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LaTeX Equation Cheat Sheet.ipynb
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latex-equation-cheat-sheet.ipynb
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"# $ \\LaTeX $ equation cheat sheet\n",
"\n",
"This cheat sheet focuses specifically on the notation needed for the equations found in *A First Course in Machine Learning*, Second Edition\n",
"\n",
"**Note**: to read this document, double-click the text cell so that the screen is split between the Markdown source code and the rendered equations and text.\n",
"\n",
"\n",
"## Chapter 1 - Linear Modelling: A Least Squares Approach\n",
"\n",
"Equations are written with surrounded by dollar signs: $ y = mx + b $\n",
"\n",
"If you want regular $ signs, you may need to escape them with two backslashes: \\\\$.\n",
"\n",
"Equations surrounded by two dollar signs are centered on the next line:\n",
"$$ y = f(x) $$\n",
"\n",
"Some mathematical operators are preceded with a single backslash `\\`: $ y = \\sin(x) $\n",
"\n",
"To get an actual backslash, escape it with another slash: \\\\.\n",
"\n",
"Some operators require arguments in curly braces `{}`: $ \\sqrt{y} = mx + c $\n",
"\n",
"Superscripts use a caret `^`: $ y = mx^2 + c $\n",
"\n",
"Subscripts are written with underscores `_`:\n",
"$$ t = f(x; w_0, w_1) = w_0 + w_1x $$\n",
"\n",
"Loss functions are defined with \"mathematical calligraphy\":\n",
"$$ \\mathcal{L}_n(t_n, f(x_n; w_0, w_1)) = (t_n - f(x_n; w_0, w_1))^2 $$\n",
"\n",
"The average loss requires fractions: $$ \\frac{1}{N} $$\n",
"and summation: $$ \\sum_{n=1}^{N} $$\n",
"*Note how `\\sum` separates its arguments with an underscore `_` for the lower argument and a caret `^` for the upper argument instead of putting them directly together the way `\\frac` does.*\n",
"\n",
"Putting it all together:\n",
"$$ \\mathcal{L} = \\frac{1}{N} \\sum_{n=1}^{N} \\mathcal{L}_n(t_n, f(x_n; w_0, w_1)) $$\n",
"\n",
"Sometimes you need to define your own operators:\n",
"$$\n",
"\\DeclareMathOperator*{\\argmin}{argmin}\n",
"\\argmin_{w_0, w_1} \\frac{1}{N} \\sum_{n=1}^{N} \\mathcal{L}_n(t_n, f(x_n; w_0, w_1))\n",
"$$\n",
"\n",
"Absolute loss uses left and right vertical bars:\n",
"$$ \\mathcal{L}_n = \\lvert t_n - f(x_n; w_0, w_1) \\rvert $$\n",
"\n",
"Use `\\dots` for mathematical ellipses: $ n = 1, \\dots, N $\n",
"\n",
"The script d for partial derivatives is called `\\partial`, and usually appears in a fraction:\n",
"$$ \\frac{\\partial \\mathcal{L}}{\\partial w_0} $$\n",
"\n",
"If you use regular parentheses `()` they'll be regular-sized. If you want larger ones, you need to prefix them with the `\\left` and `\\right` commands:\n",
"$$ \\left ( \\sum_{n=1}^{N} x_n^2 \\right ) $$\n",
"\n",
"Use `\\hat` and `\\bar` for single letters: $ \\hat{w} $, $ \\bar{x} $\n",
"\n",
"But use `\\widehat` and `\\overline` for more complicated expressions:\n",
"$$ \\widehat{w_1} = \\frac{\\overline{xt} - \\bar{x}\\bar{t}}{\\overline{x^2} - (\\bar{x})^2} $$\n",
"\n",
"Use `\\times` or a centered dot `\\cdot` as explicit multiplication operators:\n",
"$$ 2 \\times 3 = 2 \\cdot 3 $$\n",
"\n",
"Use `\\mathbf` for math symbols in bold font like vector $ \\mathbf{x} $ and matrix $ \\mathbf{X} $. You can also use `\\vec` for vectors, but it uses an arrow: $ \\vec{x} $.\n",
"\n",
"Begin matrices with `\\begin{bmatrix}` and end them with `\\end{bmatrix}`. Use `&` between items in a row and `\\\\` between rows. (The `b` is for square **b**rackets).\n",
"\n",
"You can squish them together on a single line:\n",
"$$ \\mathbf{X}= \\begin{bmatrix} 1 & 4 \\\\ 3 & 6 \\\\ -2 & 11 \\end{bmatrix} $$\n",
"\n",
"But they'll be easier to read with if you use separate lines and indent:\n",
"$$ \\mathbf{A} = \\begin{bmatrix}\n",
" a_{11} & a_{12} \\\\\n",
" a_{21} & a_{22} \\\\\n",
" a_{31} & a_{32}\n",
"\\end{bmatrix} $$\n",
"\n",
"*Note that a column vector is just an $ n \\times 1 $ matrix:*\n",
"$$ \\mathbf{y} = \\begin{bmatrix} y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\end {bmatrix} $$\n",
"\n",
"Use `\\vdots` for vertical ellipses in column vectors:\n",
"$$ \\mathbf{t} = \\begin{bmatrix} t_1 \\\\ t_2 \\\\ \\vdots \\\\ t_n \\end {bmatrix} $$\n",
"\n",
"And `\\ddots` for diagonal dots in matrices:\n",
"$$ \\mathbf{A} = \\begin{bmatrix}\n",
" a_{11} & a_{12} & \\dots & a_{1M} \\\\\n",
" a_{21} & a_{22} & \\dots & a_{2M} \\\\\n",
" \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
" a_{N1} & a_{N2} & \\dots & a_{NM}\n",
"\\end{bmatrix} $$\n",
"\n",
"You can use a capital T for transpose $ \\mathbf{x}^T $, but it looks like the textbook uses '\\top` $ \\mathbf{x}^\\top $.\n",
"\n",
"The textbook uses the word \"gradient\" interchangeably with \"slope\", but you can use `\\nabla` to get the gradient operator:\n",
"$$ \\frac{\\partial \\mathcal{L}}{\\partial \\mathbf{w}} =\n",
"\\nabla_{\\mathbf{w}} \\mathcal{L} =\n",
"\\begin{bmatrix}\n",
" \\frac{\\partial \\mathcal{L}}{\\partial w_0} \\\\\n",
" \\frac{\\partial \\mathcal{L}}{\\partial w_1}\n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
"Finally, use `\\lambda` for the lowercase Greek letter:\n",
"$$ \\mathcal{L}' = \\mathcal{L} + \\lambda \\mathbf{w}^\\top \\mathbf{w} $$\n",
"\n",
"\n",
"## Chapter 2 - Linear Modelling: A Maximum Likelihood Approach\n",
"\n",
"The ordinary `<` and `>` signs work for strict inequality, but use `\\leq` and `\\geq` for to allow equality:\n",
"$$ 0 \\leq P(Y = y) \\leq 1 $$\n",
"\n",
"Similarly, `\\neq`:\n",
"$$ P(Y \\neq 4) + P(Y = 4) = 1 $$\n",
"\n",
"You can use either `|` or `\\mid` for conditional probabilities, but they produce different spacing:\n",
"$$ P(Y = y | X = x) $$\n",
"$$ P(Y = y \\mid X = x) $$\n",
"The textbook tends to use `|`.\n",
"\n",
"`\\prod` works the same way as `\\sum`:\n",
"$$ P(y_1, y_2, \\dots, y_J) = P(y_1) \\times P(y_2) \\times \\dots \\times P(y_J) = \\prod_{j=1}^J{P(y_j)} \\$$\n",
"\n",
"Curly braces need to be escaped `\\{` and `\\}` since they are also used for arguments:\n",
"$$ \\mathbf{E}_{P(x)}\\{f(X)\\} = \\sum_{x}f(x)P(x) $$\n",
"\n",
"As with parentheses, you can prefix `\\{` and `\\}` with `\\left` and `\\right` to make them stretch to fit their contents. Some functions in the textbook use a sans-serif font, specified with `\\mathsf`:\n",
"$$\n",
"\\mathsf{var}\\{X\\} = \\mathbf{E}_{P(x)}\\left \\{(X - \\mathbf{E}_{P(x)}\\{X\\})^2\\right \\} $$\n",
"$$\n",
"\\mathsf{Tr}(\\mathbf{A}) = \\sum_{d=1}^{D}A_{dd}\n",
"$$\n",
"\n",
"Use `{ N \\choose y }` to specify a combination:\n",
"$$ P(Y = y) = P(y) = {N \\choose y} q^y(1-q)^{N-y} $$\n",
"\n",
"Integrals use `\\int`:\n",
"$$ P(x_1 \\leq X \\leq x_2) = \\int_{x_1}^{x_2}p(x) dx$$\n",
"\n",
"Specify that values are approximately equal with `\\approx`:\n",
"$$ \\mathbf{E}_{p(x)}\\{f(x)\\} \\approx \\frac{1}{S} \\sum_{s=1}^{S}f(x_s) $$\n",
"\n",
"Functions defined piecewise work like matrices, only with `cases` rather than `bmatrix`:\n",
"$$\n",
"p(y) =\n",
"\\begin{cases}\n",
" r & \\text{for $a \\leq y \\leq b$} \\\\\n",
" 0 & \\text{otherwise.}\n",
"\\end{cases}\n",
"$$\n",
"If you want the right-hand side to include non-italic text, use `\\text`. Inside `\\text`, you can switch back to math mode with `$`.\n",
"\n",
"As with `\\lambda`, other Greek letters are specified with their names (e.g. `\\alpha`, `\\beta`). Upper-case Greek letters have their named capitalized (e.g., `\\Gamma, \\Sigma`):\n",
"$$\n",
"p(r) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}r^{\\alpha - 1}(1 - r)^{\\beta - 1}\n",
"$$\n",
"\n",
"Use `\\enspace` to add a space between an equation and its conditions, `\\sim` for \"is distributed as,\" and don't forget the `\\mathcal` font:\n",
"$$\n",
"t_n = w_0 + w_1x_n + \\epsilon_n,\\enspace \\epsilon_n \\sim \\mathcal{N}(0, \\sigma^2)\n",
"$$\n",
"\n",
"$$ \\mathsf{Tr}{f} $$\n",
"\n",
"\n",
"## For more information\n",
"Check out these links for $ \\LaTeX $:\n",
" * [LaTex in Jupyter notebook](https://nbviewer.jupyter.org/github/dingran/latex-ipynb/blob/master/latex-cheatsheet.ipynb)\n",
" * [LaTeX/Mathematics](https://en.wikibooks.org/wiki/LaTeX/Mathematics) on [Wikibooks](https://en.wikibooks.org/wiki/LaTeX).\n",
"\n",
"The Google Docs Equation Editor is mostly a GUI, but it does support a subset of $ \\LaTeX $ notation:\n",
"* [Use equations in a document](https://support.google.com/docs/answer/160749)\n",
"* [Google Docs Equation Editor Shortcuts](https://equation-shortcuts.notuom.com/)"
]
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