Related papers: Symmetry Breaking in Symmetric Tensor Decompositio…
We consider the nonconvex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank-one terms. Use is made of the rich symmetry structure to construct infinite families of critical points…
We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on…
Critical points of an invariant function may or may not be symmetric. We prove, however, that if a symmetric critical point exists, those adjacent to it are generically symmetry breaking. This mathematical mechanism is shown to carry…
This paper discusses the problem of symmetric tensor decomposition on a given variety $X$: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in $X$. In this paper, we first study geometric and algebraic…
The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able…
As science and engineering have become increasingly data-driven, the role of optimization has expanded to touch almost every stage of the data analysis pipeline, from signal and data acquisition to modeling and prediction. The optimization…
In this work, we consider the optimization formulation for symmetric tensor decomposition recently introduced in the Subspace Power Method (SPM) of Kileel and Pereira. Unlike popular alternative functionals for tensor decomposition, the SPM…
Higher-order tensors are becoming prevalent in many scientific areas such as computer vision, social network analysis, data mining and neuroscience. Traditional tensor decomposition approaches face three major challenges: model selecting,…
We introduce the ``skew apolarity lemma'' and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in {$\bigwedge^dV_{\mathbb{C}}$ with $d\leq 3$ and $\dim V_{\mathbb{C}} \leq 8$}. New algorithms to…
Tensor decomposition of high-dimensional data often struggles to capture semantically or physically meaningful structures, particularly when relying on reconstruction objectives and fixed-rank constraints. We introduce a no-rank tensor…
Using symmetry as an inductive bias in deep learning has been proven to be a principled approach for sample-efficient model design. However, the relationship between symmetry and the imperative for equivariance in neural networks is not…
The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. The theory of quantum entanglement is currently leading to a paradigm shift in understanding…
Symmetry is an important feature of many constraint programs. We show that any problem symmetry acting on a set of symmetry breaking constraints can be used to break symmetry. Different symmetries pick out different solutions in each…
Artificial Neural Networks (ANN) comprise important symmetry properties, which can influence the performance of Monte Carlo methods in Neuroevolution. The problem of the symmetries is also known as the competing conventions problem or…
We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points.…
Symmetry plays a crucial role in understanding the properties of mathematical structures and optimization problems. Recent work has explored this phenomenon in the context of neural networks, where the loss function is invariant under…
One of the main issues in computing a tensor decomposition is how to choose the number of rank-one components, since there is no finite algorithms for determining the rank of a tensor. A commonly used approach for this purpose is to find a…
We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing…
Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of…
The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex analysis and optimization over spaces of tensors is now gaining…