A Guide to Artificial Intelligence by Multidimensional Regression

Or How Multidimensional Polynomial Regression Can Serve As Artificial Intelligence

One of the primary ways to establish a basic AI system is to develop a mathematical regression model that utilizes a set of features to predict an additional one. There are strict limitations, though. This article outlines two different systems based on polynomial regression: one that yields a single feature and another that yields multiple features.

Yield a Single Feature

The input is a series of variables that can be rendered as a vector X such that.

\(X \in \mathbb{R}^n\)

And an output variable y. To train this model, take a series of coordinate points provided in training data with (X, y) and plot them in a vector space of:

\(\mathbb{R}^{n+1}\)

Using the values from X, which are

\(X = [x^1, x^2,...,x^{n-1}, x^n]\)

As well as a variable y in the space. Then you run a regression on all those points to obtain a model.

\(M_\alpha(X) = \alpha \cdot X = y \)

Where alpha is a vector such that

\(\alpha \in \mathbb{R}^n\)

Once you obtain a viable alpha, the model is trained, and thus you can gain new predictions, given you have a new X vector.

What application does this system serve? Well, if you can break down something into n aspects, i.e., like a simple photo with n pixels with each pixel being fully black or fully white, then you can predict some aspect of it that can be represented with 1 more number (the y). But what if we want more than we can get from a single scalar?

Yielding Multiple Features

This section is more theoretical.

The input is again a series of variables that can be rendered as a vector X such that.

\(X\in \mathbb{R}^n\)

But also contains an output that isn’t a single variable, but a vector Y such that.

\(Y \in \mathbb{R}^m\)

Plot the points in a vector space of

\(\mathbb{R}^{m+n}\)

Using the values from X, which are

\(X = [x^1, x^2, ... , x^{n-1}, x^n] \)

And Y, which are

\(Y = [y^1, y^2,...,y^{m-1}, y^m]\)

. Then run a regression model such that with an alpha of

\(\alpha \in \mathbb{R}^n\)

So we obtain a model.

\(M_\alpha(X)= \alpha \cdot X = \beta \cdot Y \)

, such that

\(\beta \in \mathbb{R}^m\)

. To use the model, enter values for the input vector X, which will yield a function that makes.

\(M\alpha(X) = \alpha \cdot X \cdot \beta^{-1} = Y\)

This will let us obtain a point in n the vector space of

\(\mathbb{R}^m\)

Which will be what the trainer classified as the meaning of the features.

(Thumbnail: Grammarly)