# Tag Info

Accepted

### How to reverse PCA and reconstruct original variables from several principal components?

PCA computes eigenvectors of the covariance matrix ("principal axes") and sorts them by their eigenvalues (amount of explained variance). The centered data can then be projected onto these principal ...
• 105k
Accepted

### The proof of shrinking coefficients using ridge regression through "spectral decomposition"

The question appears to ask for a demonstration that Ridge Regression shrinks coefficient estimates towards zero, using a spectral decomposition. The spectral decomposition can be understood as an ...
• 323k

### Relationship between SVD and PCA. How to use SVD to perform PCA?

I wrote a Python & Numpy snippet that accompanies @amoeba's answer and I leave it here in case it is useful for someone. The comments are mostly taken from @amoeba's answer. ...

### Relationship between SVD and PCA. How to use SVD to perform PCA?

Let me start with PCA. Suppose that you have $n$ data points comprised of $d$ numbers (or dimensions) each. If you center this data (subtract the mean data point $\mu$ from each data vector $x_i$) you ...
• 511

### Why does Andrew Ng prefer to use SVD and not EIG of covariance matrix to do PCA?

amoeba already gave a good answer in the comments, but if you want a formal argument, here it goes. The singular value decomposition of a matrix $A$ is $A=U\Sigma V^T$, where the columns of $V$ are ...
• 2,251
Accepted

### Difference between scikit-learn implementations of PCA and TruncatedSVD

PCA and TruncatedSVD scikit-learn implementations seem to be exactly the same algorithm. No: PCA is (truncated) SVD on centered data (by per-feature mean substraction). If the data is already ...
• 3,769

### What is the intuition behind SVD?

Let $A$ be a real $m \times n$ matrix. I'll assume that $m \geq n$ for simplicity. It's natural to ask in which direction $v$ does $A$ have the most impact (or the most explosiveness, or the most ...
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### What are the pros and cons of applying pointwise mutual information on a word cooccurrence matrix before SVD?

according to Dan Jurafsky and James H. Martin book: "It turns out, however, that simple frequency isn’t the best measure of association between words. One problem is that raw frequency is very skewed ...
• 179
Accepted

### What does it mean when PCA does not produce a reduction in dimensionality?

Results suggest that your features are mutually orthogonal. Accounting for total variance means accounting for both variance and covariance. Orthogonality limits covariance. Standardization equates ...
• 1,514
Accepted

### Why is non-negativity important for collaborative filtering/recommender systems?

I am not a specialist in recommender systems, but as far I understand, the premise of this question is wrong. Non-negativity is not that important for collaborative filtering. The Netflix prize was ...
• 105k
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### Why cannot I obtain a valid SVD of X via eigenvalue decomposition of XX' and X'X?

Analysis of the Problem The SVD of a matrix is never unique. Let matrix $A$ have dimensions $n\times k$ and let its SVD be $$A = U D V^\prime$$ for an $n\times p$ matrix $U$ with orthonormal ...
• 323k

### Why does Andrew Ng prefer to use SVD and not EIG of covariance matrix to do PCA?

@amoeba had excellent answers to PCA questions, including this one on relation of SVD to PCA. Answering to your exact question I'll make three points: mathematically there is no difference whether ...
• 61.4k
Accepted

### Cholesky versus eigendecomposition for drawing samples from a multivariate normal distribution

The problem was studied by by Straka et.al for the Unscented Kalman Filter which draws (deterministic) samples from a multivariate Normal distribution as part of the algorithm. With some luck, the ...
• 695

### What is the intuition behind SVD?

Take an hour of your day and watch this lecture. This guy is super straight-forward; It's important not to skip any of it because it all comes together in the end. Even if it might seem a little slow ...
• 131

### Contrasting covariance calculation using R, Matlab, Pandas, NumPy cov, NumPy linalg.svd

Note that numpy.cov() considers its input data matrix to have observations in each column, and variables in each row, so to get ...
• 2,842
Accepted

### Why does the reconstruction error of truncated SVD equal the sum of squared singular values?

Let $$X = U\Sigma V^\prime$$ be the SVD of the $n\times r$ matrix $X$. Let $||\quad ||$ be any matrix norm that is left- and right-invariant under orthogonal transformations (reflections and ...
• 323k
Accepted

### Why do we need the regularization term for NMF but not for SVD?

NMF does not always include regularization -- for example, see the first two cost functions here. But, regularized NMF can be useful: If you're willing to add extra constraints beyond nonnegativity, ...
• 5,209
Accepted

### Explaining dimensionality reduction using SVD (without reference to PCA)

The SVD can be linked to dimensionality reduction from the standpoint of low rank matrix approximation. SVD for low rank matrix approximation Suppose we have a matrix $X$ and want to approximate it ...
• 32.5k

### Why does Andrew Ng prefer to use SVD and not EIG of covariance matrix to do PCA?

For Python users, I'd like to point out that for symmetric matrices (like the covariance matrix), it is better to use numpy.linalg.eigh function instead of a ...
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Accepted

• 3,210

### What does it mean when PCA does not produce a reduction in dimensionality?

PCA can be used to scale and rotate data, if we select all transformed features instead of a subset of them. My answer here gives an example of scaling and rotating the data, but without dimension ...
• 36.9k
Accepted

### Confirming an understanding of SVD

I have tried to dispel some of the popular myths regarding the SVD. A common application of SVD is to make low-rank approximations to a matrix, $A$'' This is one of the applications of the SVD ...
• 266
What you are doing is essentially PCA, even though you did not use this term. It is exactly PCA if your $X$ is centered; if not, then it is a sort of "uncentered PCA". In any case, to get from $A$ to ...