arXiv subjects

Matan Mizrachi

Publications and source records attributed to Matan Mizrachi.

4 recordsLinked to original sources

arXiv2025

Bi-Lipschitz Ansatz for Anti-Symmetric Functions

Motivated by applications to the simulation of quantum many-body systems by neural networks, researchers have suggested several models which are antisymmetric by construction, and can approximate all antisymmetric functions. However, these works either require very high computational complexity to attain universal approximation, or suffer from discontinuities. In this paper, we introduce two antisymmetric ansatzes which do not suffer from these disadvantages. The first is based on a bi-Lipschitz embedding with respect to a naturally defined metric. The second is a modular anti-symmetrizing projection framework based on the frame-averaging methodology. Both approaches yield continuous antisymmetric models which attain universal approximation guarantees with a polynomial complexity in problem size. Moreover, for both approaches, we obtain quantitative approximation results that bound the number of parameters required to approximate Lipschitz antisymmetric functions to a given accuracy $\epsilon$. We also provide preliminary experimental evidence suggesting improved performance in learning antisymmetric functions.

cs.LG↗

arXiv2022

On Unimodality of Independence Polynomials of Trees

An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $α{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for the sequence of numbers of independent sets of each size. In other words, the $k$-th coefficient of the independence polynomial equals the number of independent sets comprised of $k$ vertices. For instance, the degree of the independence polynomial of the graph $G$ is equal to $α{(G)}$. In 1987, Alavi, Malde, Schwenk, and Erd{ö}s conjectured that the independence polynomial of a tree is unimodal. In what follows, we provide support to this assertion considering trees with up to $20$ vertices. Moreover, we show that the corresponding independence polynomials are log-concave and, consequently, unimodal. The algorithm computing the independence polynomial of a given tree makes use of a database of non-isomorphic unlabeled trees to prevent repeated computations.

cs.DM↗

arXiv2022

A Linear Algorithm for Computing Independence Polynomials of Trees

An independent set in a graph is a set of pairwise non-adjacent vertices. Let $α(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x) = \sum_{k=0}^{α(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{α(G)}}x^{α(G)}, \] where $s_k$ denotes the number of independent sets of cardinality $k$ in the graph $G$. A comprehensive survey on the subject is due to Levit and Mandrescu, where some recursive formulas are allowing to calculate the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. Yosef, Mizrachi, and Kadrawi developed an efficient way for computing the independence polynomials of trees with $n$ vertices, such that a database containing all of the independence polynomials of all the trees with up to $n-1$ vertices is required. This approach is not suitable for big trees, as an extensive database is needed. On the other hand, using dynamic programming, it is possible to develop an efficient algorithm that prevents repeated calculations. In summary, our dynamic programming algorithm runs over a tree in linear time and does not depend on a database.

math.CO↗

arXiv2021

On Bi-gram Graph Attributes

We propose a new approach to text semantic analysis and general corpus analysis using, as termed in this article, a "bi-gram graph" representation of a corpus. The different attributes derived from graph theory are measured and analyzed as unique insights or against other corpus graphs. We observe a vast domain of tools and algorithms that can be developed on top of the graph representation; creating such a graph proves to be computationally cheap, and much of the heavy lifting is achieved via basic graph calculations. Furthermore, we showcase the different use-cases for the bi-gram graphs and how scalable it proves to be when dealing with large datasets.

cs.LG↗