Quantitative geometry

Naor, Assaf

doi:10.1073/pnas.1320388110

Cited by 4 publications

(3 citation statements)

References 38 publications

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“…Chapter 15 of Matoušek's monograph [44] on discrete geometry contains a beautiful account of the theory. It is also enlightening to consult [41] and [52] to get a sense of what the fast developing field of finite quantitative metric geometry is about.…”

Section: Geometry Of Metric and Banach Spaces: An Intertwined Storymentioning

confidence: 99%

Book Review: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces

Baudier¹,

Johnson²

2016

Bull. Amer. Math. Soc.

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Section: Geometry Of Metric and Banach Spaces: An Intertwined Storymentioning

confidence: 99%

Book Review: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces

Baudier¹,

Johnson²

2016

Bull. Amer. Math. Soc.

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“…We address the estimation from the quantitative geometric perspective. Quantitative geometry aims to understand geometric structures via quantitative parameters, and has connections with many mathematical disciplines, such as functional analysis and combinatorics (Naor, 2013). In computer science, quantitative geometry plays a central role in understanding the approximability of discrete optimization problems.…”

Section: Introductionmentioning

confidence: 99%

A Quantitative Geometric Approach to Neural Network Smoothness

Wang¹,

Prakriya²,

Jha³

2022

Preprint

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Fast and precise Lipschitz constant estimation of neural networks is an important task for deep learning. Researchers have recently found an intrinsic trade-off between the accuracy and smoothness of neural networks, so training a network with a loose Lipschitz constant estimation imposes a strong regularization and can hurt the model accuracy significantly. In this work, we provide a unified theoretical framework, a quantitative geometric approach, to address the Lipschitz constant estimation. By adopting this framework, we can immediately obtain several theoretical results, including the computational hardness of Lipschitz constant estimation and its approximability. Furthermore, the quantitative geometric perspective can also provide some insights into recent empirical observations that techniques for one norm do not usually transfer to another one.We also implement the algorithms induced from this quantitative geometric approach in a tool GeoLIP. These algorithms are based on semidefinite programming (SDP). Our empirical evaluation demonstrates that GeoLIP is more scalable and precise than existing tools on Lipschitz constant estimation for ∞ -perturbations. Furthermore, we also show its intricate relations with other recent SDP-based techniques, both theoretically and empirically. We believe that this unified quantitative geometric perspective can bring new insights and theoretical tools to the investigation of neural-network smoothness and robustness.

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“…Understanding how a group or a graph, viewed as a geometric object, can be faithfully embedded into certain Banach spaces is a fundamental topic with far reaching applications (see for instance [2,10,30], [45,Chapter15], [47,48,52] and [53] and the references therein) that is common to geometric group theory and theoretical computer science. Wreath products, and in particular lamplighter groups, have received a lot of attention in geometric group theory as they provide a wealth of groups with interesting algebraic and geometric properties.…”

Section: Introductionmentioning

confidence: 99%

Stochastic approximation of lamplighter metrics

Baudier

Motakis

Schlumprecht

et al. 2022

Bulletin of London Math Soc

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We observe that embeddings into random metrics can be fruitfully used to study the 𝐿 1 -embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the 𝐿 1 -distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an 𝑛-point metric space embeds bi-Lipschitzly into 𝐿 1 with distortion 𝑂(log 𝑛). In particular, for every finite group 𝐺 the lamplighter group 𝐻 = ℤ 2 ≀ 𝐺 bi-Lipschitzly embeds into 𝐿 1 with distortion 𝑂(log log |𝐻|). In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into 𝐿 1 of the lamplighter group over the 𝑑-dimensional infinite lattice ℤ 𝑑 can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite 𝑑dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

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Quantitative geometry

Cited by 4 publications

References 38 publications

Book Review: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces

Book Review: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces

A Quantitative Geometric Approach to Neural Network Smoothness

Stochastic approximation of lamplighter metrics

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