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 \(X(X^T X)^{-1} X^T\)?

Part 1)

Let \(\mathcal{X}\) be a data set of \(n\) points in \(\mathbb{R}^d\), and let \(\vec\alpha\) be the solution to the kernel ridge regression dual problem. What type of object is \(\vec\alpha\)?

Part 2)

Suppose \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) is a set of \(n\) points in \(\mathbb{R}^d\), \(y_1, \ldots, y_n\) is a set of \(n\) labels (each either \(-1\) or \(1\)), \(\vec w\) is a \(d\)-dimensional vector, and \(\lambda\) is a scalar.

Let \(\vec\phi : \mathbb{R}^d \to\mathbb{R}^k\) be a feature map.

What type of object is the following?

Part 3)

Let \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) be a set of \(n\) points in \(\mathbb{R}^d\). Let \(\vec\mu = \sum_{i=1}^n \nvec{x}{i}\) be the mean of the data set, and let \(C\) be the sample covariance matrix.

What type of object is the following?

Part 4)

Let \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) be a data set of \(n\) points in \(\mathbb{R}^d\) sampled from a multivariate Gaussian with known covariance matrix but unknown mean, \(\vec\mu\). Let \(\mathcal L(\vec\mu)\) be the likelihood function for the Gaussian's mean, \(\vec\mu\). What type of object is \(\mathcal L\)?

Part 1)

Let \(\mathcal{X}\) be a data set of \(n\) points in \(\mathbb{R}^d\), and let \(C\) be the sample covariance matrix of \(\mathcal{X}\). Let \(|C|\) denote the determinant of \(C\). What type of object is the following?

Part 2)

Suppose \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) is a set of \(n\) points in \(\mathbb{R}^d\), \(\lambda\) is a positive real number, \(\vec y \in\mathbb R^n\) is a vector of targets, and \(\vec\phi : \mathbb{R}^d \to\mathbb{R}^k\) is a feature map. Let \(K\) be the kernel matrix for this feature map on this data set, and let \(I\) be an identity matrix (the same shape as \(K\)). What type of object is the following?

Part 3)

Let \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) be a set of \(n\) points in \(\mathbb{R}^d\). Let \(\vec\mu = \frac1n \sum_{i=1}^n \nvec{x}{i}\) be the mean of the data set, and let \(C\) be the sample covariance matrix.

What type of object is the following?

Part 4)

Let \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) be a data set of \(n\) points, let \(\vec\alpha\in\mathbb{R}^n\), and let \(\kappa\) be a kernel function for a feature map \(\vec\phi : \mathbb R^d \to\mathbb R^k\). Suppose also that \(\vec z \in\mathbb{R}^d\). What type of object is the following?