Answers.Body: <p><span class="math-container">$\ell_{2,1}$</span> is a matrix norm, as stated in <a href="https://www.ijcai.org/Proceedings/11/Papers/267.pdf" rel="nofollow noreferrer">this paper</a>.
  For a certain matrix <span class="math-container">$A \in \mathbb{R}^{r\times c}$</span>,
  we have
  <span class="math-container">$$\|A\|_{2,1} = \sum_{i=1}^r \sqrt{\sum_{j=1}^c A_{ij}^2}$$</span>
  You first apply <span class="math-container">$\ell_2$</span> norm along the columns to obtain a vector with r dimensions. Then, you apply <span class="math-container">$l_1$</span> norm to that vector to obtain a real number. You can generalize this notation to every norm <span class="math-container">$\ell_{p,q}$</span>.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-01-03T19:08:31.317Z
Answers.LastActivityDate: 2022-03-10T17:57:34.277Z
Answers.LastEditDate: 2022-03-10T17:57:34.277Z
Answers.UserDisplayName: bluesisnotrock
Body: <p>I'm reading <a href="https://www.sciencedirect.com/science/article/pii/S0957417418307267" rel="nofollow noreferrer">this paper</a> and it says:</p>
  
  <blockquote>
    <p>In this paper, we present a multi-class embedded feature selection method called as sparse optimal scoring with adjustment (SOSA), which is capable of addressing the data heterogeneity issue. We propose to perform feature selection on the adjusted data obtained by estimating and removing the unknown data heterogeneity from original data. Our feature selection is formulated as a sparse optimal scoring problem by imposing <strong><span class="math-container">$\ell_{2, 1}$</span>-norm regularization</strong> on the coefficient matrix which hence can be solved effectively by proximal gradient algorithm. This allows our method can well handle the multi-class feature selection and classification simultaneously for heterogenous data</p>
  </blockquote>
  
  <p>What is the <span class="math-container">$\ell_{2, 1}$</span> norm regularization? Is it L1 regularization or L2 regularization?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-12-30T22:13:48.037Z
Comments.Text: It's a matrix norm. You have it [here](https://en.wikipedia.org/wiki/Matrix_norm#L2,1_and_Lp,q_norms).
Comments.UserDisplayName: Brale
Tags: <machine-learning><feature-selection><regularization>
Title: What is the $\ell_{2, 1}$ norm?
UserDisplayName: Gyntonic
ThreadText: # What is the $\ell_{2, 1}$ norm?
  
  (Asked by: username_0 on 2019-12-30)
  
  I'm reading [this paper](https://www.sciencedirect.com/science/article/pii/S0957417418307267) and it says:
  
  > In this paper, we present a multi-class embedded feature selection method called as sparse optimal scoring with adjustment (SOSA), which is capable of addressing the data heterogeneity issue. We propose to perform feature selection on the adjusted data obtained by estimating and removing the unknown data heterogeneity from original data. Our feature selection is formulated as a sparse optimal scoring problem by imposing **$\ell\_{2, 1}$-norm regularization** on the coefficient matrix which hence can be solved effectively by proximal gradient algorithm. This allows our method can well handle the multi-class feature selection and classification simultaneously for heterogenous data
  
  What is the $\ell\_{2, 1}$ norm regularization? Is it L1 regularization or L2 regularization?
  
  **Tags:** machine-learning, feature-selection, regularization
  > **Comments:**
  > **username_1**: It's a matrix norm. You have it [here](https://en.wikipedia.org/wiki/Matrix_norm#L2,1_and_Lp,q_norms).  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2020-01-03. Score: 5 | Accepted answer
  
  $\ell\_{2,1}$ is a matrix norm, as stated in [this paper](https://www.ijcai.org/Proceedings/11/Papers/267.pdf).
  For a certain matrix $A \in \mathbb{R}^{r\times c}$,
  we have
  $$\|A\|\_{2,1} = \sum\_{i=1}^r \sqrt{\sum\_{j=1}^c A\_{ij}^2}$$
  You first apply $\ell\_2$ norm along the columns to obtain a vector with r dimensions. Then, you apply $l\_1$ norm to that vector to obtain a real number. You can generalize this notation to every norm $\ell\_{p,q}$.
  
  
  ---
  
  