We propose a fast population game dynamics, motivated by the analogy with infection and immunization processes within a population of “players,” for finding dominant sets, a powerful graph-theoretical notion of a cluster. Each step of the proposed dynamics is shown to have a linear time/space complexity and we show that, under the assumption of symmetric … Read more
The L1 norm has been applied in numerous variations of principal component analysis (PCA). L1-norm PCA is an attractive alternative to traditional L2-based PCA because it can impart robustness in the presence of outliers and is indicated for models where standard Gaussian assumptions about the noise may not apply. Of all the previously-proposed PCA schemes … Read more
The NP-hard Maximum Monomial Agreement (MMA) problem consists of finding a single logical conjunction that best fits a weighted dataset of “positive” and “negative” binary vectors. Computing classifiers using boosting methods involves a maximum agreement subproblem at each iteration, although such subproblems are typically solved by heuristic methods. Here, we describe an exact branch and … Read more
In optimization-based classification model selection, for example when using linear programming formulations, a standard approach is to penalize the L1 norm of some linear functional in order to select sparse models. Instead, we propose a novel integer linear program for sparse classifier selection, generalizing the minimum disagreement hyperplane problem whose complexity has been investigated in … Read more
This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and Oh. The reconstruction is accomplished by minimizing the nuclear norm, or sum of the singular … Read more
In this paper we proposed quasi-Newton and limited memory quasi-Newton methods for objective functions defined on Grassmannians or a product of Grassmannians. Specifically we defined BFGS and L-BFGS updates in local and global coordinates on Grassmannians or a product of these. We proved that, when local coordinates are used, our BFGS updates on Grassmannians share … Read more
In this paper we introduce the LeGO (Learning for Global Optimization) approach for global optimization in which machine learning is used to predict the outcome of a computationally expensive global optimization run, based upon a suitable training performed by standard runs of the same global optimization method. We propose to use a Support Vector Machine … Read more
We consider a recently proposed optimization formulation of multi-task learning based on trace norm regularized least squares. While this problem may be formulated as a semidefinite program (SDP), its size is beyond general SDP solvers. Previous solution approaches apply proximal gradient methods to solve the primal problem. We derive new primal and dual reformulations of … Read more
This paper proposes to apply Gaussian graphical models to estimate the large-scale normal distribution in the context of data assimilation from a relatively small number of data from the satellite. Data assimilation is a field which fits simulation models to observation data developed mainly in meteorology and oceanography. The optimization problem tends to be huge … Read more
We formalize an algorithm for solving the L1-norm best-fit hyperplane problem derived using first principles and geometric insights about L1 projection and L1 regression. The procedure follows from a new proof of global optimality and relies on the solution of a small number of linear programs. The procedure is implemented for validation and testing. This … Read more