In addition to @paulinpaloalto‘s great summary:
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feature importance helps you in assessing and selecting your features, see also this repo:
In this repo you can also find the unsupervised methods (like: Principle Component Analysis) as mentioned by @TMosh and in addition also with the Partial Least Square transformation a great supervised method to keep a good „data to feature ratio“ using dimensionality reduction, e.g. by getting rid of redundant information in your features. If you are interested in what is meant with this data/feature ratio, feel free to check out this thread: C3_W2 - PCA Question - #3 by Christian_Simonis
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in practice you often have the trade off between: adding one or two features (with hopefully better model performance) but then also having higher maintenance and monitoring effort to ensure the quality of your ML pipeline to maintain a certain data quality which means usually more effort and higher total costs of your ML system when operating it
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one additional point: in deep learning of course you do not need to check necessarily on feature importance when it comes to big and highly unstructured data since there DL takes care of feature engineering in its own. But in classic machine learning it is definitely a good idea to check on feature importance after handcrafting your features with domain knowledge which is considered a best practice by many practitioners in the industry, that work usually rather with structured and limited data and leverage classic machine learning models!
Hope that helps!
Best regards
Christian