Codebase for Simplicial Regularizability of Pseudo-Moment Cone

Setup

Before running the scripts in this repository:

1. Pre-install dependencies in MATLAB:

• Mosek for MATLAB: https://www.mosek.com/downloads/
• msspoly (via SPOTless): https://github.com/spot-toolbox/spotless

2. Precompile the sos_sdp_conversion package by running sparse_sdp_relax_install.m in MATLAB.
3. If installation of sos_sdp_conversion fails, set if_sos_sdp_conversion = false in the scripts, and the code will automatically load pre-generated moment cone data from the constraint folder. Affected scripts: test_point_evaluation.m, test_verify_extremity.m, main_phase_transition.m.

Quick Usage

1. For the simplest usage, run test_point_evaluation.m. In this script:

• n is the variable number $n$ in the paper.
• kappa is the relaxation order $d$ in the paper.
• point_evaluation_num is $s$ in the paper.

2. To reproduce the paper's phase-transition results, run main_phase_transition.m. Warning: this experiment may take a very long time.
3. test_verify_extremity.m help to verify the extremity of an element in a spectrahedral cone.
4. recover_phase_transition.m plots $e_w$ and $e_z$ in the paper.
5. 4. plot_phase_transition.m plots scatters in the paper.

Moment Cone Properties

$d = 2, n = 3$

$d = 2, n = 4$

$d = 2, n = 5$

$d = 2, n = 6$

$d = 2, n = 7$

$d = 2, n = 8$

$d = 2, n = 9$

$d = 2, n = 10$