Feature Crosses: Encoding Nonlinearity

In Figures 1 and 2, imagine the following:

  • The blue dots represent sick trees.
  • The orange dots represent healthy trees.

Blues dots occupy the northeast quadrant; orange dots occupy the southwest quadrant.

Figure 1. Is this a linear problem?

Can you draw a line that neatly separates the sick trees from the healthy trees? Sure. This is a linear problem. The line won't be perfect. A sick tree or two might be on the "healthy" side, but your line will be a good predictor.

Now look at the following figure:

Blues dots occupy the northeast and southwest quadrants; orange dots occupy the northwest and southeast quadrants.

Figure 2. Is this a linear problem?

Can you draw a single straight line that neatly separates the sick trees from the healthy trees? No, you can't. This is a nonlinear problem. Any line you draw will be a poor predictor of tree health.

Same drawing as Figure 2, except that a horizontal line breaks the plane. Blue and orange dots are above the line; blue and orange dots are below the line.

Figure 3. A single line can't separate the two classes.

 

To solve the nonlinear problem shown in Figure 2, create a feature cross. A feature cross is a synthetic feature that encodes nonlinearity in the feature space by multiplying two or more input features together. (The term cross comes from cross product.) Let's create a feature cross named \(x_3\) by crossing \(x_1\) and \(x_2\):

$$x_3 = x_1x_2$$

We treat this newly minted \(x_3\) feature cross just like any other feature. The linear formula becomes:

$$y = b + w_1x_1 + w_2x_2 + w_3x_3$$

A linear algorithm can learn a weight for \(w_3\) just as it would for \(w_1\) and \(w_2\). In other words, although \(w_3\) encodes nonlinear information, you don’t need to change how the linear model trains to determine the value of \(w_3\).

Kinds of feature crosses

We can create many different kinds of feature crosses. For example:

  • [A X B]: a feature cross formed by multiplying the values of two features.
  • [A x B x C x D x E]: a feature cross formed by multiplying the values of five features.
  • [A x A]: a feature cross formed by squaring a single feature.

Thanks to stochastic gradient descent, linear models can be trained efficiently. Consequently, supplementing scaled linear models with feature crosses has traditionally been an efficient way to train on massive-scale data sets.

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