Related papers: On word reversing in braid groups
Variance reduction is a family of powerful mechanisms for stochastic optimization that appears to be helpful in many machine learning tasks. It is based on estimating the exact gradient with some recursive sequences. Previously, many papers…
Statistical and probabilistic characteristics of locally free group with growing number of generators are defined and their application to statistics of braid groups is given.
We study a norm for structured sparsity which leads to sparse linear predictors whose supports are unions of prede ned overlapping groups of variables. We call the obtained formulation latent group Lasso, since it is based on applying the…
We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor…
In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators…
We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an…
Recent work has explored methods for learning continuous vector space word representations reflecting the underlying semantics of words. Simple vector space arithmetic using cosine distances has been shown to capture certain types of…
We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class.…
This paper is devoted to the proof of a structural theorem, concerning certain homomorphic images of Artin braid group on $n$ strands in finite symmetric groups. It is shown that any one of these permutation groups is an extension of the…
We introduce a notion of "freely braided element" for simply laced Coxeter groups. We show that an arbitrary group element $w$ has at most $2^{N(w)}$ commutation classes of reduced expressions, where $N(w)$ is a certain statistic defined in…
Many algorithms for maximizing a monotone submodular function subject to a knapsack constraint rely on the natural greedy heuristic. We present a novel refined analysis of this greedy heuristic which enables us to: $(1)$ reduce the…
We establish several results on the word problem for just infinite groups. First, for finitely generated just infinite groups we show that the word problem is uniformly decidable for presentations with recursively enumerable sets of…
In this paper, we focus our attention on the connections between the braid group and the Nielsen fixed point theory. A new forcing relation between braids is introduced, and shown that it can be fulfilled by using Nielsen fixed point…
We prove new upper bounds for the length of laws that hold for all groups of size at most $n$ -- improving on previous results of Bou-Rabee and Kassabov-Matucci. The methods make use of the classification of finite simple groups. Stronger…
We prove that the Garside length a braid is equal to a winding-number type invariant of the curve diagram of the braid.
The higher Bruhat orders are partial orders that generalize the weak order on the symmetric group $S_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in $S_n$, where covering…
It has been shown recently that the Eigenvector Method may lead to strong rank reversal in group decision making, that is, the alternative with the highest priority according to all individual vectors may lose its position when evaluations…
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…
In natural language processing, a lot of the tasks are successfully solved with recurrent neural networks, but such models have a huge number of parameters. The majority of these parameters are often concentrated in the embedding layer,…
We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure.…