\[ \newcommand{\R}{\mathbb{R}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vw}{\mathbf{w}} \newcommand{\vz}{\mathbf{z}} \newcommand{\norm}[1]{\lVert #1 \rVert} \newcommand{\bracket}[1]{\langle #1 \rangle} \newcommand{\abs}[1]{\lvert #1 \rvert} \newcommand{\paren}[1]{\left( #1 \right)} \]
Decision Boundaries
This warmup assignment continues our development of the idea of decision boundaries in a more advanced and generalizable way.
Part A
Sketch the line in \(\R^2\) described by the equation \[ \bracket{\vw, \vx} = b\;, \tag{1}\]
where \(\vw = \paren{1, -\frac{1}{2}}^T \in \R^2\) and \(b = \frac{1}{2}\). Here, \(\bracket{\vw, \vx} = \sum_{i = 1}^p w_i x_i\) is the inner product (or dot product) between the vectors \(\vw,\vx \in \R^p\).
Part B
numpy arrays like this: x = np.array([0.2, 4.3]). The syntax x@w can also be used as a convenient shorthand for np.dot(x, w).Write a quick Python function called linear_classify(x, w, b). w and x should both be 1d numpy arrays of the same length, and b should be a scalar. The function np.dot(x, w) will compute the inner product of x and w. Argument b should be a scalar number. Your function should return 0 if if \(\bracket{\vw, \vx} < b\) and 1 if \(\bracket{\vw, \vx} \geq b\).
Verify that your function works on a few simple examples.
Part C
Let \(\phi:\mathbb{R}^2\rightarrow \mathbb{R}^5\) be the function
\[ \phi(\mathbf{x}) = \phi(x_1, x_2) = (x_1, x_2, x_1^2, x_2^2, x_1x_2)^T\;. \]
Make a sketch of the curve in \(\R^2_+\) (the nonnegative quadrant) defined by the equation
\[ \bracket{\vw, \phi(\vx)} = b\;, \]
where \(\vw = (0, 0, 1, \frac{1}{4}, 0)^T\) and \(b = 1\).
Part D (optional)
Write a Python function called quadratic_classify(x, w, b). w should be a 1d numpy array, this time of length 5. x should still have length 2, and b should still be a scalar. Your function should return 0 if if \(\bracket{\vw, \phi(\vx)} < b\) and 1 if \(\bracket{\vw, \phi(\vx)} \geq b\).
© Phil Chodrow, 2025