Related papers: Dimer lambda_3 = .453 +/- .001 and Some Other Very…
We propose three kinds of explicit formulas for the elliptic lambda function by the elliptic modular function. Further, we derive incredible cubic identities as a corollary of our explicit formulas and evaluate some singular values of the…
To help evaluate and understand the latent capabilities of language models, this paper introduces an approach using optimized input embeddings, or 'soft prompts,' as a metric of conditional distance between a model and a target behavior.…
This paper investigates the {\em nonasymptotic} properties of Bayes procedures for estimating an unknown distribution from $n$ i.i.d.\ observations. We assume that the prior is supported by a model $(\scr{S},h)$ (where $h$ denotes the…
We present a type inference algorithm for lambda-terms in Elementary Affine Logic using linear constraints. We prove that the algorithm is correct and complete.
A series transformation idea inspired by a formula of R. W. Gosper and some asymptotic expansions for the central binomial coefficients leads us to new accurate approximations for the Gamma function.
The remarkable advancements in Multimodal Large Language Models (MLLMs) have not rendered them immune to challenges, particularly in the context of handling deceptive information in prompts, thus producing hallucinated responses under such…
The Laplace approximation is a popular method for constructing a Gaussian approximation to the Bayesian posterior and thereby approximating the posterior mean and variance. But approximation quality is a concern. One might consider using…
Large Language Models (LLMs), particularly smaller variants, still struggle with complex reasoning tasks. While inference-time prompting can guide reasoning, existing methods often rely on sequential queries. Ensemble approaches offer a…
Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the…
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series which involves a product of Riemann zeta-functions of a special form.
Let $a_1$, $a_2$, and $a_3$ be distinct reduced residues modulo $q$ satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \pmod q$. We conditionally derive an asymptotic formula, with an error term that has a power savings in $q$, for…
Uncertainty estimates help to identify ambiguous, novel, or anomalous inputs, but the reliable quantification of uncertainty has proven to be challenging for modern deep networks. In order to improve uncertainty estimation, we propose…
Bayesian sequence prediction is a simple technique for predicting future symbols sampled from an unknown measure on infinite sequences over a countable alphabet. While strong bounds on the expected cumulative error are known, there are only…
The problem of f-divergence estimation is important in the fields of machine learning, information theory, and statistics. While several nonparametric divergence estimators exist, relatively few have known convergence properties. In…
In this paper, new probability bounds are derived for algebraic lattice codes. This is done by using the Dedekind zeta functions of the algebraic number fields involved in the lattice constructions. In particular, it is shown how to upper…
In this paper, we derive an asymptotic closed--form expression for the error bound on extrapolation of doubly selective mobile MIMO wireless channels. The bound shows the relationship between the prediction error and system design…
The Keiper/Li constants $\{\lambda_n\}_{n=1,2,\ldots}$ are asymptotically ($n \to \infty$) sensitive to the Riemann Hypothesis, but highly elusive analytically and difficult to compute numerically. We present quite explicit variant…
Estimation and inference in dynamic discrete choice models often relies on approximation to lower the computational burden of dynamic programming. Unfortunately, the use of approximation can impart substantial bias in estimation and results…
We make two contributions to the problem of estimating the $L_1$ calibration error of a binary classifier from a finite dataset. First, we provide an upper bound for any classifier where the calibration function has bounded variation.…
Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum…