Table 2. Common data operations expressed as MLMs.

Operation	Input Space $\mathbb{X}$	Output Space $\mathbb{Y}$	Parameters	Morphism	Empirical Risk Function
Data Encoding	Abstract Space $\mathbb{X}$	${\mathbb{R}}^m$	embedding parameters	Injective map: $F:\mathbb{X}\hookrightarrow {\mathbb{R}}^m$	trivial (one—hot encoding) or cost function, e.g. from [20]
PCA	${\mathbb{R}}^m$	${\mathbb{R}}^c$	$c\in \mathbb{N}$ $\mathbf{A}\in {\mathbb{R}}^{m\times c}$	x A	$\parallel \mathbf{X}-{\mathbf{\text{XA}}}^T{\parallel }_F^2$ Such that: AAT = I
Linear Regression	${\mathbb{R}}^m$	$\mathbb{R}$	$\mathbf{p}\in {\mathbb{R}}^m$	x ⋅ p	${\parallel \mathbf{Y}-{\mathbf{X}\mathbf{p}\parallel }_2}^2$
Logistic Regression	${\mathbb{R}}^m$	[0, 1]	$p\in {\mathbb{R}}^m$	$F(\mathbf{x}.\mathbf{p})=\frac{1}{1+\text{exp}(-\mathbf{x}·\mathbf{p})}$	Maximum Likelihood ${\prod }_{i=1}^{N}F{({\mathbf{x}}_i;\mathbf{p})}^{y_i}{(1-F({\mathbf{x}}_i;\mathbf{p}))}^{1-y_i}$
SVM	${\mathbb{R}}^m$	{−1, 1}	$\left(\mathbf{w},\mathbf{b}\right)\in ({\mathbb{R}}^m,{\mathbb{R}}^m)$ slack variables $s\in {\mathbb{R}}^m$, $c\in \mathbb{R}$	w ⋅ x − b	∥w∥2 + c∥s∥2
Decision Tree	Set X	{y1, y2, …, yk} for finite k	splitting criterion	Tree	Gini Impurity ${\sum }_{j=1}^{N}1-{\sum }_{i=0}^{1}P{(Y_j=i|X_j={\mathbf{x}}_j)}^2$ Information Gain see [21]
Standardization	${\mathbb{R}}^m$	${\mathbb{R}}^m$	$\left(\mathbf{c},\mathbf{s}\right)\in ({\mathbb{R}}^m,{\mathbb{R}}^m)$	(x − c)diag(s)−1	KL Divergence, Eq 13
Adaboost	${\mathbb{R}}^m$	$\mathbb{R}$	parameters associated with weak learners	${\sum }_{i=1}^k{F}_i\left(\mathbf{x}\right)$	Exponential Loss [22]: ${\sum }_{i=1}^N\text{exp}(-y_i{\sum }_{j=1}^k{F}_j\left({\mathbf{x}}_i\right))$
Neural Networks	${\mathbb{R}}^m$	$\mathbb{R}$	Weights in ${\mathbb{R}}^w$	F = Fk(Fk−1(Fk−2(…F1(x)))))	Loss functions, examples include Mean Squared Error: ∥Y − F(X)∥2, Cross Entropy: $\frac{-1}{N}{\sum }_{i=1}^N{y}_i\text{log}\left(F\left({\mathbf{x}}_i\right)\right)-$ (1 − yi)log(1 − F(xi)) [23]
Model Evaluation	Collection $V_{\mathbb{Y}}^k$	$\mathbb{R}$ or ${\mathbb{R}}^k$	Evaluation parameters Test/validation set	Performance Metric e.g. Accuracy, Sensitivity, etc.	Complexity Criterion or other objective e.g. Aikeke Information Criterion