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Module for ELEC 301 project

What is matrix completion

In 2006, Netflix issued a million dollar challenge to the world:

“Is there a computer algorithm that can accurately predict a user’s movie preferences?”

In the contest, a data matrix was given that contained ratings of thousands of movies from thousands of examinees, but it was only 2% completed. Contestants for this Netflix Challenge had to complete the matrix and provide the optimal algorithms for the task.

The Netflix Prize was won in 2009, but the ideas and algorithms generated to complete matrices remain vast and powerful in real world applications. Simply put, the Matrix Completion algorithm can be used for any areas that involve using a data matrix.

From a more scientific perspective, the 2008 paper, Exact Matrix Completion via Convex Optimization by Candes and Recht formalized a majorization minimization algorithm for matrix completion. Eric Chi’s 2014 article Getting to the Bottom of Matrix Completion and Nonnegative Least Squares with the MM Algorithm provides a more grounded framework for the problem and explains the mathematical concepts behind matrix completion. Matrix Completion.

A visual representation of matrix completion.

A visual representation of matrix completion.

When does matrix completion work

Given a sparse matrix with movies along one axis and users along another, the algorithm had to predict how those users would rate movies they have not seen. The solution, known as Matrix Completion, provided a good estimate of sparse data, provided it satisfied the following:

  1. The matrix must be low rank
  2. The unobserved indices in the matrix must be uniformly distributed

In terms of the Netflix Problem, the matrix was extremely sparse -- with millions of users and movies, less than 2% of the matrix was actually filled. The matrix also followed the above assumptions, specifically that there are a few “types” of people who watch Netflix (an action movie lover, a rom-com fanatic, etc.), making it low rank, and that each user’s reviews are spread uniformly throughout the matrix.

Characterizing the problem

Often, in the real world, these idealities are not upheld. It is very rare to find a matrix that is both perfectly uniform and low rank. In order to better understand matrix completions’ application to the real world, our project aimed to stretch the second requirement and better characterize the algorithm’s limits.

Specifically, we decided to focus on the requirement that the unobserved indices in the matrix must be uniformly distributed. How uniform do the unobserved entries need to be? At what point does matrix completion “stop working”?

Even more importantly, what does a plot of the error look like as a function of uniformity? We know that non-uniform data will result in a predicted matrix that is very dissimilar to the actual matrix, and we know that uniform data will result in a predicted matrix that very similar to the actual data, but what happens in between? Does a small amount of non-uniformity result in an unusable matrix, or can matrix completion continue to work under less than ideal conditions?

Real world implications

While it is important to characterize algorithms to have a better theoretical understanding of how and when they work, this research has very salient real world applications as well.

Imagine an old picture with non-uniform noise distributed throughout it -- maybe one area of the photo is particularly noisy. If matrix completion can work in the conditions described previously, it would be able to reconstruct those images.

Even more importantly, matrix completion is used to predict cancer survival rates, among other medical applications. There is no guarantee that this data is uniform, but maybe matrix completion can still be trusted in these situations despite this limitation.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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what school?
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anyone know any internet site where one can find nanotechnology papers?
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Introduction about quantum dots in nanotechnology
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what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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Source:  OpenStax, Breaking matrix completion: a stress test. OpenStax CNX. Dec 15, 2015 Download for free at http://legacy.cnx.org/content/col11934/1.1
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