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Paper Abstract and Keywords
Presentation 2019-12-13 13:00
[Invited Talk] Spectral Sparsification of Hypergraphs
Tasuku Soma (The Univ. of Tokyo), Yuichi Yoshida (NII) COMP2019-35
Abstract (in Japanese) (See Japanese page) 
(in English) For an undirected/directed hypergraph G = (V, E), its Laplacian LG : ℝv → ℝv is defined such that its “quadratic form” x⊺LG (x) captures the cut information of G. In particular, 1S⊺LG(1S) coincides with the cut size of S ⊆ V, where 1S ∊ ℝV is the characteristic vector of S.

A weighted subgraph H of a hypergraph G on a vertex set V is said to be an ∊-spectral sparsifier of G if (1 – ∊)x⊺ LH(x) ≤ x⊺ LG(x) ≤ (1 + ∊)x⊺ LH(x) holds for every x ∊ ℝV. In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph G on n vertices, constructs an ∊-spectral sparsifier of G with O(n3 log n/∊2) hyperedges/hyperarcs.

The proposed spectral sparsification can be used to improve the time and space complexities of algorithms for solving problems that involve the quadratic form, such as computing the eigenvalues of LG, computing the effective resistance between a pair of vertices in G, semi-supervised learning based on LG, and cut problems on G. In addition, our sparsification result implies that any nonnegative hypernetwork type submodular function can be concisely represented by a directed hypergraph of polynomial size, even if the original representation is of exponential size. Accordingly, we show that, for any distribution, we can properly and agnostically learn nonnegative hypernetwork type submodular functions with O(n4 log(n/∊)/∊4) samples.
Keyword (in Japanese) (See Japanese page) 
(in English) Spectral sparsification / Hypergraph / Learning theory / / / / /  
Reference Info. IEICE Tech. Rep., vol. 119, no. 340, COMP2019-35, pp. 45-45, Dec. 2019.
Paper # COMP2019-35 
Date of Issue 2019-12-06 (COMP) 
ISSN Print edition: ISSN 0913-5685  Online edition: ISSN 2432-6380
All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034)
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Conference Information
Committee COMP  
Conference Date 2019-12-13 - 2019-12-13 
Place (in Japanese) (See Japanese page) 
Place (in English) Ikaho Seminar House, Gunma University 
Topics (in Japanese) (See Japanese page) 
Topics (in English)  
Paper Information
Registration To COMP 
Conference Code 2019-12-COMP 
Language English 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) Spectral Sparsification of Hypergraphs 
Sub Title (in English)  
Keyword(1) Spectral sparsification  
Keyword(2) Hypergraph  
Keyword(3) Learning theory  
1st Author's Name Tasuku Soma  
1st Author's Affiliation The University of Tokyo (The Univ. of Tokyo)
2nd Author's Name Yuichi Yoshida  
2nd Author's Affiliation National Institute of Informatics (NII)
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Date Time 2019-12-13 13:00:00 
Presentation Time 60 
Registration for COMP 
Paper # COMP2019-35 
Volume (vol) 119 
Number (no) no.340 
Page p.45 
Date of Issue 2019-12-06 (COMP) 

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