Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
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2. Linear Algebra for Machine Learning: Basis and DimensionCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the second part which is discussing basis and dimension.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Robust Control of Uncertain Switched Linear Systems based on Stochastic Reach...Leo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of uncertain discrete-time switched linear systems based on a combination
of tree search and reachable set computations in a stochastic setting. For given Gaussian
distributions of the initial states and disturbances, state sets wich are reachable to a chosen
confidence level under the effect of time-variant hybrid control laws are computed by using
principles of the ellipsoidal calculus. The proposed algorithm iterates over sequences of the
discrete states and LMI-constrained semi-definite programming (SDP) problems to compute
stabilizing controllers, while polytopic input constraints are considered. An example for illustration is included.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
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I am Anthony F. I am a Digital Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, University of Cambridge, UK. I have been helping students with their homework for the past 8 years. I solve assignments related to Digital Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Digital Signal Processing Assignments.
2. Linear Algebra for Machine Learning: Basis and DimensionCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the second part which is discussing basis and dimension.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Robust Control of Uncertain Switched Linear Systems based on Stochastic Reach...Leo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of uncertain discrete-time switched linear systems based on a combination
of tree search and reachable set computations in a stochastic setting. For given Gaussian
distributions of the initial states and disturbances, state sets wich are reachable to a chosen
confidence level under the effect of time-variant hybrid control laws are computed by using
principles of the ellipsoidal calculus. The proposed algorithm iterates over sequences of the
discrete states and LMI-constrained semi-definite programming (SDP) problems to compute
stabilizing controllers, while polytopic input constraints are considered. An example for illustration is included.
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov –...Beniamino Murgante
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov – National Centre for Geocomputation, National University of Ireland , Maynooth (Ireland)
Intelligent Analysis of Environmental Data (S4 ENVISA Workshop 2009)
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
Digital Signal Processing[ECEG-3171]-Ch1_L03Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
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
On the Seidel’s Method, a Stronger Contraction Fixed Point Iterative Method o...BRNSS Publication Hub
In the solution of a system of linear equations, there exist many methods most of which are not fixed point iterative methods. However, this method of Sidel’s iteration ensures that the given system of the equation must be contractive after satisfying diagonal dominance. The theory behind this was discussed in sections one and two and the end; the application was extensively discussed in the last section.
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You can also call on +1 678 648 4277 for any assistance with the Computer Network Assignment.
Subgradient Methods for Huge-Scale Optimization Problems - Юрий Нестеров, Cat...Yandex
We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piecewise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, the total cost of which depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.
We show that the updating technique can be efficiently coupled with the simplest subgradient methods. Similar results can be obtained for a new non-smooth random variant of a coordinate descent scheme. We also present promising results of preliminary computational experiments.
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Opendatabay - Open Data Marketplace.pptxOpendatabay
Opendatabay.com unlocks the power of data for everyone. Open Data Marketplace fosters a collaborative hub for data enthusiasts to explore, share, and contribute to a vast collection of datasets.
First ever open hub for data enthusiasts to collaborate and innovate. A platform to explore, share, and contribute to a vast collection of datasets. Through robust quality control and innovative technologies like blockchain verification, opendatabay ensures the authenticity and reliability of datasets, empowering users to make data-driven decisions with confidence. Leverage cutting-edge AI technologies to enhance the data exploration, analysis, and discovery experience.
From intelligent search and recommendations to automated data productisation and quotation, Opendatabay AI-driven features streamline the data workflow. Finding the data you need shouldn't be a complex. Opendatabay simplifies the data acquisition process with an intuitive interface and robust search tools. Effortlessly explore, discover, and access the data you need, allowing you to focus on extracting valuable insights. Opendatabay breaks new ground with a dedicated, AI-generated, synthetic datasets.
Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
As Europe's leading economic powerhouse and the fourth-largest hashtag#economy globally, Germany stands at the forefront of innovation and industrial might. Renowned for its precision engineering and high-tech sectors, Germany's economic structure is heavily supported by a robust service industry, accounting for approximately 68% of its GDP. This economic clout and strategic geopolitical stance position Germany as a focal point in the global cyber threat landscape.
In the face of escalating global tensions, particularly those emanating from geopolitical disputes with nations like hashtag#Russia and hashtag#China, hashtag#Germany has witnessed a significant uptick in targeted cyber operations. Our analysis indicates a marked increase in hashtag#cyberattack sophistication aimed at critical infrastructure and key industrial sectors. These attacks range from ransomware campaigns to hashtag#AdvancedPersistentThreats (hashtag#APTs), threatening national security and business integrity.
🔑 Key findings include:
🔍 Increased frequency and complexity of cyber threats.
🔍 Escalation of state-sponsored and criminally motivated cyber operations.
🔍 Active dark web exchanges of malicious tools and tactics.
Our comprehensive report delves into these challenges, using a blend of open-source and proprietary data collection techniques. By monitoring activity on critical networks and analyzing attack patterns, our team provides a detailed overview of the threats facing German entities.
This report aims to equip stakeholders across public and private sectors with the knowledge to enhance their defensive strategies, reduce exposure to cyber risks, and reinforce Germany's resilience against cyber threats.
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
Machine learning and optimization techniques for electrical drives.pptx
Pydata Katya Vasilaky
1. Iterative Approach to the Tikhanov Regularization
Problem in Matrix Inversion using Numpy
Katya Vasilaky1
November 21, 2015
1
New Yorker @ Columbia University
7. Inverse problems: compute information about some “interior” properties using
“exterior” measurements.
Tomography: Xray source − > Object − > Dampening
Inference: Attributes − > Effect Size − > Outcome
Machine Learning: Features − > Predictors − > Classifier
8. X is my back.
Problem.
Don’t know x.
Figure: Xray of Spine
9. Solution.
Throw A at x (Ax)
Collect b
Infer x (x = A−1
b).
or for linear regression....
β = (XT
X)−1
(XT
Y )
10.
11. Methodology
The pseudoinverse solution, which always exists, is often computed as the
unique, minimum norm solution to the least-squares formulation
ˆx = Minx ||Ax − b||2
2
such that ||x||2
2 solves for the minimum.
Think of an underdetermined system of equations:
x + y + z = 1 (1)
x + y + 2z = 3 (2)
There are many solutions.
The pseudo inverse is not very useful for applications.
12. Lots of useful inverse problems are ill-conditioned linear systems.
Standard numerical methods x = A−1b or β = (XT X)−1XT Y produce useless
results.
That A (or X) will be singular or not full rank, so it’s not invertible, or
ill-conditioned
13. If you perturb b a little bit, x = A−1
(b + e)
Will map to something crazy!
14. Why you ask?
If A is not full rank or ill-conditioned then the solution x is unstable
A−1b will be vastly different from A−1(b + e)
The inverse of A is not a continuous mapping − > b
b gets mapped to something very different than b+e
15. What do people do then?
Regularization problems formulate a nearby problem, which has a more stable
solution:
Minx (||Ax − b||2
2 + λ||x||2
2), δ > 0
where we introduce the term λ||x||2
2 perturbing the least-squares formulation.
The regularization can be L1 (Lasso) or L2. Typically L2 (Tikhanov and
Truncated SVD) better. Most people try to choose a good lambda, and solve
the above, hoping that they’re close to the noiseless solution. Which penalizes
outliers more?
16. Problem with regularization is how to choose λ or how much to truncate.
Tikhonov and truncated SVD are equivalent in that chopping off singular
values is equivalent to choosing a λ?
Minx (||Ax − b||2
2 + λ||x||2
2), δ > 0
This solution: Instead of computing a solution we compute a sequence of
solutions which approximates the noiseless solution (A−
1b) on its way to the
noisy (A−
1(b + e) solution.
17.
18. Replace λ||x||
Minx (||Ax − b||2
2 + λ||x||2
2), δ > 0
with λ||x − x0||
Minx (||Ax − b||2
2 + λ||x − x0||2
2), δ > 0
and then plug and re-plug.......
19. Normal Equations
Remember your normal equations?
(AT
A)x = AT
b
Instead solve new normal equations with added constraint:
(AT
A + λI)x = AT
b + λx0
20. Solution for x1 is:
x1 = (AT
A + λI)−1
AT
b + λ(AT
A + λI)−1
x0
and the kth iterate is:
xk = Σk
i=1λi−1
((AT
A + λI)−1
(AT
b) + λk
(AT
A + λI)−1
x0)
21. Now we the use single value decomposition (SVD) of A: A = UΣV T
,
σ1, .., σn, 0....0) ∈ Rmxn
with σ1 ≥ σ2 ≥ · · · ≥ σn > 0, where n is the rank of A,
we then obtain after a lot of derivation, the following kth
iterate.
V
1
σ1
− 1
σ1
( λ
λ+σ2
1
)k
. . . 0 . . . 0
...
...
...
0 . . . 1
σn
− 1
σn
( λ
λ+σ2
n
)k
0 . . . 0
UT
b
For practical ease–I’m setting x0=0. It does not affect the solution.
22. Step 1
InverseProblem codes the above formula to invert A.
So inverse of A times b from InverseProblem will give us the x back.
Let’s try it out using the Inverse Problem package!
https://github.com/kathrynthegreat/InverseProblem
23. Step 2: Brace yourself. Next few steps:
Get a theoretical derivation for xk − x, the error between kith iterate and the true
x.
Show that as k goes to infinity xk approaches the noisy x solution
Need to stop k before that happens!
24. First, look at the residual error against the number of iterations.
One thing is sure. We want λ > 1
25. Next, it turns out (from experiments), that xk passes the true noiseless x when
the residual error ||Axk − b|| “turns,” a heuristic that still needs a proof.
Somehow this minimizes the error between the kith iterate and the true x:
xk − x expression.
(S1)
-V
1
σ1
( λ
λ+σ2
1
)k
. . . 0 0 . . . 0
...
...
...
0 . . . 1
σn
( λ
λ+σ2
n
)k
0 . . . 0
UT
b
(S2) V
1
σ1
− 1
σ1
( λ
λ+σ2
1
)k
. . . 0 0 . . . 0
...
...
...
0 . . . 1
σn
− 1
σn
( λ
λ+σ2
n
)k
0 . . . 0
UT
e
26.
27. Now that we’ve seen the tractable cases. Here are some other uses.
We will reconstruct an image from projections. Some reconstructions from
projections are:
1 CT scans
2 MRI
3 Oil Exploration
4 Seismic imaging
28.
29.
30.
31. Notes
Iterative approach is good for extremely ill conditioned matrices that are large.
For small matrices there are other approaches.
This approach is good for images because you can see when you’re doing
better-the image is clearer.
We need to see the graph (or the picture) to get an idea about lambda and k, the
good news is that we don’t have to know them optimally as in classical Tikhonov.
Otherwise if we know something about e (like it’s SD or norm ||e||) we can plot
that error graph (still need to package that)
Remember, of course, that if the data are extremely noisy then it is worthless, so
no recovery is possible.