Problems tagged with "object type"

Part 1)

Let \(\Phi\) be an \(n \times d\) design matrix, let \(\lambda\) be a real number, and let \(I\) be a \(d \times d\) identity matrix.

What type of object is \((\Phi^T \Phi + n \lambda I)^{-1}\)?

Part 2)

Let \(\Phi\) be an \(n \times d\) design matrix, and let \(\vec y \in\mathbb R^n\). What type of object is \(\Phi^T \vec y\)?

Part 3)

Let \(\vec w \in\mathbb R^{d+1}\), and for for each \(i \in\{1, 2, \ldots, n\}\) let \(\nvec{x}{i}\in\mathbb R^d\) and \(y_i \in\mathbb R\).

What type of object is:

Part 4)

Let \(\vec w \in\mathbb R^{d+1}\), and for for each \(i \in\{1, 2, \ldots, n\}\) let \(\nvec{x}{i}\in\mathbb R^d\) and \(y_i \in\mathbb R\). Consider the empirical risk with respect to the square loss of a linear predictor on a data set of \(n\) points:

What type of object is \(\nabla R(\vec w)\); that is, the gradient of the risk with respect to the parameter vector \(\vec w\)?

Part 1)

Let \(\vec x \in\mathbb R^d\) and let \(A\) be an \(d \times d\) matrix. What type of object is \(\vec x^T A \vec x\)?

Part 2)

Let \(A\) be an \(n \times n\) matrix, and let \(\vec x \in\mathbb R^n\). What type of object is: \((A + A^T)^{-1}x\)?

Part 3)

Suppose we train a support vector machine \(H(\vec x) = \Aug(\vec x) \cdot\vec w\) on a data set of \(n\) points in \(\mathbb R^d\). What type of object is the resulting parameter vector, \(\vec w\)?

Part 1)

For each \(i = 1, \ldots, n\), let \(\nvec{x}{i}\) be a vector in \(\mathbb R^d\) and let \(\alpha_i\) be a scalar. What type of object is:

Part 2)

Let \(\Phi\) be an \(n \times d\) matrix, let \(\vec y\) be a vector in \(\mathbb R^n\), and let \(\vec\alpha\) be a vector in \(\mathbb R^n\). What type of object is:

Part 3)

Let \(\vec x\) be a vector in \(\mathbb R^d\), and let \(A\) be a \(d \times d\) matrix. What type of object is:

Part 4)

Let \(A\) be a \(d \times n\) matrix. What type of object is \((A A^T)^{-1}\)?

Part 5)

For each \(i = 1, \ldots, n\), let \(\nvec{x}{i}\) be a vector in \(\mathbb R^d\). What type of object is:

Part 1)

Let \(f : \mathbb R^d \to\mathbb R\) be a function and let \(\nvec{x}{0}\) be a vector in \(\mathbb R^d\). What type of object is \(\frac{d}{d \vec x} f(\nvec{x}{0})\)? In other words, what type of object is the gradient of \(f\) evaluated at \(\nvec{x}{0}\)?

Part 2)

Let \(\Phi\) be an \(n \times d\) matrix and let \(\vec\alpha\) be a vector in \(\mathbb R^n\). What type of object is:

Part 3)

For each \(i = 1, \ldots, n\), let \(\nvec{x}{i}\) be a vector in \(\mathbb R^d\) and \(y_i\) be a scalar. Let \(\vec w\) be a vector in \(\mathbb R^d\). What type of object is:

Part 4)

Let \(X\) be an \(n \times d\) matrix, and assume that \(X^T X\) is invertible. What type of object is \(X(X^T X)^{-1} X^T\)?