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Absolute value

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

$|x|$ is the distance from $x$ to $0$ , and $|x - y|$ the distance between two numbers. "Close to" in every definition of calculus means a small absolute value.

Formulas

$|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}, \qquad |x + y| \le |x| + |y|$
definition and triangle inequality

Where it shows up in AI

Loss function★★★★★frequentAI and machine learning
The L1 / mean absolute error loss $\frac1n\sum|y_i - \hat y_i|$ is robust to outliers but not differentiable at 0.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Absolute and relative error→Floating point (IEEE 754)★★★★★
Absolute and relative error→Scientific computing★★★★★
Inequalities→Limit of a function→Limits at infinity→Algorithm analysis and complexity★★★★★
Inequalities→Limit of a function→Derivative→Differentiation rules→Symbolic computation (CAS)★★★★★
Inequalities→Limit of a function→Riemann sums→Definite integral→Monte Carlo methods★★★★★

ℒ AI and machine learning

Inequalities→Convexity and concavity→Loss function★★★★★
Inequalities→Convexity and concavity→Support vector machines★★★★★
Inequalities→Limit of a function→Derivative→Gradient descent★★★★★
Inequalities→Convexity and concavity→Loss function→Logistic regression★★★★★
Inequalities→Limit of a function→Derivative→Automatic differentiation★★★★★
Inequalities→Convexity and concavity→Loss function→Regularization★★★★★
+11

3D Computer graphics

Inequalities→Limit of a function→Riemann sums→Definite integral→The rendering equation★★★★★
Inequalities→Limit of a function→Continuity→Bézier curves and splines★★★★★
Inequalities→Limit of a function→Continuity→Interpolation→Lighting and shading★★★★★

⚙ Robotics and control

Inequalities→Limit of a function→Derivative→Kinematics: position, velocity, acceleration★★★★★
Inequalities→Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot Jacobian (velocity kinematics)★★★★★
Inequalities→Limit of a function→Derivative→Kinematics: position, velocity, acceleration→Robot dynamics★★★★★
Inequalities→Limit of a function→Derivative→Ordinary differential equations→Control theory★★★★★
Inequalities→Limit of a function→Derivative→PID control★★★★★
Inequalities→Convergence of sequences→Newton's method→Inverse kinematics★★★★★
+1

⚛ Physics and simulation

Absolute and relative error→Numerical stability→Physics engines★★★★★
Absolute and relative error→Numerical stability→Physics engines→N-body gravitational simulation★★★★★
Absolute and relative error→Numerical stability→Numerical differentiation→Heat equation and diffusion★★★★★
Absolute and relative error→Numerical stability→Physics engines→Fluid dynamics and CFD★★★★★
Inequalities→Limit of a function→Derivative→Ordinary differential equations→Classical mechanics★★★★★
Inequalities→Limit of a function→Derivative→Ordinary differential equations→Population and epidemic models★★★★★
+2

∿ Signals, media and vision

Inequalities→Convergence of sequences→Numerical series→Fourier series→Signal processing★★★★★
Inequalities→Limit of a function→Continuity→Interpolation→Image processing and computer vision★★★★★
Inequalities→Convergence of sequences→Numerical series→Fourier series→Media compression (JPEG, MP3, video)★★★★★

⇄ Optimization and systems

Inequalities→Convexity and concavity→Operations research and logistics★★★★★
Inequalities→Limit of a function→Derivative→Gradient descent→Recommender systems★★★★★
Inequalities→Convergence of sequences→Numerical series→Geometric series→Queueing theory and performance★★★★★

What depends on it

Exercises

1Computation

Solve $|2x - 3| < 5$ .

Solution

$-5 < 2x - 3 < 5 \iff -1 < x < 4$ , i.e. $x \in (-1, 4)$ .

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.