Related papers: On linear balancing sets
The set of linear structures of most known balanced Boolean functions is nontrivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any APN function in even dimension…
A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to be…
Complex classifiers may exhibit "embarassing" failures in cases where humans can easily provide a justified classification. Avoiding such failures is obviously of key importance. In this work, we focus on one such setting, where a label is…
In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…
In this paper we consider two pointsets in $\mathrm{PG}(2,q^n)$ arising from a linear set $L$ of rank $n$ contained in a line of $\mathrm{PG}(2,q^n)$: the first one is a linear blocking set of R\'edei type, the second one extends the…
In this paper, we continue the study of linear sets with complementary weights. We find criteria to determine the set of points of any fixed weight and use this to present particular linear sets with few points of weight more than one. We…
We investigate the scattered palindromic subwords in a finite word. We start by characterizing the words with the least number of scattered palindromic subwords. Then, we give an upper bound for the total number of palindromic subwords in a…
Large language models (LLMs), typically designed as a function of next-word prediction, have excelled across extensive NLP tasks. Despite the generality, next-word prediction is often not an efficient formulation for many of the tasks,…
A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to…
We propose new static word embeddings optimised for sentence semantic representation. We first extract word embeddings from a pre-trained Sentence Transformer, and improve them with sentence-level principal component analysis, followed by…
We present a general framework for balancing expressions (terms) in form of so called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations…
We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of…
Embeddings of word structures into matrix semigroups provide a natural bridge between combinatorics on words and linear algebra. However, low-dimensional matrix semigroups impose strong structural restrictions on possible embeddings.…
Word embeddings are a powerful natural language processing technique, but they are extremely difficult to interpret. To enable interpretable NLP models, we create vectors where each dimension is inherently interpretable. By inherently…
General-purpose language models are trained to produce varied natural language outputs, but for some tasks, like annotation or classification, we need more specific output formats. LLM systems increasingly support structured output, which…
We prove a strengthening of Chang's lemma for subsets of $\mathbb F_p^n$. The classical conclusion that the large spectrum is contained in a subspace of dimension at most $2\varepsilon^{-2}\log(1/\alpha)$ is refined to show that every…
Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and…
Supervised term weighting could improve the performance of text categorization. A way proven to be effective is to give more weight to terms with more imbalanced distributions across categories. This paper shows that supervised term…
A Boolean function of n bits is balanced if it takes the value 1 with probability 1/2. We exhibit a balanced Boolean function with a randomized evaluation procedure (with probability 0 of making a mistake) so that on uniformly random…
We study the parameterized problem of satisfying ``almost all'' constraints of a given formula $F$ over a fixed, finite Boolean constraint language $\Gamma$, with or without weights. More precisely, for each finite Boolean constraint…