Nonlinear Algebra and Applications

Breiding, Paul; Çelik, Türkü Özlüm; Timothy, Duff,; Heaton, Alexander; Maraj, Aida; Sattelberger, Anna-Laura; Venturello, Lorenzo; Yürük, Oğuzhan

doi:10.48550/arxiv.2103.16300

Cited by 4 publications

(4 citation statements)

References 176 publications

(218 reference statements)

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“…and this bound is attained for generic polynomials (Nie & Ranestad, 2009;Breiding et al, 2021). For non-complete intersections, the expression (20) does not need to yield an upper bound if some constraints are redundant.…”

Section: Appendixmentioning

confidence: 99%

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The Geometry of Memoryless Stochastic Policy Optimization in Infinite-Horizon POMDPs

Müller¹,

Montúfar²

2021

Preprint

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We consider the problem of finding the best memoryless stochastic policy for an infinitehorizon partially observable Markov decision process (POMDP) with finite state and action spaces with respect to either the discounted or mean reward criterion. We show that the (discounted) state-action frequencies and the expected cumulative reward are rational functions of the policy, whereby the degree is determined by the degree of partial observability. We then describe the optimization problem as a linear optimization problem in the space of feasible state-action frequencies subject to polynomial constraints that we characterize explicitly. This allows us to address the combinatorial and geometric complexity of the optimization problem using recent tools from polynomial optimization. In particular, we demonstrate how the partial observability constraints can lead to multiple smooth and non-smooth local optimizers and we estimate the number of critical points.

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Section: Appendixmentioning

confidence: 99%

“…We try to present the results from the mighty theory of algebraic degrees that we use here and refer the interested reader to the excellent low level introduction by Breiding et al (2021)…”

Section: D1 Introduction To Algebraic Degreesmentioning

confidence: 99%

The Geometry of Memoryless Stochastic Policy Optimization in Infinite-Horizon POMDPs

Müller¹,

Montúfar²

2021

Preprint

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“…The following examples demonstrate that our assumptions on X are not very restrictive and the results of this work can be applied in a number of different settings. For an overview of polynomial models in the sciences see also [19].…”

Section: Introductionmentioning

confidence: 99%

Algebraic compressed sensing

Breiding¹,

Gesmundo²,

Michałek³

et al. 2021

Preprint

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We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ powerful techniques from algebraic geometry to study well-posedness of sufficiently general compressed sensing problems, including existence, local recoverability, global uniqueness, and local smoothness. Our main results are summarized in thirteen questions and answers in algebraic compressed sensing. Most of our answers concerning the minimum number of required measurements for existence, recoverability, and uniqueness of algebraic compressed sensing problems are optimal and depend only on the dimension of the model.

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“…These results demonstrate connections between properties of Y and properties of the base locus of G. The goal and purpose of this paper is to extend such connections for when G is nonlinear. Therefore, our approach embodies the general and fruitful philosophy of studying nonlinear algebra [6,47].…”

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confidence: 99%