Related papers: Analyzing the Nuances of Transformers' Polynomial …
In this paper, we propose a new and simple approach to the approximation algorithms that are modified and improved from our published results. The computational and graphical examples are presented with the aid of Maple procedures.
Transformers achieve unrivalled performance in modelling language, but remain inefficient in terms of memory and time complexity. A possible remedy is to reduce the sequence length in the intermediate layers by pooling fixed-length segments…
Optimizing and certifying the positivity of polynomials are fundamental primitives across mathematics and engineering applications, from dynamical systems to operations research. However, solving these problems in practice requires large…
Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial…
This paper presents a novel algorithm for constructing a sum-of-squares (SOS) decomposition for positive semi-definite polynomials with rational coefficients. Unlike previous methods that typically yield SOS decompositions with…
The ability to learn polynomials and generalize out-of-distribution is essential for simulation metamodels in many disciplines of engineering, where the time step updates are described by polynomials. While feed forward neural networks can…
We show that a constant number of self-attention layers can efficiently simulate, and be simulated by, a constant number of communication rounds of Massively Parallel Computation. As a consequence, we show that logarithmic depth is…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a…
Transformers have demonstrated remarkable capabilities in multi-step reasoning tasks. However, understandings of the underlying mechanisms by which they acquire these abilities through training remain limited, particularly from a…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be…
As modeling and visualization applications proliferate, there arises a need to simplify large polygonal models at interactive rates. Unfortunately existing polygon mesh simplification algorithms are not well suited for this task because…
The automated generation of exercises may substantially reduce the time educators devote to manual exercise design. A major obstacle to the integration of such automation into teaching practice, however, lies in the ability to control the…
The synthesis of string transformation programs from input-output examples utilizes various techniques, all based on an inductive bias that comprises a restricted set of basic operators to be combined. A new algorithm, Transduce, is…
We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…
This document is a companion for the Maple program \textbf{Summing a polynomial function over integral points of a polygon}. It contains two parts. First, we see what this programs does. In the second part, we briefly recall the…
Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an…