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.

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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: $X$: ...
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What kind of a matrix factorization is appropriate for my problem?

I have the following problem: I measure mass concentrations of a pollutant 'p' in n stations (same pollutant in all stations). And I have many observations. Theoretically, I believe that if I assume ...
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Possible typo Logistic Matrix Factorisation?

In the paper Logistic Matrix Factorization for Implicit Feedback Data by CC. Johnson(see link). The author stated his maximum likelihood function to be (omit all the indices): $\prod p(l|X,Y,\beta)^{\...
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which algorithm is preferred for knowledge based recommendation system? [closed]

I would like to know the recommendation algorithm commonly used for knowledge based recommendation.
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Implicit feedback ALS algorithm: the alpha parameter

I'm creating a recommender system for a video streaming service. My only knowledge about the user preference on a video is the watched percentage of that video. I'm using the implicit feedback ALS ...
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How to find upper and lower bound

Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
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Kernel matrix decomposition

I had a look at the sklearn.kernel_approxiamtion.Nystroem implementation, which is also described in this post: Nystroem Method for Kernel Approximation Here, a ...
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How to predict for test set when training a recommender by decomposing the utility matrix X=UV?

This probably sounds stupid but I don't get the workflow of building a recommending system by the utility matrix: X[i,j] = how much the ith user likes the jth object. For practical issues I refer to ...
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SVD for a complex data matrix — what is the meaning of the columns of $V$?

I've read this wonderful explanation of SVD, where the writer mentions that the columns of $V$ are the principal directions (Summary, #1). Is this also true when the data matrix $X$ is complex? If I'm ...
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Convert the following expression w.r.t to the whole dataset instead of element of the dataset?

I am in the process of expressing the w in LSSVM with data points and constants. After I resolve the KKT conditions for the LSSVM I got $$w = \sum ^N _{i=1} \alpha_i x_i$$ Is it possible to convert ...
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Prove that sample covariance matrix is positive definite [duplicate]

Consider the $p \times p$ sample covariance matrix: $$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{...
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Confusion on vector, matrix notation on research papers and the need of row normalization

I attempt to understand the formulation of dictionary learning for this paper: Depression Detection via Harvesting Social Media: A Multimodal Dictionary Learning Solution Multimodal Task-Driven ...
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Human intuition behind SVD in case of recommendation system

This does not answer my question. I struggled very hard to understand the SVD from a linear-algebra point of view. But in some cases I failed to connect the dots. So, I started to see all the ...
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SVD : Why right singular matrix is written as transpose

The SVD is always written as, A = U Σ V_Transpose The question is, Why is the right singular matrix written as V_Transpose? I mean lets say, W = V_Transpose and then write SVD as A = U Σ W SVD Image ...
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Penalized Canonical Correlation in R with PMA Module

I am trying to use sparse canonical correlation analysis as implemented in the R PMA package. I'm finding that the correlations output by the package seem slightly inconsistent with the ones you would ...
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How to project a mxn matrix (m features, n samples) onto a space generated by a mxk matrix (m features, k factors)?

I got two matrices A and B, the dimension of A is m x n, where n represents the number of samples, and m represents the number of features. Then after dimensionality reduction, e.g., by NMF, I got B, ...
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Symbolic Singular Value Decomposition? U,S,V as function of the elements of M [closed]

Suppose we want to compute the SVD of $\mathbf{M} = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix}$ (...
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Interpreting PCA results of first two components

I don't like the looks of my PCA graph here. PCA coordinates should be uncorrelated, yet the variance between the coordinates of the second component increases as the first component decreases. What's ...
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Is matrix factorization in collaborative filtering equivalent to a special type of 3-layer neural networks for multi-class classification?

In Andrew Ng's Coursera course Machine Learning, he introduced a collaborative filtering algorithm, which I later checked to be called matrix factorization, where the optimization objective is $$ \...
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Use Matrix Factorization to predict probability of a recommendation system?

I have a dataset where I have a sparse utility matrix (user-product) with binary input: 1 if the user $i$ bought the product $j$, and 0 if it hasn't. However it has a different meaning on the test set,...
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Is there a version of NMF that normalizes the sum of scores of each sample?

I want to decompose a nonnegative data matrix $A \in \mathbb{R}^{n\times m}$ into nonnegative basis vectors $U \in \mathbb{R}^{n \times k}$ and a score matrix $V \in \mathbb{R}^{m \times k}$ such that ...
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ML model for Signal Decomposition [closed]

So recently I got a task which can be summarized as follows: Suppose we have 3 functions f1, f2, f3 and a certain combination of the functions gives us ...
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What is gravity in the context of “folding” in recommender systems?

What is "gravity" in the context of recommender systems? More specifically, how is it supposed to help with the "folding" problem where irrelevant queries may be returned if we don't provide ...
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Decomposition of $x^T K x$ as $y^T y$

When $x$ is a vector of size Nx1, and $K$ is a very large symmetric sparse matrix of size NxN (say N=100K), is it possible to decompose $x^T K x$ as $y^T y$? As if I could get $y = K^{1/2} x$. Edit ...
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Cold starts in factorization - WALS projections

I read here (Google's crash course on recommendations) the following: Given a new item $i_0$ not seen in training, if the system has a few interactions with users, then the system can easily compute ...
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SGD for matrix factorization: Negative sampling and gravity

I read here the following downside of using SGD for Matrix factorization: Harder to handle the unobserved entries (need to use negative sampling or gravity). What do they mean by negative sampling ...
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Does SVD provide the best low rank approximation for any matrix regardless of shape?

Wikipedia states (link below) that by the Eckart-Young-Mirsky theorem, the SVD provides the best low rank matrix approximation (on the basis of Frobenius norm of the error matrix) for any matrix A in ...
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How does multicollinearity affect the eigenvalues of a matrix?

I have been looking into ridge regression as a method to address multicollinearity in data. I am aware that multicollinearity can cause high variance in coefficient estimates. I have seen equations ...
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What is finite precision arithmetic and how does it affect SVD when computed by computers?

Was reading the paper "DETECTING AND ASSESSING THE PROBLEMS CAUSED BY MULTICOLLINEARITY:A USE OF THE SINGULAR-VALUE DECOMPOSITION" by David Belsley and Virginia Klema. After performing SVD, while ...
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How does the dimensions of my regression matrix effect its eigenvalules?

I was reading through another question answer where it discussed the inversion of the XX' matrix in ridge regression. It stated that it is impossible to have positive eigenvalues if matrix $XX'$(=$VDV'...
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Generalization Error and Matrix Factorizations?

This is more of a discussion/conceptual idea but: Is the notion of generalization error well-defined for a problem such as matrix factorization? I'm working with tensors/matrices and performing ...
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How do I approach this problem?

Let's say I have a dataset with multiple types of multiple ingredients (salt1,salt2, etc). Each n-th variation of each ingredient vs flavor may be represented by an n×k matrix that where an ingredient ...
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120 views

Moore Penrose Pseudo-Inverse Fast Algorithm in R [closed]

I want to apply Moore Penrose Pseudo-Inverse on my matrix, which is a 20,000 * 20,000 symmetric matrix with rank 19,999. I found ginv() function from the ...
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Why atoms in the dictionary of Dictionary Learning method are not required to be orthogonal?

According to Sparse Dictionary Learning (wiki), Sparse dictionary learning is a representation learning method which aims at finding a sparse representation of the input data (also known as sparse ...
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1answer
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What should I be careful when using the word “supervised” in paper writing?

I am a biologist using machine learning tool for my research. I modified matrix decomposition ($V \approx WH$) to fit my data and wanted to describe about that in my paper. If I fixed one matrix ...
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How to approximate a Hermitian matrix with a transposed cross product of a single matrix?

I have a complex square matrix, and wish to learn latent factors (equally weighted latent factors, so not SVD) from this matrix. Given a Hermitian matrix A of ...
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170 views

Cholesky decomposition or alternative for negatively correlated data simulations

I want to generate some signals that have a correlation distribution around a specific pre-defined correlation value (i.e., the distribution of the values of their correlation matrix is around a ...
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Finding the optimal number of latent factors in Symmetric Singular Value Decomposition

Consider symmetric Singular Value Decomposition of a symmetric unitary matrix A into U, D, ...
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more then one user based Collaborative filtering

I want to get a movie recommendation by selecting multiple users. Usually, it takes one user ID and gives a recommendation. I thought user-based collaborative filtering with KNN is a good idea. I have ...
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How to get recommended users for an item with a matrix factorization recommender system?

Any tutorial or guide I've seen is for recommending items to a user, but how can I recommend users for an item? I am currently using the implicit library alternating least squares model. I have ...
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Intuitions behind Singular Value Decomposing generalizing eigen decomposition [duplicate]

Let M be mxn matrix then SVD of M will be UXW^* (sorry for X, assume summation). Then how does it generalizes eigen decomposition ? Since eigen decomposition is possible for nxn matrix and that are ...
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How to create linear user embedding from some answers to binary questions?

I have each user U_i answering 10 binary questions out of a pool Q with either answer 1 or 2. I would like to learn an embedding of user profiles based on these answers to predict is answer to other ...
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How to decompose covariance matrix for a bivariate normal distribution

Let $$\boldsymbol{x} = \begin{bmatrix} x\\ y\\ \end{bmatrix}, \boldsymbol{\mu} = \begin{bmatrix} \mu_x\\ \mu_y\\ \end{bmatrix}, \boldsymbol{\Sigma}=\begin{bmatrix} \sigma_x & \rho\sigma_x\sigma_y\\...
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Row similarity in matrix vs in different factorizations

Suppose an arbitrary $m \times n$ matrix $M$ and the factorizations: Arbitrary: $M = U_a V_a^T$, where $U_a$ is $m \times k$, $V_a$ is $n \times k$ ($k < m,n$), and $rank(U_a)=rank(V_a)=k$. SVD: $...
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Collaborative filtering movie recommender: how to account for missing ratings implying information about user preference?

I'm trying to learn about recommender systems with a fairly standard data set: I have a matrix with thousands of users, thousands of movies, and the ratings that users give to each movie. Obviously, ...
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Infinitesimal generator of Markov chain using numpy

I am computing the infinitesimal generator of a continuous-time Markov chain from the transition probabilities p. I am following the methodology described here, ...
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How can this L(2,1) problem be reduced to the orthogonal procrustes problem?

NOTE: Don't take this too serious -- the question is actually due to my misreading $\|y_i - Wx_i\|^2$ as $\|y_i - Wx_i\|_2$, see the answer. Smith et al. in Offline bilingual word vectors, orthogonal ...
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Relation between eigenvalues of original and transformed matrices

Let the matrix $X$ be some data arranged in rows. Consider the following eigenvalue decomposition $X^\top X = Q \Theta Q^\top=\sum_{i=1}^n \theta_iq_iq_i^\top$ where $q_i$ are the eigenvectors and $\...
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Square root of an almost diagonal matrix

Is there an efficient way to compute square root of an almost diagonal symmetric Hessian matrix, which is diagonal with the exception of the last two columns and last two rows? Could the efficient ...
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Enforcing constraints on weight matrices using ReLU activation

In the paper 'A Deep Non-Negative Matrix Factorization Neural Network' by Flunner and Hunter, proof of Theorem 1 says that "The ReLu Activation function is a standard approximation of a non-negative ...

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