Related papers: Irredundant Triangular Decomposition
The paper surveys the topic of tensor decompositions in modern machine learning applications. It focuses on three active research topics of significant relevance for the community. After a brief review of consolidated works on multi-way…
Intrinsic image decomposition is an important and long-standing computer vision problem. Given an input image, recovering the physical scene properties is ill-posed. Several physically motivated priors have been used to restrict the…
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be…
Many complex chemical problems encoded in terms of physics-based models become computationally intractable for traditional numerical approaches due to their unfavourable scaling with increasing molecular size. Tensor decomposition…
We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system…
We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new…
One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study $n_1 \times n_2 \times n_3$ third-order tensor completion based on…
This paper presents a decomposition method for solving elliptic boundary value problems in one-dimension. The method is an improvement to an existing technique for approximating elliptic systems. It is demonstrated to be computationally…
A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which…
In this thesis, a new class of algorithms based on Sums of Squares Programming is developed. These allow to reduce a degree-$d$ homogeneous polynomial $T = \sum_{i = 1}^m \langle a_i, X \rangle^d $ to a quadratic form being close to a…
To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition which enforces one factor to belong exactly to a known dictionary. A new…
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of non-separable triangulations in which all vertices have degree at least d, for a certain…
A new class of folded subspace codes for noncoherent network coding is presented. The codes can correct insertions and deletions beyond the unique decoding radius for any code rate $R\in[0,1]$. An efficient interpolation-based decoding…
We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant under permutations and rotations. The key bottleneck is the contraction of a high-dimensional symmetric and sparse tensor…
We propose a new method for computing the eigenvalue decomposition of a dense real normal matrix $A$ through the decomposition of its skew-symmetric part. The method relies on algorithms that are known to be efficiently implemented, such as…
For a finite $\mathbb{Z}$-algebra $R$, i.e., for a ring which is not necessarily associative or unitary, but whose additive group is finitely generated, we construct a decomposition of $R/{\rm Ann}(R)$ into directly indecomposable factors…
This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints…
Image triangulation, the practice of decomposing images into triangles, deliberately employs simplification to create an abstracted representation. While triangulating an image is a relatively simple process, difficulties arise when…
We study problems related to indecomposability of modules over certain local finite dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial…