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Person# Yen-Huan Li

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Analytical engine

The analytical engine was a proposed mechanical general-purpose computer designed by English mathematician and computer pioneer Charles Babbage. It was first described in 1837 as the successor to B

Stochastic gradient descent

Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can b

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Recent advances in statistical learning and convex optimization have inspired many successful practices. Standard theories assume smoothness---bounded gradient, Hessian, etc.---and strong convexity of the loss function. Unfortunately, such conditions may not hold in important real-world applications, and sometimes, to fulfill the conditions incurs unnecessary performance degradation. Below are three examples.

- The standard theory for variable selection via L_1-penalization only considers the linear regression model, as the corresponding quadratic loss function has a constant Hessian and allows for exact second-order Taylor series expansion. In practice, however, non-linear regression models are often chosen to match data characteristics.
- The standard theory for convex optimization considers almost exclusively smooth functions. Important applications such as portfolio selection and quantum state estimation, however, correspond to loss functions that violate the smoothness assumption; existing convergence guarantees for optimization algorithms hence do not apply.
- The standard theory for compressive magnetic resonance imaging (MRI) guarantees the restricted isometry property (RIP)---a smoothness and strong convexity condition on the quadratic loss restricted on the set of sparse vectors---via random uniform sampling. The random uniform sampling strategy, however, yields unsatisfactory signal reconstruction performance empirically, in comparison to heuristic sampling approaches.

Consider the problem of minimizing a convex differentiable function on the probability simplex, spectrahedron, or set of quantum density matrices. We prove that the expo-nentiated gradient method with Armijo line search always converges to the optimum, if the sequence of the iterates possesses a strictly positive limit point (element-wise for the vector case, and with respect to the Löwner partial ordering for the matrix case). To the best of our knowledge, this is the first convergence result for a mirror descent-type method that only requires differentiability. The proof exploits self-concordant likeness of the l og-partition function, which is of independent interest.

2018Thamani Dahoun, Yen-Huan Li, Horst Vogel, Shuguang Yuan

Many central biological events rely on protein-ligand interactions. The identification and characterization of protein-binding sites for ligands are crucial for the understanding of functions of both endogenous ligands and synthetic drug molecules. G protein-coupled receptors (GPCRs) typically detect extracellular signal molecules on the cell surface and transfer these chemical signals across the membrane, inducing downstream cellular responses via G proteins or beta-arrestin. GPCRs mediate many central physiological processes, making them important targets for modern drug discovery. Here, we focus on the most recent breakthroughs in finding new binding sites and binding modes of GPCRs and their potentials for the development of new medicines.