Questions tagged [non-negative-matrix-factorization]

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Relaxed non-negative least squares

I am reconstructing a probability vector from data using non-negative least squares: $$ \sum_\alpha \left(\pi_\alpha - \sum_i W_{\alpha i}p_i\right)^2\rightarrow \min,\\ p_i\geq 0,\sum_i p_i=1 $$ ...
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2 votes
1 answer
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Examples of when PCA would be preferred over NMF

What are some specific examples of when PCA should be used instead of NMF? PCA is a widely used method for dimension reduction in data science, machine learning, and bioinformatics. NMF is also a ...
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6 votes
2 answers
418 views

ALS vs SGD in parallelization

So given the standard objective in matrix factorization for collaborative filtering of minimizing: $$ L = \sum_{u,i \in S} (r_{ui}-q_i^Tp_u)^2 + \lambda(\sum_i||q_i^2||+\sum_u||p_u^2||) $$ , where $r_{...
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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. ...
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Minimum required number of users or items to perform ALS

although matrix factorization is likely to be applied to any matrix of user data vs. items, we could think that the quality of the recommendation made by the ALS tends to vary, depending on the ...
2 votes
1 answer
134 views

Will PCA always fit a model at least as well as NMF?

If I perform PCA/NMF on a dataset, and then use the reduced models to reconstruct the original dataset, it seems to me that PCA should typically outperform NMF, simply due to the fact that NMF has the ...
0 votes
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25 views

Matrix Values to Probabilities with Logistic Regression

I have a Non Negative Matrix Factorization algorithm and I'm calculating the A-hat matrix from it. Rows of the matrix are customers, columns are my products and values are the occurrences of product ...
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1 vote
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Non-negative matrix factorization clusters

NMF can be used for clustering i.e., $V=WH$ where $W$ represents cluster centers and $H$ represents the membership of samples. But can NMF alone cluster the samples? Can we get better clusters in NMF ...
3 votes
1 answer
143 views

Is Nonnegative matrix factorization a clustering method or a dimensionality reduction method?

In the matrix factorization we have the problem of decomposing a nonnegative matrix $X$ into two lower-rank matrices $W$ and $H$. I would like to know whether this method is considered as a dimension ...
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3 votes
1 answer
6k views

What is The Main Difference between PCA and NMF and why to choose one rather than the other?

I have to develop some analyses to study cancer data. I want to use NMF and PCA. Basically these tools choose the best factorization rank and the number of components that is meaningful to your ...
0 votes
0 answers
34 views

How to choose the best model for Non Negative Matrix Factorization?

I am applying NMF with NMF R package. In the early stages, I'm comparing three algorithms (Lee, Brunet,nsNMF) visualizing how fast they converge and how much they reduce residues as in the image down ...
3 votes
1 answer
258 views

Why does NMF of a symmetric matrix yield orthogonal matrices which are not transpose identical?

Consider the non-negative factorization of a positive, real symmetric matrix A. Non-negative factorization of this matrix yields ...
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Am I interpreting correctly this NMF analysis?

I have to analyse a set of biological data and I am applying a Non-Negative Matrix Factorization (NMF) Approach. Given a 366 x 144 dataset, I am reasoning about overfitting and the correct rank r to ...
5 votes
1 answer
1k views

What is the difference between Non-Negative Matrix Factorization (NMF) and Factor Analysis (FA)?

I am performing an Exploratory Factor Analysis (EFA) for a multivariate dataset, where variables are all measurements of the same physical measure, only in different locations in space. My purpose is ...
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1 answer
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Negative Latent Factors in Factorization Machines

I'm studing a specific implementation of a recommendation system leveraging on a factorization machine algorithm. For each person_id and item_id combination, I have an implicit rating of 1 or 0 ...
1 vote
0 answers
433 views

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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2 votes
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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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2 votes
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105 views

What are the limitations of non-negative matrix factorisation when reducing the dimensions of a data set?

From what I understand NFM (non-negative matrix factorisation) is constrained by the factor that it only supports data sets with non-negative values when reducing the dimensions of a data set. ...
1 vote
0 answers
197 views

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 ...
1 vote
1 answer
121 views

Can NMF assign probabilities to the topics it outputs?

It's my understanding that only LDA can assign probabilities to words within each topic that it discovers since it's a probabilistic graphical model politicians 0.05 united states 0.10 obama 0.20 ...
2 votes
0 answers
215 views

Deriving Multiplicative Update Rules for Regularized NMF

After reading the following CrossValidated post, I cannot derived the correct multiplicative rules for regularized NMF from this paper. They obtain the coefficients $|I_u|$ and $|U_i|$ in the ...
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1 answer
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Implementation of Proximal alternating linearized minimization

The updates of the gradients are somehow wrong. I have implemented the below given algorithm. I have done something wrong ...
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Factorized matrix for recommendations, what then?

I have a dataset that looks like this: Image taken from this blog Let's assume that I have applied Matrix factorization and have learned the zero values for the items missing for every user. I now ...
1 vote
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116 views

Using complex number in non-negative matrix factorization (NMF)

In short, I wonder which kind of data can use complex number for NMF. And could an imaginary part possibly be a vector? For detail, as I saw some papers used complex number in NMF (1), I think it ...
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1 vote
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Normalizing sparse matrix by mean, should the mean be calculated excluding zero?

I have very sparse matrix (70% sparsity) which I want to normalize by mean. I tried using mean both include and exclude zero. The histogram between count (y-axis) and value (x-axis) shows The value ...
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1 vote
2 answers
1k views

Non-negative matrix factorization (NMF) on mixed data using 1-hot encoding

From a standpoint of interpretation, can I use NMF on one-hot encoded categorical data for dimension reduction? I have mixed data and was thinking about one-hot encoding the categorical features and ...
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3 votes
1 answer
924 views

Can I use word2vec vectors as input features to NMF or LDA?

I'm trying to do some topic modelling on my corpus and I want to use Word2Vec vectors as an input to my NMF and LDA models. How do I do this? Is it even possible?
1 vote
1 answer
400 views

nmf in scipy returns components with all zero weights

I'm trying to understand whether this behavior is a bug or a feature. Essentially, I have a dataset of ten thousand short pieces of text. I have used the CountVectorizer function to turn this into a ...
0 votes
0 answers
166 views

Obtaining hard, overlapping clusters using non-negative matrix factorization

From my understanding non-negative matrix factorization (NMF) provides a natural way to obtain soft clusters from a non-negative $n$x$m$ data matrix $X$. NMF decomposes $X$ into two non-negative ...
2 votes
1 answer
142 views

matrix factorization with non-negative constraint only on one of the factors

I have a 2D spectral data time series with a wavelength dimension and a time dimension, and I'd like to decompose it to the time evolution ($SV^T$ for SVD and $H$ for NNMF) of several spectral ...
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3 votes
1 answer
365 views

Distributed PCA or an equivalent

We normally have fairly large datasets to model on, just to give you an idea: over 1M features (sparse, average population of features is around 12%); over 60M rows. A lot of modeling algorithms ...
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2 votes
1 answer
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Pattern of out-of-sample reconstruction error in NMF cross-validation: why is it monotonically decreasing? [duplicate]

I am using nonnegative matrix factorization, NMF (in its variant OPNMF, which is subject to additional orthogonality and $H = W^TV$ constraints) to factorize a dataset. To find the optimal number of ...
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5 votes
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425 views

Nonnegative Matrix Factorization as Maximum Likelihood

Elements of Statistical Learning has this on such NMF loss function (section 14.6 Non-negative Matrix Factorization): The matrices $\mathbf{W}$ and $\mathbf{H}$ are found by maximizing $$ L(\mathbf{...
11 votes
2 answers
6k views

Deriving Multiplicative Update Rules for NMF

How to derive the multiplicative update rules for the non-negative matrix factorization problem given by Lee and Seung. Minimize $\left \| V - WH \right \|^2$ with respect to $W$ and $H$, subject ...
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2 votes
1 answer
432 views

Why do increasing regularization weights make objective function not monotonically decrease?

I run modified non-negative matrix factorization (NMF) and tune the regularization weight from 1e5 to 1e13. The table below ...
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1 vote
0 answers
87 views

The reason why NMF has become so popular [duplicate]

Why do we use Non-negative matrix factorization?What is the advantage and superiority of other matrix decomposition methods?
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0 answers
238 views

What can be the reasons that L1-regularized NMF gets worse result than standard NMF in sparse matrix computation?

I apply L1-norm as a group sparsity constraint [1,2] into non-negative matrix factorization $V \approx WH$ for source separation. Objective functions: Standard NMF (Kullback-Leibler divergence): $...
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7 votes
1 answer
5k views

How does LDA (Latent Dirichlet Allocation) assign a topic-distribution to a new document?

I am new to topic modeling and read about LDA and NMF (Non-negative Matrix Factorization). I understand the training process work. Let's say I have 100 documents and I want to train an LDA for these ...
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0 answers
440 views

Why does NMF perform better than LDA on shorter textual inputs

For the reading that I have done, I found that Dirichlet priors typically don't perform well when they aren't given significant amounts of data. I'm not quite sure why that is. What is it about NMF ...
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0 votes
1 answer
498 views

Geometric Interpretation of Non Negative Matrix Factorization

I'm trying to learn about the geometric interpretation of NMF. I have found the paper by Slim Essid to be very useful. I would like to make a plot like the one in Figure 1 just for a k=2 Topics (i.e. ...
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0 answers
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Derive a constant in Kullback-Liebler divergence proof

From Kullback-Liebler divergence of matrix factorization; \begin{equation*} \mathrm{X}\approx\mathbf{WH} \tag{1} \end{equation*} How equation $(2)$ is derived to constant equality in equation $(3)$? ...
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Non negative matrix factorization initial values and final values

I am planning to use initial values that are {0, 1}. How do we ensure or how does NMF ensure that the final values are also in the [0,1] range. What if we want to model a matrix of frequencies of ...
0 votes
1 answer
1k views

Deep Learning Variation of NNMF

I'm aware that there are different variations of non negative matrix factorization based on the optimization function and I have read about graph regularized NMF. Is there any method to use deep ...
0 votes
1 answer
362 views

What conclusions I can draw from matrix result after non-negative matrix factorization?

I was introduced to NMF for data analysis. I implemented some code and obtained the result of basis matrix $W$ and feature matrix $H$. From $V$ ~ $WH$, my $V$ dimension is 5100*1201. I inputted $W$ ...
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2 votes
0 answers
49 views

what is the likelihood of a levy process?

While interpreting NMF in Statistical perspective, we assume a Poisson process and to solve for the factors the using EM algorithm, the likelihood of a Poisson process is assumed to be Multinomial, I ...
3 votes
1 answer
1k views

Calculate Frobenius Norm using Non-Negative Matrix Factorization

After performing Non-Negative Matrix Factorization (using R's rnmf() function), I'm left with W, H, and the fitted matrix (W%*%H). The Frobenius norm (squared ...
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1 vote
1 answer
909 views

Nonnegative Matrix Factorization - Interpreting clustering indicator matrix

In non-negative matrix factorization (NMF), the problem is to minimize $A - WH$. Dimensions are $A$ (m x n), $W$ (m, k) and $H$ (k, n). The matrix $H$ reveals soft clustering assignments of $n$ items ...
1 vote
0 answers
303 views

Explanation of the decomposition in the Non Negative Matrix Factorization

I perform matrix factorizaition in my data using the sklearn implementation of Non Negative Matrix Factorization. In the evaluation process I am removing some values from my initial dataset and I am ...
2 votes
0 answers
108 views

How to determine mixture coefficient nonparametrically?

Problem: Given a sample $X_m$ from each of $M$ distributions $f_m$ which are all mixtures of the same $C$ unknown distributions $g_c$ but with differing mixture coefficients $\alpha_{mc}$, (when and) ...
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1 answer
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Non Negative Matrix Factorization and Projection onto other Data

Is there an equivalent in NMF to PCA projection? For example lets say you have 2 datasets of data which are generated by a highly similar process, one which is noisy (dataset 2) and one which is not (...