MultipleComponentsSoftware

Software supplement for the paper

"Sparse PCA with Multiple Principal Components"

by Ryan Cory-Wright and Jean Pauphilet for which a preprint is available here

Introduction

The software in this package is designed to provide certifiably near-optimal solutions to the problem

max tr(U'QU) s.t. ||U||_0 <=k, U'U=I

using a relax-and-round (sdp_bounds.jl), exact (exact_methods.jl), or Lagrangian relaxation (iterative_deflation3.jl) approach.

Installation and set up

To run this software, you must install a recent version of Julia from here, and a recent version of the Mosek solver (academic licenses are freely available here). To run parts of the code you will also need to install the Gurobi solver (academic licenses are freely available here). If you do not have access to Mosek, you could use another semidefinite solver, although your results may vary. The most recent version of Julia at the time this code was last tested was Julia 1.9.1 using Mosek version 10.1.11 and Gurobi version 10.0.0, the code has previously successfully run using Julia version 1.7.

Several packages must be installed in Julia before the code can be run. These packages can be found in "core_julia1p7.jl"

Correspondence between script files and figures/tables in the paper

• The file "createTable1.jl" produces the output used to generate Table 1 in our manuscript.
• The files "createTable2_x.jl" produce the output used to generate Table 2, and Tables EC.6-EC.15 in our manuscript (where the "x" denotes the method run—note that you will need to run some of the methods from the literature we benchmarked against yourself; we provide the code for our methods and those from the literature implemented in Julia in this repo).
• The file "pipeline_generate_syntheticdata.jl" and "createFigures_1_to_2.jl" generate the synthetic datasets and produce the output used to generate Figures 1-2 in our manuscript.
• The file "createFigure3.jl" produces the output used to generate Figure 3, Table 3, and Figure EC.1 in our manuscript.
• The file "createTableec2.jl" produces the output used to generate Table EC.2 in our manuscript.
• The file "createTableec3_to_ec5.jl" produces the output used to generate Tables EC.3-EC.5 in our manuscript.

Note that you may need more RAM than is available on a standard laptop to run our SDP relaxation methods on the largest datasets considered in the paper. However, the deflation method should run at scale.

Related Packages

If you are interested in computing a single sparse PC, you may want to check out our related package https://github.com/ryancorywright/ScalableSPCA.jl which computes a single provably near-optimal sparse PC via branch-and-cut or relax-and-round techniques.

Citing MultiplePCs.jl

If you use MultiplePCs.jl, we ask that you please consider citing the following preprint:

@article{cory2022sparse,
  title={Sparse PCA With Multiple Components},
  author={Cory-Wright, Ryan and Pauphilet, Jean},
  journal={arXiv preprint arXiv:2209.14790},
  year={2022}
}

Thank you

Thank you for your interest in MultiplePCs. Please let us know if you encounter any issues using this code, or have comments or questions. Feel free to email us anytime.

Ryan Cory-Wright r [dot] cory-wright [at] imperial [dot] ac [dot] uk

Jean Pauphilet jpauphilet [at] london [dot] edu