SVD and SVM solve different problems, regardless of how they work internally.
SVD is a dimensionality reduction technique that basically compresses your data. Generally machine learning, SVD is often used as a preprocessing step. In the recommendation, there are many matrix / tensor factorization methods that resemble SVD, but are often optimized for different objective functions, for example. they include the term regularization to prevent retraining (to better generalize learning data to actual prediction). They are often called SVDs or contain “SVDs” in their name, but they are not exactly what the mathematician understands as SVDs.
SVM is a kind of classifier. In recommendation systems, you can use SVM to train one classifier for each user (or group of users), which determines, based on the product attribute, whether the user will buy the product or not. Of course, you can also use other classifiers for this task, such as naive bays, logistic regression, or decision trees.
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