principal component analysis

Optimized Dimensionality Reduction for Moment-based Distributionally Robust Optimization

  • ShiyiJiang
  • Jianqiang Cheng
  • Kai Pan
  • Z.J. Max Shen
    • Categories Optimization in Data Science, Robust Optimization, Semi-definite Programming Tags , , , ,

    Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems … Read more

    Computationally Efficient Approximations for Distributionally Robust Optimization

  • Jianqiang Cheng
  • Ruiwei Jiang
  • Meysam Cheramin
  • Kai Pan
    • Categories Stochastic Programming Tags , , ,

    Distributionally robust optimization (DRO) is a modeling framework in decision making under uncertainty where the probability distribution of a random parameter is unknown while its partial information (e.g., statistical properties) is available. In this framework, the unknown probability distribution is assumed to lie in an ambiguity set consisting of all distributions that are compatible with … Read more

    Sparse PCA on fixed-rank matrices

  • Alberto Del Pia
    • Categories Global Optimization Theory, Quadratic Programming Tags , , , ,

    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

    Estimating L1-Norm Best-Fit Lines for Data

  • J. Paul Brooks
  • JH Dulá
    • Categories Applications - OR and Management Sciences, Data-Mining, Statistics Tags , , ,

    The general formulation for finding the L1-norm best-fit subspace for a point set in $m$-dimensions is a nonlinear, nonconvex, nonsmooth optimization problem. In this paper we present a procedure to estimate the L1-norm best-fit one-dimensional subspace (a line through the origin) to data in $\Re^m$ based on an optimization criterion involving linear programming but which … Read more

    Vector Transport-Free SVRG with General Retraction for Riemannian Optimization: Complexity Analysis and Practical Implementation

  • Bo Jiang
  • Shiqian Ma
  • Anthony Man-Cho So
  • Shuzhong Zhang
    • Categories Constrained Nonlinear Optimization, Stochastic Programming Tags , , , ,

    In this paper, we propose a vector transport-free stochastic variance reduced gradient (SVRG) method with general retraction for empirical risk minimization over Riemannian manifold. Existing SVRG methods on manifold usually consider a specific retraction operation, and involve additional computational costs such as parallel transport or vector transport. The vector transport-free SVRG with general retraction we … Read more

    Distributionally Robust Optimization with Principal Component Analysis

  • Richard Chen
  • Habib Najm
  • Jean-Paul Watson
  • Jianqiang Cheng
  • Ali Pinar
  • Cosmin Safta
    • Categories Robust Optimization, Stochastic Programming Tags , , ,

    Distributionally robust optimization (DRO) is widely used, because it offers a way to overcome the conservativeness of robust optimization without requiring the specificity of stochastic optimization. On the computational side, many practical DRO instances can be equivalently (or approximately) formulated as semidefinite programming (SDP) problems via conic duality of the moment problem. However, despite being … Read more

    pcaL1: An Implementation in R of Three Methods for L1-Norm Principal Component Analysis

  • J. Paul Brooks
  • Sapan Jot
    • Categories Data-Mining, Statistics Tags , ,

    pcaL1 is a package for the R environment for finding principal components using methods based on the L1 norm. The principal components derived using traditional principal component analysis (PCA) can be interpreted as an optimal solution to several optimization problems involving the L2 norm. Using the L1 norm in these problems provides an alternative that … Read more

    A Pure L1-norm Principal Component Analysis

  • E.L. Boone
  • J. Paul Brooks
  • JH Dulá
    • Categories Data-Mining, Linear Programming Tags , ,

    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

    Approximating K-means-type clustering via semidefinite programming

  • Jiming Peng
  • Yu Wei
    • Categories Approximation Algorithms, Convex Optimization, Data-Mining Tags , , ,

    One of the fundamental clustering problems is to assign $n$ points into $k$ clusters based on the minimal sum-of-squares(MSSC), which is known to be NP-hard. In this paper, by using matrix arguments, we first model MSSC as a so-called 0-1 semidefinite programming (SDP). We show that our 0-1 SDP model provides an unified framework for … Read more