3,615 research outputs found
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
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