arXiv subjects

Valentin Abadie

Publications and source records attributed to Valentin Abadie.

3 recordsLinked to original sources

arXiv2026

Recurrent neural networks approximate continuous functions

Classical approximation theorems ask for a new neural network whenever the target accuracy is improved. This paper studies the opposite possibility: can the network be chosen once and for all, and can accuracy be bought only by letting it run longer? We prove that this is possible for every continuous function on [-1,1]. More precisely, each such function is uniformly approximated by the time evolution of a single ReLU recurrent neural network with fixed weights and fixed hidden dimension. The mechanism behind the construction is a new intermediate model, the Turing machine with neural units (TMNU). This model retains the algorithmic freedom needed to implement polynomial approximation schemes, while remaining rigid enough to be simulated by RNNs with explicit bounds on hidden dimension and weight magnitude. The resulting convergence rates reflect the underlying polynomial approximation rates. We complement the construction with minimax lower bounds showing that runtime is not merely a proof artifact, but an unavoidable resource in this fixed-network approximation paradigm.

cs.LG↗

arXiv2025

A Quantifier-Reversal Approximation Paradigm for Recurrent Neural Networks

Classical neural network approximation results take the form: for every function $f$ and every error tolerance $\epsilon > 0$, one constructs a neural network whose architecture and weights depend on $\epsilon$. This paper introduces a fundamentally different approximation paradigm that reverses this quantifier order. For each target function $f$, we construct a single recurrent neural network (RNN) with fixed topology and fixed weights that approximates $f$ to within any prescribed tolerance $\epsilon > 0$ when run for sufficiently many time steps. The key mechanism enabling this quantifier reversal is temporal computation combined with weight sharing: rather than increasing network depth, the approximation error is reduced solely by running the RNN longer. This yields exponentially decaying approximation error as a function of runtime while requiring storage of only a small, fixed set of weights. Such architectures are appealing for hardware implementations where memory is scarce and runtime is comparatively inexpensive. To initiate the systematic development of this novel approximation paradigm, we focus on univariate polynomials. Our RNN constructions emulate the structural calculus underlying deep feed-forward ReLU network approximation theory -- parallelization, linear combinations, affine transformations, and, most importantly, a clocked mechanism that realizes function composition within a single recurrent architecture. The resulting RNNs have size independent of the error tolerance $\epsilon$ and hidden-state dimension linear in the degree of the polynomial.

cs.NE↗

arXiv2024

Algorithmic complexity of $\beta$-expansions and application to A/D conversion

We establish diverse relationships between the algorithmic (Kolmogorov) complexity of the prefixes of any binary expansion and $\beta$-expansions. These relationships allow to develop intuitions on the complexity behavior of $\beta$-expansions, and raise problems related to compressibility of binary sequences generated in the context of A/D conversion relying on $\beta$-expansions. Our last contribution is to solve these problems.

cs.IT↗