• Background
• Install
• Requirements
• Usage
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Consider a random vector x with a multivariate normal distribution. Let Sigma denote the covariance matrix associate with the vector x. The inverse of the covariance matrix can be used to determine the conditional independence between the random variables.

The sparsity graph of inverse of Sigma represents a graphical model capturing the conditional independence between elements of x.

Graphical lasso(GL) is one of the most commonly used techniques for estimating the inverse covariance matrix. It is known that GL is computationally expensive for large-scale problems. Therefore, we developed an explicit closed-form solution that can serve either as an approximate solution of the GL or the optimal slution of the GL with a perturbed sample covariance matix.

For a more detailed description, you can refer to this paper: Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions

The source code is currently hosted on GitHub at: https://github.com/AtomXT/GL-and-Thresholding. You can download them and simply use functions with your data.

For MATLAB users, our code is implemented with MATLAB R2020a.

For Python users, our code is implemeted with Python 3.6 and you will need:

• NumPy & SciPy

Let's say x is a n by m sample data matrix. n is the dimension of each sample and m is the number of samples.

Sigma is the sample covariance matrix of x, and lambda is the regularization parameter for thresholding.

You can use the following code:

• Python

closed_form(x, lambda)

• MATLAB

Closed_form(x, lambda)

Output:

• A : The closed form soluton.

This function will return A as the closed-form solution of GL.

There is a 12000*6000 test data, and corresponding scripts to run the test.

Here is the dataset.

With the test data, run the following test code:

• Python

  run closed_form.py

• MATLAB

  run test_closed_form.m

@Tong Xu.

Please feel free to make any suggestions! Open an issue or submit PRs.

Salar Fattahi: [email protected]

Tong Xu: [email protected]

MIT © Tong Xu