{"id":"https://openalex.org/W2605744995","doi":"https://doi.org/10.1109/tit.2017.2756880","title":"Energy Propagation in Deep Convolutional Neural Networks","display_name":"Energy Propagation in Deep Convolutional Neural Networks","publication_year":2017,"publication_date":"2017-09-26","ids":{"openalex":"https://openalex.org/W2605744995","doi":"https://doi.org/10.1109/tit.2017.2756880","mag":"2605744995"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2017.2756880","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2017.2756880","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009015123","display_name":"Thomas Wiatowski","orcid":"https://orcid.org/0000-0002-1848-6593"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Thomas Wiatowski","raw_affiliation_strings":["Department of Information Technology and Electrical Engineering, ETH Z\u00fcrich, Z\u00fcrich, Switzerland"],"raw_orcid":"https://orcid.org/0000-0002-1848-6593","affiliations":[{"raw_affiliation_string":"Department of Information Technology and Electrical Engineering, ETH Z\u00fcrich, Z\u00fcrich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5023888160","display_name":"Philipp Grohs","orcid":"https://orcid.org/0000-0001-9205-0969"},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Philipp Grohs","raw_affiliation_strings":["Faculty of Mathematics, University of Vienna, Vienna, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, University of Vienna, Vienna, Austria","institution_ids":["https://openalex.org/I129774422"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079321311","display_name":"Helmut B\u00f6lcskei","orcid":null},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Helmut Bolcskei","raw_affiliation_strings":["Department of Information Technology and Electrical Engineering, ETH Z\u00fcrich, Z\u00fcrich, Switzerland"],"raw_orcid":"https://orcid.org/0000-0003-3047-5149","affiliations":[{"raw_affiliation_string":"Department of Information Technology and Electrical Engineering, ETH Z\u00fcrich, Z\u00fcrich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.2008,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.87118509,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"64","issue":"7","first_page":"4819","last_page":"4842"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11698","display_name":"Underwater Acoustics Research","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1910","display_name":"Oceanography"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11609","display_name":"Geophysical Methods and Applications","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2212","display_name":"Ocean Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/feature","display_name":"Feature (linguistics)","score":0.6334452033042908},{"id":"https://openalex.org/keywords/shearlet","display_name":"Shearlet","score":0.5706617832183838},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.5700279474258423},{"id":"https://openalex.org/keywords/wavelet","display_name":"Wavelet","score":0.5439231395721436},{"id":"https://openalex.org/keywords/convolutional-neural-network","display_name":"Convolutional neural network","score":0.5250309109687805},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4937685430049896},{"id":"https://openalex.org/keywords/pattern-recognition","display_name":"Pattern recognition (psychology)","score":0.46848785877227783},{"id":"https://openalex.org/keywords/feature-vector","display_name":"Feature vector","score":0.46475616097450256},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4585636854171753},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4519394040107727},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4313731789588928},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3593485653400421}],"concepts":[{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.6334452033042908},{"id":"https://openalex.org/C67795661","wikidata":"https://www.wikidata.org/wiki/Q17018993","display_name":"Shearlet","level":3,"score":0.5706617832183838},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.5700279474258423},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.5439231395721436},{"id":"https://openalex.org/C81363708","wikidata":"https://www.wikidata.org/wiki/Q17084460","display_name":"Convolutional neural network","level":2,"score":0.5250309109687805},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4937685430049896},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.46848785877227783},{"id":"https://openalex.org/C83665646","wikidata":"https://www.wikidata.org/wiki/Q42139305","display_name":"Feature vector","level":2,"score":0.46475616097450256},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4585636854171753},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4519394040107727},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4313731789588928},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3593485653400421},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2017.2756880","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2017.2756880","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.4399999976158142,"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":62,"referenced_works":["https://openalex.org/W222993076","https://openalex.org/W228380312","https://openalex.org/W654435104","https://openalex.org/W1486164486","https://openalex.org/W1530034100","https://openalex.org/W1535448269","https://openalex.org/W1549358575","https://openalex.org/W1605417594","https://openalex.org/W1663973292","https://openalex.org/W1762506961","https://openalex.org/W1983989471","https://openalex.org/W1994906459","https://openalex.org/W1997146646","https://openalex.org/W2021722152","https://openalex.org/W2031621093","https://openalex.org/W2062024414","https://openalex.org/W2066462711","https://openalex.org/W2069912449","https://openalex.org/W2072072671","https://openalex.org/W2093231248","https://openalex.org/W2103835941","https://openalex.org/W2112796928","https://openalex.org/W2116741198","https://openalex.org/W2119337797","https://openalex.org/W2120731967","https://openalex.org/W2139427956","https://openalex.org/W2154579312","https://openalex.org/W2163922914","https://openalex.org/W2194775991","https://openalex.org/W2208541107","https://openalex.org/W2240077181","https://openalex.org/W2294385287","https://openalex.org/W2314948770","https://openalex.org/W2345809749","https://openalex.org/W2546302380","https://openalex.org/W2557283755","https://openalex.org/W2581711117","https://openalex.org/W2589910641","https://openalex.org/W2593628332","https://openalex.org/W2766736793","https://openalex.org/W2911938359","https://openalex.org/W2919115771","https://openalex.org/W2962957157","https://openalex.org/W2963148568","https://openalex.org/W2963803205","https://openalex.org/W2963918359","https://openalex.org/W3038420293","https://openalex.org/W3038830718","https://openalex.org/W3098103643","https://openalex.org/W3158307145","https://openalex.org/W4238924299","https://openalex.org/W4247341659","https://openalex.org/W4254734937","https://openalex.org/W4255272544","https://openalex.org/W4292024750","https://openalex.org/W4292170079","https://openalex.org/W4302311470","https://openalex.org/W6632978964","https://openalex.org/W6682610290","https://openalex.org/W6682751323","https://openalex.org/W6690017479","https://openalex.org/W6714097516"],"related_works":["https://openalex.org/W2541950815","https://openalex.org/W2767651786","https://openalex.org/W2912288872","https://openalex.org/W564581980","https://openalex.org/W2185970706","https://openalex.org/W3106036237","https://openalex.org/W2886673456","https://openalex.org/W2295021132","https://openalex.org/W2546942002","https://openalex.org/W2970216048"],"abstract_inverted_index":{"Many":[0],"practical":[1],"machine":[2],"learning":[3],"tasks":[4],"employ":[5],"very":[6],"deep":[7,127],"convolutional":[8,128],"neural":[9,129],"networks.":[10],"Such":[11],"large":[12],"depths":[13],"pose":[14],"formidable":[15],"computational":[16],"challenges":[17],"in":[18,35,60,90,96,228],"training":[19],"and":[20,133,151,158,176,192],"operating":[21],"the":[22,32,36,51,56,61,64,69,74,91,94,97,105,148,162,178,195,210,222,229],"network.":[23],"It":[24],"is":[25,48,199],"therefore":[26],"important":[27],"to":[28,68,102,201,215],"understand":[29],"how":[30],"fast":[31],"energy":[33,95,114,138,181,225],"contained":[34,227],"propagated":[37],"signals":[38],"(a.k.a.":[39],"feature":[40,52,71,98,131,136,179,230],"maps)":[41],"decays":[42,182],"across":[43],"layers.":[44],"In":[45],"addition,":[46],"it":[47],"desirable":[49],"that":[50,103,154],"extractor":[53],"generated":[54],"by":[55,85],"network":[57],"be":[58,83,202,226],"informative":[59],"sense":[62,92],"of":[63,93,104,126,170,190,209,212,221],"only":[65],"signal":[66,224],"mapping":[67],"all-zeros":[70],"vector":[72,99],"being":[73,100],"zero":[75],"input":[76,107,223],"signal.":[77,108],"This":[78,109],"\u201ctrivial":[79],"null-set\u201d":[80],"property":[81],"can":[82],"accomplished":[84],"asking":[86],"for":[87,113,118,122],"\u201cenergy":[88],"conservation\u201d":[89],"proportional":[101],"corresponding":[106,135],"paper":[110],"establishes":[111],"conditions":[112,160],"conservation":[115],"(and":[116],"thus":[117],"a":[119,123],"trivial":[120],"null-set)":[121],"wide":[124],"class":[125],"network-based":[130],"extractors":[132],"characterizes":[134],"map":[137,180],"decay":[139,197],"rates.":[140],"Specifically,":[141],"we":[142,152,205],"consider":[143],"general":[144],"scattering":[145],"networks":[146],"employing":[147],"modulus":[149],"non-linearity":[150],"find":[153],"under":[155],"mild":[156],"analyticity":[157],"high-pass":[159],"on":[161],"filters":[163],"(which":[164],"encompass,":[165],"inter":[166],"alia,":[167],"various":[168],"constructions":[169],"Weyl-Heisenberg":[171,193],"filters,":[172,194],"wavelets,":[173],"ridgelets,":[174],"(\u03b1)-curvelets,":[175],"shearlets)":[177],"at":[183,217],"least":[184,218],"polynomially":[185],"fast.":[186],"For":[187],"broad":[188],"families":[189],"wavelets":[191],"guaranteed":[196],"rate":[198],"shown":[200],"exponential.":[203],"Moreover,":[204],"provide":[206],"handy":[207],"estimates":[208],"number":[211],"layers":[213],"needed":[214],"have":[216],"((1-\u03b5)":[219],"100)%":[220],"vector.":[231]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":4}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}