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Summary of how we stretched matrix completion to discover its limits.

The stress test

The main focus of our project was not to simply implement matrix completion -- that has been done many times in the past and is proven to work. What we wanted to do was put the algorithm through a “stress test” to understand its limits.

To do this, we devised the following plan:

  1. Construct a low rank matrix
  2. Create a model of uniformity
  3. Vary the uniformity on unobserved indices in a matrix
  4. Complete the matrix
  5. Measure mean squared error between the original matrix and the completed matrix
  6. Plot mean squared error as a function of uniformity

Constructing a low rank matrix

In order to construct a low rank matrix, remove indices, and compare the results of matrix completion with the actual matrix, we first created two separate matrices of size m x r and r x n with random entries between 0 and 1, where r in the desired rank. We then multiplied those two matrices together, constructing one matrix of size m x n and rank r.

Modeling uniformity

Our second challenge was to generate a model of uniformity that provided a simple way to change how uniform the unobserved indices were. We decided to use a jointly Gaussian distribution, modifying the covariance to control uniformity. The idea is simple: low covariance results in a high concentration of indices removed from the center of the matrix, while high covariance results in an almost perfectly uniform distribution.

In the figure below, the matrix on the left shows a matrix with a very uniform distribution of unobserved entries, characterized by a jointly gaussian distribution with high covariance. The matrix on the right demonstrates a data set with a high degree of nonuniformity. The probability peaks at the center of the matrix, resulting in unobserved indices which are highly concentrated in the middle of the matrix.

A uniform and non uniform represented by a gaussian

Plotting mse and uniformity

After completing the matrix, we calculated the mean squared error between the original matrix and the prediction provided by matrix completion and stored it in an array, along with the corresponding uniformity percentage. We calculated the uniformity percentage by normalizing each covariance by the largest covariance used, providing a scale between 0 and 100%.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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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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