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We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
We outline the construction of a molecular system that could, in principle, implement a thermodynamically reversible Universal Turing Machine (UTM). By proposing a concrete-albeit idealised-design and operational protocol, we reveal…
We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves the 18th order of convergence by using only seven matrix multiplication per iteration loop. This is the record high…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
Matrix inversion problems are often encountered in experimental physics, and in particular in high-energy particle physics, under the name of unfolding. The true spectrum of a physical quantity is deformed by the presence of a detector,…
A remarkable new definition of a self-delimiting universal Turing machine is presented that is easy to program and runs very quickly. This provides a new foundation for algorithmic information theory. This new universal Turing machine is…
In recent years, a new kind of accelerated hardware has gained popularity in the Artificial Intelligence (AI) and Machine Learning (ML) communities which enables extremely high-performance tensor contractions in reduced precision for deep…
Matrix completion constantly receives tremendous attention from many research fields. It is commonly applied for recommender systems such as movie ratings, computer vision such as image reconstruction or completion, multi-task learning such…
This is the second part of a two-paper series on generalized inverses that minimize matrix norms. In Part II we focus on generalized inverses that are minimizers of entrywise p norms whose main representative is the sparse pseudoinverse for…
Deep learning still has drawbacks in terms of trustworthiness, which describes a comprehensible, fair, safe, and reliable method. To mitigate the potential risk of AI, clear obligations associated to trustworthiness have been proposed via…
We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular,…
This work explores fundamental modeling and algorithmic issues arising in the well-established MapReduce framework. First, we formally specify a computational model for MapReduce which captures the functional flavor of the paradigm by…
By considering a discrete tape where each cell corresponds to an integer, thus to a possible sum, a pseudo-polynomial solution can be given to subset sum problem, which is an NP-complete problem and a cornerstone application for this study,…
Image reconstruction in Magnetic Resonance Imaging (MRI) is fundamentally a linear inverse problem, such that the image can be recovered via explicit pseudoinversion of the encoding matrix by solving $\textbf{data} = \textbf{Encode} \times…
In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main…
Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation…
The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and…
Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are…
We propose quantum algorithms, purely quantum in nature, for calculating the determinant and inverse of an $(N-1)\times (N-1)$ matrix (depth is $O(N^2\log N)$) which is a simple modification of the algorithm for calculating the determinant…
Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation"…