On Integrals of Matrix Coefficients Associated to Spherical Models

We define a local ingredient of the Ichino-Ikeda conjecture for isometry groups, for representations arising as local components of irreducible automorphic cuspidal representations lying in generic packets, without assuming temperedness everywhere. The representations in consideration are parabolically induced from characters and a tempered representation

The Implicit Bias of Gradient Descent on Separable Data

We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization n more complex models and with other optimization methods.Comment: Final JMLR version, with improved discussions over v3. Main improvements in journal version over conference version (v2 appeared in ICLR): We proved the measure zero case for main theorem (with implications for the rates), and the multi-class cas

Genome-scale architecture of small molecule regulatory networks and the fundamental trade-off between regulation and enzymatic activity

Metabolic flux is in part regulated by endogenous small molecules that modulate the catalytic activity of an enzyme, e.g., allosteric inhibition. In contrast to transcriptional regulation of enzymes, technical limitations have hindered the production of a genome-scale atlas of small molecule-enzyme regulatory interactions. Here, we develop a framework leveraging the vast, but fragmented, biochemical literature to reconstruct and analyze the small molecule regulatory network (SMRN) of the model organism Escherichia coli, including the primary metabolite regulators and enzyme targets. Using metabolic control analysis, we prove a fundamental trade-off between regulation and enzymatic activity, and we combine it with metabolomic measurements and the SMRN to make inferences on the sensitivity of enzymes to their regulators. Generalizing the analysis to other organisms, we identify highly conserved regulatory interactions across evolutionarily divergent species, further emphasizing a critical role for small molecule interactions in the maintenance of metabolic homeostasis.P30 CA008748 - NCI NIH HHS; R01 GM121950 - NIGMS NIH HH