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The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks<jats:inline-formula><jats:alternatives><jats:tex-math>$$r\\ge 3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>r<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.<\/jats:p>","DOI":"10.1007\/s10208-022-09551-1","type":"journal-article","created":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T23:02:48Z","timestamp":1644361368000},"page":"433-491","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite"],"prefix":"10.1007","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0689-8232","authenticated-orcid":false,"given":"Carlos","family":"Beltr\u00e1n","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3747-9185","authenticated-orcid":false,"given":"Paul","family":"Breiding","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5692-4163","authenticated-orcid":false,"given":"Nick","family":"Vannieuwenhoven","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,8]]},"reference":[{"issue":"6A","key":"9551_CR1","doi-asserted-by":"publisher","first-page":"3099","DOI":"10.1214\/09-AOS689","volume":"37","author":"ES Allman","year":"2009","unstructured":"Allman ES, Matias C, Rhodes JA (2009) Identifiability of parameters in latent structure models with many observed variables. 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