving the universal approximation property in deep, narrow Multi-Layer Perceptrons (MLPs). Among these challenges, one particularly challenging task is approximating a continuous function under the uniform norm, as indicated by the significant disparity between its lower and upper bounds. To address this problem, we propose a framework that simplifies finding the minimum width for deep, narrow MLPs into determining a purely geometrical function denoted as $w(d_x, d_y)$. This function relies solely on the input and output dimensions, represented as $d_x$ and $d_y$, respectively. To achieve this, we first demonstrate that deep, narrow MLPs, when provided with a small additional width, can approximate any $C^2$-diffeomorphism. Subsequently, using this result, we prove that $w(d_x, d_y)$ equates to the optimal minimum width required for deep, narrow MLPs to achieve universality. By employing the aforementioned framework and the Whitney embedding theorem, we provide an upper bound for the minimum width, given by $\operatorname{max}(2d_x+1, d_y) + \alpha(\sigma)$, where $0 \leq \alpha(\sigma) \leq 2$ represents a constant depending explicitly on the activation function. Furthermore, we provide novel optimal values for the minimum width in several settings, including $w(2,2)=w(2,3) = 4$.

In this paper, we investigate the universal approximation property of deep, narrow multilayer perceptrons (MLPs) for $C^1$ functions under the Sobolev norm, specifically the $W^{1, \infty}$ norm. Although the optimal width of deep, narrow MLPs for approximating continuous functions has been extensively studied, significantly less is known about the corresponding optimal width for $C^1$ functions. We demonstrate that \textit{the optimal width} can be determined in a wide range of cases within the $C^1$ setting. Our approach consists of two main steps. First, leveraging control theory, we show that any diffeomorphism can be approximated by deep, narrow MLPs. Second, using the Borsuk-Ulam theorem and various results from differential geometry, we prove that the optimal width for approximating arbitrary $C^1$ functions via diffeomorphisms is $\min(n + m, \max(2n + 1, m))$ in certain cases, including $(n,m) = (8,8)$ and $(16,8)$, where $n$ and $m$ denote the input and output dimensions, respectively. Our results apply to a broad class of activation functions.

Combinatorial generalization refers to the ability to collect and assemble various attributes from diverse data to generate novel unexperienced data. This ability is considered a necessary passing point for achieving human-level intelligence. To achieve this ability, previous unsupervised approaches mainly focused on learning the disentangled representation, such as the variational autoencoder. However, recent studies discovered that the disentangled representation is insufficient for combinatorial generalization and is not even correlated. In this regard, we propose a novel framework for data generation that can robustly generalize under these distribution shift situations. Instead of representing each data, our model discovers the fundamental transformation between a pair of data by simulating a group action. To test the combinatorial generalizability, we evaluated our model in two settings: Recombination-to-Element and Recombination-to-Range. The experiments demonstrated that our method has quantitatively and qualitatively superior generalizability and generates better images than traditional models.

A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep narrow networks with bounded width and arbitrary depth are more effective than wide shallow networks with arbitrary width and bounded depth in practice; however, the universal approximation theorem for deep narrow structures has yet to be extensively studied. In this study, we prove the universality of deep narrow RNNs and show that the upper bound of the minimum width for universality can be independent of the length of the data. Specifically, we show a deep RNN with ReLU activation can approximate any continuous function or $L^p$ function with the widths $d_x+d_y+3$ and $\max\{d_x+1,d_y\}$, respectively, where the target function maps a finite sequence of vectors in $\mathbb{R}^{d_x}$ to a finite sequence of vectors in $\mathbb{R}^{d_y}$. We also compute the additional width required if the activation function is sigmoid or more. In addition, we prove the universality of other recurrent networks, such as bidirectional RNNs. Bridging a multi-layer perceptron and an RNN, our theory and technique can shed light on further research on deep RNNs.