## Sparse PCA on fixed-rank matrices

Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational complexity of sparse PCA with respect to the rank of the covariance matrix. We show that, if … Read more

## Subset selection in sparse matrices

In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using mainly tools from discrete geometry, we show that some sparsity conditions on the original data matrix allow … Read more

## A Deterministic Rescaled Perceptron Algorithm

The perceptron algorithm is a simple iterative procedure for finding a point in a convex cone $F$. At each iteration, the algorithm only involves a query of a separation oracle for $F$ and a simple update on a trial solution. The perceptron algorithm is guaranteed to find a feasible point in $F$ after $\Oh(1/\tau_F^2)$ iterations, … Read more

## An Information Geometric Approach to Polynomial-time Interior-point Algorithms: Complexity Bound via Curvature Integral

In this paper, we study polynomial-time interior-point algorithms in view of information geometry. Information geometry is a differential geometric framework which has been successfully applied to statistics, learning theory, signal processing etc. We consider information geometric structure for conic linear programs introduced by self-concordant barrier functions, and develop a precise iteration-complexity estimate of the polynomial-time … Read more