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We say that $$\\mathcal {S}$$<\/jats:tex-math>\n S<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>can be compressed at rate<\/jats:italic>s<\/jats:italic> if we can achieve an error of at most $$\\mathcal {O}(R^{-s})$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n R<\/mml:mi>\n \n -<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for encoding the given signal class; the supremal compression rate is denoted by $$s^*(\\mathcal {S})$$<\/jats:tex-math>\n \n \n s<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n S<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Given a fixed coding scheme, there usually are some<\/jats:italic> elements of $$\\mathcal {S}$$<\/jats:tex-math>\n S<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that are compressed at a higher rate than $$s^*(\\mathcal {S})$$<\/jats:tex-math>\n \n \n s<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n S<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> by the given coding scheme; in this paper, we study the size of this set of signals. We show that for certain \u201cnice\u201d signal classes $$\\mathcal {S}$$<\/jats:tex-math>\n S<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, a phase transition<\/jats:italic> occurs: We construct a probability measure $$\\mathbb {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on $$\\mathcal {S}$$<\/jats:tex-math>\n S<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that for every<\/jats:italic> coding scheme $$\\mathcal {C}$$<\/jats:tex-math>\n C<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and any $$s > s^*(\\mathcal {S})$$<\/jats:tex-math>\n \n s<\/mml:mi>\n ><\/mml:mo>\n \n s<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n S<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the set of signals encoded with error $$\\mathcal {O}(R^{-s})$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n R<\/mml:mi>\n \n -<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> by $$\\mathcal {C}$$<\/jats:tex-math>\n C<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> forms a $$\\mathbb {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-null-set. In particular, our results apply to all unit balls in Besov and Sobolev spaces that embed compactly into $$L^2 (\\varOmega )$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03a9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for a bounded Lipschitz domain $$\\varOmega $$<\/jats:tex-math>\n \u03a9<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. As an application, we show that several existing sharpness results concerning function approximation using deep neural networks are in fact generically sharp<\/jats:italic>. In addition, we provide quantitative and non-asymptotic bounds on the probability that a random $$f\\in \\mathcal {S}$$<\/jats:tex-math>\n \n f<\/mml:mi>\n \u2208<\/mml:mo>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be encoded to within accuracy $$\\varepsilon $$<\/jats:tex-math>\n \u03b5<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using R<\/jats:italic> bits. This result is subsequently applied to the problem of approximately representing $$f\\in \\mathcal {S}$$<\/jats:tex-math>\n \n f<\/mml:mi>\n \u2208<\/mml:mo>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to within accuracy $$\\varepsilon $$<\/jats:tex-math>\n \u03b5<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> by a (quantized) neural network with at most W<\/jats:italic> nonzero weights. We show that for any $$s > s^*(\\mathcal {S})$$<\/jats:tex-math>\n \n s<\/mml:mi>\n ><\/mml:mo>\n \n s<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n S<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> there are constants c<\/jats:italic>,\u00a0C<\/jats:italic> such that, no matter what kind of \u201clearning\u201d procedure is used to produce such a network, the probability of success is bounded from above by $$\\min \\big \\{1, 2^{C\\cdot W \\lceil \\log _2 (1+W) \\rceil ^2 - c\\cdot \\varepsilon ^{-1\/s}} \\big \\}$$<\/jats:tex-math>\n \n min<\/mml:mo>\n \n {<\/mml:mo>\n <\/mml:mrow>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n 2<\/mml:mn>\n \n C<\/mml:mi>\n \u00b7<\/mml:mo>\n W<\/mml:mi>\n \n \n \u2308<\/mml:mo>\n \n log<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n W<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2309<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n -<\/mml:mo>\n c<\/mml:mi>\n \u00b7<\/mml:mo>\n \n \u03b5<\/mml:mi>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n \n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10208-021-09546-4","type":"journal-article","created":{"date-parts":[[2021,11,16]],"date-time":"2021-11-16T15:02:56Z","timestamp":1637074976000},"page":"329-392","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Phase Transitions in Rate Distortion Theory and Deep Learning"],"prefix":"10.1007","volume":"23","author":[{"given":"Philipp","family":"Grohs","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klotz","sequence":"additional","affiliation":[]},{"given":"Felix","family":"Voigtlaender","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,11,16]]},"reference":[{"key":"9546_CR1","unstructured":"Adams, R., Fournier, J.: Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, second edn. 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