The document discusses non-negative matrix factorization and algorithms for solving it. It introduces non-negative matrix factorization as factorizing a non-negative matrix A into non-negative matrices W and H such that A = W×H. It then presents a simple algorithm for solving the exact non-negative matrix factorization problem in polynomial time by modeling it as a satisfiability problem over polynomial constraints. It also discusses an approach for simplicial factorization that reduces the number of variables by exploiting the rank of the matrix.
Linear Combination, Span And Linearly Independent, Dependent SetDhaval Shukla
In this presentation, the topic of Linear Combination, Span and Linearly Independent and Linearly Dependent Sets have been discussed. The sums for each topic have been given to understand the concept clearly for viewers.
Linear Combination, Span And Linearly Independent, Dependent SetDhaval Shukla
In this presentation, the topic of Linear Combination, Span and Linearly Independent and Linearly Dependent Sets have been discussed. The sums for each topic have been given to understand the concept clearly for viewers.
Recurrence relation of Bessel's and Legendre's functionPartho Ghosh
This presentation tells about use recurrence relation in finding the solution of ordinary differential equations, with special emphasis on Bessel's and Legendre's Function.
Recurrence relation of Bessel's and Legendre's functionPartho Ghosh
This presentation tells about use recurrence relation in finding the solution of ordinary differential equations, with special emphasis on Bessel's and Legendre's Function.
Presentation of the paper "An output-sensitive algorithm for computing (projections of) resultant polytopes" in the Annual Symposium on Computational Geometry (SoCG 2012)
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
Basic algebra, trig and calculus needed for physics.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
Bch and reed solomon codes generation in frequency domainMadhumita Tamhane
Digital signal processing is permeated with application of Fourier Transforms. When time variable is continuous, study of real-valued or complex valued signals rely heavily on Fourier transforms. Fourier Transforms also exist on the vector space of n-tuples over the Galois field GF(q) for many values of n, i.e. code-words. Cyclic codes can be defined as codes whose code-words have certain specific spectral components equal to zero. Conjugacy constraints provide an analogous condition for a finite field. BCH and Reed Solomon codes can be easily generated in frequency domain based on conjugacy constraints.
Guarding Terrains though the Lens of Parameterized ComplexityAkankshaAgrawal55
The Terrain Guarding problem is a well-studied visibility problem in Discrete and Computational Geometry. So far, the understanding of the parameterized complexity of Terrain Guarding has been very limited, and, more generally, exact (exponential-time) algorithms for visibility problem are extremely scarce. In this talk we will look at two results regarding Terrain Guarding, from the viewpoint of parameterized complexity. Both of these results will utilize new and known structural properties of terrains. The first result that we will see is a polynomial kernel for Terrain Guarding, when parameterized by the number of reflex vertices. (A reflex vertex is a vertex of the terrain where the angle is at least 180 degrees.) The next result will be regarding a special version of Terrain Guarding, called Orthogonal Terrain Guarding. We will consider the above problem when parameterized by the number of minima in the input terrain, and obtain a dynamic programming based XP algorithm for it.
This presentation is the one that I gave at the Parameterized Complexity Seminar (https://sites.google.com/view/pcseminar).
The Art Gallery problem is a fundamental visibility problem in Computational Geometry, introduced by Klee in 1973. The input consists of a simple polygon P, (possibly infinite) sets X and Y of points within P, and an integer k, and the objective is to decide whether at most k guards can be placed on points in X so that every point in Y is visible to at least one guard. In the classic formulation of Art Gallery, X and Y consist of all the points within P. Other well-known variants restrict X and Y to consist either of all the points on the boundary of P or of all the vertices of P. The above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16]. Given the above result, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016].
``Is Art Gallery FPT with respect to the number of reflex vertices?''
In this talk, we will obtain a positive answer to the above question, for some variants of the Art Gallery problem. By utilising the structural properties of ``almost convex polygons'', we design a two-stage reduction from (Vertex,Vertex)-Art Gallery to a new CSP problem where constraints have arity two and involve monotone functions. For the above special version of CSP, we obtain a polynomial time algorithm. Sieving these results, we obtain an FPT algorithm for (Vertex,Vertex)-Art Gallery, when parameterized by the number of reflex vertices. We note that our approach also extends to (Vertex,Boundary)-Art Gallery and (Boundary,Vertex)-Art Gallery.
This slides are from a talk that I gave at the Algorithms Seminar at Tel-Aviv University.
This is the talk that I gave at the Dagstuhl seminar ''New Horizons in Parameterized Complexity'', 2019. This is about a polynomial kernel for Interval Vertex Deletion. The main focus of this talk is to obtain a polynomial kernel for a slightly larger parameter, which is the vertex cover number of the input graph.
This is concerned with designing an exact exponential time algorithm that is better than the well-known 2^n algorithm for the problem Path Contraction. This answers an open question of van't Hof et. al [TCS 2009]. This is based on the article that appeared in ICALP 2019.
Kernelization of Cycle Packing with Relaxed Disjointness ConstraintsAkankshaAgrawal55
This is a talk given at ICALP 2016. Here, I talk about two natural generalizations of the Cycle Packing problem in the realm of Parameterized Complexity.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
6. n x m
= xA W H
Minimize k
n x k
k x m
Factor of a Matrix
7. n x m
= xA W H
Minimize k
k is the rank, r of A
n x k
k x m
Factor of a Matrix
8. n x m n x k
= x
A basis
k ≤ r
k x m
W H
Factor of a Matrix
9. n x m
= x
k ≤ r
n x r
r x m
H
Factor of a Matrix
A basis
10. n x m
= x
k ≤ r
a1 a2 ar+ + +
n x r
r x m
H
Factor of a Matrix
A basis
j
11. n x m
= x
k ≤ r
a1 a2 ar+ + +
j
a1
a2
ar
n x r
r x m
Factor of a Matrix
A basis
j
12. n x m
= x
r ≤ k
n x k
k x m
A H
Factor of a Matrix
13. n x m
= x
r ≤ k
n x k
k x m
Can obtain a generating
set of the vector space
spanned by columns of A
A H
Factor of a Matrix
14. Non-Negative Matrix
1 2 3
1 55 119 11
2 -112 456 154
3 513 33 223
4 324 123 543
4 x 3
All elements are non-negative
15. Non-Negative Matrix
1 2 3
1 55 119 11
2 -112 456 154
3 513 33 223
4 324 123 543
4 x 3
All elements are non-negative
16. Non-Negative Matrix
1 2 3
1 55 119 11
2 112 456 154
3 513 33 223
4 324 123 543
4 x 3
All elements are non-negative
17. Non-negative (Exact) Factor of a Non-negative
Matrix
n x m
= xA W H
Minimize k
n x k
k x m
non-negative non-negative non-negative
18. Non-negative (Exact) Factor of a Non-negative
Matrix
n x m
= xA W H
Minimize k
n x k
k x m
non-negative non-negative non-negative
Non-negative
rank
19. Decision Version of the Problem
Exact Non-negative Matrix Factorization (ENMF)
Input:
Question:
An n x m non-negative matrix A and an
integer k.
Are there non-negative matrices W and H
such that A = W x H, W is of order
n x k, and H is of order k x m?
20. (A,k)
n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
(Cohen and Rothblum)
A Simple Algorithm for ENMF
21. A Simple Algorithm for ENMF
= W
x
n x k
w11 w12 w1k
w21 w22 w2k
wn1 wn2 wnk
k x m
h11 h12 h1m
h21 h22 h2m
wk1 wk2 wkm
Create variables
n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
H
22. A Simple Algorithm for ENMF
n x m
A
W
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
x
n x k
w11 w12 w1k
w21 w22 w2k
wn1 wn2 wnk
k x m
h11 h12 h1m
h21 h22 h2m
wk1 wk2 wkm
Create variables
=
H
Create polynomial constraints:
[Const(A,k)]
1. For all i,j wij, hij ≥ 0.
23. A Simple Algorithm for ENMF
n x m
A
W
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
x
n x k
w11 w12 w1k
w21 w22 w2k
wn1 wn2 wnk
H
k x m
h11 h12 h1m
h21 h22 h2m
wk1 wk2 wkm
Create variables
Create polynomial constraints:
[Const(A,k)]
1. For all i,j wij, hij ≥ 0.
2. For all i,j, aij = wik hkj.∑
k
=
24. A Simple Algorithm for ENMF
(A,k)
n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Number of variables:
Number of polynomial
constraints:
nk + km
nm + nk + km
(A,k) is a yes-instance of ENMF if and only if
Const(A,k) is satisfiable (over reals)
25. A Simple Algorithm for ENMF
(A,k)
n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Number of variables:
Number of polynomial
constraints:
nk + km
nm + nk + km
We can find a solution to
a set of polynomial
inequalities in time (Dp)O(x)
26. A Simple Algorithm for ENMF
(A,k)
n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Number of variables:
Number of polynomial
constraints:
nk + km
nm + nk + km
We can find a solution to
a set of polynomial
inequalities in time (Dp)O(x)
x: number of variables
p: number of inequalities
D: Maximum degree of a
polynomial inequality
27. A Simple Algorithm for ENMF
(A,k)
n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Number of variables:
Number of polynomial
constraints:
nk + km
nm + nk + km
We can decide if Const(A,k) is
satisfiable in time O((nm)O(k(n+m)))
28. An Illustration of Variable Reduction
Simplicial Factorization
Input:
Question:
An n x m non-negative matrix A of rank k.
Are there non-negative matrices W and H
such that A = W x H, W is of order n x k,
and H is of order k x m?
29. n x m
xA W H
n x k
k x m
Rank k
Simplicial Factorization
=
34. Simplicial Factorization
Goal: To design an algorithm for Simplicial
Factorization that runs in time O((nm)O(r )).2
Follow similar approach as the algorithm for ENMF, but
apply with reduced number of variables.
35. Pseudo Inverse
Consider a full column (or row) rank
matrix Mp,q of rank p (q).
M+ has all real entries;
M+ has order q x p;
M+ x M = Iq,q and M x M+ = Ip,p.
The (unique) pseudo inverse M+, of M satisfies
the following:
37. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
H+Pseudo inverse:
38. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
H;i
W+ A;i = W+ W H;i = H;i
39. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
H;i
W+ A;i = W+ W H;i = H;i
40. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
H;i
k x k k x 1
W+ A;i = W+ W H;i = H;i
41. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
H;i
W+ A;i = W+ W H;i = H;i
42. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
W+ A;i = W+ W H;i = H;i
H;i
43. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
H;i
aji Wj;
W+ A;i = W+ W H;i = H;i
44. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
H;i
W+ A;i = W+ W H;i = H;i
45. Simplicial Factorization
n x m
= xA W H
n x k
k x m
W+ has order k x n and H+ has order m x k;
W+ x W = Ik,k and H x H+ = Ik,k.
A;i
H;i
W+ A;i = W+ W H;i = H;i
Aj; H+ = Wj; H H+ = Wj;
46. Simplicial Factorization
C = {U1, U2,…, Uk} : A column basis for A.
R = {V1, V2,…, Vk} : A row basis for A.
A
Columns of A
expressed in basic C
a1U1 + a2U2 + … + akUk
j AC
k x mn x m
a1
a2
ak
j
47. Simplicial Factorization
AC AR
Columns of A
expressed in basic C
Rows of A
expressed in basic R
n x kk x m
C = {U1, U2,…, Uk} : A column basis for A.
R = {V1, V2,…, Vk} : A row basis for A.
48. Simplicial Factorization
TC AC and AR TR are non-negative;
AR TR TC AC = A.
Lemma: A has a simplicial factor if and only if the for
every column and row basis C and R of A
there are k x k matrices TC and TR such that:
49. Simplicial Factorization
Lemma: A has a simplicial factor if and only if the for
every column and row basis C and R of A
there are k x k matrices TC and TR such that:
TC AC and AR TR are non-negative;
AR TR TC AC = A.
A has a simplicial factors
by the two conditions and the
construction of AC and AR.
n x k k x m
50. Simplicial Factorization
TC AC and AR TR are non-negative;
AR TR TC AC = A.
Lemma: A has a simplicial factor if and only if the for
every column and row basis C and R of A
there are k x k matrices TC and TR such that:
A = W x H
n x k k x m
U and V be column and
row basis respectively
52. Simplicial Factorization
A = W x H
n x k k x m
U and V be column and
row basis respectively
U
n x k
V
k x m
TC = W+ x U
TR = V x H+
53. Simplicial Factorization
A = W x H
n x k k x m
U and V be column and
row basis respectively k x k
TC = W+ x U
TR = V x H+
54. Simplicial Factorization
A = W x H
n x k k x m
U and V be column and
row basis respectively
TC = W+ x U
TR = V x H+
k x k
TC x AC = W+ x U x AC = H
W+ A;i = W+ W H;i = H;i
Aj; H+ = Wj; H H+ = Wj;
(non -ve)
55. Simplicial Factorization
A = W x H
n x k k x m
U and V be column and
row basis respectively
TC = W+ x U
TR = V x H+
k x k
TC x AC = W+ x U x AC = H
W+ A;i = W+ W H;i = H;i
Aj; H+ = Wj; H H+ = Wj;
AR x TR = AR x V x H+ = W
(non -ve)
(non -ve)
56. Simplicial Factorization
A = W x H
n x k k x m
U and V be column and
row basis respectively
TC = W+ x U
TR = V x H+
k x k
TC x AC = W+ x U x AC = H (non -ve)
AR x TR = AR x V x H+ = W (non -ve)
TC AC and AR TR are non-negative;
AR TR TC AC = A.
57. Simplicial Factorization
TC AC and AR TR are non-negative;
AR TR TC AC = A.
Lemma: A has a simplicial factor if and only if the for
every column and row basis C and R of A
there are k x k matrices TC and TR such that:
58. A Simple Algorithm for ENMF
n x m
A
W
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
x
n x k
w11 w12 w1k
w21 w22 w2k
wn1 wn2 wnk
H
k x m
h11 h12 h1m
h21 h22 h2m
wk1 wk2 wkm
Create variables
Create polynomial constraints:
[Const(A,k)]
1. For all i,j wij, hij ≥ 0.
2. For all i,j, aij = wik hkj.∑
k
=
59. Simplicial Factorization
TC AC and AR TR are non-negative;
AR TR TC AC = A.
Lemma: A has a simplicial factor if and only if the for
every column and row basis C and R of A
there are k x k matrices TC and TR such that:
60. U n x k V k x m
n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Simplicial Factorization
61. n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Simplicial Factorization
AR AC
k x mn x k k x k
w11 w12 w1k
w21 w22 w2k
wk1 wk2 wkk
TR
k x k
h11 h12 h1k
h21 h22 h2k
hk1 hk2 hkk
TC
=
62. n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Number of variables:
Number of polynomial
constraints:
2k2
poly(n,m,k)
We can find a solution to
a set of polynomial
inequalities in time (Dp)O(x)
x: number of variables
p: number of inequalities
D: Maximum degree of a
polynomial inequality
Simplicial Factorization
63. n x m
A
a11 a12 a1m
a21 a22 a2m
an1 an2 anm
Number of variables:
Number of polynomial
constraints:
2k2
poly(n,m,k)
Simplicial Factorization
We can solve Simplicial Factorization
in time O((nm)O(k )).
2
65. Other Results on ENMF
[Vavasis] ENMF is known to be NP-Hard.
[Arora et al.] Assuming ETH, there is no algorithm for
ENMF running in time O((nm)o(k)).
66. Other Results on ENMF
2
[Vavasis] ENMF is known to be NP-Hard.
[Arora et al.] Assuming ETH, there is no algorithm for
ENMF running in time O((nm)o(k)).
[Moitra] EMNF admits an algorithm running in time
O((nm)O(k )).
67. n x m
= xA W H
n x k
k x m
non-negative non-negative non-negative
Exact Non-negative Matrix Factorization
For most applications, close
approximation is good enough.
68. n x m
xA W H
n x k
k x m
non-negative non-negative non-negative
Non-negative Matrix Factorization
For most applications, close
approximation is good enough.
≈
70. Example: Distance Function
Square of Euclidean distance:
For matrices A and B (of same order)
|| A - B ||2 = (Aij - Bij)2∑
i,j
|| A - B ||2 = 0 if and only if A = B
71. Example: Divergence Function
For matrices A and B (of same order)
D(A || B ) = (Aij log (Aij/Bij) - Aij + Bij)∑
i,j
D(A || B ) = 0 if and only if A = B
72. General Scheme of Algorithm: Non-negative
Matrix Factorization
Input:
Output:
A, W(0), H(0), and t=1.
W and H.
73. General Scheme of Algorithm: Non-negative
Matrix Factorization
1. Fix H(t-1) and find W(t), such that D(A, W(t)H(t-1)) ≤
D(A, W(t-1)H(t-1)).
2. Fix W(t) and find H(t), such that D(A, W(t)H(t)) ≤ D(A,
W(t)H(t-1)).
3. If convergence satisfied return W and H.
4. t=t+1.
Input:
Output:
A, W(0), H(0), and t=1.
W and H.
While true
74. Main Challenges in Designing Better NMF
Algorithms
Getting a good seeding for initialisation of W and H.
75. Main Challenges in Designing Better NMF
Algorithms
Getting a good seeding for initialisation of W and H.
Devising updating rules for W and H at subsequent
iterations.
76. Main Challenges in Designing Better NMF
Algorithms
Getting a good seeding for initialisation of W and H.
Devising updating rules for W and H at subsequent
iterations.
Selecting distance/ divergence norms based on the
application.
77. Main Challenges in Designing Better NMF
Algorithms
Getting a good seeding for initialisation of W and H.
Devising updating rules for W and H at subsequent
iterations.
Selecting distance/ divergence norms based on the
application.
Proving/ giving enough evidences for convergence of
the algorithm.
79. Origin of Non-negative Matrix Factorization
Evolved from Principal Component Analysis, which is
used for dimension reduction.
Disadvantage: Both positive and
negative elements appear in
principal components and
coefficients in linear combinations.
Hard to interpret results in
applications like storing pixel
brightness
81. Applications
Image Processing.
Data represented as a non-negative matrix of
pixels.
NMF can find A W x H≈
W is the basis matrix, its
column can be regarded as
parts like nose, ear, eye,
etc.
89. Applications
Financial Data Mining
The stock price fluctuations seem to be
dominated by several underlying factors. NMF
has been used to obtain underlying trends
from the stock market data.