1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ³ 2, n ³ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
Fast Algorithm for Computing the Discrete Hartley Transform of Type-IIijeei-iaes
The generalized discrete Hartley transforms (GDHTs) have proved to be an efficient alternative to the generalized discrete Fourier transforms (GDFTs) for real-valued data applications. In this paper, the development of direct computation of radix-2 decimation-in-time (DIT) algorithm for the fast calculation of the GDHT of type-II (DHT-II) is presented. The mathematical analysis and the implementation of the developed algorithm are derived, showing that this algorithm possesses a regular structure and can be implemented in-place for efficient memory utilization.The performance of the proposed algorithm is analyzed and the computational complexity is calculated for different transform lengths. A comparison between this algorithm and existing DHT-II algorithms shows that it can be considered as a good compromise between the structural and computational complexities.
1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ³ 2, n ³ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
Fast Algorithm for Computing the Discrete Hartley Transform of Type-IIijeei-iaes
The generalized discrete Hartley transforms (GDHTs) have proved to be an efficient alternative to the generalized discrete Fourier transforms (GDFTs) for real-valued data applications. In this paper, the development of direct computation of radix-2 decimation-in-time (DIT) algorithm for the fast calculation of the GDHT of type-II (DHT-II) is presented. The mathematical analysis and the implementation of the developed algorithm are derived, showing that this algorithm possesses a regular structure and can be implemented in-place for efficient memory utilization.The performance of the proposed algorithm is analyzed and the computational complexity is calculated for different transform lengths. A comparison between this algorithm and existing DHT-II algorithms shows that it can be considered as a good compromise between the structural and computational complexities.
N-gram IDF: A Global Term Weighting Scheme Based on Information Distance (WWW...Masumi Shirakawa
A deck of slides for "N-gram IDF: A Global Term Weighting Scheme Based on Information Distance" (Shirakawa et al.) that was presented at 24th International World Wide Web Conference (WWW 2015).
The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found
applications in optimization, navigation, network theory, image processing, pattern recognition etc.
Several other authors have studied metric dimension of various standard graphs. In this paper we
introduce a real valued function called generalized metric G X × X × X ® R+ d : where X = r(v /W) =
{(d(v,v1),d(v,v2 ),...,d(v,v ) / v V (G))} k Î , denoted d G and is used to study metric dimension of graphs. It
has been proved that metric dimension of any connected finite simple graph remains constant if d G
numbers of pendant edges are added to the non-basis vertices.
N-gram IDF: A Global Term Weighting Scheme Based on Information Distance (WWW...Masumi Shirakawa
A deck of slides for "N-gram IDF: A Global Term Weighting Scheme Based on Information Distance" (Shirakawa et al.) that was presented at 24th International World Wide Web Conference (WWW 2015).
The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found
applications in optimization, navigation, network theory, image processing, pattern recognition etc.
Several other authors have studied metric dimension of various standard graphs. In this paper we
introduce a real valued function called generalized metric G X × X × X ® R+ d : where X = r(v /W) =
{(d(v,v1),d(v,v2 ),...,d(v,v ) / v V (G))} k Î , denoted d G and is used to study metric dimension of graphs. It
has been proved that metric dimension of any connected finite simple graph remains constant if d G
numbers of pendant edges are added to the non-basis vertices.
Seminar: Visualisasi Data Interaktif Data Terbuka Pemerintah Provinsi DKI Jak...Nadiar AS
Slide persentasi seminar tanggal 9 Mei 2016. Visualisasi Data Interaktif Data Terbuka Pemerintah Provinsi DKI Jakarta Topik Ekonomi dan Keuangan Daerah
Briefly reviews International Conference on Weblogs and Social Media (ICWSM12) from my perspective.
The latter part written in Japanese, sorry for that.
Fuzzy clustering algorithm can not obtain good clustering effect when the sample characteristic is not obvious and need to determine the number of clusters firstly. For thi0s reason, this paper proposes an adaptive fuzzy kernel clustering algorithm. The algorithm firstly use the adaptive function of clustering number to calculate the optimal clustering number, then the samples of input space is mapped to highdimensional feature space using gaussian kernel and clustering in the feature space. The Matlab simulation results confirmed that the algorithm's performance has greatly improvement than classical clustering algorithm and has faster convergence speed and more accurate clustering results.
Fuzzy clustering algorithm can not obtain good clustering effect when the sample characteristic is not
obvious and need to determine the number of clusters firstly. For thi0s reason, this paper proposes an
adaptive fuzzy kernel clustering algorithm. The algorithm firstly use the adaptive function of clustering
number to calculate the optimal clustering number, then the samples of input space is mapped to highdimensional
feature space using gaussian kernel and clustering in the feature space. The Matlab simulation
results confirmed that the algorithm's performance has greatly improvement than classical clustering algorithm and has faster convergence speed and more accurate clustering results
Hierarchical matrix techniques for maximum likelihood covariance estimationAlexander Litvinenko
1. We apply hierarchical matrix techniques (HLIB, hlibpro) to approximate huge covariance matrices. We are able to work with 250K-350K non-regular grid nodes.
2. We maximize a non-linear, non-convex Gaussian log-likelihood function to identify hyper-parameters of covariance.
Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)paperpublications3
Abstract: Ramanujan’s lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously many authors proved the first six of Ramanujan’s tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan’s tenth order mock theta function identities which are expressed by mock theta functions and also a definite integral. The properties of modular forms are used for the proofs of theta function identities and L. J. Mordell’s transformation formula for the definite integral.Keywords: Mock Theta Functions from Ramanujan’s Lost Notebook.
Title: Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)
Author: MOHAMMADI BEGUM JEELANI SHAIKH
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Information-theoretic clustering with applicationsFrank Nielsen
Information-theoretic clustering with applications
Abstract: Clustering is a fundamental and key primitive to discover structural groups of homogeneous data in data sets, called clusters. The most famous clustering technique is the celebrated k-means clustering that seeks to minimize the sum of intra-cluster variances. k-Means is NP-hard as soon as the dimension and the number of clusters are both greater than 1. In the first part of the talk, we first present a generic dynamic programming method to compute the optimal clustering of n scalar elements into k pairwise disjoint intervals. This case includes 1D Euclidean k-means but also other kinds of clustering algorithms like the k-medoids, the k-medians, the k-centers, etc.
We extend the method to incorporate cluster size constraints and show how to choose the appropriate number of clusters using model selection. We then illustrate and refine the method on two case studies: 1D Bregman clustering and univariate statistical mixture learning maximizing the complete likelihood. In the second part of the talk, we introduce a generalization of k-means to cluster sets of histograms that has become an important ingredient of modern information processing due to the success of the bag-of-word modelling paradigm.
Clustering histograms can be performed using the celebrated k-means centroid-based algorithm. We consider the Jeffreys divergence that symmetrizes the Kullback-Leibler divergence, and investigate the computation of Jeffreys centroids. We prove that the Jeffreys centroid can be expressed analytically using the Lambert W function for positive histograms. We then show how to obtain a fast guaranteed approximation when dealing with frequency histograms and conclude with some remarks on the k-means histogram clustering.
References: - Optimal interval clustering: Application to Bregman clustering and statistical mixture learning IEEE ISIT 2014 (recent result poster) http://arxiv.org/abs/1403.2485
- Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms.
IEEE Signal Process. Lett. 20(7): 657-660 (2013) http://arxiv.org/abs/1303.7286
http://www.i.kyoto-u.ac.jp/informatics-seminar/
K-means Clustering Algorithm with Matlab Source codegokulprasath06
K-means algorithm
The most common method to classify unlabeled data.
Also Checkout: http://bit.ly/2Mub6xP
Any Queries, Call us@ +91 9884412301 / 9600112302
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
9. How to Find good Clustering?
Minimize the sum of
distance within clusters
C1
C2
C3
C4
C5
,
6 2
,
1 1,
arg min
j i j
n
i j i j
j iC m
m x C
,
6
,
1
1 the j-th cluster
0 the j-th cluster
1
any a single cluster
i
i j
i
i j
j
i
x
m
x
m
x
10. How to Efficiently Clustering Data?
,
6 2
,
1 1,
arg min
j i j
n
i j i j
j iC m
m x C
,Memberships and centers are correlated.i j jm C
,
1
,
,
1
Given memberships ,
n
i j i
i
i j j n
i j
i
m x
m C
m
2
,
1 arg min( )
Given centers { },
0 otherwise
i j
kj i j
j x C
C m
11. K-means for Clustering
K-means
Start with a random
guess of cluster
centers
Determine the
membership of each
data points
Adjust the cluster
centers
12. K-means for Clustering
K-means
Start with a random
guess of cluster
centers
Determine the
membership of each
data points
Adjust the cluster
centers
13. K-means for Clustering
K-means
Start with a random
guess of cluster
centers
Determine the
membership of each
data points
Adjust the cluster
centers
15. K-means
1. Ask user how many clusters
they’d like. (e.g. k=5)
2. Randomly guess k cluster
Center locations
16. K-means
1. Ask user how many clusters
they’d like. (e.g. k=5)
2. Randomly guess k cluster
Center locations
3. Each datapoint finds out
which Center it’s closest to.
(Thus each Center “owns” a
set of datapoints)
17. K-means
1. Ask user how many clusters
they’d like. (e.g. k=5)
2. Randomly guess k cluster
Center locations
3. Each datapoint finds out
which Center it’s closest to.
4. Each Center finds the
centroid of the points it
owns
18. K-means
1. Ask user how many clusters
they’d like. (e.g. k=5)
2. Randomly guess k cluster
Center locations
3. Each datapoint finds out
which Center it’s closest to.
4. Each Center finds the
centroid of the points it
owns
Any Computational Problem?
Computational Complexity: O(N)
where N is the number of points?
19. Improve K-means
Group points by region
KD tree
SR tree
Key difference
Find the closest center for
each rectangle
Assign all the points within a
rectangle to one cluster
20. Improved K-means
Find the closest center for
each rectangle
Assign all the points within
a rectangle to one cluster
30. A Gaussian Mixture Model for Clustering
Assume that data are
generated from a
mixture of Gaussian
distributions
For each Gaussian
distribution
Center: i
Variance: i (ignore)
For each data point
Determine membership
: if belongs to j-th clusterij iz x
31. Learning a Gaussian Mixture
(with known covariance)
Probability ( )ip x x
2
/ 2 2
2
( ) ( , ) ( ) ( | )
1
( ) exp
22
j j
j
i i j j i j
i j
j d
p x x p x x p p x x
x
p
32. Learning a Gaussian Mixture
(with known covariance)
Probability ( )ip x x
2
/ 2 2
2
( ) ( , ) ( ) ( | )
1
( ) exp
22
j j
j
i i j j i j
i j
j d
p x x p x x p p x x
x
p
Log-likelihood of data
Apply MLE to find optimal parameters
2
/ 2 2
2
1
log ( ) log ( ) exp
22j
i j
i j d
i i
x
p x x p
( ),j j j
p
33. Learning a Gaussian Mixture
(with known covariance)
2
2
2
2
1
( )
2
1
( )
2
1
( )
( )
i j
i n
x
j
k x
n
n
e p
e p
[ ] ( | )ij j iE z p x x E-Step
1
( | ) ( )
( | ) ( )
i j j
k
i n j
n
p x x p
p x x p
34. Learning a Gaussian Mixture
(with known covariance)
1
1
1
[ ]
[ ]
m
j ij im
i
ij
i
E z x
E z
M-Step
1
1
( ) [ ]
m
j ij
i
p E z
m
43. Mixture Model for Doc Clustering
A set of language models
1 2, ,..., K
1 2{ ( | ), ( | ),..., ( | )}i i i V ip w p w p w
44. Mixture Model for Doc Clustering
A set of language models
1 2, ,..., K
1 2{ ( | ), ( | ),..., ( | )}i i i V ip w p w p w
( )ip d d
( , )
1
( ) ( , )
( ) ( | )
( ) ( | )
j
j
k i
j
i i j
j i j
V tf w d
j k j
k
p d d p d d
p p d d
p p w
Probability
45. Mixture Model for Doc Clustering
A set of language models
1 2, ,..., K
1 2{ ( | ), ( | ),..., ( | )}i i i V ip w p w p w
( )ip d d
( , )
1
( ) ( , )
( ) ( | )
( ) ( | )
j
j
k i
j
i i j
j i j
V tf w d
j k j
k
p d d p d d
p p d d
p p w
Probability
46. Mixture Model for Doc Clustering
A set of language models
1 2, ,..., K
1 2{ ( | ), ( | ),..., ( | )}i i i V ip w p w p w
( )ip d d
( , )
1
( ) ( , )
( ) ( | )
( ) ( | )
j
j
k i
j
i i j
j i j
V tf w d
j k j
k
p d d p d d
p p d d
p p w
Probability
Introduce hidden variable zij
zij: document di is generated by the
j-th language model j.
47. Learning a Mixture Model
( , )
1
( , )
1 1
( | ) ( )
( | ) ( )
k i
k i
V tf w d
m j j
m
VK
tf w d
m n n
n m
p w p
p w p
1
[ ] ( | )
( | ) ( )
( | ) ( )
ij j i
i j j
K
i n n
n
E z p d d
p d d p
p d d p
E-Step
K: number of language models
48. Learning a Mixture Model
M-Step
1
1
( ) [ ]
N
j ij
i
p E z
N
1
1
[ ] ( , )
( | )
[ ]
N
ij i k
k
i j N
ij k
k
E z tf w d
p w
E z d
N: number of documents
50. Other Mixture Models
Probabilistic latent semantic index (PLSI)
Latent Dirichlet Allocation (LDA)
51. Problems (I)
Both k-means and mixture models need to compute
centers of clusters and explicit distance measurement
Given strange distance measurement, the center of
clusters can be hard to compute
E.g., ' ' '
1 1 2 2' max , ,..., n nx x x x x x x x
x y
z
x y x z
52. Problems (II)
Both k-means and mixture models look for compact
clustering structures
In some cases, connected clustering structures are more desirable
54. 2-way Spectral Graph Partitioning
Weight matrix W
wi,j: the weight between two
vertices i and j
Membership vector q
1 Cluster
-1 Cluster
i
i A
q
i B
[ 1,1]
2
,
,
arg min
1
4
n
i j i j
i j
CutSize
CutSize J q q w
q
q
55. Solving the Optimization Problem
Directly solving the above problem requires
combinatorial search exponential complexity
How to reduce the computation complexity?
2
,
[ 1,1] ,
1
argmin
4n
i j i j
i j
q q w
q
q
56. Relaxation Approach
Key difficulty: qi has to be either –1, 1
Relax qi to be any real number
Impose constraint 2
1
n
ii
q n
2 2 2
, ,
, ,
2
, ,
,
1 1
2
4 4
1 1
2 2
4 4
i j i j i j i j i j
i j i j
i i j i j i j
i j i j
J q q w q q q q w
q w q q w
,i i j
j
d w
2
, , ,
,
1 1 1
2 2 2
i i i j i j i i i j i j j
i i j i
q d q q w q d w q
,i i jD d
( )T
J q D W q
58. Relaxation Approach
Solution: the second minimum eigenvector for D-W
2
* argmin argmin ( )
subject to
T
k
k
J
q n
q q
q q D W q
2( )D W q q
59. Graph Laplacian
L is semi-positive definitive matrix
For Any x, we have xTLx 0, why?
Minimum eigenvalue 1 = 0 (what is the eigenvector?)
The second minimum eigenvalue 2 gives the best bipartite
graph
, , ,: ,i j i j i jj
w w L D W W D
1 2 30 ... k
60. Recovering Partitions
Due to the relaxation, q can be any number (not just
–1 and 1)
How to construct partition based on the eigenvector?
Simple strategy: { | 0}, { | 0}i iA i q B i q
62. Normalized Cut (Shi & Malik, 1997)
Minimize the similarity between clusters and meanwhile
maximize the similarity within clusters
,( , ) , ,
( , ) ( , )
i j A i B i
i A j B i A i B
A B
s A B w d d d d
s A B s A B
J
d d
,
( , ) ( , ) B A
i j
i A j BA B A B
d ds A B s A B
J w
d d d d
j
j
d d
2
,
B A
i j
i A j B A B
d d
w
d d d
Biddd
Aiddd
iq
BA
AB
if
if
/
/
)(
2
,i j i j
i j
w q q
63. Normalized Cut
2
, ( - )
/ if
/ if
T
i j i j
i j
B A
i
A B
J w q q
d d d i A
q
d d d i B
q D W q
64. Normalized Cut
Relax q to real value under the constraint
2
, ( - )
/ if
/ if
T
i j i j
i j
B A
i
A B
J w q q
d d d i A
q
d d d i B
q D W q
0,1 DeqDqq TT
Solution: DqqWD )(