I wrote an article titled “Linear Regression with Pseudo-Inverse Training Using C#” in the December 2025 edition of Microsoft Visual Studio Magazine. See https://visualstudiomagazine.com/articles/2025/12/15/linear-regression-with-pseudo-inverse-training-using-csharp.aspx.

The goal of a machine learning regression problem is to predict a single numeric value. For example, you might want to predict the bank account balance of an employee based on his age, height, and years of work experience.

There are roughly a dozen main regression techniques, including nearest neighbors regression, and neural network regression. Linear regression is the most fundamental technique.

The form of a linear regression prediction equation is y’ = (w0 * x0) + (w1 * x1) + . . + (wn * xn) + b where y’ is the predicted value, the xi are predictor values, the wi are constants called model weights, and b is a constant called the bias. For example, y’ = predicted balance = (-0.54 * age) + (0.38 * height) + (0.11 * experience) + 0.72. Training the model is the process of finding the values of the weights and bias so that predicted y values are close to known correct target y values in a set of training data.

There are three main techniques to train a linear regression model: stochastic gradient descent (SGD), pseudo-inverse, and closed form training. My article explains how to implement the pseudo-inverse training technique.

The output of my demo program is:

Begin C# linear regression using pseudo-inverse training

Loading synthetic train (200) and test (40) data
Done

First three train X:
 -0.6046  0.7260  0.9668 -0.6723  0.1947
  0.9341  0.0945  0.9454  0.4296  0.3955
 -0.9820 -0.2269 -0.9117  0.9133 -0.1277

First three train y:
  0.7180
  0.2507
  0.5698

Creating and training Linear Regression model 
 using QR p-inverse
Done

Coefficients/weights:
-0.2500  -0.0220  0.0272  -0.1434  0.0511
Bias/constant: 0.4938

Evaluating model

Accuracy train (within 0.10) = 0.7500
Accuracy test (within 0.10) = 0.7750

MSE train = 0.0013
MSE test = 0.0011

Predicting for x =
  -0.6046   0.7260   0.9668  -0.6723   0.1947

Predicted y = 0.7616

End demo

There are several algorithms to compute a pseudo-inverse. The demo uses QR decomposition via the Householder algorithm.

For large datasets, training with SGD is often best, but it requires values for a learning rate parameter and a maximum epochs parameter, and those must be determined by trial and error. For medium size datasets, training with pseudo-inverse works well, but it is the most complicated technique. For small datasets, closed form training is simpler than pseudo-inverse, but it can fail more easily.

I’m a big fan of 1950s science fiction movies. Two of my favorites featured spacecraft with tubes.

Right: In “This Island Earth” (1955), scientists from Earth are recruited by aliens from Metaluna to help in a war with the Zagons. The Metaluna spacecraft has some sort of conditioning tubes to deal with the differences between the atmospheres of Earth and Metaluna.