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   Cai, H ; Hamm, K ; Huang, L ; Needell, D. ( January 2021 , Journal of machine learning research)
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   https://doi.org/10.1093/imrn/rnx207  

   Pechenik, Oliver ; Searles, Dominic ( September 2017 , International Mathematics Research Notices)

   null (Ed.)

   Abstract We investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $K$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $n$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $\beta$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $K$-theory, and we state our results in this more general context. 

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5. On Estimation of Modal Decompositions

   https://doi.org/10.1109/ISIT44484.2020.9174093  

   Makur, A ; Wornell, G. W. ; Zheng, L ( June 2020 , IEEE International Symposium on Information Theory)

   A modal decomposition is a useful tool that deconstructs the statistical dependence between two random variables by decomposing their joint distribution into orthogonal modes. Historically, modal decompositions have played important roles in statistics and information theory, e.g., in the study of maximal correlation. They are defined using the singular value decompo- sitions of divergence transition matrices (DTMs) and conditional expectation operators corresponding to joint distributions. In this paper, we first characterize the set of all DTMs, and illustrate how the associated conditional expectation operators are the only weak contractions among a class of natural candidates. While modal decompositions have several modern machine learning applications, such as feature extraction from categorical data, the sample complexity of estimating them in such scenarios has not been analyzed. Hence, we also establish some non-asymptotic sample complexity results for the problem of estimating dominant modes of an unknown joint distribution from training data. 

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