chunk_uid stringlengths 40 40 | chunk_type stringclasses 2
values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9218fdac9bacece77e709fe79d2294851132c84e | subsection | 34 | 52 | Proof of Things | Now consider the function \varepsilon f^{\prime } with \varepsilon >0, it's clear that\tau _P(\mathbb {P}\Vert \mathbb {Q})\ge \tau _P(\mathbb {P}\Vert \mathbb {Q};\varepsilon f^{\prime })
=\varepsilon (\underbrace{\mathbb {E}_{\mathbb {P}}[f^{\prime }]-\mathbb {E}_{\mathbb {Q}}[f^{\prime }]}_{\ge \frac{\mu }{2}})
-\va... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.04924669489264488,
0.024928469210863113,
-0.04592086374759674,
-0.03063425049185753,
-0.014531436376273632,
-0.02665240876376629,
0.0605056956410408,
-0.003465041983872652,
0.03838435187935829,
0.05809523165225983,
-0.018627699464559555,
-0.008268807083368301,
-0.03505852073431015,
0.01... | |
e00bb285e7794391b98b3fa440694c9dbb95c7e3 | subsection | 35 | 52 | Proof of Things | Every f^*\in \operatorname{OC}_{\tau _F}(\mathbb {P},\mathbb {Q}^{\prime }_{\theta _0}) is in \operatorname{OC}_{\tau _P}(\mathbb {P},\mathbb {Q}^{\prime }_{\theta _0})\subseteq C^1(X),
therefore f^* the gradient \nabla _\theta \mathbb {E}_{\mathbb {Q}_\theta }[f^*]|_{\theta _0} exists. Further Lemma REF
shows that th... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01636967621743679,
0.012380234897136688,
-0.03902481868863106,
-0.019863296300172806,
0.013875321485102177,
-0.01861230656504631,
0.028147298842668533,
0.04326598346233368,
0.021678758785128593,
0.014538956806063652,
-0.0514737032353878,
0.009885880164802074,
-0.031732454895973206,
0.00... | |
05c44ee984d55eca52ff298ce9d7d43176d8e31c | subsection | 36 | 52 | Proof of Things | Thenthere exists f^*\in \operatorname{OC}_{\tau _P}(\mathbb {P},\mathbb {Q}) so that \tau _F(\mathbb {P}\Vert \mathbb {Q};f^*)=\tau _P(\mathbb {P}\Vert \mathbb {Q};f^*),
\tau _P(\mathbb {P}\Vert \mathbb {Q})=\tau _F(\mathbb {P}\Vert \mathbb {Q}),
\emptyset \ne \operatorname{OC}_{\tau _F}(\mathbb {P},\mathbb {Q}),
\o... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01775958761572838,
0.0054850270971655846,
-0.05059346556663513,
-0.0340544655919075,
0.011877868324518204,
-0.04012690484523773,
0.034390129148960114,
0.04207984730601311,
0.0339629240334034,
0.025495078414678574,
-0.016111791133880615,
-0.00955874752253294,
-0.013022172264754772,
-0.00... | |
aa5686e341fa911ef6cca8e2a1483f0387357c46 | subsection | 37 | 52 | Proof of Things | Then Lemma REF tells us there is a f\in \operatorname{OC}_{\tau _P}(\mathbb {P},\mathbb {Q}) (and thus f\in C^1(X)) such that\forall x^{\prime }\in \Omega :\quad \mathbb {E}_{\tilde{x}\sim \mathbb {P}}\left[\frac{f(\tilde{x})-f(x^{\prime })}{\Vert \tilde{x}-x^{\prime }\Vert }\right]=\frac{1}{2\lambda }and thus, because... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.05102638900279999,
0.03613352030515671,
-0.032837554812431335,
-0.06475956737995148,
0.029236411675810814,
-0.022049380466341972,
0.006324891932308674,
0.029999366030097008,
0.02467394433915615,
-0.0008573699742555618,
-0.01712069660425186,
-0.014427467249333858,
-0.012291194871068,
-0.... | |
e07ae57d38d9d4a14d58c4d27c5a700d4769ac2c | subsection | 38 | 52 | Proof of Things | The claims are a direct result of Claim (1);
for every \mathbb {P},\mathbb {Q}\in \mathcal {P}(X) there exists af^*\in \operatorname{OC}_{\tau _P}(\mathbb {P},\mathbb {Q})such that G(\mathbb {P},\mathbb {Q};f^*)=0. Therefore\tau _P(\mathbb {P}\Vert \mathbb {Q})\ge \tau _F(\mathbb {P}\Vert \mathbb {Q})\ge \tau _F(\mathb... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02880995348095894,
0.03683645650744438,
-0.02150064893066883,
-0.04370323568582535,
0.016526049003005028,
-0.019791584461927414,
0.01905912719666958,
0.051271952688694,
0.0019122072262689471,
0.03802669793367386,
-0.004268847871571779,
-0.012886655516922474,
-0.017334802076220512,
-0.00... | |
bd0c6dc44952472cb68a41ed1eeb4c97cb45bf20 | subsection | 39 | 52 | Proof of Things | This claim is a direct result of claim (2); since \tau _P(\mathbb {P}\Vert \mathbb {Q})=\tau _F(\mathbb {P}\Vert \mathbb {Q}),
that means that if f^*\in \operatorname{OC}_{\tau _F}(\mathbb {P}\Vert \mathbb {Q}), then\tau _F(\mathbb {P}\Vert \mathbb {Q})=\tau _F(\mathbb {P}\Vert \mathbb {Q};f^*)=\tau _P(\mathbb {P}\Vert... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.022328678518533707,
0.03266122564673424,
-0.03797249123454094,
-0.023107053712010384,
0.0020108020398765802,
-0.016284825280308723,
0.0298529714345932,
0.0704505667090416,
0.044199489057064056,
0.027807828038930893,
-0.014735707081854343,
-0.010202817618846893,
-0.007020637392997742,
-0... | |
b08861c30f1337f220e5f5b32d2ab0645fd27c97 | subsection | 40 | 52 | Proof of Things | REF we see,&\mathbb {E}_{x\sim \mathbb {P}}\left[\frac{f^*(x)-f^*(x^{\prime }+\varepsilon v)}{\Vert x-(x^{\prime }+\varepsilon v)\Vert }\right]\\
=&\varepsilon \left\langle v,\nabla _{\hat{x}}\mathbb {E}_{x\sim \mathbb {P}}\left[\frac{f^*(x)-f^*(\hat{x})}{\Vert x-\hat{x}\Vert }\right]\bigg |_{x^{\prime }}\right\rangle ... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.022793134674429893,
-0.003917092923074961,
-0.004756197798997164,
-0.002669878536835313,
-0.008413931354880333,
0.030375588685274124,
0.047111913561820984,
0.06014091894030571,
0.027278529480099678,
0.02540198713541031,
-0.03746983781456947,
0.026256347075104713,
-0.019970690831542015,
... | |
74529356f8141255fb94f9ae16c2a803ca92f724 | subsection | 41 | 52 | Proof of Things | \ref {E:slope} and definition of }\mathbb {Q}^{\prime }}
\frac{f^*(\tilde{x})-f^*(x^{\prime })}{\Vert x^{\prime }-\tilde{x}\Vert ^3}\right]}{\left\Vert \mathbb {E}_{\tilde{x}\sim \mathbb {P}}\left[(\tilde{x}- x^{\prime })\frac{f^*(\tilde{x})-f^*(x^{\prime })}{\Vert x^{\prime }-\tilde{x}\Vert ^3}\right]\right\Vert }
+\m... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.050770435482263565,
-0.004881772678345442,
-0.01495805662125349,
-0.03081618994474411,
0.01421053521335125,
0.02204425446689129,
0.039725422859191895,
0.025827627629041672,
0.022440897300839424,
0.013714729808270931,
-0.016979414969682693,
0.006224260199815035,
-0.008115947246551514,
-0... | |
55be67bdb097dfd7c66cf15a05aa24a08fa5c5e1 | subsection | 42 | 52 | Proof of Things | REF gives us\forall x^{\prime }\in \operatorname{supp}(\mathbb {Q}^{\prime }):\quad \nabla _x\mathbb {E}_{\tilde{x}\sim \mathbb {P}}\left[\frac{f^*(\tilde{x})-f^*(x)}{\Vert \tilde{x} - x\Vert }\right]\bigg |_{x^{\prime }}=0Lemma 7
Let \mathbb {P} and (\mathbb {Q}_\theta )_{\theta \in \Theta } in \mathcal {P}(X) and fu... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0024803003761917353,
0.01776658184826374,
-0.023932991549372673,
-0.0265277661383152,
-0.009867779910564423,
0.011256747879087925,
0.05058286339044571,
0.038707949221134186,
0.016026556491851807,
0.026100391522049904,
-0.007898802869021893,
-0.005376528017222881,
-0.02338350936770439,
0... | |
d338cfd7cb15f5484f805e75a400ff9eb85b50a9 | subsection | 43 | 52 | Proof of Things | Further, if \mathbb {P},\mathbb {Q}_\theta are such that there exits an f with f(x)-f(x^{\prime })=\Vert x-x^{\prime }\Vert for all x\in \operatorname{supp}(\mathbb {P})
and x^{\prime }\in \operatorname{supp}(\mathbb {Q}) then\nabla _\theta \tau _F(\mathbb {P}\Vert \mathbb {Q}^{\prime }_\theta )=-\frac{1}{2}\nabla _\th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1111/1468-0262.00296",
"end": 1577,
"openalex_id": "https://openalex.org/W2129462326",
"raw": "Milgrom, P. and Segal, I. Envelope theorems for arbitrary choice sets. Econometrica, 70(2):583–601, 2002.",
"source_ref_id": "68c45c772e... | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.005965966731309891,
0.0033053134102374315,
-0.030577486380934715,
-0.04128876328468323,
0.012748248875141144,
-0.0019759403076022863,
0.03158453106880188,
0.0786714181303978,
0.030043449252843857,
0.027144385501742363,
-0.018004707992076874,
0.011596252210438251,
-0.031065750867128372,
... | |
1b8f3e7f3db2fd08558225f0c43d8f20923d405d | subsection | 44 | 52 | Proof of Things | Note that if \mathbb {P} and \mathbb {Q}_\theta
are such that f^*(x)-f^*(x^{\prime })=c\Vert x - x^{\prime }\Vert is possible everywhere, then this term is equal to zero.\nabla _\theta \tau _F(\mathbb {P}\Vert \mathbb {Q}^{\prime }_\theta )|_{\theta _0}
&=\nabla _\theta \mathbb {E}_{\mathbb {P}\otimes \mathbb {Q}^{\pr... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.008764770813286304,
0.012594114057719707,
-0.007044618017971516,
-0.0020557926036417484,
0.003909438848495483,
-0.006564042996615171,
0.04287033528089523,
0.05962180346250534,
0.03493703529238701,
0.017193902283906937,
0.0016619882080703974,
-0.01051162276417017,
0.008871565572917461,
-... | |
ff0d456bac83bb0da4dfcd2a5d9f1706ab8530e4 | subsection | 45 | 52 | Proof of Things | Therefore if we set g_\theta (\cdot )=g(\theta ,\cdot ) we get\nabla _\theta \tau _F(\mathbb {P}\Vert \mathbb {Q}^{\prime })|_{\theta _0}
=\,&\nabla _\theta \mathbb {E}_{x,\tilde{x}\sim \mathbb {P}, z\sim \mathbb {Z},\alpha \sim \mathcal {U}([0,\varepsilon ])}\left[(f^*(x)-f^*(\alpha \tilde{x} + (1-\alpha )g_\theta (z)... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.023578617721796036,
-0.02394488826394081,
-0.009790086187422276,
-0.030797185376286507,
0.0007830924587324262,
-0.013445153832435608,
0.019870135933160782,
0.028828484937548637,
0.008393682539463043,
0.04132744297385216,
-0.03571130335330963,
0.024066977202892303,
-0.022220367565751076,
... | |
5a072c4788552dcde39924adea13bec3be7cb660 | subsection | 46 | 52 | Proof of Things | REF , let the target distribution be the Dirac distribution \delta _0
and the family of generated distributions be the uniform distributions \mathcal {U}([0,\theta ]) with \theta > 0. Then there is no
C\in \mathbb {R} that fulfills Eq. REF for all \theta > 0.For convenience, we'll restrict ourselves to the \lambda =1 c... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.05687575414776802,
0.03655426576733589,
-0.03887323662638664,
-0.0027327975258231163,
0.018277132883667946,
0.014852077700197697,
-0.016904059797525406,
0.0006154992734082043,
0.02582903765141964,
0.020214693620800972,
-0.011190548539161682,
0.0036691573914140463,
-0.019146746024489403,
... | |
6dae6026e72ae31ecc7478668bdd720fb02e29b9 | subsection | 47 | 52 | Proof of Things | This gives us u(\theta )=\int _0^\theta u^{\prime }(x)\;dx and thus\int _0^\theta u^{\prime }(x)\;dx+\frac{2}{\theta }\int _0^\theta u^{\prime }(x)(f^{\prime }(x)+1)\;dx = \frac{2}{\theta }\int _0^\theta u^{\prime }(x)\left(\frac{\theta }{2} + f^{\prime }(x)+1\right)\;dx.Therefore, for the optimal critic it holds f^{\p... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02098127454519272,
0.03561476245522499,
-0.02636774070560932,
-0.005191911943256855,
-0.008880801498889923,
-0.0029335636645555496,
0.0030766178388148546,
0.011802921071648598,
0.012832910753786564,
0.02413991093635559,
-0.03250190243124962,
0.04180995747447014,
-0.014610596932470798,
-... | |
14727331a164e53310da2c31819ecbb0c290c5b8 | subsection | 48 | 52 | CelebA | The parameters used for CelebA training were:'batch_size': 64,'beta1': 0.5,'c_dim': 3,'calculate_slope': True,'checkpoint_dir': 'logs/1127_220919_.0001_.0001/checkpoints','checkpoint_name': None,'counter_start': 0,'data_path': 'celebA_cropped/','dataset': 'celebA','discriminator_batch_norm': False,'epoch': 81,'fid_batc... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
0.0008104291628114879,
-0.02062494494020939,
-0.06214943155646324,
-0.014652559533715248,
0.0014015657361596823,
-0.01108285691589117,
0.009236985817551613,
0.03076961264014244,
-0.019679127261042595,
0.011159133166074753,
-0.032340891659259796,
-0.026681235060095787,
0.015758557245135307,
... | |
65031682f104580a8509b98af41c72a41c5a0cfc | subsection | 49 | 52 | CIFAR-10 | The parameters used for CIFAR-10 training were:BATCH_SIZE: 64BETA1_D: 0.0BETA1_G: 0.0BETA2_D: 0.9BETA2_G: 0.9BN_D: TrueBN_G: TrueCHECKPOINT_STEP: 5000CRITIC_ITERS: 1DATASET: cifar10DATA_DIR: /data/cifar10/DIM: 32D_LR: 0.0003FID_BATCH_SIZE: 200FID_EVAL_SIZE: 50000FID_SAMPLE_BATCH_SIZE: 1000FID_STEP: 5000GRADIENT_PENALTY... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0064200893975794315,
-0.04479948431253433,
-0.07379097491502762,
-0.004405943676829338,
-0.001077644294127822,
-0.03686496987938881,
0.02317793481051922,
0.029006751254200935,
-0.01156607922166586,
0.027770796790719032,
-0.04638638719916344,
-0.013717553578317165,
-0.02719096839427948,
... | |
357241173c300a6a7984fcdab18d9ff7a0d526b0 | subsection | 50 | 52 | LSUN | The parameters used for LSUN Bedrooms training were:BATCH_SIZE: 64BETA1_D: 0.0BETA1_G: 0.0BETA2_D: 0.9BETA2_G: 0.9BN_D: TrueBN_G: TrueCHECKPOINT_STEP: 4000CRITIC_ITERS: 1DATASET: lsunDATA_DIR: /data/lsunDIM: 64D_LR: 0.0003FID_BATCH_SIZE: 200FID_EVAL_SIZE: 50000FID_SAMPLE_BATCH_SIZE: 1000FID_STEP: 4000GRADIENT_PENALTY: ... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.015849173069000244,
-0.019922060891985893,
-0.06327465921640396,
-0.042528871446847916,
-0.01495679933577776,
-0.02143223211169243,
-0.006292381323873997,
0.00819153618067503,
-0.0240559633821249,
0.015193240717053413,
-0.039355985820293427,
0.025154270231723785,
-0.029166139662265778,
... | |
a3a7e6a2d172289063f32e7eb24885a975304b27 | subsection | 51 | 52 | Billion Word | The parameters used for the Billion Word training were one run with the following settings, followed by
a second run using initialized with the best saved model from the first run and learning rates divided by 10.
Samples from our method and the WGAN-GP baseline can be found in figure REF'activation_d': 'relu','batch_n... | {
"cite_spans": []
} | 1802.04591 | First Order Generative Adversarial Networks | [
"Calvin Seward",
"Thomas Unterthiner",
"Urs Bergmann",
"Nikolay Jetchev",
"Sepp Hochreiter"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.023006334900856018,
0.0004972564638592303,
-0.039452508091926575,
-0.014462868683040142,
-0.0059651704505085945,
-0.02091623656451702,
0.024196317419409752,
-0.0006841447902843356,
-0.023494532331824303,
0.007216177880764008,
-0.05342717841267586,
-0.0005330131389200687,
-0.00722380587831... | |
6af1f5cf0322c6c29a34e02c43ce95605db177bc | abstract | 0 | 34 | Abstract | To fix a software bug, you must first find it. As software grows in size and
complexity, finding bugs is becoming harder. To solve this problem, measures
have been developed to rank lines of code according to their "suspiciousness"
wrt being faulty. Engineers can then inspect the code in descending order of
suspiciousn... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.07257583737373352,
0.01915111020207405,
-0.026597917079925537,
-0.019227409735322,
0.013039540499448776,
-0.028154421597719193,
0.011635635048151016,
-0.013840682804584503,
0.0086904838681221,
0.011902681551873684,
-0.028047602623701096,
0.02760506607592106,
-0.013306587934494019,
0.038... | |
674ae8a3c11e1749840cc9f36f8d84036d07f5ec | subsection | 1 | 34 | Introduction | Software fault localisation is the problem of quickly identifying the parts of the code that caused an error. Accordingly,
the development of effective and efficient
methods for fault localisation has the potential to greatly reduce costs,
wasted programmer time, and the possibility of catastrophe .
In this paper, we f... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/2001420.2001445",
"end": 1347,
"openalex_id": "https://openalex.org/W2067436653",
"raw": "Chris Parnin and Alessandro Orso. 2011. Are Automated Debugging Techniques Actually Helping Programmers?. In International Symposium on Softwa... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.06334000825881958,
0.030220750719308853,
-0.05177649110555649,
-0.004477431066334248,
0.008855702355504036,
-0.007375937886536121,
0.030907239764928818,
-0.018397893756628036,
0.028359604999423027,
0.029061349108815193,
-0.04353863000869751,
0.039145100861787796,
0.011525379493832588,
0... | |
be235442cb359d36159da48fb2103573f6175eac | subsection | 2 | 34 | Preliminaries | To reconstruct statistical fault localisation (sfl) from the ground up, we must precisely define our terms. sfl conventionally assumes a number of artifacts are available. This includes a program (to perform fault localisation on), a test suite (to test the program on), and some units under test located inside the prog... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/2483760.2483767",
"end": 353,
"openalex_id": "https://openalex.org/W2059018540",
"raw": "Friedrich Steimann, Marcus Frenkel, and Rui Abreu. 2013. Threats to the Validity and Value of Empirical Assessments of the Accuracy of Coverage... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.053749844431877136,
0.014025016687810421,
-0.05646633356809616,
-0.009332206100225449,
0.04309754818677902,
-0.019366426393389702,
0.021365640684962273,
0.022128699347376823,
-0.004875944461673498,
0.02701227366924286,
0.020663626492023468,
0.029438799247145653,
0.025882946327328682,
0.... | |
e7e7a81790ba2b26078c51fb5b783230619fbd8b | subsection | 3 | 34 | Preliminaries | We let
U^* = U - \lbrace e\rbrace and U_{|U|} = e.In Figure REF ,
the uuts are the statements labeled in comments
marked u1, \dots , u4. Accordingly, the set of units is U = \lbrace u_1, u_2, u_3, u_4, e\rbrace .Coverage Matrices. A useful way to represent the coverage details of a test suite is in the form of a covera... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1607.04347",
"end": 1024,
"openalex_id": "https://openalex.org/W2514300030",
"raw": "Higor Amario de Souza, Marcos Lordello Chaim, and Fabio Kon. 2016. Spectrum-based Software Fault Localization: A Survey of Techniques, Advan... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.056553978472948074,
0.036990515887737274,
-0.04150750860571861,
-0.007572831120342016,
0.009545200504362583,
-0.06058264896273613,
0.009239998646080494,
0.059575483202934265,
-0.0029261268209666014,
0.028887394815683365,
-0.016648784279823303,
0.031985197216272354,
0.03268716111779213,
... | |
98b1bb8fb966357dd6fa89a1cc2980bf561305cd | subsection | 4 | 34 | Preliminaries | For each u \in U^* s(u_i) is called u_i's degree of suspiciousness, and is defined as a function of u_i's spectrum, which is the vector \langle c_{ef}^i,c_{nf}^i,c_{ep}^i,c_{np}^i \rangle .The intuition behind sbh's is that s(u_i) > s(u_j) just in case u_i is more "suspicious" wrt being faulty than u_j.
In spectrum-bas... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/2000791.2000795",
"end": 998,
"openalex_id": "https://openalex.org/W2010833880",
"raw": "Lee Naish, Hua Jie Lee, and Kotagiri Ramamohanarao. 2011. A Model for Spectra-based Software Diagnosis. ACM Trans. Softw. Eng. Methodol. (2011)... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.058388881385326385,
0.02389884553849697,
-0.062448326498270035,
0.00489117531105876,
-0.00027088410570286214,
0.010965083725750446,
0.03268922492861748,
-0.01568838767707348,
0.02086189202964306,
0.024295633658766747,
-0.030262716114521027,
0.05765634775161743,
0.03598561882972717,
0.01... | |
9bb0a652ae119818ad249e07530b5c8f007737a9 | subsection | 5 | 34 | Body | We present DoricGiven our goal of
providing a simple foundation to statistical fault localisation, we name our
framework after this simple type of Greek column, our formal framework based on
probability theory. We proceed
in four steps. First, we define a set of models to represent the universe of
possibilities. Each m... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.02817796915769577,
-0.01824626512825489,
0.005522698629647493,
-0.0010069009149447083,
0.014729739166796207,
0.0017773326253518462,
0.03167160972952843,
-0.0022826900240033865,
0.045768219977617264,
0.04286956787109375,
-0.04052013158798218,
0.03768250346183777,
0.011190329678356647,
0.... | |
cf12bf31dbd8407e1d5c4dfeea7fe3334909e45c | subsection | 6 | 34 | The Models of Doric | [Figure: Causal Models.]In our framework, classical probabilities are defined in terms of the proportion of models in which a given formula is true. To achieve this, we first define a set of models for our system. We first describe some notation used in the forthcoming definition of models here. Let j \in \mathbb {N} a... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.057657644152641296,
-0.029561370611190796,
-0.05567365884780884,
-0.013269804418087006,
0.008111444301903248,
-0.031713228672742844,
-0.013796323910355568,
0.04312877357006073,
0.007096560206264257,
0.0252881720662117,
-0.04450229927897453,
0.030110782012343407,
0.007008806802332401,
0.... | |
97e2521b30774819190bca5e2cdeb3c0b964adee | subsection | 7 | 34 | The Syntax and Semantics of Doric | What sort of hypotheses does the engineer want to estimate the likelihood of? In
this section, we present a language fundamental to the fault localisation task.
This language includes hypotheses about which line of code was faulty, which
caused the error in which test case, etc. We develop such a language as follows.
F... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.030141983181238174,
0.014712340198457241,
-0.019855555146932602,
0.017047390341758728,
0.021915892139077187,
-0.03839859738945961,
0.00478837825357914,
0.0005150845390744507,
0.01503283716738224,
0.005562912672758102,
-0.056865330785512924,
0.005883410107344389,
0.00367999286390841,
0.0... | |
64049ff9cca3dfe2f5b96afefab8ad1aec1f5f63 | subsection | 8 | 34 | The Syntax and Semantics of Doric | Then the set of valuations is a set V = \lbrace v_1,\dots , v_{|V|} \rbrace , where for each t_k \in T there is some v_k \in V with signature v_k : L \rightarrow 2^M, defined inductively as follows:v_k(u_i) = \lbrace m^j \in M| m_{i,k}^j \in \lbrace 1, \bullet \rbrace \rbrace , for u_i \in U
v_k(h_i) = \lbrace m^j \in... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.05141785368323326,
0.003732371609658003,
-0.041561491787433624,
0.022901542484760284,
0.003019082359969616,
-0.004344579763710499,
-0.02602933533489704,
0.03979162126779556,
0.022077636793255806,
-0.0007252091309055686,
-0.018156452104449272,
0.010001306422054768,
0.02828744798898697,
0... | |
e0d7050ac5014e8f52ac5e52315fbefbe27d51e2 | subsection | 9 | 34 | The Probability Theory of Doric | We want to determine the probability of a given hypothesis.
We do this by presenting our theory of probability. The theory is based around the following assumptions:
We assume the engineer does not always know which hypotheses are true of each test case. Accordingly, we want our probabilities about hypotheses to take a... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.03655867278575897,
0.015548116527497768,
-0.011580982245504856,
0.009383799508213997,
-0.004722416400909424,
-0.04745303466916084,
0.04702580347657204,
0.007106511387974024,
0.032194823026657104,
0.021987082436680794,
-0.051969464868307114,
-0.003726817900314927,
0.014525816775858402,
0... | |
5a2bfd05c6ae80f34513743682c94a045d85e2b4 | subsection | 10 | 34 | The Probability Theory of Doric | We then have the following result:For all \phi \in L^*, P(\phi ) = \frac{f(\phi )}{|T|}See Appendix.Using this result, we can identify many sbh's with an intuitive probabilistic expression stated within Doric. For example, P(u_i \wedge e) = c_{ef}^i/|T|, P(u_i \wedge \lnot e) = c_{ep}^i/|T|, P(\lnot u_i \wedge \lnot e)... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1002/smr.1616",
"end": 523,
"openalex_id": "https://openalex.org/W2147699889",
"raw": "Lucia, David Lo, Lingxiao Jiang, Ferdian Thung, and Aditya Budi. 2014. Extended comprehensive study of association measures for fault localization. Jo... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.037452757358551025,
0.027303580194711685,
-0.03690332919359207,
-0.004052040632814169,
-0.03931471332907677,
-0.042641811072826385,
0.008775605820119381,
-0.020344143733382225,
0.027288317680358887,
0.01791749708354473,
-0.014956684783101082,
0.0565912090241909,
-0.004963940475136042,
0... | |
1dd784525c1498ab49d64459f12ea080f7b03817 | subsection | 11 | 34 | Classical Interpretation | What conditions hold on the relative likelihood function? The question here is which causal models are more likely than others. To illustrate our framework, we will impose conditions on the relative likelihood function to give us a classical interpretation of probability. Informally, probability has a classical interpr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 473,
"openalex_id": "",
"raw": "E. T. Jaynes. 2003. Probability theory: The logic of science. Cambridge University Press, Cambridge.",
"source_ref_id": "2de1cc4e5c9a51654860837270ac4987f1ab227a",
"start": 273
}
]... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.07616499066352844,
-0.007643962744623423,
-0.017896942794322968,
0.003184984438121319,
0.00977999996393919,
-0.00964268296957016,
0.022611482068896294,
0.014189391396939754,
0.0589546337723732,
0.005042573902755976,
-0.07390689849853516,
-0.0017412519082427025,
0.00017987056344281882,
0... | |
42f123b3da83f542555113f7c19fdb3e2e115377 | subsection | 12 | 34 | Classical Interpretation | Let i \in [0, |U|] and m, t \in [0, |T|] be free variables, and let
\rho ^k = {\sum _{u_i \in U^*}} c_{j,k} , then:~
P(f_i) = P(\bigvee \limits _{k = 1}^{|T|} \Diamond _k h_i)~
P(\bigvee \limits _{k = m}^{|T|} \Diamond _k h_i) = (1 - P_k(h_i)P(\bigvee \limits _{j = k+1}^{|T|} \Diamond _j h_i)) + P_k(h_i)~
P_t(h_i) = {\... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.01014935877174139,
0.03604930266737938,
-0.06281919032335281,
-0.01889459602534771,
-0.010034892708063126,
0.020161358639597893,
-0.00029975903453305364,
0.02896764501929283,
0.0012839321279898286,
0.015491127036511898,
-0.045756056904792786,
-0.0022378191351890564,
-0.0001390290417475626... | |
d91d737a53c423cb86d7d50c261c2decb357e0e3 | subsection | 13 | 34 | Measure for Fault Localisation | To develop an efficient fault localisation method based on our framework, we need to do two things. First, we need to identify a probabilistic expression which tells us which unit should be investigated first when looking for faults. Second, we need to identify an efficient way to compute this. In this section, we addr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4324/9780203090732-9",
"end": 1498,
"openalex_id": "https://openalex.org/W1565693361",
"raw": "Karl Popper. 2005. The logic of scientific discovery. Routledge.",
"source_ref_id": "56ec966e2a743c7649d422c0d12de7a7b395ca54",
"s... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.05257898569107056,
0.020735589787364006,
-0.051571957767009735,
0.01090182363986969,
0.007644245866686106,
-0.016097163781523705,
0.011611320078372955,
-0.005237300414592028,
0.016707483679056168,
0.016203969717025757,
-0.026533620432019234,
0.014716317877173424,
0.01526560541242361,
0.... | |
fe2663d5996a0f237fea3fc5fe56859e098c8953 | subsection | 14 | 34 | Measure for Fault Localisation | Let t \in {[1,|T|]} and i \in {[1,|U|]}, then:P(H_i|u_i) = \frac{P(H_i \wedge u_i)}{P(u_i)}P(H_i \wedge u_i) = P(H_i)P(H_i) = \sum _{k} \frac{P_k(H_i)}{|T|}P_t(H_i) = {\left\lbrace \begin{array}{ll}
\end{array}{1}/{(2^{\rho _t}-1)} \:\:\:\:\:\:\:\:\: \textrm {if} \: c_{i,t}, e_{t} = 1 \\
\right.0 \:\:\:\:\:\:\:\:\:\:\:... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.015339896082878113,
0.05690483748912811,
-0.04866660386323929,
-0.02863549068570137,
-0.0028032879345119,
0.016537491232156754,
0.014142300933599472,
0.02611825242638588,
0.005701925605535507,
0.005923137534409761,
-0.043327007442712784,
-0.0008047533920034766,
0.027231939136981964,
0.0... | |
80db85a1955766b766a5d9a55da2392d6fbf76a9 | subsection | 15 | 34 | Measure for Fault Localisation | Thus, P_k(H_i) = c_{i,k}e_k / 2^{\rho _k -1}.To illustrate cl, we find P(H_i|u_i) for each of u_1,\dots ,u_4 for the the running example of minmax.c.
We begin with P(H_1|u_1).
We begin by evaluating the numerator of Eq. REF , which is equal to (0 x 0) / (2^{2}- 1) + (0 \times 0) / (2^{2}- 1) + (0 \times 0) / (2^{1}- 1)... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.0753733441233635,
0.011344145983457565,
-0.027570774778723717,
-0.02123880386352539,
0.007503004278987646,
0.006232795305550098,
0.014037140645086765,
0.01830931380391121,
0.006782074924558401,
0.019163748249411583,
-0.05334113538265228,
0.028913456946611404,
0.013197964057326317,
0.031... | |
8ecedebc54b45c0d21ff11c5b79068a0eb9d25c4 | subsection | 16 | 34 | Semi-Automated Methods | We now address the question of how to use Eq REF in a fault localisation method. We present two such methods. We then compare our methods to sbfl.Our first method is similar to the sbfl procedure discussed in Section . Here, each u_i \in U is associated with a causal likelihood, as determined by using Eq. REF . The eng... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.0811348557472229,
0.010294480249285698,
-0.03321095556020737,
-0.009485574439167976,
-0.011103386990725994,
0.030234789475798607,
0.034554045647382736,
-0.012179385870695114,
0.03174576535820961,
-0.0028540664352476597,
-0.017200710251927376,
0.0380033440887928,
0.013545368798077106,
0.... | |
6d801237b759fd1793aea717cac9cecb5ab6b393 | subsection | 17 | 34 | Semi-Automated Methods | We think it is desirable for a fault localisation method to satisfy a similar property. Accordingly, in this section we show that conditioning on causal likelihood (as per the method of cl_u ) satisfies a similar property, as follows:Let c be a coverage matrix where c_{i,k} = c_{j,k} = 1 for some t_k \in T. Then P(H_i|... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.06823023408651352,
0.02218398079276085,
-0.04516133666038513,
-0.027234116569161415,
0.018812138587236404,
-0.023190956562757492,
0.03579341247677803,
0.0032211781945079565,
0.02477770671248436,
-0.014547748491168022,
-0.01565389521420002,
0.007312017027288675,
0.0373496450483799,
0.024... | |
64ab758af23340ec7185be3e6e05582b28421a61 | subsection | 18 | 34 | Semi-Automated Methods | At step one, for all u_i \in U, P(H_i|u_i) = 1/3. Suppose the engineer learns u_1 is not faulty. Thus, the engineer evaluates all u_i \in U - \lbrace u_1\rbrace next. P(H_2|u_2 \wedge \lnot h_1) = 1, and remain P(H_3|u_3 \wedge \lnot h_1) = P(H_3|u_3 \wedge \lnot h_1) = 1/3. | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.027626309543848038,
0.010663053952157497,
-0.03304173797369003,
-0.003178703598678112,
0.012768206186592579,
-0.007577785290777683,
0.013637725263834,
0.024987241253256798,
0.0011984494049102068,
0.022882089018821716,
-0.026527969166636467,
-0.007932458072900772,
0.037984270602464676,
-... | |
5c63e60fe4a427b3728f42adb6275381b21f55d1 | subsection | 19 | 34 | Empirical Evaluation | In this section, we compare the performance of our new methods with all known 127 sbhs on large faulty programs. The goal of the experiment is to establish whether cl_n and cl_u are effective at fault localisation than sbhs . | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.019888153299689293,
0.011340674012899399,
-0.028511950746178627,
-0.021963970735669136,
-0.012859378010034561,
0.019155513495206833,
0.009844865649938583,
-0.033304646611213684,
0.014141499996185303,
0.029656702652573586,
-0.017140749841928482,
0.014393345452845097,
-0.0029725388158112764... | |
8047e6349d421b403269c6d96d7097e3a6d80b13 | subsection | 20 | 34 | Setup | We first present the benchmarks used in our experiment, then describe the methods compared in the experiment, the methods we used to evaluate the performance of the different methods. Finally, we present some research questions for our experiment to answer.We first describe the benchmarks used in our experiments. We us... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/2610384.2628055",
"end": 2136,
"openalex_id": "https://openalex.org/W2156723666",
"raw": "René Just, Darioush Jalali, and Michael D. Ernst. 2014. Defects4J: A Database of Existing Faults to Enable Controlled Testing Studies for Java... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.03527703508734703,
-0.0007610043394379318,
-0.05697423964738846,
0.008132637478411198,
0.00716374022886157,
-0.02394016645848751,
0.05160334333777428,
-0.00123877776786685,
0.027525851503014565,
0.014823372475802898,
-0.037474218755960464,
0.05193902179598808,
-0.02526763267815113,
0.02... | |
cc1aa52930ca444cd9acfad2a12d6dedfee6f2cc | subsection | 21 | 34 | Setup | These measures are described in Landsberg , and is an attempt at an exhaustive list of sbhs available in the literatureFollowing established conventions on avoiding divisions by zero with the sbhs, we added 0.5 to each of the elements of a spectrum , .
As a baseline for our comparison, we also compared the constant mea... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 252,
"openalex_id": "",
"raw": "David Landsberg. 2016. Methods and Measures for Statistical Fault Localisation (doctoral thesis). Ph.D. Dissertation. University of Oxford.",
"source_ref_id": "ef185b5e063b8749d760e408ab9a5f00... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.042040809988975525,
0.015711912885308266,
-0.046708621084690094,
-0.0009610199485905468,
0.01504835207015276,
-0.0410340242087841,
0.05473237484693527,
0.00284111057408154,
0.008427992463111877,
0.03328485041856766,
-0.011311052367091179,
0.04365776479244232,
-0.00633434159681201,
0.035... | |
6e70a846dae6cc73f3ba2012c107996195ed0b5b | subsection | 22 | 34 | Results | We first directly answer our two research questions. First,
Which method is the most accurate?
For Defects4j cl_n has the best accuracy score (213.6). For Steimann's benchmarks, cl_u has the best accuracy score (4.9) (cl_n came second with 5.02).
Second, Do our new techniques have the best n-scores for any n \in [0,10]... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.051659539341926575,
0.005748001392930746,
-0.046075548976659775,
-0.045892469584941864,
0.01844547688961029,
-0.015867076814174652,
0.01824713870882988,
-0.019132032990455627,
-0.008734520524740219,
0.05272751674056053,
-0.010206802748143673,
0.030864516273140907,
-0.025097444653511047,
... | |
59e79644817625b2ce1aabcda4072f266e88169b | subsection | 23 | 34 | Results | For each n \in [0,10], and for the set of 2-fault programs, D3 outperformed cl_u or cl_n at every value of n. For each n \in [0,10], and for each of the sets of 4, 8, 16, 32 fault programs, cl_u or cl_n outperformed all sbfl methods at every value of n. cl_u and cl_n n-scores were always similar (+/- 2 percent of on on... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/2001420.2001445",
"end": 1187,
"openalex_id": "https://openalex.org/W2067436653",
"raw": "Chris Parnin and Alessandro Orso. 2011. Are Automated Debugging Techniques Actually Helping Programmers?. In International Symposium on Softwa... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.03207116201519966,
-0.003667509648948908,
-0.04867126792669296,
-0.021055281162261963,
0.011328657157719135,
-0.003114426974207163,
0.02235216461122036,
-0.017256176099181175,
-0.0036999317817389965,
0.03068273328244686,
-0.011847410351037979,
0.01421231497079134,
-0.027661757543683052,
... | |
17fd0f65cdb2073abe981d3f70955df6bbd9a887 | subsection | 24 | 34 | Results | We investigated reasons why for this, and after investigating the cases where cl_n outperforms cl_u in terms of accuracy on Defects4j, we discovered there were two major outliers that made cl_u 's overall accuracy score lower (in Chart-5 cl_u had to investigate 3177 more lines of code, and Math-6 cl_u had to investigat... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.06047964096069336,
0.023099439218640327,
-0.05040987208485603,
-0.03475596010684967,
0.00667122146114707,
-0.019666563719511032,
-0.023450354114174843,
-0.005168384872376919,
0.006514835171401501,
0.02764609083533287,
-0.016477802768349648,
0.015898028388619423,
-0.031048452481627464,
0... | |
b7c43181a158e267c41236e44cbf7ce2cb39a0d4 | subsection | 25 | 34 | Threats to Validity | Our threats are informed by the recent work of , who perform a similar test on the Defects4j benchmarks, and by Steimann et al., who perform similar experiments on the Steimann benchmarks .The main threat is wrt how well our results generalize to practical instances of fault localisation. Given the variety of programs,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/2610384.2628055",
"end": 189,
"openalex_id": "https://openalex.org/W2156723666",
"raw": "René Just, Darioush Jalali, and Michael D. Ernst. 2014. Defects4J: A Database of Existing Faults to Enable Controlled Testing Studies for Java ... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.052047789096832275,
0.025657575577497482,
-0.033701326698064804,
-0.016072236001491547,
-0.02033069171011448,
0.019109629094600677,
0.025703366845846176,
-0.00644110469147563,
0.011432197876274586,
0.011279565282166004,
-0.03397606313228607,
0.06544894725084305,
-0.029931295663118362,
0... | |
61a26265480d393e2c987332934b7fc866a30a2a | subsection | 26 | 34 | Related Work | The recent survey of Wong et al. identifies the most prominent fault localisation methods to be spectrum based , , , , , , , , , slice based , , , , , model based , , ,
and mutation-based , . For reasons of space, we discuss closely related statistical approaches.We first discuss sbh, which is one of the most lightw... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/tse.2016.2521368",
"end": 194,
"openalex_id": "https://openalex.org/W2343875716",
"raw": "W. Eric Wong, Ruizhi Gao, Yihao Li, Rui Abreu, and Franz Wotawa. 2016b. A Survey on Software Fault Localization. IEEE Trans. Softw. Eng. 42, 8... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.05477200821042061,
0.004126971587538719,
-0.06304121017456055,
0.0026718517765402794,
0.01919308863580227,
-0.005603069439530373,
0.053765058517456055,
0.008337855339050293,
0.033992208540439606,
0.00008230342791648582,
-0.05736566707491875,
0.044549934566020966,
0.01958976499736309,
0.... | |
3ccaa10a2d036cc69978fab33128105b9c3a2d41 | subsection | 27 | 34 | Related Work | Secondly, in fully-automated fault
localisation subroutines within algorithms which inductively synthesize (such as
cegis ) or repair programs (such as GenProg ).
Thirdly, as a technique combined with other methods , , , . Finally, as a potential substitute for heavyweight methods which cannot scale to large programs. ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4204/eptcs.157.10",
"end": 162,
"openalex_id": "https://openalex.org/W2017565015",
"raw": "Susmit Jha and Sanjit A. Seshia. 2014. Are There Good Mistakes? A Theoretical Analysis of CEGIS. In 3rd Workshop on Synthesis (SYNT). 84–99.",
... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.03100501373410225,
-0.01457174587994814,
-0.04092295467853546,
0.03195103257894516,
0.019744334742426872,
-0.012244843877851963,
0.028868841007351875,
0.007526192348450422,
0.030730362981557846,
0.009032956324517727,
-0.04681268706917763,
0.03283601626753807,
0.006046130321919918,
0.009... | |
a3ac701eceef07ed24d5549e5486f18a3a274016 | subsection | 28 | 34 | Conclusion | In this paper, we have demonstrated there is a principled formal foundation (Doric) available for statistical fault localisation that does not require recourse to spectrum-based heuristics. In general, Doric opens up a world of different meaningful probabilities which can be reported to the engineer to aid in understan... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.infsof.2014.06.014",
"end": 1648,
"openalex_id": "https://openalex.org/W2065111847",
"raw": "Tihana Galinac Grbac and Darko Huljenić. 2015. On the probability distribution of faults in complex software systems. Information and Sof... | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.047418177127838135,
0.0030933190137147903,
-0.0034880891907960176,
-0.01795727014541626,
-0.008437017910182476,
-0.021359538659453392,
0.033595502376556396,
-0.022763166576623917,
0.00824630819261074,
0.006636714097112417,
-0.04207829385995865,
0.024548213928937912,
-0.0073232706636190414... | |
7bdd18f01c43e5608f1bc0f7c9ad08362f5a67a9 | subsection | 29 | 34 | Proofs | In this appendix, we present the proofs supporting the main text. To simplify, we have put a proof later in our order of presentation if a part of that proof relies on a part of an earlier proof.To aid in the proofs we introduce some
notation. For each m^j \in M, m^j_{i,k} is the value at the ith column and kth row. m^... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.005736520513892174,
0.008902286179363728,
-0.03911818563938141,
-0.0019490438280627131,
-0.0027061577420681715,
-0.01897933892905712,
0.02637884020805359,
0.01312076486647129,
0.023510579019784927,
0.017102764919400215,
-0.051994845271110535,
-0.02640935219824314,
0.0027576491702347994,
... | |
d8d88787cd6645995505e8f53075c8e8b2e1398f | subsection | 30 | 34 | Proofs | 3 and 2).
=
\lbrace m^j|m^j_{1,k} = 2\rbrace \cap \lbrace m^j|m^j_{1,k} \in \lbrace 1,0\rbrace \rbrace \cap \dots
=
\lbrace m^j|m^j_{1,k} = 2 \wedge m^j_{1,k} \in \lbrace 1,0\rbrace \wedge \dots \rbrace
=
\lbrace m^j|m^j_{1,k} \in \lbrace 1,2\rbrace \wedge m^j_{1,k} = 2 \wedge m^j_{1,k} \in \lbrace 1,0\rbrace \wedge ... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.024222856387495995,
0.008043758571147919,
-0.04884256049990654,
-0.035594016313552856,
-0.005082678981125355,
0.033304519951343536,
-0.022162310779094696,
0.014179605059325695,
0.028832372277975082,
0.003481940133497119,
-0.058397386223077774,
-0.017507005482912064,
0.01102010253816843,
... | |
460a2b81d7935643895149e01f638cc364ee0cb6 | subsection | 31 | 34 | Proofs | REF and H_i), and so P_1(H_i) = 0.Equation REF follows using the defs.We must show P(u_i) = \sum _k c_{i,k}/|T|. P(u_i) is equal to f(u_i)/|T| (by proposition REF ). The latter is equal to \sum _k f_k(u_i)/|T|. It remains to show f_k(u_i) = c_{i,k}, for both cases when c_{i,k} is 1 or 0 (given c is a Boolean matrix). A... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.02620355226099491,
0.022495070472359657,
-0.03815310448408127,
-0.0482560470700264,
-0.0007153707556426525,
0.0211062990128994,
0.022189846262335777,
-0.0008159995777532458,
0.04263991117477417,
0.026035679504275322,
-0.02815699204802513,
-0.00821817945688963,
0.01424636971205473,
0.021... | |
0316e03cf46a30fdab36107567985254db89d007 | subsection | 32 | 34 | Proofs | Given indifference, let w(\lbrace m^k\rbrace ) = x \times |\lbrace m^k\rbrace |. Then P_k(\phi ) = \sum _{i=1}^{j} x|\lbrace m^i\rbrace | / \sum _{k=1}^{|M|} x|\lbrace m^k\rbrace | (by substitution). Thus, P_k(\phi ) = x\sum _{i=1}^{j} |\lbrace m^i\rbrace | / x\sum _{k=1}^{|M|} |\lbrace m^k\rbrace | (by distribution). ... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.03301168605685234,
0.02120436355471611,
-0.04573430120944977,
-0.057511113584041595,
-0.027672456577420235,
-0.01141832023859024,
0.030647169798612595,
0.0022767994087189436,
0.02887759730219841,
0.03475074842572212,
-0.03499482572078705,
-0.025277432054281235,
-0.00004945936598232947,
... | |
003a916fdde83bbea1294fcf8aee9ea07abc391f | subsection | 33 | 34 | Proofs | Equivalently, \sum _k |v_k(u_i)| > \sum _k |v_k(u_i) \cap (M - v(h_j)))| (by def. REF ).
It is sufficient to show M - v(h_j) \subset v_k(u_i). v_k(u_i) = M (given c_{i,k} = 1).
Thus it suffices to show
M - v(h_j) \subset M.
To prove this, it is sufficient to show v(h_j) \ne \emptyset . This holds given c_{j,k} = 1 (giv... | {
"cite_spans": []
} | 1810.00798 | Doric: Foundations for Statistical Fault Localisation | [
"David Landsberg",
"Earl Barr"
] | [
"cs.SE"
] | 2,018 | en | Computer Science | [
-0.0347214937210083,
0.021540751680731773,
-0.05031260475516319,
-0.042074643075466156,
-0.021784840151667595,
0.03502660617232323,
-0.022547613829374313,
0.014607131481170654,
0.004065588116645813,
0.00045790307922288775,
-0.040274493396282196,
-0.013699430041015148,
-0.016491184011101723,
... | |
b70ad9a8fb1c325c2d0fef10226ed014b0db47c4 | abstract | 0 | 110 | Abstract | The AdS/Ricci-flat (AdS/RF) correspondence is a map between families of
asymptotically locally AdS solutions on a torus and families of asymptotically
flat spacetimes on a sphere. The aim of this work is to perturbatively extend
this map to general AdS and asymptotically flat solutions. A prime application
for such map... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.05494561791419983,
0.052991997450590134,
-0.018391519784927368,
-0.0013202211121097207,
-0.019688846543431282,
-0.024053970351815224,
0.038309305906295776,
-0.05296147242188454,
0.04682587832212448,
0.04072080925107002,
0.009684165008366108,
0.025259722024202347,
0.0031822670716792345,
... |
1558a3e4d3a7fb0357100efa570453e1e89ade12 | subsection | 1 | 110 | Introduction | The advent of the holographic principle , , and its precise realization in the form of the AdS/CFT correspondence , has provided us with deep insights into the nature of spacetime and gravitational forces. This was enabled by special properties of anti-de Sitter (AdS) gravity, that grant better control on the
asymptoti... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 205,
"openalex_id": "https://openalex.org/W2963584573",
"raw": "G. 't Hooft, “Dimensional reduction in quantum gravity,” Salamfest 1993:0284-296 [gr-qc/9310026].",
"source_ref_id": "4c1407d0603270e21b4f1c55b3b9c05a1d350e0c",... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.0610336996614933,
0.0126187177374959,
-0.002700741169974208,
0.011230201460421085,
-0.005390038713812828,
-0.08233446627855301,
0.0023440755903720856,
-0.030837276950478554,
0.044066332280635834,
0.03332440182566643,
-0.000635131960734725,
0.013579998165369034,
-0.0011253089178353548,
0... |
756d2abb5618a25ba6940ae032415e502e349923 | subsection | 2 | 110 | Introduction | It can be used to map families of solutions of AdS gravity to families of solution of vacuum Einstein gravity, and has therefore the potential to enlighten us upon the formulation of holography on Ricci-flat manifolds by mapping to them the well-known holographic tools developed in the context of AdS/CFT. This line of ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1103/physrevd.87.061502",
"end": 672,
"openalex_id": "https://openalex.org/W2046095207",
"raw": "M. M. Caldarelli, J. Camps, B. Goutéraux and K. Skenderis, “AdS/Ricci-flat correspondence,” JHEP 1404 (2014) 071 [arXiv:1312.7874 [hep-th]].... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.04278096184134483,
0.02435036189854145,
-0.012510837987065315,
-0.02146676741540432,
0.00553070567548275,
-0.025815045461058617,
-0.01360935065895319,
-0.00016019975009839982,
0.06105899065732956,
0.031139779835939407,
-0.016645517200231552,
0.021634595468640327,
0.008208329789340496,
0... |
ef64b7612464b151632da5d64a1f2b9948641702 | subsection | 3 | 110 | Introduction | The map works in the other direction as well: starting from a family of (n+p+3)-dimensional Ricci-flat manifolds that have a round sphere {\mathcal {S}}^{n+1} warped over a (p+2)-dimensional base, one casts their metrics in the form (REF ), reads off the metric \tilde{g}_{ab}(y;n) and the scalar field \tilde{\phi }(y;n... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1126-6708/2009/04/062",
"end": 1226,
"openalex_id": "https://openalex.org/W3105542986",
"raw": "I. Kanitscheider and K. Skenderis, “Universal hydrodynamics of non-conformal branes,” JHEP 0904 (2009) 062 doi:10.1088/1126-6708/2009/04... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.048580415546894073,
0.01882796362042427,
-0.023832477629184723,
-0.008490893058478832,
-0.0007247392786666751,
-0.01946878433227539,
0.031079871580004692,
-0.028470810502767563,
0.030408533290028572,
0.027219681069254875,
-0.012824070639908314,
0.016417251899838448,
0.0036637475714087486,... |
553f0055c7e047c4a555f7f8768dca0ef19d5143 | subsection | 4 | 110 | Introduction | Moreover, the study of these linear perturbations is the first, necessary step towards translating the AdS/CFT dictionary to AF spacetimes, since these fluctuations of the metric determine the correlation functions of the dual CFT operators .To be more precise, recall that the AdS/RF correspondence maps AdS_{d+1} on th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/0264-9381/19/22/306",
"end": 242,
"openalex_id": "https://openalex.org/W2153274301",
"raw": "K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067].",
"source_ref_id":... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.02406912110745907,
0.018696686252951622,
-0.044841520488262177,
-0.021474478766322136,
-0.012461913749575615,
-0.012935054488480091,
0.011492738500237465,
-0.06959746032953262,
0.035165030509233475,
0.049603451043367386,
-0.0247254129499197,
-0.007207763381302357,
0.025580119341611862,
0... |
ab2d7f4c6ed1327d6886dc579fee27f73caae06e | subsection | 5 | 110 | Notation and conventions | On the RF side of the correspondence we consider D=n+p+3 dimensional spacetimes on which define coordinates X^A, and use early capital latin letters A, B, ... for tensor indices. We will use a bar for all quantities defined in these spacetimes, so the metric is \bar{g}_{AB}, the associated covariant derivative \bar{\na... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.008771752938628197,
0.01113631296902895,
-0.03411067649722099,
-0.032707199454307556,
0.01688753254711628,
-0.015247595496475697,
-0.004881671164184809,
-0.0027230572886765003,
0.04716914892196655,
0.05150163173675537,
-0.027276339009404182,
0.009496375918388367,
-0.0069678230211138725,
... |
9b7f34898648118ba4c2430d091f1811a0c52c64 | subsection | 6 | 110 | Kaluza-Klein reductions | Our aim is to investigate whether it is possible to perturbatively unfreeze the compact manifolds in the AdS/RF map, i.e. we will start with an AdS/RF pair and ask whether we can map across perturbations that depend on the compact manifolds.
We will discuss this in the simplest case where the pair is AdS on a torus/Min... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1126-6708/2006/05/057",
"end": 1012,
"openalex_id": "https://openalex.org/W3104946429",
"raw": "K. Skenderis and M. Taylor, “Kaluza-Klein holography,” JHEP 0605 (2006) 057 [hep-th/0603016].",
"source_ref_id": "c1458093dcf6d452... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.02876252867281437,
0.03860434889793396,
-0.0008182443561963737,
-0.01901226118206978,
0.000869742245413363,
-0.06198057904839516,
0.05248970910906792,
-0.03845176473259926,
0.037139520049095154,
0.05255074426531792,
-0.017501655966043472,
-0.004337267484515905,
0.004867504816502333,
0.0... |
9b9b0338c9f2b9544dcf6fe7de9021792af572c7 | subsection | 7 | 110 | Vacuum Einstein gravity and deformations of Minkowski spacetime | On the one side of the correspondence we have vacuum Einstein gravity in n+p+3 dimensions, by which we simply mean General Relativity with no cosmological constant nor external matter. The equations of motion are \bar{R}_{AB}=0, so that the solutions are Ricci-flat manifolds.
Its vacuum is Minkowski spacetime whose met... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.048724543303251266,
0.009656429290771484,
0.011937055736780167,
-0.047443121671676636,
0.0037565494421869516,
-0.01589573360979557,
0.021143462508916855,
-0.007223253138363361,
0.0588843896985054,
-0.002305415226146579,
-0.03003714047372341,
0.026528485119342804,
-0.007413940969854593,
... |
256402979fdc435e549e4073f2346b2a1595a13a | subsection | 8 | 110 | AdS gravity and deformations of AdS spacetime | On the other side of the correspondence we have AdS gravity in d+1 dimensions, by which we mean General Relativity in presence of the negative cosmological constant (REF ) and no external matter fields. The field equations are thus
\check{G}_{MN}+\Lambda \check{g}_{MN}=0.
The vacuum solution in this case is given by Ad... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.08985568583011627,
-0.01276567205786705,
0.016496941447257996,
-0.06977245211601257,
0.0067834327928721905,
-0.032536059617996216,
0.01657324656844139,
-0.02954493835568428,
0.046240270137786865,
0.026752209290862083,
-0.03317701444029808,
-0.006508737802505493,
-0.0030101959127932787,
... |
fcd4b4133e863d9b398cafd6c971b948ee66e4f3 | subsection | 9 | 110 | AdS/RF correspondence for perturbations respecting the original Ansätze | Let us pause before performing the full KK reduction to consider how the AdS/RF map acts on these perturbed solutions. When the perturbations respect the Ansätze (REF ) and (REF ), the map goes through in a straightforward way, and it is easy to see how the various components get mapped into one another. This is done i... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.026345616206526756,
0.026513421908020973,
-0.012471071444451809,
-0.00008610807708464563,
-0.011502371169626713,
-0.02355392649769783,
0.08341506868600845,
-0.0063423216342926025,
0.030281441286206245,
0.05583379045128822,
-0.029488174244761467,
-0.006533010862767696,
0.006109680980443954... |
6cf838a2aa438b6341a215eef51a4b4ac3700911 | subsection | 10 | 110 | Harmonic decomposition of the fields | In order to perform the KK reductions, we first decompose the fields in scalar, vector and tensor components with respect to the reduced manifold, and expand them in harmonics of the compactification space. Thus, the perturbation of the Minkowski spacetime is expanded in spherical harmonics, while the expansion of the ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.014579696580767632,
0.0229796152561903,
-0.021957281976938248,
0.004928562790155411,
-0.0229643564671278,
-0.024047724902629852,
0.01899709366261959,
-0.025680407881736755,
0.034148991107940674,
0.052581511437892914,
-0.016448888927698135,
-0.024627557024359703,
-0.03707866370677948,
0.... |
621561908dc25fec89562dabac21ebdbb8b494f9 | subsection | 11 | 110 | Mode decomposition on | We can use the spherical symmetry of the metric Ansatz (REF ) to decompose the metric perturbation h_{AB}(y^a,\theta ^i) of Minkowski into scalar h_{ab}, vectorial h_{ai}, and tensorial h_{ij} components under the rotation group \mathsf {SO}(n+2).
We further decompose the latter into a symmetric traceless part h_{(ij)}... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.0007029736298136413,
0.02208310179412365,
-0.012292977422475815,
-0.004383808467537165,
0.014414298348128796,
0.013529143296182156,
0.014780569821596146,
-0.003496745601296425,
0.05106431245803833,
0.03427688032388687,
-0.05625315383076668,
0.03900788351893425,
-0.009553574025630951,
0.... |
4fa69b02831c3d86a72d133d9aab72a6dc769284 | subsection | 12 | 110 | Mode decomposition on | The resulting decomposition of the metric perturbation readsh_{ab}&=
\sum _{{I_\mathsf {s}}}
h_{ab}^{I_\mathsf {s}}(y)\mathbb {S}^{I_\mathsf {s}}(\theta ),
\\
h_{ai}&=r^2\left(
\sum _{{I_\mathsf {v}}}
B^{I_\mathsf {v}}_{\mathsf {(v)}a}(y)\mathbb {V}^{I_\mathsf {v}}_i(\theta )+
\sum _{{I_\mathsf {s}}}
B^{I_\mathsf {s}}... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.0025524632073938847,
0.0013782919850200415,
-0.05136975273489952,
-0.010881353169679642,
-0.002327358117327094,
-0.03183520585298538,
-0.00839375052601099,
0.020129740238189697,
0.013338432647287846,
0.023624591529369354,
-0.07801609486341476,
0.01924457959830761,
0.018481511622667313,
0... |
a2a77000cf6072b89e7c34352d8a8f6d19f6069d | subsection | 13 | 110 | Mode decomposition on | The same remark holds when applied to the AdS mode \hat{\psi }^{(k,l,{\mathbf {m}_\mathsf {t}})}_{\mathsf {t}} introduced in the next paragraph.
h^{I_\mathsf {s}}_{ab}, B^{I_\mathsf {v}}_{\mathsf {(v)}a}, B^{I_\mathsf {s}}_{\mathsf {(s)}a}, \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}, \phi ^{I_\mathsf {v}}_{\mathsf {v}},... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.019789552316069603,
-0.01924026571214199,
-0.01380081009119749,
0.00567595474421978,
-0.019896358251571655,
-0.006408336106687784,
0.02226134017109871,
0.005485230591148138,
0.023848164826631546,
0.0348186269402504,
-0.06921003013849258,
0.005542447790503502,
0.004176862072199583,
0.007... |
6305187af3d969d1699933069b7e308a9c26a470 | subsection | 14 | 110 | Mode decomposition on | In addition, we use parentheses to indicate symmetric traceless combinations on the tangent space to the sphere,A_{(ij)}\equiv \frac{1}{2}\left(A_{ij}+A_{ji}\right)-\frac{1}{n+1}A^k{}_k\sigma _{ij},so that the full tensorial part of the perturbation is given byh_{ij}=r^2\left(
\hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}\... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.013911907561123371,
0.02289668098092079,
-0.01583394780755043,
-0.0058195097371935844,
-0.0019544551614671946,
-0.030066195875406265,
0.01664242520928383,
0.011883087456226349,
0.02533736638724804,
0.053359489887952805,
-0.05052218958735466,
0.0005472474731504917,
0.00568222114816308,
0.... |
ddb1335cc1ad307c270d185f6cbe5e5f7e308d19 | subsection | 15 | 110 | Mode decomposition on | We hope this will not lead to any confusion, since it will always be clear which case we are discussing, and the comparison of the two KK reductions will always be performed using the modes obtained from the harmonic expansion, which are easily differentiated by their notation.
with respect of the group of isometries o... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.004208473954349756,
0.026784665882587433,
-0.04954018443822861,
0.027105165645480156,
-0.012545295991003513,
-0.015597679652273655,
0.02243501879274845,
0.02788352407515049,
0.03156164661049843,
0.06416109949350357,
-0.03269102796912193,
-0.01501009613275528,
-0.029211310669779778,
0.04... |
5cf8cce5e2cc60f51fec6d9b7ae9c0b7af2663eb | subsection | 16 | 110 | Mode decomposition on | \\
&\hphantom{\frac{\ell ^2}{r^2}}\qquad \qquad \qquad \qquad \left.
+\sum _{{\mathbf {m}_\mathsf {s}}}\psi ^{\mathbf {m}_\mathsf {s}}_{\mathsf {s}}(y){\partial }_{(i}{\partial }_{j)}\mathbb {S}^{\mathbf {m}_\mathsf {s}}(\chi )\right),
\\
h^i{}_{i}&\equiv \delta ^{ij}h_{ij}=(d-p-1)\frac{\ell ^2}{r^2}\sum _{{\mathbf {m... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.007836381904780865,
0.011537107639014721,
-0.03171178698539734,
-0.004440871067345142,
-0.02330312691628933,
-0.008271312341094017,
0.01461977418512106,
0.031437091529369354,
0.0030883890576660633,
0.03784659877419472,
-0.049749962985515594,
0.04623999446630478,
0.004498098511248827,
0.... |
3f20f66f76f0ca78042eb770fcaf3d6e22023acc | subsection | 17 | 110 | Mode decomposition on | The overall factors of \ell ^2/r^2 in the definitions (REF )-() ensure that these fields transform covariantly in the reduced theory.
When referring to scalar/vector/tensor modes, we will mean that from the perspective of the (p+2)-dimensional theory.
In addition, we have used parentheses to indicate symmetric traceles... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.04104333743453026,
0.00850619375705719,
-0.02059795893728733,
-0.0018719348590821028,
-0.01367857027798891,
-0.017546409741044044,
0.005523304454982281,
0.055538199841976166,
0.03335343301296234,
0.053157988935709,
-0.03149199113249779,
0.017927853390574455,
-0.00664856331422925,
0.0217... |
9bac2453bddb8b747184be41868d992893dc5c8e | subsection | 18 | 110 | Gauge invariant variables | Not all the fluctuations are independent, as some modes are diffeomorphic to each other or to the background solution.
Indeed, under a change of coordinates\delta X^A \equiv X^{A^{\prime }}-X^A= -\xi ^A,the linearized perturbations transform asIn general the right hand side of (REF ) contains terms which are higher ord... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1126-6708/2006/05/057",
"end": 621,
"openalex_id": "https://openalex.org/W3104946429",
"raw": "K. Skenderis and M. Taylor, “Kaluza-Klein holography,” JHEP 0605 (2006) 057 [hep-th/0603016].",
"source_ref_id": "c1458093dcf6d4525... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.03609466180205345,
-0.006399708800017834,
-0.029382213950157166,
-0.006273850332945585,
-0.009778816252946854,
-0.019084708765149117,
0.006392080802470446,
0.04320375248789787,
0.018581274896860123,
0.03380632773041725,
-0.035911593586206436,
-0.027627823874354362,
0.011929850094020367,
... |
5c0fdd4047345e3fd3c9e018749aeae73b997578 | subsection | 19 | 110 | Gauge invariant variables at linear order for fluctuations of Minkowski | Expand the diffeomorphism generator \xi ^A=\lbrace \xi ^a, \xi ^i \rbrace in spherical harmonics,\xi _a(y,\theta ) \equiv \eta _{ab} \xi ^b =\xi _a^{I_\mathsf {s}}(y)\mathbb {S}^{I_\mathsf {s}}(\theta ),\qquad \xi _i(y,\theta )\equiv \sigma _{ij} \xi ^i =\xi _\mathsf {v}^{I_\mathsf {v}}(y)\mathbb {V}^{I_\mathsf {v}}_i(... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.020154109224677086,
0.031489841639995575,
-0.025677036494016647,
-0.0073003340512514114,
-0.01632467657327652,
-0.01846061460673809,
0.010611039586365223,
0.026745006442070007,
0.04647192731499672,
0.023571612313389778,
-0.024456501007080078,
0.0027957914862781763,
-0.017346875742077827,
... |
9a7a707737477ec8ed234973dd58fb0cee5dd89c | subsection | 20 | 110 | Gauge invariant variables at linear order for fluctuations of Minkowski | On the other hand, \phi ^{I_\mathsf {s}}_{\mathsf {s}} and \phi ^{I_\mathsf {v}}_{\mathsf {v}} are pure gauge field, and they can be complemented by an additional pure gauge field \tilde{B}^{I_\mathsf {s}}_{\mathsf {(s)}a} given by\tilde{B}^{I_\mathsf {s}}_{\mathsf {(s)}a}=r^2\left(B^{I_\mathsf {s}}_{\mathsf {(s)}a}-\f... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.04326575621962547,
0.03104577399790287,
-0.027247052639722824,
-0.0037434184923768044,
-0.006197712849825621,
-0.04369292035698891,
0.024989178404211998,
0.05144292116165161,
0.0333189032971859,
0.04192323610186577,
-0.049581702798604965,
-0.03362402319908142,
0.011571606621146202,
0.01... |
43e8192720b285942032f4ae6a28315ba44ed1f8 | subsection | 21 | 110 | Gauge invariant variables at linear order for fluctuations of Minkowski | Armed with them, we can compensate for the variations in the remaining fields and define the diffeomorphism invariant quantities \hat{h}^{I_\mathsf {s}}_{ab}, \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}a}, and \hat{\pi }^{I_\mathsf {s}} as\hat{h}^{{I_\mathsf {s}}}_{ab}&=h^{I_\mathsf {s}}_{ab}-{\partial }_a\tilde{B}^{I_\math... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.03151755779981613,
0.0029080514796078205,
-0.031548067927360535,
0.008047197945415974,
0.0008314166916534305,
-0.03075478971004486,
0.05168513208627701,
0.056170206516981125,
0.017238548025488853,
0.03829093277454376,
-0.06431656330823898,
0.005865682847797871,
-0.017925038933753967,
0.... |
1dd58b7966e5e888191f337c689f70847efc81d0 | subsection | 22 | 110 | Gauge invariant variables at linear order for fluctuations of Minkowski | As a consequence \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}a} is only defined for l\ge 2 and we have to work with the unhatted variable B^{I_\mathsf {v}}_{\mathsf {(v)}a} when l=1.
The latter field transforms nevertheless as a gauge field from the perspective of the (p+2)-dimensional theory,\delta {B}^{(l=1)}_{(\mathsf {v}... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.0637926533818245,
0.024280888959765434,
-0.00389165710657835,
0.011667340993881226,
0.00042350386502221227,
-0.02449454739689827,
0.00905764102935791,
0.06055723875761032,
0.030644891783595085,
0.02872195653617382,
-0.031346917152404785,
0.0014670020900666714,
0.0162686537951231,
-0.002... |
25cd7a43d678dfa1a954b79858a1a3d8a168b962 | subsection | 23 | 110 | Gauge invariant variables at linear order for fluctuations of AdS | The same approach can be applied to perturbations of the AdS metric.
Consider the diffeomorphism \delta X^M\equiv X^{M^{\prime }}-X^M=-\xi ^M generated by the vector field \xi ^M. Its Fourier decomposition on {\mathcal {T}}^{d-p-1} is\xi _a(y,\chi )&= \check{g}_{ab} \xi ^b = \frac{\ell ^2}{r^2} \eta _{ab} \xi ^b= \frac... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.026390664279460907,
-0.019571805372834206,
-0.035268910229206085,
-0.01166223268955946,
-0.03096708096563816,
-0.0008351958822458982,
-0.007413793355226517,
0.013851284980773926,
0.0381062887609005,
0.043536968529224396,
-0.06297148019075394,
0.005365847609937191,
-0.010068114846944809,
... |
58173d44b84ddb31d9222adc2f37913dcd01b587 | subsection | 24 | 110 | Gauge invariant variables at linear order for fluctuations of AdS | Then the variation (REF ) of the metric perturbation assumes the compact form\delta h_{mn}={\partial }_m\xi _n+{\partial }_n\xi _m-\frac{2}{r}\left(\eta _{mn}\xi _r-\delta _{m}{}^r\xi _n-\delta _{n}{}^r\xi _m\right),from which we can read off how the modes change under this transformation,&\delta h_{ab}^{\mathbf {m}_\m... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.028049105778336525,
-0.0019800716545432806,
-0.032169487327337265,
-0.00614623399451375,
-0.018358582630753517,
0.007576921489089727,
0.019381048157811165,
0.06235508620738983,
0.054022762924432755,
0.04056285321712494,
-0.021532801911234856,
0.0035194915253669024,
0.000670038687530905,
... |
21f885ce4a9f6cf53755299bc157531ede9ffd3c | subsection | 25 | 110 | Gauge invariant variables at linear order for fluctuations of AdS | The remaining gauge degrees of freedom are encoded in the field \tilde{C}^{\mathbf {m}_\mathsf {s}}_{\mathsf {(s)}a},\tilde{C}^{\mathbf {m}_\mathsf {s}}_{\mathsf {(s)}a}=C^{\mathbf {m}_\mathsf {s}}_{\mathsf {(s)}a}-\frac{1}{2}{\partial }_a\psi ^{\mathbf {m}_\mathsf {s}}_{\mathsf {s}},that transforms under diffeomorphis... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.030401986092329025,
0.01379758957773447,
-0.04689197242259979,
-0.0201205313205719,
-0.004580128472298384,
-0.03795290365815163,
0.027274835854768753,
0.05638020113110542,
0.03764781728386879,
0.024361249059438705,
-0.030157914385199547,
-0.008420418947935104,
0.032064713537693024,
0.01... |
4ee6bf2156c958d17868b3e32d52bb390cafed96 | subsection | 26 | 110 | Gauge invariant variables at linear order for fluctuations of AdS | As before, we use these pure gauge fields to compensate for the variations of the remaining modes by defining the gauge invariant fields \hat{h}^{\mathbf {m}_\mathsf {s}}_{ab}, \hat{C}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}, and \hat{\varpi }^{\mathbf {m}_\mathsf {s}},&\hat{h}^{{\mathbf {m}_\mathsf {s}}}_{ab}... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.02385847456753254,
-0.009061339311301708,
-0.05766306445002556,
0.025567010045051575,
-0.010304603725671768,
-0.029365450143814087,
0.021234652027487755,
0.055527396500110626,
0.024926308542490005,
0.05140860751271248,
-0.02367541752755642,
-0.003266429528594017,
0.006464212667196989,
0... |
5f041d41283edf513f518fe08a203eba3b5834a6 | subsection | 27 | 110 | Gauge invariant variables at linear order for fluctuations of AdS | As a consequence \tilde{C}^{\mathbf {m}_\mathsf {s}}_{\mathsf {(s)}a} (and the hatted fields) can be defined for {\mathbf {m}^2_\mathsf {s}}\ne 0 only, and one has to work with the unhatted fields h_{ab}^{\mathbf {m}_\mathsf {s}}, \varpi ^{\mathbf {m}_\mathsf {s}}, and C^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.008671869523823261,
-0.003628533100709319,
-0.027838455513119698,
0.012866705656051636,
-0.016992898657917976,
0.0028563018422573805,
0.0010830303654074669,
0.05097869411110878,
0.05244307592511177,
0.05656164139509201,
-0.02397920750081539,
-0.039263661950826645,
0.02825031243264675,
-... |
aa92d157259a910cb3d963acc06ce5a868c5f3a0 | subsection | 28 | 110 | A note on the De Donder-Lorentz gauge | To perform the Kaluza-Klein reduction it is common to fix the gauge by imposing the De Donder-Lorentz (DDL) gauge fixing condition on the perturbations,ih_{(ij)}=0,\qquad ih_{ia}=0.In the case of the dimensional reduction of Einstein vacuum gravity on a sphere, this gauge condition kills the components B^{I_\mathsf {s}... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.05321420729160309,
0.026210438460111618,
-0.01847545988857746,
-0.013196757063269615,
-0.010542149655520916,
0.011335480026900768,
0.03176375478506088,
0.025569669902324677,
0.04045988246798515,
0.024837365373969078,
-0.0656634047627449,
-0.007212447468191385,
0.005774534773081541,
0.02... |
2b7e40e442f5ca02b62802ab504f91e031a9e14a | subsection | 29 | 110 | A note on the De Donder-Lorentz gauge | We will however not impose this gauge, but rather work with gauge invariant perturbations, as the cancellation of all gauge dependence is a useful consistency check. | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.012162626720964909,
0.027011102065443993,
-0.05884452536702156,
0.01041529793292284,
-0.010209280997514725,
-0.03195550665259361,
0.02600390836596489,
0.029254397377371788,
0.01568780466914177,
0.02189883217215538,
-0.05344230681657791,
-0.013696307316422462,
0.021395234391093254,
0.011... |
97d8e3597336b5d4f7c7adf1c84c14a18f6cd09e | subsection | 30 | 110 | Equation of motion for Kaluza-Klein modes | In this section we derive the field equations that the Kaluza-Klein modes satisfy by substituting the KK expansion in the linearized field equations. | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.032612983137369156,
0.04652460291981697,
-0.03462650999426842,
-0.024787697941064835,
-0.03212485834956169,
0.008938825689256191,
0.04329076409339905,
-0.007619357202202082,
0.04615850746631622,
-0.0037124345544725657,
-0.01337010320276022,
-0.010174397379159927,
-0.033070605248212814,
... |
bc890a471154cd9f4ac631c3dd6faf1b0b2437b4 | subsection | 31 | 110 | Perturbations of Minkowski reduced on a sphere | We first decompose the linearized field equations (REF ) into their scalar ab, vector ai, and tensor ij components under the rotation group of the sphere. | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.007005789317190647,
0.009930905885994434,
-0.017924446612596512,
-0.017192214727401733,
-0.005278177559375763,
0.018778719007968903,
-0.000756545108743012,
-0.010167356580495834,
0.019541461020708084,
0.009496143087744713,
-0.024880656972527504,
-0.009648691862821579,
-0.03828966990113258... |
d580b1d60f476a2b2c976d3fd238ed4074e1079a | subsection | 32 | 110 | Perturbations of Minkowski reduced on a sphere | After evaluating all covariant derivatives on the background metric (REF ), one obtainsE^{(0)}_{ab} = {}& {\partial }_a{\partial }^ch_{bc}+{\partial }_b{\partial }^ch_{ac}-\Box h_{ab}-\frac{1}{r^2}\Box _\theta h_{ab}-{\partial }_a{\partial }_b h^c{}_c-\frac{1}{r^2}{\partial }_a{\partial }_b h^i{}_i\\
&+\frac{1}{r^2}\le... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.06203038990497589,
0.02141924947500229,
-0.012296236120164394,
-0.02015301212668419,
0.0017734955763444304,
0.011899583041667938,
0.014691407792270184,
0.04112984240055084,
0.017651047557592392,
0.02788774110376835,
-0.023692375048995018,
0.02927602455019951,
-0.013867590576410294,
0.00... |
ef8c1b3122444640f0d3f819692ea3fe51053649 | subsection | 33 | 110 | Perturbations of Minkowski reduced on a sphere | The scalar equation becomes simply\left.E^{(0)}_{ab}\right|_{\mathbb {S}^{I_\mathsf {s}}}={}
&
{\partial }_a{\partial }^c\hat{h}^{I_\mathsf {s}}_{bc}+{\partial }_b{\partial }^c\hat{h}^{I_\mathsf {s}}_{ac}-\Box \hat{h}^{I_\mathsf {s}}_{ab}-{\partial }_a{\partial }_b\hat{h}^{I_\mathsf {s}}+\frac{n+1}{r}\left({\partial }_... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.026984013617038727,
0.05079343542456627,
-0.03391316533088684,
-0.0048153032548725605,
0.006860471796244383,
0.010905020870268345,
0.01958172395825386,
0.06230132281780243,
0.019154375419020653,
0.014430646784603596,
-0.041544388979673386,
0.022741051390767097,
-0.01868123933672905,
0.0... |
d79ebf922a68690a22488820e61988225a715d44 | subsection | 34 | 110 | Special cases | When l=0, \mathbb {S}^{I_\mathsf {s}} is constant, and its derivatives vanish. Hence, the perturbations that respect the {\mathcal {S}}^{n+1} are described by the fields h_{ab}^{l=0} and \pi ^{l=0}. These fields verify the two equations \left.E^{(0)}_{ab}\right|_{\mathbb {S}^{I_\mathsf {s}}}=0 and \left.E^{(0)}_{ij}\ri... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.057829342782497406,
0.003761196043342352,
-0.002033181954175234,
-0.002504285192117095,
-0.0009774913778528571,
-0.019362911581993103,
0.031066106632351875,
0.028167009353637695,
0.02091926895081997,
0.02764822356402874,
-0.04107561707496643,
0.014587031677365303,
0.003658201778307557,
... |
86af777910f908863d2c8bd3bb0da5b61a5d08a6 | subsection | 35 | 110 | Special cases | Defining the field strengthsG^{(l=1)}_{ab}={\partial }_a B^{(l=1)}_{(\mathsf {v})b}-{\partial }_b B^{(l=1)}_{(\mathsf {v})a}associated to the l=1 vectors B^{(l=1)}_{(\mathsf {v})a}, equation (REF ), for l=1, takes the form,\partial ^a {G}^{(l=1)}_{ab} + (n+3) \frac{1}{r} {G}^{(l=1)}_{rb} =0,and this equation is manifes... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.025913065299391747,
0.002186128869652748,
-0.031101783737540245,
-0.042455919086933136,
0.0012695189798250794,
-0.03363509848713875,
0.051642999053001404,
0.041112955659627914,
0.055244579911231995,
0.02119743637740612,
-0.058144159615039825,
0.022174136713147163,
0.002594358753412962,
... |
9514d1b37e0dbb50094930512fb436c5036f3a0e | subsection | 36 | 110 | Independent equations | There are linear differential relations between the field equations for these modes,&\left[2\delta _a^b{\partial }^c-\eta ^{bc}{\partial }_a+\frac{2(n+1)}{r}\delta _a^b\delta _r^c\right]
\!\left.E_{ab}^{(0)}\right|_{\mathbb {S}^{I_\mathsf {s}}}
\!\!-\frac{n+1}{r^2}{\partial }_a\!\left.E_{ij}^{(0)}\right|_{\sigma _{ij}\... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.05389155074954033,
0.02561832219362259,
-0.006660153158009052,
-0.011168916709721088,
-0.0314316526055336,
0.007594711147248745,
0.023451674729585648,
0.05141974240541458,
0.045804765075445175,
-0.01618882641196251,
-0.023329610005021095,
0.01940828189253807,
-0.0015181793132796884,
0.0... |
ec48b7befd5d714218bd976d0e81a86d2bcf7954 | subsection | 37 | 110 | Independent equations | \\
&\left[2{\partial }^a{\partial }^b-\eta ^{ab}\Box +\frac{4}{r}(n+2)\delta _r^a{\partial }^b-\frac{n+3}{r}\eta ^{ab}{\partial }_r
+\frac{2}{r^2}(n+1)(n+2)\delta _r^a\delta _r^b
+\frac{\Lambda ^{I_\mathsf {s}}}{r^2}\eta ^{ab}\right]\!\left.E_{ab}^{(0)}\right|_{\mathbb {S}^{I_\mathsf {s}}}\\
&\qquad \qquad -\frac{n+1}{... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.03823027014732361,
0.05790986120700836,
-0.017787907272577286,
-0.024851202964782715,
0.010419503785669804,
-0.024500325322151184,
0.034507930278778076,
0.05803190544247627,
0.02816164493560791,
0.012036586180329323,
-0.05668942257761955,
0.015347028151154518,
-0.016125058755278587,
0.0... |
36b6f7c7b346df14d0cc8e1c49fb55af8e46f5c7 | subsection | 38 | 110 | Independent equations | This will become obvious in the Lagrangian analysis in § .There are linear differential relations between the field equations for the scalar modes,&\left[\delta _a^b{\partial }^c-\frac{1}{2}\eta ^{bc}{\partial }_a-\frac{d-1}{r}\delta _a^b\delta _r^c\right]
\!\left.E_{bc}^{(\Lambda )}\right|_{\mathbb {S}^{\mathbf {m}_\m... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.032989367842674255,
0.013870183378458023,
-0.0598142109811306,
-0.032134879380464554,
-0.010391193442046642,
0.005325295962393284,
0.03625473752617836,
0.032653674483299255,
0.044311344623565674,
0.010703996755182743,
-0.04013045132160187,
0.01771538145840168,
0.01152796857059002,
0.005... |
6e966c86eee151552566390bc0a3ddc863504836 | subsection | 39 | 110 | Independent equations | Note however that they are obtained with {\mathbf {m}^2_\mathsf {s}} prefactors; this was expected, as the latter equations hold for the {\mathbf {m}^2_\mathsf {s}}\ne 0 modes only.
Again, these extra equations are a consequence of the gauge freedom that allows to choose arbitrary fields C^{\mathbf {m}_\mathsf {s}}_{\m... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.002506899880245328,
0.014514836482703686,
-0.07008635997772217,
0.004712361376732588,
-0.015537437982857227,
-0.03189906105399132,
0.013759331777691841,
0.04044618085026741,
0.037485212087631226,
0.009493404999375343,
-0.04697861894965172,
0.0007039924385026097,
0.04801648110151291,
-0.0... |
2218c9bfbc05c13ab351b10406d13f623ae85beb | subsection | 40 | 110 | Perturbations of AdS reduced on a torus | Since the Poincaré metric is conformal to the Minkowski metric, it is quicker to evaluate the linearized equations (REF ) for a small metric perturbation introducing once more the coordinates \zeta ^m=\lbrace r,z^\alpha \rbrace =\lbrace r, \lbrace x^\mu , \chi ^i\rbrace \rbrace and working with the background metric ds... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.027685491368174553,
0.010027214884757996,
0.0105308648198843,
-0.048075687140226364,
0.006287994794547558,
-0.008630730211734772,
0.02716657891869545,
-0.0004964959225617349,
0.023259475827217102,
0.022221650928258896,
-0.057812921702861786,
0.024923047050833702,
-0.0006691486923955381,
... |
73acc534c8db6b681e89766e6724211f280860e6 | subsection | 41 | 110 | Perturbations of AdS reduced on a torus | \\
&\hphantom{+}+\frac{1}{r}\left[2\delta _{m}{}^r{\partial }^p h_{pn}+2\delta _{n}{}^r{\partial }^p h_{pm}
-2\eta _{mn}{\partial }^p h_{p r}-(d-3)\left({\partial }_m h_{n r}+{\partial }_n h_{m r}\right)\right.\\
&\qquad \qquad \left.
+(d-5){\partial }_rh_{mn}-2\left(\delta _{m}{}^r{\partial }_n h^p{}_p+\delta _{n}{}^r... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.04361902177333832,
0.019917013123631477,
-0.007566175889223814,
-0.007032003253698349,
-0.0005732818390242755,
0.01169075258076191,
0.012331760488450527,
0.049601759761571884,
0.03693423047661781,
0.026327086612582207,
-0.05738541856408119,
0.0347975417971611,
-0.04166547581553459,
0.01... |
645476df3059912290c1c10ebf2f8886770d5139 | subsection | 42 | 110 | Perturbations of AdS reduced on a torus | Next, we project these equations on the Fourier basis of the torus {\mathcal {T}}^{d-p-1}, and rewrite all fields in terms of the hatted, gauge invariant fields. | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.015993086621165276,
0.04727727547287941,
-0.0247984416782856,
0.018022743985056877,
-0.006142382975667715,
0.004234810825437307,
0.027148570865392685,
0.0309942364692688,
0.029178228229284286,
0.021410591900348663,
-0.040379490703344345,
0.0025466091465204954,
-0.002872803946956992,
-0.0... |
8e28dc0d6ef79a8fa30e0adfaaef09b0628abe34 | subsection | 43 | 110 | Perturbations of AdS reduced on a torus | The scalar equation becomes simply\left.E_{ab}^{(\Lambda )}\right|_{\mathbb {S}^{\mathbf {m}_\mathsf {s}}}=
\frac{1}{2}&\left\lbrace
-\Box \hat{h}^{\mathbf {m}_\mathsf {s}}_{ab}+{\mathbf {m}^2_\mathsf {s}}\hat{h}^{\mathbf {m}_\mathsf {s}}_{ab}
+{\partial }_a{\partial }^c\hat{h}^{\mathbf {m}_\mathsf {s}}_{bc}+{\partial... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.01835322566330433,
-0.007456474471837282,
-0.03142780065536499,
-0.016598761081695557,
-0.0036042805295437574,
0.008268867619335651,
0.014882436953485012,
0.040520504117012024,
0.04430404305458069,
0.010908192954957485,
-0.021892666816711426,
0.030512427911162376,
-0.0041191778145730495,
... |
62d6dc434636739ce4bf0f69120ed8d1e9e94f37 | subsection | 44 | 110 | Perturbations of AdS reduced on a torus | \\
&\qquad \qquad \qquad \qquad \qquad \left.
-\frac{d-1}{r}\left({\partial }_a\hat{C}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}r}-{\partial }_r\hat{C}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}\right)+{\mathbf {m}^2_\mathsf {v}}\hat{C}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}
\right),
\\
&\left.E_{... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.047640930861234665,
0.043703917413949966,
-0.01889156736433506,
-0.01881526969373226,
-0.026094168424606323,
0.03170975670218468,
0.03041267767548561,
0.04129287600517273,
0.026506181806325912,
0.026307804509997368,
-0.03619612008333206,
0.009621270932257175,
-0.015267377719283104,
0.00... |
a0da2e54ffd9587322cefc2e3cf7c8f974293746 | subsection | 45 | 110 | Perturbations of AdS reduced on a torus | \\
&\qquad \left.\qquad \qquad -\frac{2}{r}{\partial }^a\hat{h}^{\mathbf {m}_\mathsf {s}}_{ar}+\frac{1}{r}{\partial }_r\hat{h}^{\mathbf {m}_\mathsf {s}}+\frac{2d}{r^2}\hat{h}^{\mathbf {m}_\mathsf {s}}_{rr}
+\frac{{\mathbf {m}^2_\mathsf {s}}}{d-p-1}
\hat{h}^{\mathbf {m}_\mathsf {s}}\right\rbrace .Again, the equations of... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.013991917483508587,
0.05599818751215935,
-0.013625716790556908,
-0.018798300996422768,
-0.027190400287508965,
0.02090395614504814,
0.044523898512125015,
0.048918306827545166,
-0.01812693290412426,
-0.0048788608983159065,
-0.04211307689547539,
-0.0009236076730303466,
-0.020095262676477432,... |
4f863550bf5e2063817e842bf2280113417bb8c4 | subsection | 46 | 110 | Quadratic action for the perturbations | We start with the Einstein-Hilbert action in presence of a cosmological constant,S=\int d^DX\sqrt{-g}\left(R-2\Lambda \right).We consider perturbations h_{AB} on top of a background \check{g}_{AB} solving the field equations, g_{AB}=\check{g}_{AB}+h_{AB}, and expand the action up to quadratic order in the perturbation.... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
0.006466674618422985,
-0.019747203215956688,
-0.007340343203395605,
-0.0021269447170197964,
0.013177518732845783,
-0.029193507507443428,
0.017625980079174042,
-0.008889293298125267,
0.013604815118014812,
0.016466176137328148,
-0.05188600346446037,
-0.00001692971636657603,
0.01751915551722049... |
aae8f0a27c7a8555114ae8143b9178955e1ee71b | subsection | 47 | 110 | Quadratic action for the perturbations | Expanding the measure we obtain,\sqrt{-g}=\sqrt{-\check{g}}\left[1+\frac{1}{2}h^A{}_A-\frac{1}{4}\left(h^{AB}h_{AB}-\frac{1}{2}(h^A{}_A)^2\right)+\mathcal {O}(h^3)\right],while the Ricci tensor can be expanded asR_{AB}=\check{R}_{AB}+R^{(1)}_{AB}+R^{(2)}_{AB}+\mathcal {O}(h^3),with \check{R}_{AB} the Ricci tensor of th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1126-6708/2006/05/057",
"end": 836,
"openalex_id": "https://openalex.org/W3104946429",
"raw": "K. Skenderis and M. Taylor, “Kaluza-Klein holography,” JHEP 0605 (2006) 057 [hep-th/0603016].",
"source_ref_id": "c1458093dcf6d4525... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.019713854417204857,
0.01326717995107174,
-0.02494748681783676,
-0.01705889217555523,
0.009483096189796925,
-0.01145143061876297,
0.007903852500021458,
0.0191035196185112,
0.014121650718152523,
0.018706800416111946,
-0.041563887149095535,
0.011405655182898045,
-0.007442285306751728,
-0.0... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.