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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Machine learning and statistics typically focus on building models that capture the vast majority of the data, possibly ignoring a small subset of data as "noise" or "outliers." By contrast, here we consider the problem of jointly identifying a significant (but perhaps small) segment of a population in which there is a highly sparse linear regression fit, together with the coefficients for the linear fit. We contend that such tasks are of interest both because the models themselves may be able to achieve better predictions in such special cases, but also because they may aid our understanding of the data. We give algorithms for such problems under the sup norm, when this unknown segment of the population is described by a k-DNF condition and the regression fit is s-sparse for constant k and s. For the variants of this problem when the regression fit is not so sparse or using expected error, we also give a preliminary algorithm and highlight the question as a challenge for future work.

Brendan Juba. Conditional Sparse Linear Regression. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{juba:LIPIcs.ITCS.2017.45, author = {Juba, Brendan}, title = {{Conditional Sparse Linear Regression}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.45}, URN = {urn:nbn:de:0030-drops-81518}, doi = {10.4230/LIPIcs.ITCS.2017.45}, annote = {Keywords: linear regression, conditional regression, conditional distribution search} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

AC^0 o MOD_2 circuits are AC^0 circuits augmented with a layer of parity gates just above the input layer. We study AC^0 o MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC^0 o MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an ~Omega(n^2) lower bound for the special case of depth-4 AC^0 o MOD_2. Our proof of the depth-4 lower bound employs a new "moment-matching" inequality for bounded, nonnegative integer-valued random variables that may be of independent interest: we prove an optimal bound on the maximum difference between two discrete distributions’ values at 0, given that their first d moments match.

Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, and Ning Xie. AC^0 o MOD_2 Lower Bounds for the Boolean Inner Product. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 35:1-35:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cheraghchi_et_al:LIPIcs.ICALP.2016.35, author = {Cheraghchi, Mahdi and Grigorescu, Elena and Juba, Brendan and Wimmer, Karl and Xie, Ning}, title = {{AC^0 o MOD\underline2 Lower Bounds for the Boolean Inner Product}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {35:1--35:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.35}, URN = {urn:nbn:de:0030-drops-63150}, doi = {10.4230/LIPIcs.ICALP.2016.35}, annote = {Keywords: Boolean analysis, circuit complexity, lower bounds} }

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