Wasserstein distance to independence models

Çelik, Türkü Özlüm; Jamneshan, Asgar; Montúfar, Guido; Sturmfels, Bernd; Venturello, Lorenzo

doi:10.1016/j.jsc.2020.10.005

Cited by 16 publications

(8 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As an example, consider the so-called Wasserstein distance induced by a metric on a finite space, which is a main concept in optimal transport and machine learning. The algebraic treatment of this optimization problem has been described recently in [43,44].…”

Section: Polynomial Optimization By Lorenzo Venturellomentioning

confidence: 99%

“…Invariant theory has shown useful in computing maximum likelihood estimates and degrees [19,67,75]. Another approach for tackling 3.2 is to compute the point in a model that minimizes some distance of the sample point from the model [43,74]. Model selection [231,Chapter 17] and identifiability [17,120] are other topics of current importance in algebraic statistics.…”

Section: Algebraic Statistics By Aida Marajmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear algebra and applications

Breiding¹,

Çelik²,

Duff³

et al. 2023

NACO

Self Cite

View full text Add to dashboard Cite

<p style='text-indent:20px;'>We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.</p>

show abstract

Section: Polynomial Optimization By Lorenzo Venturellomentioning

confidence: 99%

Section: Algebraic Statistics By Aida Marajmentioning

confidence: 99%

Nonlinear algebra and applications

Breiding¹,

Çelik²,

Duff³

et al. 2023

NACO

Self Cite

View full text Add to dashboard Cite

show abstract

“…The answer depends on the chosen metric. One might choose the Euclidean distance, a 𝑝-norm [29], or polyhedral norms, such as those arising in optimal transport [15]. In all of these cases, the solution 𝑥 * can be found by solving a system of polynomial equations.…”

Section: Nearest Points On Algebraic Varietiesmentioning

confidence: 99%

“…Another , where 𝐵 is the cube and the crosspolytope respectively. In optimal transport theory, one uses a Wasserstein norm [15] whose unit ball 𝐵 is the polar dual of a Lipschitz polytope.…”

Section: Nearest Points On Algebraic Varietiesmentioning

confidence: 99%

Beyond Linear Algebra

Sturmfels¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Our title challenges the reader to venture beyond linear algebra in designing models and in thinking about numerical algorithms for identifying solutions. This article accompanies the author's lecture at the International Congress of Mathematicians 2022. It covers recent advances in the study of critical point equations in optimization and statistics, and it explores the role of nonlinear algebra in the study of linear PDE with constant coefficients.

show abstract

“…Although neural networks are usually presented in a parameterized form, it is in principle possible to characterize the representable functions implicitily as the solutions to certain constraints. Such descriptions have been particularly fruitful in algebraic statistics, e.g., [34,32,2,12,13]. In the case of fully-connected linear networks, the function space is characterized by rank constraints depending on the layer widths.…”

Section: Introductionmentioning

confidence: 99%

Geometry of Linear Convolutional Networks

Kohn¹,

Merkh²,

Montúfar³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study the family of functions that are represented by a linear convolutional neural network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions represented by fully-connected linear networks form algebraic sets. We observe that the functions represented by LCNs can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the resulting function space. We further study the optimization of an objective function over an LCN, analyzing critical points in function space and in parameter space, and describing dynamical invariants for gradient descent. Overall, our theory predicts that the optimized parameters of an LCN will often correspond to repeated filters across layers, or filters that can be decomposed as repeated filters. We also conduct numerical and symbolic experiments that illustrate our results and present an in-depth analysis of the landscape for small architectures.

show abstract

Wasserstein distance to independence models

Cited by 16 publications

References 8 publications

Nonlinear algebra and applications

Nonlinear algebra and applications

Beyond Linear Algebra

Geometry of Linear Convolutional Networks

Contact Info

Product

Resources

About