I wrote an article titled “Quadratic Regression with Pseudo-Inverse Training Using C#” in the May 2026 edition of Microsoft Visual Studio Magazine. See https://visualstudiomagazine.com/articles/2026/05/01/quadratic-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 a person’s credit rating based on age, annual salary, bank account balance, and so on. There are approximately a dozen common regression techniques. The most basic technique is called linear regression, or sometimes multiple linear regression, where the “multiple” indicates two or more predictor variables.

The form of a basic linear regression prediction model is y’ = (w0 * x0) + (w1 * x1) + . . + (wn * xn) + b, where y’ is the predicted value, the xi are n predictor values, the wi are model weights, and b is the bias.

Quadratic regression extends linear regression. The form of a quadratic regression model is y’ = (w0 * x0) + . . + (wn * xn) + (wj * x0 * x0) + . . + (wk * x0 * x1) + . . . + b. There are derived predictors that are the square of each original predictor, and interaction terms that are the multiplication product of all possible pairs of original predictors.

Compared to basic linear regression, quadratic regression can handle more complex data. Compared to the most powerful regression techniques such as gradient boosting regression, quadratic regression often has slightly worse prediction accuracy, but has much better model interpretability.

The output of the demo program presented in the article is:

Begin C# quadratic regression with pseudo-inverse
 (QR-Householder) training

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

First three train X:
 -0.1660  0.4406 -0.9998 -0.3953 -0.7065
  0.0776 -0.1616  0.3704 -0.5911  0.7562
 -0.9452  0.3409 -0.1654  0.1174 -0.7192

First three train y:
  0.4840
  0.1568
  0.8054

Creating quadratic regression model

Starting pseudo-inverse training
Done

Model base weights:
 -0.2630  0.0354 -0.0421  0.0341 -0.1124

Model quadratic weights:
  0.0655  0.0194  0.0051  0.0047  0.0243

Model interaction weights:
  0.0043  0.0249  0.0071  0.1081 -0.0012 -0.0093
  0.0362  0.0085 -0.0568  0.0016

Model bias/intercept:   0.3220

Evaluating model
Accuracy train (within 0.10) = 0.8850
Accuracy test (within 0.10) = 0.9250

MSE train = 0.0003
MSE test = 0.0005

Predicting for x =
  -0.1660   0.4406  -0.9998  -0.3953  -0.7065

Predicted y = 0.4843

End demo

The model accuracy is quite good compared to many other regression techniques applied to the synthetic dataset. The model achieves 88.50% accuracy on the training data (177 out of 200 correct) and 92.50% accuracy on the test data (37 out of 40 correct). A prediction is scored as correct if it’s within 10% of the true target value.

Behind the scenes, the demo program trains the quadratic regression model using a technique called relaxed Moore-Penrose pseudo-inverse with QR inverse via the Householder algorithm. This is just one of many possible training algorithms. The different algorithms have different pros and cons and no algorithm works the best all the time.

Quadratic regression has a nice balance of prediction power and interpretability. The model weights/coefficients are easy to interpret. If the predictor values have been normalized to the same scale, larger magnitudes mean larger effect, and the sign of the weights indicate the direction of the effect.

Quadratic regression is not always effective — if it were, it would be used far more often than it is. Compared to basic linear regression, quadratic regression can sometimes provide a big improvement in model prediction accuracy for a relatively small investment in effort, and so it’s usually worth exploring.

Quadratic regression was developed in the 1950s and 1960s, about the same time as early space exploration. Galaxy Science Fiction was published from 1950 to 1980. It was the leading science fiction magazine of its time. I really enjoy looking at older covers to get a sense of the optimism and excitement in those days. Here are three nice 1952 covers by artist Jack Coggins (1911-2006).