Related papers: Improved method for finding optimal formulae for b…
In this paper, we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is…
Optimal control of bilinear systems has been a well-studied subject in the area of mathematical control. However, techniques for solving emerging optimal control problems involving an ensemble of structurally identical bilinear systems are…
Biclustering structures in data matrices were first formalized in a seminal paper by John Hartigan (1972) where one seeks to cluster cases and variables simultaneously. Such structures are also prevalent in block modeling of networks. In…
We develop an optimization framework for identifying ideal Mixed Binary Linear Programs (MBLP) which is linear when using known input data and nonconvex quadratic over parametric input data. These techniques are applied to various…
We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…
In this paper, we propose novel algorithms for inferring the Maximum a Posteriori (MAP) solution of discrete pairwise random field models under multiple constraints. We show how this constrained discrete optimization problem can be…
Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose…
We propose an exact iterative algorithm for minimization of a class of continuous cell-wise linear convex functions on a hyperplane arrangement. Our particular setup is motivated by evaluation of so-called rank estimators used in robust…
Computing an optimal classification tree that provably maximizes training performance within a given size limit, is NP-hard, and in practice, most state-of-the-art methods do not scale beyond computing optimal trees of depth three.…
Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved…
We introduce an efficient and scalable method for density-based multi-material topology optimization, integrating classical mirror descent techniques with point-wise polytopal design constraints. Such constraints arise naturally in this…
The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over…
Understanding how systems built out of modular components can be jointly optimized is an important problem in biology, engineering, and machine learning. The backpropagation algorithm is one such solution and has been instrumental in the…
While the trapezoidal formula can attain exponential convergence when applied to infinite integrals of bilateral rapidly decreasing functions, it is not capable of this in the case of unilateral rapidly decreasing functions. To address this…
Convergence guarantees for optimization over bounded-rank matrices are delicate to obtain because the feasible set is a non-smooth and non-convex algebraic variety. Existing techniques include projected gradient descent, fixed-rank…
We analyze different ways of constructing binary extended formulations of mixed-integer problems with bounded integer variables and compare their relative strength with respect to split cuts. We show that among all binary extended…
We exploit the versatile framework of Riemannian optimization on quotient manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank matrix completion. The underlying search space of fixed-rank matrices is endowed with a…
A line search in gradient-based optimization algorithm solves the problem of determining the optimal learning rate for a given gradient or search direction in a single iteration. For most problems, this is determined by evaluating different…
Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these…
We consider the problem of matrix completion on an $n \times m$ matrix. We introduce the problem of Interpretable Matrix Completion that aims to provide meaningful insights for the low-rank matrix using side information. We show that the…