Dear Ankit,

PCA and NMF are both dimensionality reduction techniques, but they work differently and have different assumptions about the data.

PCA aims to find orthogonal linear combinations of the original features that capture the maximum variance in the data. These linear combinations are called principal components. The first principal component captures the largest variance, the second one captures the next largest variance, and so on. By selecting the top k principal components, you can reduce the dimensionality of the data while still preserving most of the variability.

NMF, on the other hand, is a factorization method that decomposes a non-negative data matrix into two non-negative matrices: the basis matrix (W) and the coefficient matrix (H). The goal of NMF is to find a lower-dimensional representation of the data by approximating the original data matrix as the product of the two smaller matrices. NMF is particularly useful for data that has a natural additive structure, such as images or documents with word counts.

Unlike PCA, NMF does not explicitly optimize for the axes of variability. Instead, it learns the key components or features of the data by constraining all factors to be non-negative. This leads to a parts-based representation of the data, which can be more interpretable than PCA.

Determining the optimal number of factors in NMF can be more challenging than in PCA. One common approach is to use cross-validation or other model selection techniques to find the number of factors that best balances between fitting the data well and minimizing complexity. Another approach is to use a pre-specified number of factors based on prior knowledge or to use methods such as the Bayesian Information Criterion (BIC) or the Akaike Information Criterion (AIC) to determine the optimal number of factors.

Kind regards,

Wadie

Dear Wadie,

Thanks for your very detailed explanation. So NMF is particularly useful for the data that has a natural additive structure and it learns the key components or features of the data.

Is there any way to know that particular NMF factor most important? Can we know the rank of NMF factors based on the key components?

Thanks.

Yes, it is possible to assess the importance of individual NMF factors and determine their rank based on the key components they represent. To do this, you can use a combination of quantitative and qualitative methods:

You might need to combine several of these methods to obtain a comprehensive understanding of the importance of each factor.

Viel Gluck,

Wadie

Thanks a lot Wadie for very concise information.

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Hi Wadie Abu Dahoud

I hope you are doing well. I need some more suggestion about finding the stability of NMF if you know it.

I have two dataset A and B each having 10 clusters. Assume that cluster i in A gives similar information as cluster i in B. Where i is any cluster id between 1 and 10.

I have implemented an integrated NMF approach to combined the cluster i of A and B simulatenously.

But I want to know how the factors are stable and consistent across different runs. And what is approach to find the best k.

Do you know any approach which I can follow in this case?

Thank you.