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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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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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