My research interests are on nonlinear optimization and machine learning. I look at exploiting the two fundamental structures of least-squares and symmetry in optimization problems. A specific focus has been on developing efficient numerical algorithms for large-scale problems with manifold constraints, which are ubiquitous in machine learning applications.

Below are some specific projects that I have been actively working on with my collaborators.

#### Research at Amazon.com

**Competitive pricing of products.**We estimate the competitive prices of 3p unique products, i.e., Amazon is not a retail player for these products. The large-scale nature of the project demands to work on simple algorithmic implementations that are readily scalable. Currently, we are generating price estimates for all of the Amazon marketplaces. The model is currently deployed. (Patent filed.)**Tensor decomposition.**We exploit the problem structure to propose natural gradient learning algorithms for tensor decomposition. The algorithm is part of ongoing production efforts.**Demand forecasting.**We study low-rank tensor factorization algorithms in demand forecasting problems. This is especially critical for capturing “global” trends of data.

#### Research themes

**Optimization on Riemannian manifolds.**Understanding the (Riemannian) geometry of structured constraints is of particular interest in machine learning. Conceptually, it allows to translate a constrained optimization problem into an unconstrained optimization problem on a nonlinear search space (manifold). Building upon this point of view, one research interest is to exploit manifold geometry in nonlinear optimization.

We have been actively involved in both matrix and tensor applications. Specific papers include [arXiv:1211.1550][arXiv:1306.2672][arXiv:1405.6055][arXiv:1605.08257][arXiv:1712.01193].**Decenetralized and stochastic optimization algorithms.**We explore recent advances in stochastic gradient algorithms on manifolds. Specific papers include [arXiv:1605.07367][arXiv:1603.04989][arXiv:1703.04890].

We exploit consensus learning on manifolds in the context of large-scale distributed algorithms on problems like matrix factorization and multitask learning. An initial work is in [arXiv:1605.06968][arXiv:1705.00467].**Low-rank optimization with structure.**We develop efficient algorithms for problems in big data systems by exploiting low-rank and sparse decomposition. Papers include [arXiv:1604.04325][arXiv:1607.07252].

The work [arXiv:1704.07352] proposes a generic framework for tackling low-rank optimization with constraints by exploiting a variational characterization of nuclear norm.**Deep learning.**We study geometric algorithms for modern deep networks that are robust to invariances of the parameters. An initial work is in [arXiv:1511.01754]. Currently, we are exploring the duality between “complex architecture and simple algorithms” and “simple architecture and complex algorithms”.**Manopt.**I am also involved in the development and promotion of the Matlab toolbox Manopt for optimization on manifolds.