Uniform Distribution
\begin{equation} x \in U:P(x) = \frac{1}{|U|} \end{equation} Where \(|U|\) is the size of the universe (set). In Probability Theory, a uniform distribution assigns an equal probability to each element of a given set.
Math
Point Distribution
x0: P(x) = 1,∀ x ≠ x0: P(x) = 0 In Probability Theory, a point distribution is a distribution which assigns all the probability to a given point (set element).
Gradient Descent
An optimization algorithm for finding the local minimum of a differentiable function. (The red arrows show the minimums of \(J(\Theta_{0},\Theta_{1})\), i.e. the cost function) To find the minimum of the cost function, we take its derivative and “move along” the tangential line of steepest (negative) descent. Each “step” is determined by the coefficient \(\alpha\), which is called the Learning Rate. \begin{equation} \Theta_{j_{new}} := \Theta_{j_{old}} - \alpha\frac{\partial}{\partial\Theta_{j}}J(\Theta_{0},\Theta_{1}) \end{equation} ...
Covectors
A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar) \begin{equation} \alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R} \end{equation} Simplistically, covectors can be thought of as “row vectors”, or: \begin{equation} \begin{bmatrix} 1 & 2 \end{bmatrix} \end{equation} This might look like a standard vector, which would be true in an orthonormal basis, but it is not true generally. ...
Differential Geometry
Tensors Tensor Product
Linear Algebra
Bases Bases Transformation Coordinate Transformation Covectors Dual Space Identity Matrix Invertible Matrix Invertible Matrix Orthonormal Basis Tensor Product Tensors Vector Space Axioms (Vector Space)
Linear Mapping
A mapping from \(\mathbf{V} \rightarrow \mathbf{W}\) that preserves the operations of addition and scalar multiplication. Also known as # Linear Map Linear Transformation Linear Function
Multilinear Map
A function of several variables that is linear, separately, in each variable. A multilinear map of one variable is a standard linear mapping.
Tensor Product
Dual Space
The space of all linear functionals \(f:V\rightarrow \mathbb{R}\), noted as \(V^{*}\) The dual space has the same dimension as the corresponding vector space or, given a space \(V\), with bases \((v_{1},…,v_{n})\), there exists a dual space \(V^{*}\) with a dual basis \((v^{*}_{1},…,v^{*}_{n})\).