<< Chapter < Page Chapter >> Page >
A brief discussion of NMF (Non-negative Matrix Factorization).

Non-negative matrix factorization

NMF provides an alternative approach to decomposition that assumes that the data and the components are non-negative. The model for continuous data looks like this: min W , H 1 2 X W H 2 2 s u b j e c t t o W i j 0 , H i j 0 There are two views for NMF, and they provide different insights.

Archetypal analysis

PCA tends to group both positively correlated and negatively correlated components together (it’s only looking for variables with strong correlation). On the other hand, NMF, by forcing W and H to be positive, finds patterns with the same direction of correlation.

Fuzzy clustering

W is sparse and it gives probabilistic cluster memberships (columns of W corresponds to kth cluster) and columns of H gives variables that define the kth cluster.

For count data, the model looks slightly different (related to Poisson):

maximize W , H i = 1 n j = 1 p [ X i j log ( W i H j ) W i H j ] s u b j e c t t o W i j 0 , H i j 0

Nmf: discussion on image dataset

When used on non-negative data such as images, NMF learns a parts-based representation of the dataset, resulting in interpretable models; while PCA learns the holistic representation.Example of facial recognition:

Basis image given by nmf

A dog sitting on a bed

Basis image given by pca

A dog sitting on a bed
Here positive values are denoted with black pixels and negative values with red pixels.

All three algorithms (PCA, ICA and NMF) learn to represent a face using linear combination of basis images. We see that NMF picks out individual components like nose, eyes and mouth, which corresponds well with our intuitive notion of faces. The reason for this is that first, only additive combinations are allowed in NMF and so this is compatible with the intuitive notion of combining parts to form a whole. Second, the basis images are sparse (It is an advantage because they are non-global and should contain several versions of mouths, noses etc. The variability of the face is generated by combining these different parts). PCA, on the other hand, gives out noisy components that doesn’t offer much interpretability. The reason for this is because first, PCA lets each face to be approximated by a linear combination of all the basis images (the vectors aren't sparse) and secondly, it allows the entries of the factorized vectors to be of arbitrary sign. And since these generally involve complex cancellations between positive and negative numbers, many basis images will lack intuitive meaning. ICA basis images (which isn's shown here), are independent holistic representations. The independence assumption made by ICA is ill-suited for learning parts-based representation because various parts are likely to occur together.

This is a biconvex optimization problem (convex in H when W fixed, convex in W when H fixed). For implementation, we used the nnmf command in Matlab (this algorithm uses Alternating Least Squares for solving).

Nmf: other application

More generally, NMF models the directly observable variables from the hidden variables, where each hidden variable activates a subset of visible variables ("part"). Activation of a collection of hidden variables combine these parts additively to generate a whole. Seen from this perspective, another often used application of NMF is semantic analysis of text document (think text mining).

Reference: "Learning the Parts of Objects by Nonnegative Matrix Factorization", D.D.Lee, H.S.Seung.

Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elec 301 projects fall 2013. OpenStax CNX. Sep 14, 2014 Download for free at http://legacy.cnx.org/content/col11709/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elec 301 projects fall 2013' conversation and receive update notifications?

Ask