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The self-attention mechanism, a cornerstone of Transformer-based state-of-the-art deep learning architectures, is largely heuristic-driven and fundamentally challenging to interpret. Establishing a robust theoretical foundation to explain…

The computational complexity of reasoning within the Dempster-Shafer theory of evidence is one of the main points of criticism this formalism has to face. To overcome this difficulty various approximation algorithms have been suggested that…

The aim of this work is to develop a fast algorithm for approximating the matrix function $f(A)$ of a square matrix $A$ that is symmetric and has hierarchically semiseparable (HSS) structure. Appearing in a wide variety of applications,…

We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…

A common method to study deep learning systems is to use simplified model representations--for example, using singular value decomposition to visualize the model's hidden states in a lower dimensional space. This approach assumes that the…

A wide range of symbolic analysis and optimization problems can be formalized using polyhedra. Sub-classes of polyhedra, also known as sub-polyhedral domains, are sought for their lower space and time complexity. We introduce the Strided…

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous…

Parameterized complexity theory has lead to a wide range of algorithmic breakthroughs within the last decades, but the practicability of these methods for real-world problems is still not well understood. We investigate the practicability…

Distance metric learning is a branch of machine learning that aims to learn distances from the data, which enhances the performance of similarity-based algorithms. This tutorial provides a theoretical background and foundations on this…

This paper proposes a new model for extracting an interpretable sentence embedding by introducing self-attention. Instead of using a vector, we use a 2-D matrix to represent the embedding, with each row of the matrix attending on a…

Semidefinite programs (SDP) are one of the most versatile frameworks in numerical optimization, serving as generalizations of many conic programs and as relaxations of NP-hard combinatorial problems. Their main drawback is their…

Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality…

With the growing adoption of privacy-preserving machine learning algorithms, such as Differentially Private Stochastic Gradient Descent (DP-SGD), training or fine-tuning models on private datasets has become increasingly prevalent. This…

Decomposition is a fundamental skill in algorithmic programming, requiring learners to break down complex problems into smaller, manageable parts. However, current self-study methods, such as browsing reference solutions or using LLM…

In semidefinite programming (SDP), a number of pre-processing techniques have been developed including chordal-completion procedures, which reduce the dimension of individual constraints by exploiting sparsity therein, and facial reduction,…

As robots and other intelligent agents move from simple environments and problems to more complex, unstructured settings, manually programming their behavior has become increasingly challenging and expensive. Often, it is easier for a…

This paper explores the relationship between matrix factorizations and linear matrix equations. It shows that every matrix factorization defines two hidden projectors, one for the column space and one for the row space of a matrix, and how…

Program decomposition is essential for developing maintainable and efficient software, yet it remains a challenging skill to teach and learn in introductory programming courses. What does program decomposition for procedural CS1 programs…

A major focus in designing methods for learning distributions defined on manifolds is to alleviate the need to implicitly learn the manifold so that learning can concentrate on the data distribution within the manifold. However,…

In the arena of privacy-preserving machine learning, differentially private stochastic gradient descent (DP-SGD) has outstripped the objective perturbation mechanism in popularity and interest. Though unrivaled in versatility, DP-SGD…