Related papers: Multivariate growth and cogrowth
Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $\gamma_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that…
In this paper, a new theory is developed for first-order stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal…
While large models achieve impressive results, their learning dynamics are far from understood. Many domains of interest, such as natural language syntax, coding languages, arithmetic problems, are captured by context-free grammars (CFGs).…
Consider the set of those binary words with no non-empty factors of the form $xxx^R$. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the…
Neural dependency parsing has achieved remarkable performance for low resource morphologically rich languages. It has also been well-studied that morphologically rich languages exhibit relatively free word order. This prompts a fundamental…
We present new large-scale algorithms for fitting a subgradient regularized multivariate convex regression function to $n$ samples in $d$ dimensions -- a key problem in shape constrained nonparametric regression with applications in…
In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an N-series of subgroups. The asymptotics of the Poincare series of this algebra give estimates on the growth of the group…
Accelerating large language model pre-training is a critical issue in present research. In this paper, we focus on speeding up pre-training by progressively growing from a small Transformer structure to a large one. There are two main…
Let $A$ be an algebra with fixed set of generators $a_1,\dots,a_s$. $V_A(n)$ be dimension of the space, generated by worlds of length $\le n$ over $a_i$, $T_A(n)=V_A(n)-V_A(n-1)$. If $T_A(n)<\mbox{Const}$, algebra $A$ is a {\it boundary…
A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all…
We propose a new stochastic L-BFGS algorithm and prove a linear convergence rate for strongly convex and smooth functions. Our algorithm draws heavily from a recent stochastic variant of L-BFGS proposed in Byrd et al. (2014) as well as a…
Let $\widehat{\mathcal {S}}_g^{\alpha, \beta}(\mathbb{B}^n)$ be a subclass of normalized biholomorphic mappings defined on the unit ball in $\mathbb{C}^n,$ which is closely related to the starlike mappings. Firstly, we obtain the growth…
In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with…
The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for…
The curse of multilinguality phenomenon is a fundamental problem of multilingual Large Language Models (LLMs), where the competition between massive languages results in inferior performance. It mainly comes from limited capacity and…
It is argued that the present log-normal distribution of language sizes is, to a large extent, a consequence of demographic dynamics within the population of speakers of each language. A two-parameter stochastic multiplicative process is…
We present a new model for the cross-covariance between galaxy redshift-space distortions and peculiar velocities. We combine this with the auto-covariance models of both probes in a fully self-consistent, maximum-likelihood method,…
Viable models of modified gravity designed to produce cosmic acceleration at the current epoch, closely mimic the $\Lambda$CDM model at the level of background cosmology. However, this degeneracy is generically broken at the level of linear…
This study focuses on improving the performance of lightweight Large Language Models (LLMs) in mathematical reasoning tasks. We introduce a novel method for measuring mathematical logic similarity and design an automatic screening mechanism…
Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks the…