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In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on…
Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph structured linear systems, in the general setting, the bit complexity of solving an $n \times n$ linear…
In multi-objective optimization, computing the entire non-dominated set (also known as the Pareto front or the Pareto frontier) is often intractable. However, for any multiplicative factor greater than one, an approximation set can be…
We present a faster interior-point method for optimizing sum-of-squares (SOS) polynomials, which are a central tool in polynomial optimization and capture convex programming in the Lasserre hierarchy. Let $p = \sum_i q^2_i$ be an…
Obeying constraints imposed by classical physics, we give optimal fine-grained algorithms for matrix multiplication and problems involving graphs and mazes, where all calculations are done in 3-dimensional space. We assume that whatever the…
We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst $n$ given points in $d$ dimensions. Previously, the best algorithms known have running time…
Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, and this work was performed by individuals from many…
Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as LCP (largest common point set…
In this paper, we obtain a number of new simple pseudo-polynomial time algorithms on the well-known knapsack problem, focusing on the running time dependency on the number of items $n$, the maximum item weight $w_\mathrm{max}$, and the…
A fundamental problem in shape matching and geometric similarity is computing the maximum area overlap between two polygons under translation. For general simple polygons, the best-known algorithm runs in $O((nm)^2 \log(nm))$ time [Mount,…
The matrix-completion problem has attracted a lot of attention, largely as a result of the celebrated Netflix competition. Two popular approaches for solving the problem are nuclear-norm-regularized matrix approximation (Candes and Tao,…
Solving linear systems of equations is an important problem in science and engineering. Many quantum algorithms, such as the Harrow-Hassidim-Lloyd (HHL) algorithm (for quantum-gate computers) and the box algorithm (for quantum-annealing…
Gradient-based (a.k.a. `first order') optimization algorithms are routinely used to solve large scale non-convex problems. Yet, it is generally hard to predict their effectiveness. In order to gain insight into this question, we revisit the…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
This paper presents and analyzes the first matrix optimization model which allows general coordinate and spectral constraints. The breadth of problems our model covers is exemplified by a lengthy list of examples from the literature,…
The control and sensing of large-scale systems results in combinatorial problems not only for sensor and actuator placement but also for scheduling or observability/controllability. Such combinatorial constraints in system design and…
We consider the problem of reconstructing a rank-$k$ $n \times n$ matrix $M$ from a sampling of its entries. Under a certain incoherence assumption on $M$ and for the case when both the rank and the condition number of $M$ are bounded, it…
What is the value of input information in solving linear programming? The celebrated ellipsoid algorithm tells us that the full information of input constraints is not necessary; the algorithm works as long as there exists an oracle that,…
Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these…