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11 views

Matrix factorization: Item vector clustering.

I tried to run k-means clustering (with euclidean distance) on top of item vectors that come from a matrix factorization algorithm. The results make absolutely no sense. Most (95%) items are in the ...
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0answers
14 views

Recommender; Matrix factorization: How to compute item to item recommendations

I am trying to compute item to item recommendations. Basically to output related/similar movies. Not depending on the user but only on the movie. My first thought was to compute the dot product for ...
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1answer
49 views

Optimization with both L1 and L2 regularization

After doing some research I suppose the hard part is that, L2 regularized problem is often solved by gradient descent, while L1 regularized problem is often solved by coordinate descent. But which ...
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0answers
19 views

The orthogonal complement of the range

I am reading about the pseudo inverse, and there is a concept that I am not quite able to understand. It would be nice if someone could give me a clue of how to work it out, or an easy example on how ...
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1answer
19 views

Prediction of unknown triples for relational learning tasks using RESCAL

Background: I am focusing on a relational learning task, where links between entities are predicted across several relations. An example of a relation in this task is if two entities have the same ...
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1answer
62 views

What is UV decomposition?

As I'm reading about different matrix decomposition methods, I see a reference to a decomposition method that is known as UV method where: U: has small number of columns V: has small number of rows ...
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1answer
45 views

Markov Cluster Algorithm transition matrix

I am reading the notes on Markov Cluster Algorithm by Kathy Macropol (http://www.cs.ucsb.edu/~xyan/classes/CS595D-2009winter/MCL_Presentation2.pdf) On slide 14/46 the author talks about inflation and ...
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0answers
6 views

How to determine if a lower rank approximation exists?

In all the literature I read on low-rank approximations, I have yet to run across situations in which people first check to see if a lower rank exists. My understanding is that a matrix whose ...
1
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1answer
53 views

Mahalanobis distance with LDL decomposition

I've got an extended Kalman filter with innovation covariance defined as $\mathbf{W}=\mathbf{H}\mathbf{P}\mathbf{H}^\textrm{T} + \mathbf{R}$. I want to know the squared Mahalanobis distance $\|z\|^2$ ...
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0answers
42 views

Confirming an understanding of SVD

I'm trying to sift through the concepts of low-rank approximations, matrix factorizations and SVD. There's a lot of info out there and the rabbit hole is deep so I just want to make sure my high level ...
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0answers
22 views

Comparison of two Products of Variance

Let $A$, $B$ and $B'$ be random variables, and consider $\text{Var}(AB) \text{ and} \text{ Var}(AB')$, such that $\text{Cov}(A,B)\ne 0$, $\text{Cov}(A,B')\ne 0$ and $\text{Var}(B)\geq ...
6
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1answer
110 views

Explain how `eigen` helps inverting a matrix

My question relates to a computation technique exploited in geoR:::.negloglik.GRF or geoR:::solve.geoR. In a linear mixed model ...
6
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0answers
135 views

State-of-the-art in Collaborative Filtering

I am working on a project for collaborative filtering (CF), i.e. completing a partially observed matrix or more generally tensor. I am a newbie to the field, and for this project eventually I have to ...
7
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1answer
195 views

Updating SVD decomposition after adding one new row to the matrix

Suppose that I have a dense matrix $ \textbf{A}$ of $m \times n$ size, with SVD decomposition $$\mathbf{A}=\mathbf{USV}^\top.$$ In R I can calculate the SVD as ...
4
votes
1answer
49 views

When do we use matrix norm?

When do we use matrix norm? matrix norm is one of the property of a matrix, but I am not sure when I will use it. Do we use it for calculating a upper bound of a matrix? ...
2
votes
1answer
98 views

Overfitting in Matrix Factorization models used in Recommender Systems (Collaborative Filtering)

I'm wondering if I should check if a Matrix Factorization model I built for recommendation by Collaborative Filtering is overfitting. I trained a model using MLlib ALS (Alternating Least Squares) ...
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0answers
68 views

Calculation for variance of slope for linear model in R predict function

I'm looking through the code for the predict.lm function in R, and am confused about one of the calculations. I believe it is the calculation used to determine the ...
0
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1answer
29 views

Convergence of eigenvectors and eigenvalues of matrix that converges

For each random variable $X=x$, there is a symmetric positive definite matrix $M(x)$. Suppose there is a set of samples of random matrix $M_1,M_2,...,M_n$, where each $M_i$ is calculated based on the ...
2
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0answers
29 views

eigen problem of random matrix

Suppose I need to find the eigen vectors of a matrix $M(X)$ for any sample $X$ to be my dimension reduction projector. Then since I have samples $X_1,...,X_n$, so we propose to use the eigen vectors ...
2
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1answer
364 views

Scaling a Covariance Matrix by a factor?

I have a $n\times n$ covariance matrix representing the covariance between n random processes. I want to increase all of the covariances by a certain factor, say $1.5x$. Multiplying each covariance ...
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0answers
6 views

“Tucker decomposition often assumes Gaussian noises”. Are the noises here input or output?

I saw in some papers saying " Tucker decomposition methods have implicitly or explicitly assumed the noises are Gaussian" in ...
0
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2answers
55 views

optimizing over a set of symmetric matrices

I need to minimize a complicated loss function, $f\left(\Lambda\right)$ over a set of symmetric matrices, $S_{p}$ of dimension p, such that all the eigenvalues of $\Lambda \in \left[0,1\right]$. I ...
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1answer
113 views

Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
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0answers
22 views

Shrinkage of Schafer and Strimmer

As we all know that the sample covariance matrix $(S = (s_{ij}))$ is postive definite when the number of observations is smaller than the number of samples, that is n>p. But, the sample covariance ...
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0answers
57 views

Interpretation of matrix factorization results

Matrix factorization methods are known to give good results pertaining to problems like movie recommendation. The method reduces the feature space, which is then used for recommendations. For example ...
0
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1answer
89 views

How to find unknown correlation coefficients in a correlation matrix from known correlation coefficients? [duplicate]

I have a correlation matrix A given below. Here A should be a positive-definite matrix so that we can perform Cholesky decomposition of A. ...
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0answers
116 views

Non-negative matrix factorization in recommender systems

As i understand, in NMF we should have our three matrices elements non-negative. But i can't understand how to do it so far. Shouldn't we just initialize our factor matrices at the start with random ...
2
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2answers
66 views

Singular value and eigen-decomposition of a square symmetric matrix should be identical, but differ in sign

As far as I know, singular value decomposition (SVD) and eigendecomposition give the same result for symmetric square matrices. But when I check the results in R, that's not what I see. Please see ...
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1answer
194 views

What is the relationship between SVD and UV decomposition?

I am aware that both are Matrix decomposition techniques. SVD decomposes the Matrix as $\mathbf C = \mathbf U_1 \mathbf L \mathbf V_1^\top$. UV decomposes the Matrix as $\mathbf C = \mathbf ...
2
votes
0answers
113 views

orthogonalized impulse response's contradictory forms in a VAR(p) model

I have so far discovered three different ways of utilizing the Cholesky decomposition for calculating the OIRFs of a VAR(k). The different methods seem contradictory so I would like some input on ...
2
votes
1answer
105 views

Decomposition of inverse covariance matrix

Let $\Sigma$ be a covariance matrix and let $$x^T \Sigma^{-1} x = \|Ax\|_2^2.$$ What is the interpretation of matrix $A$? I tried solving for $A$ with an eigenvalue decomposition of $\Sigma$ as ...
3
votes
1answer
192 views

Derivative of $x^T A^Ty$ with respect to $\Sigma$ where $A$ is (an upper triangle matrix and ) Cholesky decomposition of $\Sigma$

I would like to evaluate: $$ \frac{ \partial x^T A^Ty}{\partial \Sigma} $$ where $A$ is a Cholesky decomposition of $\Sigma$ and an upper triangle matrix such that $\Sigma = A^T A$, $x$ and $y$ are a ...
0
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1answer
167 views

Relationship between Poisson generation and generalized Kullback-Leibler divergence

I have read that, in the context of matrix factorization, performing maximum likelihood estimation under the assumption that the entries are Poisson generated is equivalent to minimizing the ...
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0answers
87 views

What is the advantage of non-negativity in matrix factorization?

I am wondering why matrix factorization techniques in the machine learning domain almost always expect the provided matrix to be non-negative. What is the advantage of this constraint? Background: I ...
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1answer
707 views

What norm of the reconstruction error is minimized by the low-rank approximation matrix obtained with PCA?

Given a PCA (or SVD) approximation of matrix $X$ with a matrix $\hat X$, we know that $\hat X$ is the best low-rank approximation of $X$. Is this according to the induced $\parallel \cdot ...
1
vote
2answers
201 views

Maximally reducing the rank of a matrix by removing some rows or columns

I have a $N \times M$ matrix, and the rank of matrix, $r$, is near $\min(M,N)$. I want to minimize the rank by removing some of the rows or columns to get $r \ll \min(M,N)$. The goal is to achieve ...
0
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1answer
35 views

Independent component analysis with nonnegative mixing matrix

In independent component(s?) analysis, I have the observed signal, $O$, the mixing matrix, $A$, and the source matrix, $S$, with $O ≈ AS$ I've found some literature on ICA with the sources assumed to ...
0
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1answer
288 views

Oaxaca Decomposition: Unexplained Constant

I am doing an Oaxaca decomposition of the Log Wage Differential between Whites and non-Whites. I would like to find out if there is any interpretation for the constant term under the unexplained ...
3
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1answer
48 views

WH=V matrix decomposition that allows negative values

I'm looking for a matrix factorization method that is able to decompose a matrix: V => W * H V has dimensions ...
4
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2answers
779 views

How to choose an optimal number of latent factors in non-negative matrix factorization?

Given a matrix $\mathbf V^{m \times n}$, Non-negative Matrix Factorization (NMF) finds two non-negative matrices $\mathbf W^{m \times k}$ and $\mathbf H^{k \times n}$ (i.e. with all elements $\ge 0$) ...
1
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1answer
148 views

Intuition behind matrix factorization formulations?

I'm reading this paper about matrix factorization. In the paper they propose to use this factorization for the adjacency (or similarity) matrix $G$ using the following formulation: $G = U \Lambda U^T$ ...
2
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1answer
156 views

Matrix Factorization Recommendation Systems with Only “Like” Ratings

I'm trying to build a recommendation system, but I only have data on what my users have "liked", i.e. all non-missing data has the same numeric value. Is it possible for me to use matrix ...
2
votes
1answer
108 views

Cholesky decomposition and confidence ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. (This question succeeds this one.) What I'm ...
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0answers
260 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
0
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0answers
37 views

clustering versus projection - what are the best example/scenario to explain their differences

I am dealing with non statistician who know are crunching data but don't have a deep understanding of statistics. I am trying to introduce them to non-matrix factorization methods but it has been ...
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0answers
76 views

Is there a way to perform SVD in a sequential manner?

My neurology experiment has a spike detector outputting 40 sample long spike waveforms. I'm using a dictionary method for sorting the spikes in real time. To ...
1
vote
1answer
205 views

Trying to understand non-negative matrix factorization (NMF)

I'm trying to understand how NMF is derived, and I got the basic idea of NMF, that is, it tries to approximate the original matrix $V$ with $WH$, where $V$ are non-negative, and $W,H$ are constrained ...
3
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1answer
596 views

Evaluating matrix factorization algorithms for Netflix

I've been trying to implement Simon Funk's movie recommendation algorithm explained here. I understand how the user and item factors are computed. However the evaluation method is not clearly ...
2
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0answers
119 views

Matrix decomposition

I have a symmetric matrix $V$ of order 10. I want to decompose $V$ in such a way that $$V=SS'$$ with $S$ being non triangular. The matrix $S$ has some restrictions that 45 cells have ‘0’ values and ...
2
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1answer
700 views

Plotting error ellipsoid from 3x3 covariance matrix in R?

I'm hoping to be able to take a 3x3 covariance matrix and turn this into an error ellipsoid but so far I haven't been able to achieve this. I'm very new to R (in fact turned to it to attempt to solve ...