{"id":"https://openalex.org/W2592411267","doi":"https://doi.org/10.1137/151002526","title":"An Automatic Inequality Prover and Instance Optimal Identity Testing","display_name":"An Automatic Inequality Prover and Instance Optimal Identity Testing","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2592411267","doi":"https://doi.org/10.1137/151002526","mag":"2592411267"},"language":"en","primary_location":{"id":"doi:10.1137/151002526","is_oa":false,"landing_page_url":"https://doi.org/10.1137/151002526","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","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/A5079503799","display_name":"Gregory Valiant","orcid":"https://orcid.org/0000-0002-2211-1073"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Gregory Valiant","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5036230157","display_name":"Paul Valiant","orcid":"https://orcid.org/0000-0001-5523-2974"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Paul Valiant","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5079503799"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":8.107,"has_fulltext":false,"cited_by_count":91,"citation_normalized_percentile":{"value":0.97886669,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"46","issue":"1","first_page":"429","last_page":"455"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T12072","display_name":"Machine Learning and Algorithms","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.989799976348877,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9769999980926514,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6853854060173035},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6226101517677307},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.581972599029541},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5457271337509155},{"id":"https://openalex.org/keywords/identity","display_name":"Identity (music)","score":0.4710542857646942},{"id":"https://openalex.org/keywords/total-variation","display_name":"Total variation","score":0.4443149268627167},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4400067627429962},{"id":"https://openalex.org/keywords/probability-distribution","display_name":"Probability distribution","score":0.4397466480731964},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4367048740386963},{"id":"https://openalex.org/keywords/property-testing","display_name":"Property testing","score":0.4246782958507538},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4184151589870453},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3507577180862427},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2102811336517334},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1576576828956604},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.109398752450943}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6853854060173035},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6226101517677307},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.581972599029541},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5457271337509155},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.4710542857646942},{"id":"https://openalex.org/C62100291","wikidata":"https://www.wikidata.org/wiki/Q1936288","display_name":"Total variation","level":2,"score":0.4443149268627167},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4400067627429962},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.4397466480731964},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4367048740386963},{"id":"https://openalex.org/C123842658","wikidata":"https://www.wikidata.org/wiki/Q2918980","display_name":"Property testing","level":2,"score":0.4246782958507538},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4184151589870453},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3507577180862427},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2102811336517334},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1576576828956604},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.109398752450943},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/151002526","is_oa":false,"landing_page_url":"https://doi.org/10.1137/151002526","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Reduced inequalities","score":0.44999998807907104,"id":"https://metadata.un.org/sdg/10"}],"awards":[{"id":"https://openalex.org/G6851672665","display_name":null,"funder_award_id":"CCF-1351108","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306151","display_name":"Alfred P. Sloan Foundation","ror":"https://ror.org/052csg198"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1501167079","https://openalex.org/W1928812145","https://openalex.org/W2000163531","https://openalex.org/W2002001845","https://openalex.org/W2076381458","https://openalex.org/W2102942501","https://openalex.org/W2104648144","https://openalex.org/W2114771311","https://openalex.org/W2170990775","https://openalex.org/W2419099043","https://openalex.org/W2952058015"],"related_works":["https://openalex.org/W621782861","https://openalex.org/W2073966617","https://openalex.org/W267063485","https://openalex.org/W2185560008","https://openalex.org/W2064609684","https://openalex.org/W2785084575","https://openalex.org/W2964084133","https://openalex.org/W2029005240","https://openalex.org/W2397544591","https://openalex.org/W2135813035"],"abstract_inverted_index":{"We":[0,61,304],"consider":[1],"the":[2,6,12,44,49,52,85,170,180,184,189,226,234,265,271,306,321],"problem":[3],"of":[4,8,14,132,147,173,192,197,216,220,228,236,238,250,309],"verifying":[5],"identity":[7],"a":[9,15,18,81,130,213,217],"distribution:":[10],"Given":[11],"description":[13],"distribution":[16,87,191],"over":[17],"discrete":[19],"finite":[20,248],"or":[21,287],"countably":[22],"infinite":[23],"support,":[24],"$p=(p_1,p_2,\\ldots)$,":[25],"how":[26],"many":[27],"samples":[28,104,167],"(independent":[29],"draws)":[30],"must":[31],"one":[32,322],"obtain":[33],"from":[34,48,99,115],"an":[35,70,289],"unknown":[36],"distribution,":[37],"$q$,":[38],"to":[39,66,270,285,295,315],"distinguish,":[40],"with":[41,105],"high":[42],"probability,":[43],"case":[45,50,185],"that":[46,51,94,162,186,245],"$p=q$":[47,98,114],"total":[53],"variation":[54],"distance":[55],"($L_1$":[56],"distance)":[57],"$\\|p-q\\|_1":[58],"\\ge":[59,261],"\\epsilon$?":[60],"resolve":[62],"this":[63,158,209],"question,":[64],"up":[65],"constant":[67],"factors,":[68],"on":[69,84],"instance":[71,73],"by":[72,129,156,302],"basis:":[74],"there":[75],"exist":[76],"universal":[77],"constants":[78],"$c,c'$":[79],"and":[80,89,141,178,225,254,280,300,317],"function":[82,125],"$f(p,\\epsilon)$":[83,103,126],"known":[86],"$p$":[88,187],"error":[90],"parameter":[91],"$\\epsilon$,":[92],"such":[93,164],"our":[95,198,310],"tester":[96,111],"distinguishes":[97],"$\\|p-q\\|_1\\ge":[100,116],"\\epsilon$":[101,118],"using":[102],"success":[106],"probability":[107],"$>2/3$,":[108],"but":[109,134],"no":[110],"can":[112],"distinguish":[113],"c\\cdot":[117],"when":[119],"given":[120],"$c'\\cdot":[121],"f(p,\\epsilon)$":[122],"samples.":[123],"The":[124,195],"is":[127,135,188,278],"upper-bounded":[128],"multiple":[131],"$\\frac{\\|p\\|_{2/3}}{\\epsilon^2}$":[133],"more":[136],"complicated.":[137],"This":[138],"result":[139,159],"generalizes":[140],"tightens":[142],"previous":[143,171],"results:":[144],"since":[145],"distributions":[146],"support":[148,193],"at":[149],"most":[150],"$n$":[151],"have":[152,294],"$L_{2/3}$":[153],"norm":[154],"bounded":[155],"$\\sqrt{n},$":[157],"immediately":[160],"shows":[161],"for":[163,183,241,246],"distributions,":[165],"$O(\\sqrt{n}/{\\epsilon^2})$":[166],"suffice,":[168],"tightening":[169],"bound":[172],"$O(\\frac{\\sqrt{n}\\":[174],"{\\rm":[175],"polylog}\\":[176],"n}{\\epsilon^4})$":[177],"matching":[179],"(tight)":[181],"results":[182],"uniform":[190],"$n$.":[194],"analysis":[196],"very":[199],"simple":[200],"testing":[201],"algorithm":[202],"involves":[203],"several":[204],"hairy":[205],"inequalities.":[206],"To":[207],"facilitate":[208,318],"analysis,":[210],"we":[211,232],"give":[212],"complete":[214],"characterization":[215,277,311],"general":[218],"class":[219],"inequalities---generalizing":[221],"Cauchy--Schwarz,":[222],"H\u00f6lder's":[223],"inequality,":[224,290],"monotonicity":[227],"$L_p$":[229],"norms.":[230],"Specifically,":[231],"characterize":[233],"set":[235],"sequences":[237,249],"triples":[239,272],"$(a,b,c)_i=(a_1,b_1,c_1),\\ldots,(a_r,b_r,c_r)$":[240],"which":[242,291],"it":[243],"holds":[244],"all":[247],"positive":[251],"numbers":[252],"$(x)_j=x_1,\\ldots$":[253],"$(y)_j=y_1,\\ldots,$":[255],"$\\prod_{i=1}^r":[256],"(":[257],"\\sum_j":[258],"x_j^{a_i}":[259],"y_j^{b_i})^{c_i}":[260],"1.$":[262],"For":[263],"example,":[264],"standard":[266],"Cauchy--Schwarz":[267],"inequality":[268],"corresponds":[269],"$(a,b,c)_i=(1,0,\\frac{1}{2}),$":[273],"$\\,(0,1,\\frac{1}{2}),$":[274],"$\\,(\\frac{1}{2},\\frac{1}{2},-1)$.":[275],"Our":[276],"constructive":[279],"algorithmic,":[281],"leveraging":[282],"linear":[283],"programming":[284],"prove":[286],"refute":[288],"would":[292],"otherwise":[293],"be":[296,313],"investigated,":[297],"through":[298],"trial":[299],"error,":[301],"hand.":[303],"hope":[305],"computational":[307],"nature":[308],"will":[312],"useful":[314],"others":[316],"analyses":[319],"like":[320],"here.":[323]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":13},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":20},{"year":2020,"cited_by_count":15},{"year":2019,"cited_by_count":16},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}