3.5 Gradients of Matrices

We will encounter situations where we need to take gradients of matrices with respect to vectors ( or other matrices), which results in multidimensional tensor.We can think of this tensor as a multidimensional array that collects partial derivatives. For example if we compute the gradient of an $m \times n$ matrix $A$ with respect to a $p \times q$ matrix $B$, the resulting Jacobian would be $(m \times n) \times ( p \times q)$.i.e, a four dimensional tensor $J$, whose entries are given as $J_{ijkl}=\frac{\partial A_{ij}}{\partial B_{kl}}$

Since matrices represent linear mappings, we can exploit the fact that there is a vector-space isomorphism (linear, invertible mapping) between the space $R^{m \times n}$ of $m \times n$ matrices and the space $R^{mn}$ of $mn$ vectors.Therefore, we can re-shape our matrices into vectors of lengths $mn$ and

$pq$, respectively. The gradient using these $mn$ vectors results in a Jacobian Matrices can be of size $mn \times pq$. The following Figure visualizes both approaches.

Example Problems:

Compute the derivatives $\frac{\mathrm{d} f}{\mathrm{d} x}$ of the following functions by using the chain rule. Provide the dimensions of every single partial derivative. Describe your steps in detail.

$f(z)=sin(z), \quad z=Ax+b , \quad A \in \mathbb{R}^{E \times D }, x \in \mathbb{R}^D, b \in \mathbb{R}^E$

$f(z)=exp(\frac{-1}{2}z)$

$z=g(y)=y^TS^{-1}y$

$y=h(x)=x-\mu$

where $x,\mu \in \mathbb{R}^D$, $S \in \mathbb{R}^{D \times D}$

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Mathematics for Machine Learning- CST 284 - KTU Minor Notes - Dr Binu V P

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4.3 Sum Rule, Product Rule, and Bayes’ Theorem

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Since machine learning algorithms are implemented on a computer, the mathematical formulations are expressed as numerical optimization methods.Training a machine learning model often boils down to finding a good set of parameters. The notion of “good” is determined by the objective function or the probabilistic model. Given an objective function, finding the best value is done using optimization algorithms. There are two main branches of continuous optimization constrained and unconstrained. By convention, most objective functions in machine learning are intended to be minimized, that is, the best value is the minimum value. Intuitively finding the best value is like finding the valleys of the objective function, and the gradients point us uphill. The idea is to move downhill (opposite to the gradient) and hope to find the deepest point. For unconstrained optimization, this is the only concept we need,but there are several design choices. For constrained optimization, we need to intr

Mathematics for Machine Learning with Python- CST284 KTU Minor - Dr. Binu V P -9847390760

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