A Discrete Signal Processing Framework for Meet/join Lattices with Applications to Hypergraphs and Trees

Püschel, Markus

doi:10.1109/icassp.2019.8682729

Cited by 8 publications

(8 citation statements)

References 23 publications

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“…We provide the necessary background on lattice theory [18] and discrete lattice SP (DLSP) following [9].…”

Section: Discrete Lattice Signal Processingmentioning

confidence: 99%

“…There are other non-Euclidean index domains besides graphs and first steps towards associated SP frameworks. Examples include simplicial complexes [5], powersets [6,7], hypergraphs [8], and lattices [9][10][11]. A lattice is a partially ordered set with a meet and join operation that returns the greatest lower bound and smallest upper bound for any two elements, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Lattice SP [9,11] uses the meet (or join) operation to define a shift operation and then uses the general procedure from algebraic signal processing [16] to obtain an associated notion of convolution, Fourier transform, and frequency response.…”

Section: Introductionmentioning

confidence: 99%

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Wiener Filter on Meet/Join Lattices

Seifert

Wendler

Püschel

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Recent work introduced a framework for signal processing (SP) on meet/join lattices. Such a lattice is partially ordered and supports a meet (or join) operation that returns the greatest lower bound and the smallest upper bound of two elements, respectively. Lattices appear in various domains and can be used, for example, to express rankings in social choice theory or multisets in combinatorial auctions. Discrete lattice SP (DLSP) uses the meet operation as shift and derives associated notions of convolution and Fourier transform for signals indexed by lattices. In this paper we extend DLSP with Wiener filtering for denoising and demonstrate it on two prototypical applications.

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“…We provide the necessary background on lattice theory [18] and discrete lattice SP (DLSP) following [9].…”

Section: Discrete Lattice Signal Processingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Wiener Filter on Meet/Join Lattices

Seifert

Wendler

Püschel

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…ASP provides a theoretical framework for deriving a complete set of basic signal processing concepts, including convolution, for novel index domains, using as starting point a chosen shift to which convolutions should be equivariant. To date the approach was used for index domains including graphs [34,44,45], powersets (set functions) [36], meet/join lattices [37,61], and a collection of more regular domains, e.g., [39,46,49].…”

Section: Related Workmentioning

confidence: 99%

Powerset Convolutional Neural Networks

Wendler,

Alistarh,

Püschel

2019

Preprint

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We present a novel class of convolutional neural networks (CNNs) for set functions, i.e., data indexed with the powerset of a finite set. The convolutions are derived as linear, shift-equivariant functions for various notions of shifts on set functions. The framework is fundamentally different from graph convolutions based on the Laplacian, as it provides not one but several basic shifts, one for each element in the ground set. Prototypical experiments with several set function classification tasks on synthetic datasets and on datasets derived from real-world hypergraphs demonstrate the potential of our new powerset CNNs.Preprint. Under review.

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“…Parallel efforts, especially by Püschel et al, have been made at directly processing sets [4] and meet/join lattices [13]. A convolutional neural information processing model was proposed for the former [14], and an architecture employing the convolution of the later was recently used to classify multidimensional persistence modules [15].…”

Section: Introductionmentioning

confidence: 99%

Quiver Signal Processing (QSP)

Parada-Mayorga¹,

Riess²,

Ribeiro³

2020

Preprint

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In this paper we state the basics for a signal processing framework on quiver representations. A quiver is a directed graph and a quiver representation is an assignment of vector spaces to the nodes of the graph and of linear maps between the vector spaces associated to the nodes. Leveraging the tools from representation theory, we propose a signal processing framework that allows us to handle heterogeneous multidimensional information in networks. We provide a set of examples where this framework provides a natural set of tools to understand apparently hidden structure in information. We remark that the proposed framework states the basis for building graph neural networks where information can be processed and handled in alternative ways.

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A Discrete Signal Processing Framework for Meet/join Lattices with Applications to Hypergraphs and Trees

Cited by 8 publications

References 23 publications

Wiener Filter on Meet/Join Lattices

Wiener Filter on Meet/Join Lattices

Powerset Convolutional Neural Networks

Quiver Signal Processing (QSP)

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