Derived equivalences and sl_2-categorifications for U_q(gl_n)

We give a construction of sl_2-categorifications (in the sense of Chuang-Rouquier) for representations of U_q(gl_n), for generic q and for q a root of unity.Comment: 15 pages, references added, exposition streamlined, all comments welcom

A note on Hecke patterns in Category O

We study the family of derived auto-equivalences of the BGG-category O that correspond to the action of the standard generators of the Hecke algebra. Both the non-graded and graded situations are considered.Comment: 20 pages, some typos in previous version fixed, proof of Thm. 9.6 expanded, acknowledgments added, all comments welcom

A remark on some bases in the Hecke algebra

We consider some bases in the Hecke algebra and exhibit certain dualities between them.Comment: 6 pages, exposition streamlined, all comments welcom

Spectral Analysis of Kernel and Neural Embeddings: Optimization and Generalization

We extend the recent results of (Arora et al. 2019). by spectral analysis of the representations corresponding to the kernel and neural embeddings. They showed that in a simple single-layer network, the alignment of the labels to the eigenvectors of the corresponding Gram matrix determines both the convergence of the optimization during training as well as the generalization properties. We generalize their result to the kernel and neural representations and show these extensions improve both optimization and generalization of the basic setup studied in (Arora et al. 2019). In particular, we first extend the setup with the Gaussian kernel and the approximations by random Fourier features as well as with the embeddings produced by two-layer networks trained on different tasks. We then study the use of more sophisticated kernels and embeddings, those designed optimally for deep neural networks and those developed for the classification task of interest given the data and the training labels, independent of any specific classification model