# Questions tagged [matrix-decomposition]

Matrix decomposition refers to the process of factorizing a matrix into a product of smaller matrices. By decomposing a large matrix, one can efficiently perform many matrix algorithms.

283 questions
Filter by
Sorted by
Tagged with
5 views

### State and Action spaces for MDP: Matrix as a state? Mixed action space? And more [closed]

I am pretty new to Reinforcement Learning (RL) and currently formulating the problem into a proper MDP. The goal is to restore the loads in a distribution system after an outage. The restoration is ...
1 vote
17 views

### General matrix decomposition downgrading algorithm for sampling

I would like to sample from a multivariate Gaussian distribution with covariance matrix $\Sigma - uu^T$, where $u$ is a vector and $\Sigma - uu^T$ is PSD. I have knowledge of a non-Cholesky ...
1 vote
33 views

### Latent Semantic Indexing vs. PCA

I am trying to understand how Latent Semantic Analysis works, reading demonstrations based on singular value decomposition. Let's denote $X$ a $N \times D$ document-term matrix. The $D$ rows of $X$ ...
• 249
16 views

### Understanding Leverage Score Sampling to get representative sample

I was reading about Leverage Score Sampling. If I am not wrong then what I know that Leverage Score Sampling help us to select representative sample. But I didn't understand how the whole process is ...
• 121
28 views

### Correct NMF usage in context of recommender systems

I am trying to teach myself about the NMF models (in the context of recommender systems), and I have come across different suggestions on how to set up such a workflow, but I'm not sure if both are ...
• 101
1 vote
35 views

### Implementing eigen decomposition [closed]

Question Please help understand why the eigen vectors do not match below. If there are misunderstandings or incorrect place, please correct too. It would be much appreciated. Eigen decomposition ...
• 1,186
1 vote
20 views

### Eckart–Young–Mirsky theorem for $n \gg m$

It has been proven that the best reconstruction error in the $k$ rank matrix estimation problem in terms of Frobenius or $L2$ norm is given by the $k$-truncated SVD as shown here. I've read in ...
1 vote
180 views

### Why PCA is invariant under rotation

Lets say that we have a matrix of variables (the columns are variables and rows are the observations) called X whenre X = [x1, x2, ...., xp] where ...
18 views

### Generate dataset that adhere to a given correlation matrix [duplicate]

I'm newbie in statistics and this question might be naive so kindly excuse me in advance. I have a correlation matrix of 40 features and I want to generate a dataset of hundreds of observations that ...
• 101
1 vote
31 views

### Decomposition analysis for data between zero and one

I want to analyze latent components of data that has values between zero and one (including zero and one). In detail, the data structure is n x m and I'm looking to find the r underlying components. ...
• 31
42 views

### How to find a decomposition of multivariate X along which y varies the most?

I'm looking for an existing algorithm which carries out the task shown in the title. My use-case in other words: I have a set of continuous independent variables (X) and a continuous dependent ...
• 123
1 vote
22 views

1 vote
126 views

### Understanding the decomposed matrices in Singular Value Decomposition [duplicate]

I am trying to get some intuition regarding the U and V matrices in SVD ($M=UEV^T$). I think these are orthonormal basis vectors, but I am struggling to get an intuition if they represent anything ...
• 249
1 vote
300 views

### Why is there a discrepancy between the eigenvalues of the covariance matrix (PCA) and the eigenvalues of the kernel matrix (kernel PCA)?

I've done PCA on my data matrix $\mathbf{X}$ which gives me i.a. the eigenvalues $\lambda$ and eigenvectors $v$ of the data covariance matrix $C=\mathbf{X}^T \mathbf{X}$. I'm now extending my ...
• 21
5k views

### Relationship between Cholesky decomposition and matrix inversion?

I've been reviewing Gaussian Processes and, from what I can tell, there's some debate whether the "covariance matrix" (returned by the kernel), which needs to be inverted, should be done so ...
• 2,113
1 vote
206 views

### Alternating least squares --- what are the limitations?

I am taking an introductory course to Machine Learning and we learned alternating least squares for recommender systems. I learned that this method has some advantages --- easy to parallelize, and ...
• 11
41 views

### Convex Optimization Algorithm for Additive Matrix Decomposition

Given an m $\times$ n matrix $Y = S + L + E$, where $S$ is a sparse matrix, $L$ is a low rank matrix and $E$ is a noise matrix, I want to recover $S$, $L$. One of the techniques given in Agarwal et al ...
1 vote
177 views

355 views

### Covariance Matrix Decomposition - Data Decorrelation

So I recently found out about Mahalanobis distance. Given a r.v $x$ in N-dimensional space, an associated metric is defined by $$M(x) = \sqrt{(x-\mu)^T S^{-1}(x-\mu)}$$ where $\mu$ and $S$ are mean ...
• 1,749
1 vote
84 views

### How to compute the left singular eigenvector matrix (U) from the output of prcomp() for PCA in R?

I am examining the output of the prcomp() function in R for PCA in light of the singular value decomposition equation: $X = U \cdot \Sigma \cdot V^{T}$, where: $X$: ...
• 1,085
1 vote