Codes, Cubes, and Graphical Designs

Babecki, Catherine

doi:10.1007/s00041-021-09852-z

Cited by 8 publications

(3 citation statements)

References 23 publications

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“…Graphical designs lie at an exciting intersection between contemporary applications and mathematical theory. The appearance of the graph Laplacian suggests further ties to spectral graph theory, [1] connects to coding theory, and extremal combinatorics appears in [4]. Graphical designs in Cayley graphs are rooted in group representations, and probabilistic tools were used in [7].…”

Section: Communicated By Notices Associate Editor Emilie Purvinementioning

confidence: 99%

What is... a Graphical Design?

Babecki¹

2022

Notices Amer. Math. Soc.

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Graphical designs are quadrature rules for graphs 𝐺 = (𝑉, 𝐸). Broadly speaking, a graphical design is a relatively small subset of vertices that captures the global behavior of functions 𝑓 ∶ 𝑉 → ℝ. We motivate the precise definition through numerical integration on the sphere.The regular icosahedron (Figure 1) inscribed in the unit sphere 𝕊 2 ⊂ ℝ 3 has the following amazing property: for any polynomial 𝑝(𝑥, 𝑦, 𝑧) of degree at most 5, the average of 𝑝 on the vertices of the icosahedron is equal to the average of 𝑝 over 𝕊 2 . By exploiting symmetry, these 12 points exactly average the 36-dimensional vector space of polynomials with degree at most 5. More generally, a spherical 𝑡-design [3] is a finite subset of points 𝑋 ⊂ 𝕊 2 chosen so that for any polynomial 𝑝(𝑥, 𝑦, 𝑧) of degree at most 𝑡,

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Section: Communicated By Notices Associate Editor Emilie Purvinementioning

confidence: 99%

What is... a Graphical Design?

Babecki¹

2022

Notices Amer. Math. Soc.

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“…Golubev in [Gol20] introduced extremal designs and connected them to extremal combinatorics. Babecki in [Bab21] refined the definition of graphical designs to make sense when the eigenvalues of a graph Laplacian have multiplicity, connected linear codes in the Boolean cube to graphical designs in the edge graphs of hypercubes, and distinguished graphical designs from a handful of related concepts in the adjacent literature. She also hosts a database of examples and code at https://sites.math.washington.edu/ ~GraphicalDesigns/.…”

Section: Introductionmentioning

confidence: 99%

Graphical Designs and Gale Duality

Babecki¹,

Thomas²

2022

Preprint

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A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in regular graphs are in bijection with the faces of a generalized eigenpolytope of the graph. This connection can be used to organize, compute and optimize designs. We illustrate the power of this tool on three families of Cayley graphs -cocktail party graphs, cycles, and graphs of hypercubes -by computing or bounding the smallest designs that average all but the last eigenspace in frequency order. We also prove that unless NP = coNP, there cannot be an efficient description of all minimal designs that average a fixed number of eigenspaces in a graph.

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“…It is impossible to improve on the linear algebra setting in the continuous setting, see [6,25]. Graphical designs were further explored by Babecki [1] and Golubev [11]. The first general Existence Theorem was recently proven by Babecki & Thomas [2] who established a bijection between positively weighted graphical designs and the faces of a generalized eigenpolytope of the graph.…”

Section: Introduction and Statementmentioning

confidence: 99%

Random Walks, Equidistribution and Graphical Designs

Steinerberger¹,

Thomas²

2022

Preprint

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Let G = (V, E) be a d-regular graph on n vertices and let µ 0 be a probability measure on V . The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on V given by µ k+1 = AD −1 µ k , where A is the adjacency matrix and D is the diagonal matrix of vertex degrees of G. Ordering the eigenvalues ofit is well-known that the graphs for which |λ 2 | is small are those in which the random walk process converges quickly to the uniform distribution: for all initial probability measures µ 0 and all k ≥ 0,One could wonder whether this rate can be improved for specific initial probability measures µ 0 . We show that if G is regular, then for any 1 ≤ ≤ n, there exists a probability measure µ 0 supported on at most vertices so thatThe result has applications in the graph sampling problem: we show that these measures have good sampling properties for reconstructing global averages.

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Codes, Cubes, and Graphical Designs

Cited by 8 publications

References 23 publications

What is... a Graphical Design?

What is... a Graphical Design?

Graphical Designs and Gale Duality

Random Walks, Equidistribution and Graphical Designs

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