Answers.Body: <p>Without looking in too much detail:</p>
  <ul>
  <li>We may suppose that <span class="math-container">$S$</span> is a matrix with dimensions indexed as <span class="math-container">$(i, j)$</span>. So <span class="math-container">$s_{ij}$</span> is the cell of <span class="math-container">$S$</span> indexed to the <span class="math-container">$i$</span>th row and <span class="math-container">$j$</span>th column, and <span class="math-container">$s_{ji}$</span> is the cell vice versa. Therefore, symmetrizing <span class="math-container">$S$</span> is a matter of ensuring that <span class="math-container">$s_{ij} == s_{ji}$</span>, which you can achieve by multiplying the (ranked?) entries. Symmetric matrices are nice for doing certain kinds of math, and yield a handy representation of undirected graphs, which the authors appear to be using for <a href="https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm" rel="nofollow noreferrer">kNN</a>.</li>
  <li><span class="math-container">$max(x, 0)$</span> is just saying that you want either <span class="math-container">$x$</span> if <span class="math-container">$x$</span> is non-negative, or <span class="math-container">$0$</span> if it is negative. Thus, you make the whole matrix non-negative, which makes some handy matrix math possible. You can see how this works computationally by inspecting e.g. the Python <code>max()</code> <a href="https://docs.python.org/3/library/functions.html#max" rel="nofollow noreferrer">function</a>.</li>
  </ul>
  
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Answers.CreationDate: 2022-11-20T22:22:30.313Z
Answers.LastActivityDate: 2022-11-20T22:22:30.313Z
Answers.UserDisplayName: Maximilian Press
Body: <p>I'm reading through this paper: Baran, Y., Bercovich, A., Sebe-Pedros, A., Lubling, Y., Giladi, A., Chomsky, E., ... &amp; Tanay, A. (2019). <a href="https://genomebiology.biomedcentral.com/articles/10.1186/s13059-019-1812-2" rel="nofollow noreferrer">MetaCell: analysis of single-cell RNA-seq data using K-nn graph partitions</a>. Genome biology, 20(1), 1-19.</p>
  <p><a href="https://i.sstatic.net/BMITn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BMITn.png" alt="enter image description here" /></a></p>
  <p>Maybe I am not familiar with this notation but I don't understand the regularization procedure as follows:</p>
  <ol>
  <li>What is happening in these max functions? What does the 0 indicate? They have some terms <span class="math-container">$x$</span> and then in the max function they write <span class="math-container">$max( x, 0)$</span>.</li>
  <li>I assume <span class="math-container">$K$</span> is some regularization parameter, like a cost. Is <span class="math-container">$s_{ij} * s_{ji}$</span> the rank product?  I.e. the geometric mean?</li>
  </ol>
  
Tags: <scrnaseq><r><graphs>
Title: Understanding regularization step in Meta-cell pre processing
ThreadText: # Understanding regularization step in Meta-cell pre processing
  
  (Asked by: username_0 on 2022-11-20)
  
  I'm reading through this paper: Baran, Y., Bercovich, A., Sebe-Pedros, A., Lubling, Y., Giladi, A., Chomsky, E., ... & Tanay, A. (2019). [MetaCell: analysis of single-cell RNA-seq data using K-nn graph partitions](https://genomebiology.biomedcentral.com/articles/10.1186/s13059-019-1812-2). Genome biology, 20(1), 1-19.
  
  [![enter image description here](https://i.sstatic.net/BMITn.png)](https://i.sstatic.net/BMITn.png)
  
  Maybe I am not familiar with this notation but I don't understand the regularization procedure as follows:
  
  1. What is happening in these max functions? What does the 0 indicate? They have some terms $x$ and then in the max function they write $max( x, 0)$.
  2. I assume $K$ is some regularization parameter, like a cost. Is $s\_{ij} \* s\_{ji}$ the rank product? I.e. the geometric mean?
  
  **Tags:** scrnaseq, r, graphs
  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2022-11-20. Score: 2 | Accepted answer
  
  Without looking in too much detail:
  
  * We may suppose that $S$ is a matrix with dimensions indexed as $(i, j)$. So $s\_{ij}$ is the cell of $S$ indexed to the $i$th row and $j$th column, and $s\_{ji}$ is the cell vice versa. Therefore, symmetrizing $S$ is a matter of ensuring that $s\_{ij} == s\_{ji}$, which you can achieve by multiplying the (ranked?) entries. Symmetric matrices are nice for doing certain kinds of math, and yield a handy representation of undirected graphs, which the authors appear to be using for [kNN](https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm).
  * $max(x, 0)$ is just saying that you want either $x$ if $x$ is non-negative, or $0$ if it is negative. Thus, you make the whole matrix non-negative, which makes some handy matrix math possible. You can see how this works computationally by inspecting e.g. the Python `max()` [function](https://docs.python.org/3/library/functions.html#max).
  
  
  ---
  
  