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The document discusses singular value decomposition (SVD), which is a way to decompose a matrix A into three matrices: A = UΣV^T. U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of A. SVD can be used to perform dimensionality reduction by approximating A using only the top k singular values/vectors in Σ, U, and V^T. This reduces the number of parameters needed to represent A while retaining most of its information.

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Flow based generative models

NICE: Non-linear Independent Components Estimation
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Density estimation using Real NVP
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논문 리뷰 자료

Singular Value Decompostion (SVD)

(i) Singular Value Decomposition (SVD) factorizes an m x n matrix A into the product of three matrices: A = USV^T, where U and V are orthogonal matrices and S is a diagonal matrix containing the singular values of A.
(ii) The matrices A^TA and AA^T are symmetric and their eigenvalues are real and non-negative.
(iii) In an example, the singular values of a 5x3 matrix are found to be √5, √2, and 1 by computing the eigenvalues of A^TA.

Clustering training

What is clustering?
Distance: Similarity and dissimilarity
Data types in cluster analysis
Clustering methods
Evaluation of clustering
Summary

Knn Algorithm presentation

k-Nearest Neighbors (k-NN) is a simple machine learning algorithm that classifies new data points based on their similarity to existing data points. It stores all available data and classifies new data based on a distance function measurement to find the k nearest neighbors. k-NN is a non-parametric lazy learning algorithm that is widely used for classification and pattern recognition problems. It performs well when there is a large amount of sample data but can be slow and the choice of k can impact performance.

Analysis of Feature Selection Algorithms (Branch & Bound and Beam search)

Branch & Bound and Beam search algorithms were illustrated according to the feature selection domain. Presentation is structured as follows,
- Motivation
- Introduction
- Analysis
- Algorithm
- Pseudo Code
- Illustration of examples
- Applications
- Observations and Recommendations
- Comparison between two algorithms
- References

High Dimensional Data Visualization using t-SNE

Review of the t-SNE algorithm which helps visualizing the high dimensional data on manifold by projecting them onto 2D or 3D space with metric preserving.

Two dimentional transform

This document discusses 2D geometric transformations including translation, rotation, scaling, and composite transformations. It provides definitions and formulas for each type of transformation. Translation moves objects by adding offsets to coordinates without deformation. Rotation rotates objects around an origin by a certain angle. Scaling enlarges or shrinks objects by multiplying coordinates by scaling factors. Composite transformations apply multiple transformations sequentially by multiplying their matrices. Homogeneous coordinates are also introduced to represent transformations in matrix form.

Anomaly detection Full Article

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Flow based generative models

NICE: Non-linear Independent Components Estimation
Laurent Dinh, David Krueger, Yoshua Bengio. 2014.
Density estimation using Real NVP
Laurent Dinh, Jascha Sohl-Dickstein, Samy Bengio. 2017.
Glow: Generative Flow with Invertible 1x1 Convolutions
Diederik P. Kingma, Prafulla Dhariwal. 2018.
논문 리뷰 자료

Singular Value Decompostion (SVD)

(i) Singular Value Decomposition (SVD) factorizes an m x n matrix A into the product of three matrices: A = USV^T, where U and V are orthogonal matrices and S is a diagonal matrix containing the singular values of A.
(ii) The matrices A^TA and AA^T are symmetric and their eigenvalues are real and non-negative.
(iii) In an example, the singular values of a 5x3 matrix are found to be √5, √2, and 1 by computing the eigenvalues of A^TA.

Clustering training

What is clustering?
Distance: Similarity and dissimilarity
Data types in cluster analysis
Clustering methods
Evaluation of clustering
Summary

Knn Algorithm presentation

k-Nearest Neighbors (k-NN) is a simple machine learning algorithm that classifies new data points based on their similarity to existing data points. It stores all available data and classifies new data based on a distance function measurement to find the k nearest neighbors. k-NN is a non-parametric lazy learning algorithm that is widely used for classification and pattern recognition problems. It performs well when there is a large amount of sample data but can be slow and the choice of k can impact performance.

Analysis of Feature Selection Algorithms (Branch & Bound and Beam search)

Branch & Bound and Beam search algorithms were illustrated according to the feature selection domain. Presentation is structured as follows,
- Motivation
- Introduction
- Analysis
- Algorithm
- Pseudo Code
- Illustration of examples
- Applications
- Observations and Recommendations
- Comparison between two algorithms
- References

High Dimensional Data Visualization using t-SNE

Review of the t-SNE algorithm which helps visualizing the high dimensional data on manifold by projecting them onto 2D or 3D space with metric preserving.

Two dimentional transform

This document discusses 2D geometric transformations including translation, rotation, scaling, and composite transformations. It provides definitions and formulas for each type of transformation. Translation moves objects by adding offsets to coordinates without deformation. Rotation rotates objects around an origin by a certain angle. Scaling enlarges or shrinks objects by multiplying coordinates by scaling factors. Composite transformations apply multiple transformations sequentially by multiplying their matrices. Homogeneous coordinates are also introduced to represent transformations in matrix form.

Anomaly detection Full Article

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Recent Progress on Object Detection_20170331

This slide provides a brief summary of recent progress on object detection using deep learning.
The concept of selected previous works(R-CNN series/YOLO/SSD) and 6 recent papers (uploaded to the Arxiv between Dec/2016 and Mar/2017) are introduced in this slide.
Most papers are focusing on improving the performance of small object detection.

2-Approximation Vertex Cover

This document presents an approximation algorithm for the vertex cover problem in graphs. It begins with definitions of the vertex cover problem and shows that finding an optimal solution is NP-complete. It then presents a 2-approximation algorithm that finds a vertex cover of size at most twice the optimal. The time complexity of the algorithm is O(V+E). Applications of the vertex cover problem and some open questions are also discussed.

From decision trees to random forests

This document discusses decision trees and random forests for classification problems. It explains that decision trees use a top-down approach to split a training dataset based on attribute values to build a model for classification. Random forests improve upon decision trees by growing many de-correlated trees on randomly sampled subsets of data and features, then aggregating their predictions, which helps avoid overfitting. The document provides examples of using decision trees to classify wine preferences, sports preferences, and weather conditions for sport activities based on attribute values.

Image compression using singular value decomposition

Singular value decomposition (SVD) can be used to compress images by decomposing images into orthogonal matrices and a diagonal matrix. This decomposition allows images to be approximated using only the first few terms in the decomposition series, reducing memory usage. However, there are limits to the compression rate that can be achieved while still saving memory. The rank of the approximation must be less than mn/(m+n+1) for memory savings, where m and n are the image dimensions. Higher ranks improve quality but also increase memory usage.

Gradient Boosted Regression Trees in scikit-learn

Slides of the talk "Gradient Boosted Regression Trees in scikit-learn" by Peter Prettenhofer and Gilles Louppe held at PyData London 2014.
Abstract:
This talk describes Gradient Boosted Regression Trees (GBRT), a powerful statistical learning technique with applications in a variety of areas, ranging from web page ranking to environmental niche modeling. GBRT is a key ingredient of many winning solutions in data-mining competitions such as the Netflix Prize, the GE Flight Quest, or the Heritage Health Price.
I will give a brief introduction to the GBRT model and regression trees -- focusing on intuition rather than mathematical formulas. The majority of the talk will be dedicated to an in depth discussion how to apply GBRT in practice using scikit-learn. We will cover important topics such as regularization, model tuning and model interpretation that should significantly improve your score on Kaggle.

Chap8 basic cluster_analysis

- Hierarchical clustering produces nested clusters organized as a hierarchical tree called a dendrogram. It can be either agglomerative, where each point starts in its own cluster and clusters are merged, or divisive, where all points start in one cluster which is recursively split.
- Common hierarchical clustering algorithms include single linkage (minimum distance), complete linkage (maximum distance), group average, and Ward's method. They differ in how they calculate distance between clusters during merging.
- K-means is a partitional clustering algorithm that divides data into k non-overlapping clusters based on minimizing distance between points and cluster centroids. It is fast but sensitive to initialization and assumes spherical clusters of similar size and density.

Clustering: A Survey

This document provides an overview of clustering techniques. It defines clustering as grouping a set of similar objects into classes, with objects within a cluster being similar to each other and dissimilar to objects in other clusters. The document then discusses partitioning, hierarchical, and density-based clustering methods. It also covers mathematical elements of clustering like partitions, distances, and data types. The goal of clustering is to minimize a similarity function to create high similarity within clusters and low similarity between clusters.

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Feature selection is the process of selecting a subset of relevant features for model construction. It reduces complexity and can improve or maintain model accuracy. The curse of dimensionality means that as the number of features increases, the amount of data needed to maintain accuracy also increases exponentially. Feature selection methods include filter methods (statistical tests for correlation), wrapper methods (using the model to select features), and embedded methods (combining filter and wrapper approaches). Common filter methods include linear discriminant analysis, analysis of variance, chi-square tests, and Pearson correlation. Wrapper methods use techniques like forward selection, backward elimination, and recursive feature elimination. Embedded methods dynamically select features based on inferences from previous models.

Visualization using tSNE

This document discusses t-Distributed Stochastic Neighbor Embedding (t-SNE), a technique for visualizing high-dimensional data. It begins by overviewing dimension reduction techniques before focusing on t-SNE. t-SNE is an improvement on Stochastic Neighbor Embedding (SNE) that converts similarities between data points to joint probabilities and minimizes the divergence between a high-dimensional and low-dimensional distribution. The document explains how t-SNE addresses issues like the "crowding problem" to better separate clusters in low dimensions. Optimization methods for t-SNE are also covered.

=SLAM ppt.pdf

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Computer Graphic - Transformations in 2D

The document discusses 2D geometric transformations using matrices. It defines a general transformation equation [B] = [T] [A] where [T] is the transformation matrix and [A] and [B] are the input and output point matrices. It then explains various transformation matrices for scaling, reflection, rotation and translation. It also discusses representing transformations in homogeneous coordinates using 3x3 matrices. Finally, it provides examples of applying multiple transformations and conditions when the order of transformations can be changed.

Reinforcement learning, Q-Learning

Reinforcement learning is a machine learning technique where an agent learns how to behave in an environment by receiving rewards or punishments for its actions. The goal of the agent is to learn an optimal policy that maximizes long-term rewards. Reinforcement learning can be applied to problems like game playing, robot control, scheduling, and economic modeling. The reinforcement learning process involves an agent interacting with an environment to learn through trial-and-error using state, action, reward, and policy. Common algorithms include Q-learning which uses a Q-table to learn the optimal action-selection policy.

Tensor Train decomposition in machine learning

The document discusses using Tensor Train (TT) decomposition to efficiently represent tensors and apply it to machine learning models. Some key points:
- TT decomposition provides a compact representation of tensors that allows efficient linear algebra operations.
- It has been used to compress the weights matrix of neural networks without loss of accuracy.
- Exponential machines model all feature interactions using a TT-formatted weight tensor, controlling complexity with TT-rank. This outperforms other models on classification tasks involving interactions.

Mask R-CNN

Mask R-CNN extends Faster R-CNN by adding a branch for predicting segmentation masks in parallel with bounding box recognition and classification. It introduces a new layer called RoIAlign to address misalignment issues in the RoIPool layer of Faster R-CNN. RoIAlign improves mask accuracy by 10-50% by removing quantization and properly aligning extracted features. Mask R-CNN runs at 5fps with only a small overhead compared to Faster R-CNN.

SVM

This presentation covers the basics of support vector machine along with the kernels and the code for image classification in the end.

Introduction to Linear Discriminant Analysis

This document provides an introduction and overview of linear discriminant analysis (LDA). It discusses that LDA is a dimensionality reduction technique used to separate classes of data. The document outlines the 5 main steps to performing LDA: 1) calculating class means, 2) computing scatter matrices, 3) finding linear discriminants using eigenvalues/eigenvectors, 4) determining the transformation subspace, and 5) projecting the data onto the subspace. Examples using the Iris dataset are provided to illustrate how LDA works step-by-step to find projection directions that separate the classes.

A brief survey of tensors

Because of deep learning we now talk a lot about tensors, yet tensors remain relatively unknown objects. In this presentation I will introduce tensors and the basics of multilinear algebra, then describe tensor decompositions and give some examples of how they are used in representation learning for understanding/compressing data. I will also briefly describe how tensor decompositions are used in 1) the method of moments for training latent variable models, and 2) deep learning for understanding why deep convolutional networks are such excellent classifiers.

Gradient descent method

This method gives the Artificial Neural Network, its much required tradeoff between cost function and processing powers

Rev1.0

This document provides information and equations for computing the stiffness matrix of a finite element made up of 4 identical triangles. It defines the geometry and material properties of the element. It then shows the calculations to derive the elasticity matrix coefficients and the individual components of the stiffness matrix based on the geometry, properties, and equations provided. The stiffness matrix is then computed for a given example where N=5. Forces on the element are also calculated based on the pressures and geometry.

Algebra

This document contains solutions to exercises on linear transformations. It begins with objectives to analyze and calculate linear transformations and develop skills from class. It then shows work solving 7 exercises involving determining if functions define linear transformations, calculating outputs of transformations given inputs, and determining an output given transformation definitions on other inputs. References include YouTube videos on linear transformations.

4 Dimensionality reduction (PCA & t-SNE)

The fourth lecture from the Machine Learning course series of lectures. This lecture first introduces a problem of visualising multi-dimensional data on fewer dimensions and later discusses one of the most popular methods for reducing dimensionality - principal component analysis (PCA). Later, also t-SNE is mentioned briefly as a non-linear alternative to PCA. A link to my github (https://github.com/skyfallen/MachineLearningPracticals) with practicals that I have designed for this course in both R and Python. I can share keynote files, contact me via e-mail: dmytro.fishman@ut.ee.

Recent Progress on Object Detection_20170331

This slide provides a brief summary of recent progress on object detection using deep learning.
The concept of selected previous works(R-CNN series/YOLO/SSD) and 6 recent papers (uploaded to the Arxiv between Dec/2016 and Mar/2017) are introduced in this slide.
Most papers are focusing on improving the performance of small object detection.

2-Approximation Vertex Cover

This document presents an approximation algorithm for the vertex cover problem in graphs. It begins with definitions of the vertex cover problem and shows that finding an optimal solution is NP-complete. It then presents a 2-approximation algorithm that finds a vertex cover of size at most twice the optimal. The time complexity of the algorithm is O(V+E). Applications of the vertex cover problem and some open questions are also discussed.

From decision trees to random forests

This document discusses decision trees and random forests for classification problems. It explains that decision trees use a top-down approach to split a training dataset based on attribute values to build a model for classification. Random forests improve upon decision trees by growing many de-correlated trees on randomly sampled subsets of data and features, then aggregating their predictions, which helps avoid overfitting. The document provides examples of using decision trees to classify wine preferences, sports preferences, and weather conditions for sport activities based on attribute values.

Image compression using singular value decomposition

Singular value decomposition (SVD) can be used to compress images by decomposing images into orthogonal matrices and a diagonal matrix. This decomposition allows images to be approximated using only the first few terms in the decomposition series, reducing memory usage. However, there are limits to the compression rate that can be achieved while still saving memory. The rank of the approximation must be less than mn/(m+n+1) for memory savings, where m and n are the image dimensions. Higher ranks improve quality but also increase memory usage.

Gradient Boosted Regression Trees in scikit-learn

Slides of the talk "Gradient Boosted Regression Trees in scikit-learn" by Peter Prettenhofer and Gilles Louppe held at PyData London 2014.
Abstract:
This talk describes Gradient Boosted Regression Trees (GBRT), a powerful statistical learning technique with applications in a variety of areas, ranging from web page ranking to environmental niche modeling. GBRT is a key ingredient of many winning solutions in data-mining competitions such as the Netflix Prize, the GE Flight Quest, or the Heritage Health Price.
I will give a brief introduction to the GBRT model and regression trees -- focusing on intuition rather than mathematical formulas. The majority of the talk will be dedicated to an in depth discussion how to apply GBRT in practice using scikit-learn. We will cover important topics such as regularization, model tuning and model interpretation that should significantly improve your score on Kaggle.

Chap8 basic cluster_analysis

- Hierarchical clustering produces nested clusters organized as a hierarchical tree called a dendrogram. It can be either agglomerative, where each point starts in its own cluster and clusters are merged, or divisive, where all points start in one cluster which is recursively split.
- Common hierarchical clustering algorithms include single linkage (minimum distance), complete linkage (maximum distance), group average, and Ward's method. They differ in how they calculate distance between clusters during merging.
- K-means is a partitional clustering algorithm that divides data into k non-overlapping clusters based on minimizing distance between points and cluster centroids. It is fast but sensitive to initialization and assumes spherical clusters of similar size and density.

Clustering: A Survey

This document provides an overview of clustering techniques. It defines clustering as grouping a set of similar objects into classes, with objects within a cluster being similar to each other and dissimilar to objects in other clusters. The document then discusses partitioning, hierarchical, and density-based clustering methods. It also covers mathematical elements of clustering like partitions, distances, and data types. The goal of clustering is to minimize a similarity function to create high similarity within clusters and low similarity between clusters.

Gradient descent method

The document discusses gradient descent methods for unconstrained convex optimization problems. It introduces gradient descent as an iterative method to find the minimum of a differentiable function by taking steps proportional to the negative gradient. It describes the basic gradient descent update rule and discusses convergence conditions such as Lipschitz continuity, strong convexity, and condition number. It also covers techniques like exact line search, backtracking line search, coordinate descent, and steepest descent methods.

Feature selection

Feature selection is the process of selecting a subset of relevant features for model construction. It reduces complexity and can improve or maintain model accuracy. The curse of dimensionality means that as the number of features increases, the amount of data needed to maintain accuracy also increases exponentially. Feature selection methods include filter methods (statistical tests for correlation), wrapper methods (using the model to select features), and embedded methods (combining filter and wrapper approaches). Common filter methods include linear discriminant analysis, analysis of variance, chi-square tests, and Pearson correlation. Wrapper methods use techniques like forward selection, backward elimination, and recursive feature elimination. Embedded methods dynamically select features based on inferences from previous models.

Visualization using tSNE

This document discusses t-Distributed Stochastic Neighbor Embedding (t-SNE), a technique for visualizing high-dimensional data. It begins by overviewing dimension reduction techniques before focusing on t-SNE. t-SNE is an improvement on Stochastic Neighbor Embedding (SNE) that converts similarities between data points to joint probabilities and minimizes the divergence between a high-dimensional and low-dimensional distribution. The document explains how t-SNE addresses issues like the "crowding problem" to better separate clusters in low dimensions. Optimization methods for t-SNE are also covered.

=SLAM ppt.pdf

This document discusses visual simultaneous localization and mapping (SLAM) and visual odometry (VO). It provides an overview of different approaches including geometric formulations, error formulations, geometry parameterizations, sparse vs dense models, optimization approaches, and sensor combinations. It analyzes two example systems - ORB-SLAM which uses an indirect, sparse model optimized using graph optimization, and Direct Sparse Odometry (DSO) which uses a direct, sparse model optimized using information filtering. It discusses important details in SLAM/VO systems like point selection, keyframe selection, residual selection, parameter initialization, and optimization strategies. It concludes with discussing evaluating SLAM/VO on a wide range of datasets to avoid overfitting.

Computer Graphic - Transformations in 2D

The document discusses 2D geometric transformations using matrices. It defines a general transformation equation [B] = [T] [A] where [T] is the transformation matrix and [A] and [B] are the input and output point matrices. It then explains various transformation matrices for scaling, reflection, rotation and translation. It also discusses representing transformations in homogeneous coordinates using 3x3 matrices. Finally, it provides examples of applying multiple transformations and conditions when the order of transformations can be changed.

Reinforcement learning, Q-Learning

Reinforcement learning is a machine learning technique where an agent learns how to behave in an environment by receiving rewards or punishments for its actions. The goal of the agent is to learn an optimal policy that maximizes long-term rewards. Reinforcement learning can be applied to problems like game playing, robot control, scheduling, and economic modeling. The reinforcement learning process involves an agent interacting with an environment to learn through trial-and-error using state, action, reward, and policy. Common algorithms include Q-learning which uses a Q-table to learn the optimal action-selection policy.

Tensor Train decomposition in machine learning

The document discusses using Tensor Train (TT) decomposition to efficiently represent tensors and apply it to machine learning models. Some key points:
- TT decomposition provides a compact representation of tensors that allows efficient linear algebra operations.
- It has been used to compress the weights matrix of neural networks without loss of accuracy.
- Exponential machines model all feature interactions using a TT-formatted weight tensor, controlling complexity with TT-rank. This outperforms other models on classification tasks involving interactions.

Mask R-CNN

Mask R-CNN extends Faster R-CNN by adding a branch for predicting segmentation masks in parallel with bounding box recognition and classification. It introduces a new layer called RoIAlign to address misalignment issues in the RoIPool layer of Faster R-CNN. RoIAlign improves mask accuracy by 10-50% by removing quantization and properly aligning extracted features. Mask R-CNN runs at 5fps with only a small overhead compared to Faster R-CNN.

SVM

This presentation covers the basics of support vector machine along with the kernels and the code for image classification in the end.

Introduction to Linear Discriminant Analysis

This document provides an introduction and overview of linear discriminant analysis (LDA). It discusses that LDA is a dimensionality reduction technique used to separate classes of data. The document outlines the 5 main steps to performing LDA: 1) calculating class means, 2) computing scatter matrices, 3) finding linear discriminants using eigenvalues/eigenvectors, 4) determining the transformation subspace, and 5) projecting the data onto the subspace. Examples using the Iris dataset are provided to illustrate how LDA works step-by-step to find projection directions that separate the classes.

A brief survey of tensors

Because of deep learning we now talk a lot about tensors, yet tensors remain relatively unknown objects. In this presentation I will introduce tensors and the basics of multilinear algebra, then describe tensor decompositions and give some examples of how they are used in representation learning for understanding/compressing data. I will also briefly describe how tensor decompositions are used in 1) the method of moments for training latent variable models, and 2) deep learning for understanding why deep convolutional networks are such excellent classifiers.

Gradient descent method

This method gives the Artificial Neural Network, its much required tradeoff between cost function and processing powers

4 Dimensionality reduction (PCA & t-SNE)

4 Dimensionality reduction (PCA & t-SNE)

Recent Progress on Object Detection_20170331

Recent Progress on Object Detection_20170331

2-Approximation Vertex Cover

2-Approximation Vertex Cover

From decision trees to random forests

From decision trees to random forests

Image compression using singular value decomposition

Image compression using singular value decomposition

Gradient Boosted Regression Trees in scikit-learn

Gradient Boosted Regression Trees in scikit-learn

Chap8 basic cluster_analysis

Chap8 basic cluster_analysis

Clustering: A Survey

Clustering: A Survey

Gradient descent method

Gradient descent method

Feature selection

Feature selection

Visualization using tSNE

Visualization using tSNE

=SLAM ppt.pdf

=SLAM ppt.pdf

Computer Graphic - Transformations in 2D

Computer Graphic - Transformations in 2D

Reinforcement learning, Q-Learning

Reinforcement learning, Q-Learning

Tensor Train decomposition in machine learning

Tensor Train decomposition in machine learning

Mask R-CNN

Mask R-CNN

SVM

SVM

Introduction to Linear Discriminant Analysis

Introduction to Linear Discriminant Analysis

A brief survey of tensors

A brief survey of tensors

Gradient descent method

Gradient descent method

Rev1.0

This document provides information and equations for computing the stiffness matrix of a finite element made up of 4 identical triangles. It defines the geometry and material properties of the element. It then shows the calculations to derive the elasticity matrix coefficients and the individual components of the stiffness matrix based on the geometry, properties, and equations provided. The stiffness matrix is then computed for a given example where N=5. Forces on the element are also calculated based on the pressures and geometry.

Algebra

This document contains solutions to exercises on linear transformations. It begins with objectives to analyze and calculate linear transformations and develop skills from class. It then shows work solving 7 exercises involving determining if functions define linear transformations, calculating outputs of transformations given inputs, and determining an output given transformation definitions on other inputs. References include YouTube videos on linear transformations.

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Visual - various maths sites (credits to original creator)
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This document contains an instructor's resource manual with solutions to problems involving rational numbers, decimals, and operations with fractions and radicals. It provides step-by-step workings for 53 problems involving simplifying expressions, evaluating expressions, determining if numbers are rational or irrational, and approximating values of expressions using decimals. The problems cover basic concepts relating to rational numbers, decimals, fractions, and radicals that are often encountered in pre-algebra and beginning algebra courses.

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Espressioni

−3
3
−2
3
3
+ + ( −3) 2 ⋅ − + ( −3) −3
3 6
2
Se
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Math 5

1. The document discusses addition, subtraction, and multiplication of numbers with 7, 8, or 9 digits. It explains that the placement of ones, thousands, and millions should be under their respective places.
2. Examples are provided to demonstrate addition, subtraction, and multiplication of multi-digit numbers. Diagrams are used to illustrate multiplication.
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Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para admi...

Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para administración y economía. (12ª edición). año de edición 2012. editorial pearson

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The exercises provide practice on basic math skills for young learners such as counting, addition, subtraction, comparison, and word problems involving money.

8 points on the unit circle the wrapping function w(t)

This document provides examples and explanations for using the wrapping function W(t) to find points on the unit circle. It begins with examples of finding points at various angle values. It then explains how the wrapping function maps real numbers to corresponding points on the unit circle. The document gives examples of naming points, identifying quadrants, checking if a point is on the unit circle, and performing operations like W(-t) on the wrapping function. It concludes with an assignment on further wrapping function problems.

Solving ode problem using the Galerkin's method

1) The document demonstrates Galerkin's method for solving ordinary differential equations (ODEs) numerically. It considers an ODE with boundary conditions and varies parameters like a, b, n to observe how the error ε changes.
2) The data shows that as n increases, ε decreases at a rate close to 0.25, 0.1111, or 0.0625 depending on the parameters. It also shows that as parameter a increases, the ratio of ε values approaches 1.
3) The document claims that for sufficiently large n and a, the error ratio will converge to these values and ratios multiplied by h will approach 1, where h is a real number.

Maths Y4 Week 1 days 1, 2, 3, 4 and 5

The document provides examples and practice problems for rounding numbers to the nearest 10, 100, 1000, and decimal numbers to the nearest whole number. It includes the rules for rounding (look at the digit in the relevant place value and if it is 4 or less, round down, if 5 or more, round up). Various multiplication tables are also provided as examples. The document supports learning and practicing skills in rounding and multiplication tables.

คณิตศาสตร์ 60 เฟรม กาญจนรัตน์

1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem.
2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem.
3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.

Rev1.0

Rev1.0

Algebra

Algebra

SUEC 高中 Adv Maths (Matrix) (Part 3).pptx

SUEC 高中 Adv Maths (Matrix) (Part 3).pptx

0. preliminares

0. preliminares

Kunci Jawaban kalkulus edisi 9[yunusFairVry.blogspot.com].pdf

Kunci Jawaban kalkulus edisi 9[yunusFairVry.blogspot.com].pdf

Modul linus numerasi tahun 3

Modul linus numerasi tahun 3

Dmxchart

Dmxchart

TRANSFORMACIONES LINEALES

TRANSFORMACIONES LINEALES

POTENCIAS Y RADICALES

POTENCIAS Y RADICALES

Espressioni

Espressioni

Math 5

Math 5

Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para admi...

Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para admi...

Solucionario de matemáticas para administación y economia

Solucionario de matemáticas para administación y economia

31350052 introductory-mathematical-analysis-textbook-solution-manual

31350052 introductory-mathematical-analysis-textbook-solution-manual

Sol mat haeussler_by_priale

Sol mat haeussler_by_priale

Ppt 1stelj Getallen

Ppt 1stelj Getallen

8 points on the unit circle the wrapping function w(t)

8 points on the unit circle the wrapping function w(t)

Solving ode problem using the Galerkin's method

Solving ode problem using the Galerkin's method

Maths Y4 Week 1 days 1, 2, 3, 4 and 5

Maths Y4 Week 1 days 1, 2, 3, 4 and 5

คณิตศาสตร์ 60 เฟรม กาญจนรัตน์

คณิตศาสตร์ 60 เฟรม กาญจนรัตน์

Convolutional neural neworks

The document introduces convolutional neural networks and how they are used for image recognition through a series of examples using simple arithmetic operations on matrices to represent images and applying filters. It explains how convolutional neural networks use convolution layers to apply filters to images to extract features, pooling layers to downsample images, and fully connected layers to classify images. The networks are trained on labeled image data using gradient descent to minimize errors and improve the ability of the network to accurately classify new images.

Linear regression

This document provides an overview of linear regression. It begins by showing an example of housing prices based on number of rooms. It then explains how to move a linear regression line closer to data points by changing the slope and y-intercept. An algorithm for linear regression is presented using gradient descent to iteratively minimize the distance between the line and points. Finally, it discusses using an absolute value approach instead of squares to handle different types of errors.

Support vector machines (SVM)

1. The document introduces support vector machines (SVM) and provides a friendly introduction through a series of videos.
2. It explains the SVM algorithm which starts with a line and two parallel lines, picks a learning rate and number of repetitions, then moves the lines to correctly classify points while keeping the margin between the lines as large as possible.
3. Different error functions for SVM are discussed, including classification error, margin error, and focusing on minimizing their sum. The C parameter allows balancing focusing on margin versus classification.

Logistic regression

The document provides an introduction and overview of logistic regression and the perceptron algorithm. It explains what logistic regression and the perceptron algorithm are, provides an example of using them to classify email as spam or ham, and describes the algorithms in detail. It discusses how the perceptron algorithm works to iteratively adjust the separating line to better classify points by moving the line towards misclassified points. It also introduces the concept of using gradient descent to minimize the log-loss error function in logistic regression classification.

Restricted Boltzmann Machines (RBM)

The document discusses Restricted Boltzmann Machines (RBM), including:
1) RBMs use hidden and visible layers with weights to model joint probabilities between inputs and outputs.
2) Training an RBM involves using contrastive divergence to adjust the weights to maximize the probability of training data by running Gibbs sampling.
3) Exact computation of probabilities is intractable, so approximate methods like Gibbs sampling are used to sample from the distribution.

Generative Adversarial Networks (GANs)

This document provides a friendly introduction to generative adversarial networks (GANs). It explains the general idea of GANs which involve a discriminator and generator playing a game, with the goal of the generator being to generate fake images that cannot be distinguished from real images by the discriminator. The document then walks through building the simplest GAN with a 1-layer neural network discriminator and generator. It explains how to train the GAN by having the discriminator and generator update through backpropagation to minimize their loss functions. Code examples are provided to demonstrate how to implement the GAN.

Bayes theorem and Naive Bayes algorithm

The document discusses the Naive Bayes classifier algorithm for spam detection. It shows how the algorithm calculates the probability that an email is spam based on the presence of words like "buy" and "cheap". It begins with small amounts of sample data that indicate high spam probabilities for individual words. However, it then collects more representative sample data that shows the words actually occur independently, leading to a lower calculated spam probability when both words are present. The document is an example walkthrough of how Naive Bayes modeling works for spam filtering.

PCA (Principal Component Analysis)

Principal Component Analysis (PCA) is a technique used to reduce the dimensionality of data by transforming correlated variables into a smaller number of uncorrelated variables called principal components. The document discusses PCA concepts like projections, dimensionality reduction, and applications to housing data. It explains how PCA finds the directions of maximum variance in high-dimensional data and projects it onto a new coordinate system.

Matrix factorization

How does Netflix recommend movies? In this presentation we go over a very common technique for recommendations called matrix factorization to predict what rating a user will give a movie. It sounds like a complicated mathematical concept, but all it consists of is finding a set of intermediate features such as action, comedy, etc., and using them to help us determine the ratings.

Convolutional neural neworks

Convolutional neural neworks

Linear regression

Linear regression

Support vector machines (SVM)

Support vector machines (SVM)

Logistic regression

Logistic regression

Restricted Boltzmann Machines (RBM)

Restricted Boltzmann Machines (RBM)

Generative Adversarial Networks (GANs)

Generative Adversarial Networks (GANs)

Bayes theorem and Naive Bayes algorithm

Bayes theorem and Naive Bayes algorithm

PCA (Principal Component Analysis)

PCA (Principal Component Analysis)

Matrix factorization

Matrix factorization

BREEDING METHODS FOR DISEASE RESISTANCE.pptx

Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes

Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...

Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor

Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/Nucleophilic Addition of carbonyl compounds.pptx

Nucleophilic addition is the most important reaction of carbonyls. Not just aldehydes and ketones, but also carboxylic acid derivatives in general.
Carbonyls undergo addition reactions with a large range of nucleophiles.
Comparing the relative basicity of the nucleophile and the product is extremely helpful in determining how reversible the addition reaction is. Reactions with Grignards and hydrides are irreversible. Reactions with weak bases like halides and carboxylates generally don’t happen.
Electronic effects (inductive effects, electron donation) have a large impact on reactivity.
Large groups adjacent to the carbonyl will slow the rate of reaction.
Neutral nucleophiles can also add to carbonyls, although their additions are generally slower and more reversible. Acid catalysis is sometimes employed to increase the rate of addition.

The binding of cosmological structures by massless topological defects

Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.

Eukaryotic Transcription Presentation.pptx

ukaryotic Transcription Presentation and RNA Precessing

Shallowest Oil Discovery of Turkiye.pptx

The Petroleum System of the Çukurova Field - the Shallowest Oil Discovery of Türkiye, Adana

20240520 Planning a Circuit Simulator in JavaScript.pptx

Evaporation step counter work. I have done a physical experiment.
(Work in progress.)

Topic: SICKLE CELL DISEASE IN CHILDREN-3.pdf

Sickle cell in children

Bob Reedy - Nitrate in Texas Groundwater.pdf

Presented at June 6-7 Texas Alliance of Groundwater Districts Business Meeting

3D Hybrid PIC simulation of the plasma expansion (ISSS-14)

3D Particle-In-Cell (PIC) algorithm,
Plasma expansion in the dipole magnetic field.

What is greenhouse gasses and how many gasses are there to affect the Earth.

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.

EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...

Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.

The debris of the ‘last major merger’ is dynamically young

The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.

Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...

Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.

The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptx

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).

ESR spectroscopy in liquid food and beverages.pptx

With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.

Leaf Initiation, Growth and Differentiation.pdf

Leaf initiation, growth and differentiation, genetic control of leaf development.

原版制作(carleton毕业证书)卡尔顿大学毕业证硕士文凭原版一模一样

原版纸张【微信：741003700 】【(carleton毕业证书)卡尔顿大学毕业证】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
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3:国家专业人才认证中心颁发入库证书
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5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
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ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptx

Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.

BREEDING METHODS FOR DISEASE RESISTANCE.pptx

BREEDING METHODS FOR DISEASE RESISTANCE.pptx

Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...

Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...

Nucleophilic Addition of carbonyl compounds.pptx

Nucleophilic Addition of carbonyl compounds.pptx

The binding of cosmological structures by massless topological defects

The binding of cosmological structures by massless topological defects

Eukaryotic Transcription Presentation.pptx

Eukaryotic Transcription Presentation.pptx

Shallowest Oil Discovery of Turkiye.pptx

Shallowest Oil Discovery of Turkiye.pptx

20240520 Planning a Circuit Simulator in JavaScript.pptx

20240520 Planning a Circuit Simulator in JavaScript.pptx

Topic: SICKLE CELL DISEASE IN CHILDREN-3.pdf

Topic: SICKLE CELL DISEASE IN CHILDREN-3.pdf

Bob Reedy - Nitrate in Texas Groundwater.pdf

Bob Reedy - Nitrate in Texas Groundwater.pdf

3D Hybrid PIC simulation of the plasma expansion (ISSS-14)

3D Hybrid PIC simulation of the plasma expansion (ISSS-14)

What is greenhouse gasses and how many gasses are there to affect the Earth.

What is greenhouse gasses and how many gasses are there to affect the Earth.

EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...

EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...

The debris of the ‘last major merger’ is dynamically young

The debris of the ‘last major merger’ is dynamically young

Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...

Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...

The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptx

The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptx

ESR spectroscopy in liquid food and beverages.pptx

ESR spectroscopy in liquid food and beverages.pptx

Leaf Initiation, Growth and Differentiation.pdf

Leaf Initiation, Growth and Differentiation.pdf

Chapter 12 - climate change and the energy crisis

Chapter 12 - climate change and the energy crisis

原版制作(carleton毕业证书)卡尔顿大学毕业证硕士文凭原版一模一样

原版制作(carleton毕业证书)卡尔顿大学毕业证硕士文凭原版一模一样

ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptx

ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptx

- 1. Singular Value Decomposition Luis Serrano
- 5. https://www.manning.com/books/grokking-machine-learning Discount code: serranoyt Grokking Machine Learning By Luis G. Serrano
- 7. Transformations Stretch (or compress) horizontally
- 8. Transformations Stretch (or compress) horizontally
- 9. Transformations Stretch (or compress) horizontally
- 10. Transformations Stretch (or compress) horizontally
- 11. Transformations Stretch (or compress) vertically
- 12. Transformations Stretch (or compress) vertically
- 13. Transformations Stretch (or compress) vertically
- 14. Transformations Stretch (or compress) vertically
- 16. Puzzle (easy)
- 17. Puzzle (easy)
- 18. Puzzle (easy)
- 19. Puzzle (easy)
- 20. Puzzle (easy)
- 21. Puzzle (hard)
- 22. Puzzle (hard)
- 23. Puzzle (hard)
- 24. Puzzle (hard)
- 25. Puzzle (hard)
- 26. Solution
- 27. Solution
- 28. Solution
- 29. Solution
- 30. Solution
- 31. Solution
- 32. Solution
- 34. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 What does this have to do with matrices? (p,q) (3p+0q, 4p+5q) (1,0) (3, 4) (0,1) (0, 5) (-1,0) (0,-1) (-3, -4) (0, -5) 3 0 4 5[ ]A =
- 35. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Rotation matrices cos(θ) −sin(θ) sin(θ) cos(θ)[ ] θ
- 36. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Stretching matrices σ1 0 0 σ2 [ ] σ1 σ2
- 37. Stretching matrices σ1 0 0 σ2 [ ] -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 σ1 σ2
- 38. Stretching matrices σ1 0 0 σ2 [ ] -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 σ1 σ2
- 39. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 What does this have to do with matrices? 3 0 4 5[ ]A = cos(θ) sin(θ) −sin(θ) cos(θ)[ ] cos(ϕ) sin(ϕ) −sin(ϕ) cos(ϕ)[ ] σ1 0 0 σ2 [ ]
- 40. Singular value decomposition 3 0 4 5[ ] cos(θ) sin(θ) −sin(θ) cos(θ)[ ] cos(ϕ) sin(ϕ) −sin(ϕ) cos(ϕ)[ ] σ1 0 0 σ2 [ ]= A = UΣV†
- 41. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 SVD A = UΣV† 3 5 0 0 5[ ] 1/ 10 −3/ 10 3/ 10 1/ 10[ ] 0.7071 0.7071 −0.7071 0.7071[ ] 6.708 0 0 2.236[ ] 0.316 −0.949 0.949 0.316[ ] 1/ 2 1/ 2 −1/ 2 1/ 2[ ] Rotation of θ = − π 4 = − 45o 3 0 4 5[ ] =
- 42. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Rotation 1/ 2 1/ 2 −1/ 2 1/ 2[ ] 3 5 0 0 5[ ] 1/ 10 −3/ 10 3/ 10 1/ 10[ ] 0.7071 0.7071 −0.7071 0.7071[ ] 6.708 0 0 2.236[ ] 0.316 −0.949 0.949 0.316[ ] A = UΣV† Rotation of θ = − π 4 = − 45o 3 0 4 5[ ] =
- 43. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Rotation 1/ 2 1/ 2 −1/ 2 1/ 2[ ] 3 5 0 0 5[ ] 1/ 10 −3/ 10 3/ 10 1/ 10[ ] 0.7071 0.7071 −0.7071 0.7071[ ] 6.708 0 0 2.236[ ] 0.316 −0.949 0.949 0.316[ ] A = UΣV† Horizontal scaling by 3 5 3 0 4 5[ ] =
- 44. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Scaling 1/ 2 1/ 2 −1/ 2 1/ 2[ ] 3 5 0 0 5[ ] 1/ 10 −3/ 10 3/ 10 1/ 10[ ] 0.7071 0.7071 −0.7071 0.7071[ ] 6.708 0 0 2.236[ ] 0.316 −0.949 0.949 0.316[ ] A = UΣV† Horizontal scaling by 3 5 Vertical scaling by 5 3 0 4 5[ ] =
- 45. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Rotation 1/ 2 1/ 2 −1/ 2 1/ 2[ ] 3 5 0 0 5[ ] 1/ 10 −3/ 10 3/ 10 1/ 10[ ] 0.7071 0.7071 −0.7071 0.7071[ ] 6.708 0 0 2.236[ ] 0.316 −0.949 0.949 0.316[ ] A = UΣV† Vertical scaling by 5 3 0 4 5[ ] =
- 46. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Rotation 1/ 2 1/ 2 −1/ 2 1/ 2[ ] 3 5 0 0 5[ ] 1/ 10 −3/ 10 3/ 10 1/ 10[ ] 0.7071 0.7071 −0.7071 0.7071[ ] 6.708 0 0 2.236[ ] 0.316 −0.949 0.949 0.316[ ] A = UΣV† Rotation of θ = arctan(3) = 71.72o 3 0 4 5[ ] =
- 47. -7 -5 -3 -1 1 3 5 7 -7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 Rotation 1/ 2 1/ 2 −1/ 2 1/ 2[ ] 3 5 0 0 5[ ] 1/ 10 −3/ 10 3/ 10 1/ 10[ ] 0.7071 0.7071 −0.7071 0.7071[ ] 6.708 0 0 2.236[ ] 0.316 −0.949 0.949 0.316[ ] A = UΣV† Rotation of θ = arctan(3) = 71.72o 3 0 4 5[ ] =
- 48. -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 A = UΣV† A Σ V† U
- 50. 1 2 3 4 -1 -2 -3 -4 2 4 6 8 10 20 30 40 1 2 3 4 1 -1 2 10 Difference between these two matrices?
- 51. 1 2 3 4 -1 -2 -3 -4 2 4 6 8 10 20 30 40 1 2 3 4 1 -1 2 10 Difference between these two matrices? 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 ? ? ? ? ? ? ? ?
- 52. = 1 2 3 4 -1 -2 -3 -4 2 4 6 8 10 20 30 40 1 2 3 4 1 -1 2 10 Rank 1 matrices 16 numbers 8 numbers
- 53. = 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 ? ? ? ? ? ? ? ? Higher rank matrices 16 numbers
- 54. Rank of a matrix = Rank 1 = Rank 2 = Rank 3 = Rank 4
- 55. ∼ 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 ? ? ? ? ? ? ? ? Approximation by a rank one matrix
- 56. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4σ4 = U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 0 1 2 3 4 0 1 2 3 4 A
- 57. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 + = 0 1 2 3 4 0 1 2 3 4 A
- 58. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 + = 0 1 2 3 4 0 1 2 3 4 A
- 59. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 + = + 0 1 2 3 4 0 1 2 3 4 A
- 60. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4V1 V2 V3 V4σ1 σ2 σ3 σ4+ = + + Rank 1 Rank 1 Rank 1 Rank 1 TinySmallMediumLarge 0 1 2 3 4 0 1 2 3 4 A
- 61. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2V1 V2σ1 σ2+ = Rank 1 Rank 1 MediumLarge 0 1 2 3 4 0 1 2 3 4 A
- 62. 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 0.15+ Tiny 0.79 -0.29 -0.54 -0.04 -0.89 -0.23 0.15 0.36 0 1 2 3 4 0 1 2 3 4 21.2 6.4 4.9 0.15 = -0.55 -0.52 -0.49 -0.43 0.26 -0.4 0.65 -0.59 0.07 0.7 -0.22 -0.68 0.79 -0.29 -0.54 -0.04 -0.21 0.37 -0.13 -0.89 -0.52 -0.7 0.43 -0.23 -0.48 -0.21 -0.84 0.15 -0.67 0.57 0.31 0.36 4.9+ 0.07 0.7 -0.22 -0.68 -0.13 0.43 -0.84 0.31 Small 6.4+ 0.26 -0.4 0.65 -0.59 0.37 -0.7 -0.21 0.57 Medium 21.2 -0.55 -0.52 -0.49 -0.43 -0.21 -0.52 -0.48 -0.67 Large 2.51 2.37 2.22 1.97 6.07 5,72 5.37 4.77 5.63 5,31 4.99 4.43 7.88 7.43 6.98 6.19 3.15 1.41 3.79 0.56 4.87 7.53 2.44 7.43 5.28 5.85 4.12 5.22 8.85 5.96 9.36 4.03 3.1 0.96 3.93 0.99 5.03 8.99 1.98 6 4.98 3.01 5.01 8 8.96 7.02 9.03 3 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3
- 63. Rank of a matrix = Rank 1 = Rank 2 = Rank 3 = Rank 4
- 65. -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 1.8 1.2 4.4 4.6[ ] A -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 A = UΣV† 0.316 −0.949 0.949 0.316[ ]U 6.71 0 0 0.44[ ]Σ 0.7071 0.7071 −0.7071 0.7071[ ] V† 6.71 0.44
- 66. -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 6.71 0 0 0.44[ ] 1.8 1.2 4.4 4.6[ ] A = UΣV† A Σ -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 6.71 0.44 0 0 0.7071 0.7071 −0.7071 0.7071[ ] V† 0.316 −0.949 0.949 0.316[ ]U
- 67. -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 1.8 1.2 4.4 4.6[ ] A = UΣV† A Σ -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 6.71 [ 6.71 0 0 0.44 ] 0.44 0 0 0.316 −0.949 0.949 0.316[ ]U 0.7071 0.7071 −0.7071 0.7071[ ] V†
- 68. -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 1.8 1.2 4.4 4.6[ ] A = UΣV† A Σ -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 6.71 [ 6.71 0 0 0.44 ] 0.44 0 0 1.5 1.5 4.5 4.5[ ] 0.316 −0.949 0.949 0.316[ ]U 0.7071 0.7071 −0.7071 0.7071[ ] V†
- 69. -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 1.5 1.5 4.5 4.5[ ] Rank 1
- 70. -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 1.8 1.2 4.4 4.6[ ] Rank 2 -7 -5 -3 -1 1 3 5 7 -7 -5 -3 -1 1 3 5 7 1.5 1.5 4.5 4.5[ ] Rank 1
- 71. 1 2 3 4 -1 -2 -3 -4 2 4 6 8 10 20 30 40 1 2 3 4 1 -1 2 10 Rank 1 matrices
- 72. = 1 2 3 4 -1 -2 -3 -4 2 4 6 8 10 20 30 40 1 2 3 4 1 -1 2 10 Rank 1 matrices 16 numbers 8 numbers
- 73. = 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 ? ? ? ? ? ? ? ? Higher rank matrices 16 numbers
- 74. Approximation by rank one matrices = + + … 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3
- 75. U 0 1 2 3 4 0 1 2 3 4 V† Σ= U1 U2 U3 U4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 0 1 2 3 4 0 1 2 3 4 A
- 76. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4σ4 = U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 0 1 2 3 4 0 1 2 3 4 A
- 77. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 + = 0 1 2 3 4 0 1 2 3 4 A
- 78. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 + = 0 1 2 3 4 0 1 2 3 4 A
- 79. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4 V1 V2 V3 V4 σ1 σ2 σ3 σ4 + = + 0 1 2 3 4 0 1 2 3 4 A
- 80. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2 U3 U4V1 V2 V3 V4σ1 σ2 σ3 σ4+ = + + Rank 1 Rank 1 Rank 1 Rank 1 TinySmallMediumLarge 0 1 2 3 4 0 1 2 3 4 A
- 81. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4 0 1 2 3 4 0 1 2 3 4 V1 V2 V3 V4 0 1 2 3 4 0 1 2 3 4 σ1 σ2 σ3 σ4 U1 U2V1 V2σ1 σ2+ = Rank 1 Rank 1 MediumLarge 0 1 2 3 4 0 1 2 3 4 A
- 83. 0 1 2 3 4 0 1 2 3 4 U1 U2 U3 U4= 0 1 2 3 4 0 1 2 3 4 5 6 A V1 V2 V3 V4 V5 V6 σ1 σ2 σ3 σ4 0 0 0 0 0 0 0 0 No square matrix? No problem!
- 85. 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 Rank 4 www.github.com/luisguiserrano/singular_value_decomposition
- 86. Thank you!
- 87. Similar videos on dimensionality reduction Matrix Factorization Principal Component Analysis
- 88. https://www.manning.com/books/grokking-machine-learning Discount code: serranoyt Grokking Machine Learning By Luis G. Serrano
- 89. Thank you! @luis_likes_math Subscribe, like, share, comment! youtube.com/c/LuisSerrano http://serrano.academy