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Thanks to recent advancements in machine learning, vector-based methods have been adopted in many modern information retrieval (IR) systems. While showing promising retrieval performance, these approaches typically fail to explain why a…

Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are…

We present a general class of machine learning algorithms called parametric matrix models. In contrast with most existing machine learning models that imitate the biology of neurons, parametric matrix models use matrix equations that…

Dynamic arrays, also referred to as vectors, are fundamental data structures used in many programs. Modeling their semantics efficiently is crucial when reasoning about such programs. The theory of arrays is widely supported but is not…

Machine learning on graphs is an important and ubiquitous task with applications ranging from drug design to friendship recommendation in social networks. The primary challenge in this domain is finding a way to represent, or encode, graph…

A subgraph is constructed by using a subset of vertices and edges of a given graph. There exist many graph properties that are hereditary for subgraphs. Hence, researchers from different communities have paid a great deal of attention in…

In ranking problems, the goal is to learn a ranking function from labeled pairs of input points. In this paper, we consider the related comparison problem, where the label indicates which element of the pair is better, or if there is no…

Matrix--vector algorithms, particularly Krylov subspace methods, are widely viewed as the most effective algorithms for solving large systems of linear equations. This paper establishes lower bounds on the worst-case number of…

We study the problem of recovering the structure underlying large Gaussian graphical models or, more generally, partial correlation graphs. In high-dimensional problems it is often too costly to store the entire sample covariance matrix. We…

Finding a Hadamard matrix (H-matrix) among the set of all binary matrices of corresponding order is a hard problem, which potentially can be solved by quantum computing. We propose a method to formulate the Hamiltonian of finding H-matrix…

Graphs are ubiquitous data structures for representing interactions between entities. With an emphasis on the use of graphs to represent chemical molecules, we explore the task of learning to generate graphs that conform to a distribution…

Graph embedding provides a feasible methodology to conduct pattern classification for graph-structured data by mapping each data into the vectorial space. Various pioneering works are essentially coding method that concentrates on a…

The modern data analyst must cope with data encoded in various forms, vectors, matrices, strings, graphs, or more. Consequently, statistical and machine learning models tailored to different data encodings are important. We focus on data…

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…

The application of the max-algebra to describe queueing systems by both linear scalar and vector equations is discussed. It is shown that these equations may be handled using ordinary algebraic manipulations. Examples of solving the…

Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially…

Consider the following generalization of the classic binary search problem: A searcher is required to find a hidden target vertex $x$ in a graph $G$. To do so, they iteratively perform queries to an oracle, each about a chosen vertex $v$.…

Let $\boldsymbol{\Lambda}\,(=\mathbb{F}^{n^{3}})$, where $\mathbb{F}$ is a field with $|\mathbb{F}|>2$, be the space of structure vectors of algebras having the $n$-dimensional $\mathbb{F}$-space $V$ as the underlying vector space. Also let…

A classical vector retrieval problem typically considers a \emph{single} query embedding vector as input and retrieves the most similar embedding vectors from a vector database. However, complex reasoning and retrieval tasks frequently…

Let $\mathcal{T}$ be a set of $n$ flat (planar) semi-algebraic regions in $\mathbb{R}^3$ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess $\mathcal{T}$ into a data structure so that for a query…