Sign function

As explained in Math and Architectures of Deep Learning

The positive, negative, or zero indicator of a numeric expression, used here to determine which side of a decision boundary a point lies on.

The sign function reports whether a numeric expression is negative, zero, or positive. In binary classification, applying sign to a decision-boundary function reveals which side of the boundary contains an input point. section 3.1.4

For a boundary defined by φ(x, b) = 0, interpret its sign as follows:

• φ(x, b) < 0 places the point on one side of the decision boundary. section 3.1.4
• φ(x, b) = 0 places the point on the boundary itself. section 7.2.2
• φ(x, b) > 0 places the point on the opposite side. section 3.1.4

A two-dimensional separator

The line y + 2x + 1 = 0 divides the plane into two half-planes. Evaluating y₀ + 2x₀ + 1 for a point (x₀, y₀) determines its side: a negative result identifies one half-plane, a positive result the other, and zero means the point lies on the line. section 3.1.4

Figure 3.3 Given a point (x₀, y₀) and a separator y + 2x + 1 = 0, we can tell which side of the separator the point lies on from the sign of y₀ + 2x₀ + 1.

Use the separator diagram to connect the sign of the surface function with the geometric side of a decision boundary. section 3.1.4

How Math and Architectures of Deep Learning covers it

How deep it goes

Defines negative, zero, and positive values geometrically, extending from lines to higher-dimensional surfaces.

Examples

Uses the line y + 2x + 1 = 0 and classifier planes to interpret point locations.

What you can do after

Evaluate a surface function’s sign to predict a binary class for an input vector.

How it connects

Connects binary classification with decision boundaries, input space, classifier functions, and hyperplanes.

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Also explained in

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  Machine Learning Algorithms in DepthA sign function returns the sign of a numeric input, commonly mapping positive and negative values to +1 and −1.

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Related concepts

• floating-point arithmeticpart of · Floating-point arithmetic is computation using finite-precision representations of real numbers, which can introduce rounding and range errors.
• decision boundaryuses · A decision boundary is the locus in input or feature space that separates regions assigned different classes or decision outcomes by a predictive model.
• sign bitimplemented by · A sign bit is a designated bit in a numeric representation that indicates whether the represented value is negative or nonnegative.

This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.