cutnorm 0.1.5

Cutnorm approximation via Gaussian Rounding and Optimization with Orthogonality Constraints

Approximation via Gaussian Rounding and Optimization with Orthogonality Constraints

This package computes the approximations to the cutnorm using some of the techniques detailed by Alon and Noar [ALON2004] and a fast optimization algorithm by Wen and Yin [WEN2013].

Read the documentation.

Installation

Use pip to install the package. Install from terminal as follows:

$ pip install cutnorm

Example Usage

Below is an example of using the cutnorm package and tools. Given two graphs A and B, we wish to compute a norm for the difference matrix (A - B) between the two graphs. An obvious example to represent the advantage of using a cutnorm over l1 norm is to consider A and B as Erdos-Renyi random graphs. Under a fixed vertex set, an Erdos-Renyi random graph is one where a fixed probability determines the presence of an edge.

Given two Erdos-Renyi random graphs with fix n and p=0.5, the edit distance (l1 norm) of the difference (after normalization) is 1/2 with large probability. However, these two graphs have the same global structure. The edit distance fails as a notion of ‘distance’ between the two graphs in the perspective of global structural similarity as discussed by Lovasz [LOVASZ2009]. The cutnorm is a measure of distance that reflects global structural similarity. In fact, the cutnorm of the difference for this example approaches 0 as n grows.

import numpy as np
from cutnorm import cutnorm, tools

# Generate Erdos Renyi Random Graph
n = 100
p = 0.5
erdos_renyi_a = tools.sbm.erdos_renyi(n, p)
erdos_renyi_b = tools.sbm.erdos_renyi(n, p)

# Compute l1 norm
normalized_diff = (erdos_renyi_a - erdos_renyi_b) / n**2
l1 = np.linalg.norm(normalized_diff.flatten(), ord=1)

# Compute cutnorm
cutn_round, cutn_sdp, info = cutnorm(erdos_renyi_a, erdos_renyi_b)

print("l1 norm: ", l1)                 # prints l1 norm value near ~0.5
print("cutnorm rounded: ", cutn_round) # prints cutnorm rounded solution near ~0
print("cutnorm sdp: ", cutn_sdp)       # prints cutnorm sdp solution near ~0

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[ALON2004]

Noga Alon and Assaf Naor. 2004. Approximating the cut-norm via Grothendieck’s inequality. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (STOC ‘04). ACM, New York, NY, USA, 72-80. DOI: http://dx.doi.org/10.1145/1007352.1007371

[WEN2013]

Zaiwen Wen and Wotao Yin. 2013. A feasible method for optimization with orthogonality constraints. Math. Program. 142, 1-2 (December 2013), 397-434. DOI: https://doi.org/10.1007/s10107-012-0584-1

[LOVASZ2009]

Lovasz, L. 2009. Very large graphs. ArXiv:0902.0132 [Math]. Retrieved from http://arxiv.org/abs/0902.0132