This document discusses K-means clustering, Expectation Maximization (EM), and Gaussian mixture models (GMM). It begins with an overview of unsupervised learning and introduces K-means as a simple clustering algorithm. It then describes EM as a general algorithm for maximum likelihood estimation that can be applied to problems like GMM. GMM is presented as a density estimation technique that models data using a weighted sum of Gaussian distributions. EM is described as a method for estimating the parameters of a GMM from data.
Lecture 18: Gaussian Mixture Models and Expectation Maximizationbutest
This document discusses Gaussian mixture models (GMMs) and the expectation-maximization (EM) algorithm. GMMs model data as coming from a mixture of Gaussian distributions, with each data point assigned soft responsibilities to the different components. EM is used to estimate the parameters of GMMs and other latent variable models. It iterates between an E-step, where responsibilities are computed based on current parameters, and an M-step, where new parameters are estimated to maximize the expected complete-data log-likelihood given the responsibilities. EM converges to a local optimum for fitting GMMs to data.
The document discusses fuzzy sets and fuzzy relations. It defines a fuzzy set as a membership function mapping elements to degrees of membership between 0 and 1. A fuzzy relation is defined as a membership function mapping ordered pairs of elements to degrees of membership. Fuzzy relations can represent concepts like closeness or dependence between elements. The max-min composition is introduced as a way to combine multiple fuzzy relations. Examples are provided to demonstrate fuzzy sets, relations, and their composition.
This document provides an overview of Independent Component Analysis (ICA). It discusses:
1. ICA is a technique that decomposes linear mixtures of signals into their underlying independent components.
2. The ICA model assumes observed data is generated by some independent source signals that are linearly combined using a mixing matrix. ICA estimates the inverse of this mixing matrix to retrieve the original source signals.
3. For ICA to work, the source signals must be independent and non-Gaussian. The document demonstrates ICA on some toy signals and verifies the recovered components match this criteria.
In machine learning, support vector machines (SVMs, also support vector networks[1]) are supervised learning models with associated learning algorithms that analyze data and recognize patterns, used for classification and regression analysis. The basic SVM takes a set of input data and predicts, for each given input, which of two possible classes forms the output, making it a non-probabilistic binary linear classifier.
Machine Learning With Logistic RegressionKnoldus Inc.
Machine learning is the subfield of computer science that gives computers the ability to learn without being programmed. Logistic Regression is a type of classification algorithm, based on linear regression to evaluate output and to minimize the error.
A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data (supervised learning), the algorithm outputs an optimal hyperplane which categorizes new examples. In two dimentional space this hyperplane is a line dividing a plane in two parts where in each class lay in either side.
A brief presentation given on the basics of Ensemble Methods. Given as a 'Lightning Talk' during the 7th Cohort of General Assembly's Data Science Immersive Course
Lecture 4 Decision Trees (2): Entropy, Information Gain, Gain RatioMarina Santini
attribute selection, constructing decision trees, decision trees, divide and conquer, entropy, gain ratio, information gain, machine leaning, pruning, rules, suprisal
Lecture 18: Gaussian Mixture Models and Expectation Maximizationbutest
This document discusses Gaussian mixture models (GMMs) and the expectation-maximization (EM) algorithm. GMMs model data as coming from a mixture of Gaussian distributions, with each data point assigned soft responsibilities to the different components. EM is used to estimate the parameters of GMMs and other latent variable models. It iterates between an E-step, where responsibilities are computed based on current parameters, and an M-step, where new parameters are estimated to maximize the expected complete-data log-likelihood given the responsibilities. EM converges to a local optimum for fitting GMMs to data.
The document discusses fuzzy sets and fuzzy relations. It defines a fuzzy set as a membership function mapping elements to degrees of membership between 0 and 1. A fuzzy relation is defined as a membership function mapping ordered pairs of elements to degrees of membership. Fuzzy relations can represent concepts like closeness or dependence between elements. The max-min composition is introduced as a way to combine multiple fuzzy relations. Examples are provided to demonstrate fuzzy sets, relations, and their composition.
This document provides an overview of Independent Component Analysis (ICA). It discusses:
1. ICA is a technique that decomposes linear mixtures of signals into their underlying independent components.
2. The ICA model assumes observed data is generated by some independent source signals that are linearly combined using a mixing matrix. ICA estimates the inverse of this mixing matrix to retrieve the original source signals.
3. For ICA to work, the source signals must be independent and non-Gaussian. The document demonstrates ICA on some toy signals and verifies the recovered components match this criteria.
In machine learning, support vector machines (SVMs, also support vector networks[1]) are supervised learning models with associated learning algorithms that analyze data and recognize patterns, used for classification and regression analysis. The basic SVM takes a set of input data and predicts, for each given input, which of two possible classes forms the output, making it a non-probabilistic binary linear classifier.
Machine Learning With Logistic RegressionKnoldus Inc.
Machine learning is the subfield of computer science that gives computers the ability to learn without being programmed. Logistic Regression is a type of classification algorithm, based on linear regression to evaluate output and to minimize the error.
A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data (supervised learning), the algorithm outputs an optimal hyperplane which categorizes new examples. In two dimentional space this hyperplane is a line dividing a plane in two parts where in each class lay in either side.
A brief presentation given on the basics of Ensemble Methods. Given as a 'Lightning Talk' during the 7th Cohort of General Assembly's Data Science Immersive Course
Lecture 4 Decision Trees (2): Entropy, Information Gain, Gain RatioMarina Santini
attribute selection, constructing decision trees, decision trees, divide and conquer, entropy, gain ratio, information gain, machine leaning, pruning, rules, suprisal
This document discusses unsupervised learning and clustering. It defines unsupervised learning as modeling the underlying structure or distribution of input data without corresponding output variables. Clustering is described as organizing unlabeled data into groups of similar items called clusters. The document focuses on k-means clustering, describing it as a method that partitions data into k clusters by minimizing distances between points and cluster centers. It provides details on the k-means algorithm and gives examples of its steps. Strengths and weaknesses of k-means clustering are also summarized.
1. This document outlines an introduction to machine learning lecture by Dr. Varun Kumar. It discusses examples of machine learning, attributes in machine learning applications, and examples such as classification, regression, supervised vs unsupervised learning.
2. Machine learning can analyze large amounts of data from sciences, the world wide web, and adapt to changes without needing every situation predefined. It involves programming computers to optimize performance using example data.
3. Attributes in machine learning applications map the input to the output through mathematical functions. Examples given include factors that influence data transmission rates in wireless communication.
This document discusses neural networks and their biological and technical underpinnings. It covers how natural neural networks operate using electrochemical signals and thresholds. It also discusses early artificial neural network models like McCulloch-Pitts networks and perceptrons. Perceptrons are defined as single-layer feedforward networks and can only represent linearly separable functions. The document introduces the concept of adding hidden layers to networks to increase their computational power and ability to represent more complex functions like XOR.
The document provides an overview of perceptrons and neural networks. It discusses how neural networks are modeled after the human brain and consist of interconnected artificial neurons. The key aspects covered include the McCulloch-Pitts neuron model, Rosenblatt's perceptron, different types of learning (supervised, unsupervised, reinforcement), the backpropagation algorithm, and applications of neural networks such as pattern recognition and machine translation.
This document discusses machine learning and artificial intelligence. It defines machine learning as a branch of AI that allows systems to learn from data and experience. Machine learning is important because some tasks are difficult to define with rules but can be learned from examples, and relationships in large datasets can be uncovered. The document then discusses areas where machine learning is influential like statistics, brain modeling, and more. It provides an example of designing a machine learning system to play checkers. Finally, it discusses machine learning algorithm types and provides details on the AdaBoost algorithm.
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.
This document provides an overview of decision trees, including definitions, key terms, algorithms, and advantages/limitations. It defines a decision tree as a model that classifies instances by sorting them from the root to a leaf node. Important terms are defined like root node, branches, and leaf nodes. Popular algorithms like CART and C5.0 are described. Advantages are that decision trees are fast, robust, and require little experimentation. Limitations include class imbalance and overfitting with too many records and few attributes.
Complete Presentation on Mycin - An Expert System. ,mycin - an expert system ,mycin ,mycin expert system ,mycin system ,mycin expert ,expert system mycin ,mycin presentation ,how mycin work ,mycin architecture ,components of mycin ,tasks of mycin ,how mycin became successful ,is mycin used today? ,user interface of mycin
This document compares and contrasts boosting with other ensemble methods such as bagging and random forests. It discusses two specific boosting algorithms - AdaBoost, which fits models on weighted labels, and gradient boosting, which fits models on residuals from previous models. Both aim to produce low bias, low variance predictions by building models sequentially. The document provides pseudocode for AdaBoost classification and gradient boosting regression, and explains how boosting methods work to improve upon previous predictions at each step of the ensemble.
Supervised machine learning uses labeled training data to build models that can predict outputs. There are two main types: regression predicts continuous variables, while classification predicts categorical variables. Supervised learning algorithms include linear regression, which finds a linear relationship between variables, and logistic regression or decision trees for classification. The process involves collecting labeled data, training an algorithm on part of the data, and evaluating its accuracy on test data.
The document discusses procedural versus declarative knowledge representation and how logic programming languages like Prolog allow knowledge to be represented declaratively through logical rules. It also covers topics like forward and backward reasoning, matching rules to facts in working memory, and using control knowledge to guide the problem solving process. Logic programming represents knowledge through Horn clauses and uses backward chaining inference to attempt to prove goals.
Defuzzification is the process of producing a quantifiable result in Crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems.
Dynamic programming is used to solve optimization problems by combining solutions to overlapping subproblems. It works by breaking down problems into subproblems, solving each subproblem only once, and storing the solutions in a table to avoid recomputing them. There are two key properties for applying dynamic programming: overlapping subproblems and optimal substructure. Some applications of dynamic programming include finding shortest paths, matrix chain multiplication, the traveling salesperson problem, and knapsack problems.
Probability for Machine Learning
Here is a scant introduction to an important subject in Machine Learning. However, we are looking to work with our Probability Professor, Ofelia Begovich to write a series of notes in basic probability for something else, to improve this introduction. Nevertheless, there are still several things like:
1.- Linear Algebra
2.- Topology
3.- Mathematical Analysis
4.- Optimization
That need to be addressed, thus I am working in a class for intelligent systems for that.
This document summarizes a machine learning workshop on feature selection. It discusses typical feature selection methods like single feature evaluation using metrics like mutual information and Gini indexing. It also covers subset selection techniques like sequential forward selection and sequential backward selection. Examples are provided showing how feature selection improves performance for logistic regression on large datasets with more features than samples. The document outlines the workshop agenda and provides details on when and why feature selection is important for machine learning models.
This document discusses unsupervised learning techniques for clustering data, specifically K-Means clustering and Gaussian Mixture Models. It explains that K-Means clustering groups data by assigning each point to the nearest cluster center and iteratively updating the cluster centers. Gaussian Mixture Models assume the data is generated from a mixture of Gaussian distributions and uses the Expectation-Maximization algorithm to estimate the parameters of the mixture components.
This document provides an overview of dimensionality reduction techniques. It discusses how increasing dimensionality can negatively impact classification accuracy due to the curse of dimensionality. Dimensionality reduction aims to select an optimal set of features of lower dimensionality to improve accuracy. Feature extraction and feature selection are two common approaches. Principal component analysis (PCA) is described as a popular linear feature extraction method that projects data to a lower dimensional space while preserving as much variance as possible.
This document discusses unsupervised learning and clustering. It defines unsupervised learning as modeling the underlying structure or distribution of input data without corresponding output variables. Clustering is described as organizing unlabeled data into groups of similar items called clusters. The document focuses on k-means clustering, describing it as a method that partitions data into k clusters by minimizing distances between points and cluster centers. It provides details on the k-means algorithm and gives examples of its steps. Strengths and weaknesses of k-means clustering are also summarized.
1. This document outlines an introduction to machine learning lecture by Dr. Varun Kumar. It discusses examples of machine learning, attributes in machine learning applications, and examples such as classification, regression, supervised vs unsupervised learning.
2. Machine learning can analyze large amounts of data from sciences, the world wide web, and adapt to changes without needing every situation predefined. It involves programming computers to optimize performance using example data.
3. Attributes in machine learning applications map the input to the output through mathematical functions. Examples given include factors that influence data transmission rates in wireless communication.
This document discusses neural networks and their biological and technical underpinnings. It covers how natural neural networks operate using electrochemical signals and thresholds. It also discusses early artificial neural network models like McCulloch-Pitts networks and perceptrons. Perceptrons are defined as single-layer feedforward networks and can only represent linearly separable functions. The document introduces the concept of adding hidden layers to networks to increase their computational power and ability to represent more complex functions like XOR.
The document provides an overview of perceptrons and neural networks. It discusses how neural networks are modeled after the human brain and consist of interconnected artificial neurons. The key aspects covered include the McCulloch-Pitts neuron model, Rosenblatt's perceptron, different types of learning (supervised, unsupervised, reinforcement), the backpropagation algorithm, and applications of neural networks such as pattern recognition and machine translation.
This document discusses machine learning and artificial intelligence. It defines machine learning as a branch of AI that allows systems to learn from data and experience. Machine learning is important because some tasks are difficult to define with rules but can be learned from examples, and relationships in large datasets can be uncovered. The document then discusses areas where machine learning is influential like statistics, brain modeling, and more. It provides an example of designing a machine learning system to play checkers. Finally, it discusses machine learning algorithm types and provides details on the AdaBoost algorithm.
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.
This document provides an overview of decision trees, including definitions, key terms, algorithms, and advantages/limitations. It defines a decision tree as a model that classifies instances by sorting them from the root to a leaf node. Important terms are defined like root node, branches, and leaf nodes. Popular algorithms like CART and C5.0 are described. Advantages are that decision trees are fast, robust, and require little experimentation. Limitations include class imbalance and overfitting with too many records and few attributes.
Complete Presentation on Mycin - An Expert System. ,mycin - an expert system ,mycin ,mycin expert system ,mycin system ,mycin expert ,expert system mycin ,mycin presentation ,how mycin work ,mycin architecture ,components of mycin ,tasks of mycin ,how mycin became successful ,is mycin used today? ,user interface of mycin
This document compares and contrasts boosting with other ensemble methods such as bagging and random forests. It discusses two specific boosting algorithms - AdaBoost, which fits models on weighted labels, and gradient boosting, which fits models on residuals from previous models. Both aim to produce low bias, low variance predictions by building models sequentially. The document provides pseudocode for AdaBoost classification and gradient boosting regression, and explains how boosting methods work to improve upon previous predictions at each step of the ensemble.
Supervised machine learning uses labeled training data to build models that can predict outputs. There are two main types: regression predicts continuous variables, while classification predicts categorical variables. Supervised learning algorithms include linear regression, which finds a linear relationship between variables, and logistic regression or decision trees for classification. The process involves collecting labeled data, training an algorithm on part of the data, and evaluating its accuracy on test data.
The document discusses procedural versus declarative knowledge representation and how logic programming languages like Prolog allow knowledge to be represented declaratively through logical rules. It also covers topics like forward and backward reasoning, matching rules to facts in working memory, and using control knowledge to guide the problem solving process. Logic programming represents knowledge through Horn clauses and uses backward chaining inference to attempt to prove goals.
Defuzzification is the process of producing a quantifiable result in Crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems.
Dynamic programming is used to solve optimization problems by combining solutions to overlapping subproblems. It works by breaking down problems into subproblems, solving each subproblem only once, and storing the solutions in a table to avoid recomputing them. There are two key properties for applying dynamic programming: overlapping subproblems and optimal substructure. Some applications of dynamic programming include finding shortest paths, matrix chain multiplication, the traveling salesperson problem, and knapsack problems.
Probability for Machine Learning
Here is a scant introduction to an important subject in Machine Learning. However, we are looking to work with our Probability Professor, Ofelia Begovich to write a series of notes in basic probability for something else, to improve this introduction. Nevertheless, there are still several things like:
1.- Linear Algebra
2.- Topology
3.- Mathematical Analysis
4.- Optimization
That need to be addressed, thus I am working in a class for intelligent systems for that.
This document summarizes a machine learning workshop on feature selection. It discusses typical feature selection methods like single feature evaluation using metrics like mutual information and Gini indexing. It also covers subset selection techniques like sequential forward selection and sequential backward selection. Examples are provided showing how feature selection improves performance for logistic regression on large datasets with more features than samples. The document outlines the workshop agenda and provides details on when and why feature selection is important for machine learning models.
This document discusses unsupervised learning techniques for clustering data, specifically K-Means clustering and Gaussian Mixture Models. It explains that K-Means clustering groups data by assigning each point to the nearest cluster center and iteratively updating the cluster centers. Gaussian Mixture Models assume the data is generated from a mixture of Gaussian distributions and uses the Expectation-Maximization algorithm to estimate the parameters of the mixture components.
This document provides an overview of dimensionality reduction techniques. It discusses how increasing dimensionality can negatively impact classification accuracy due to the curse of dimensionality. Dimensionality reduction aims to select an optimal set of features of lower dimensionality to improve accuracy. Feature extraction and feature selection are two common approaches. Principal component analysis (PCA) is described as a popular linear feature extraction method that projects data to a lower dimensional space while preserving as much variance as possible.
Anomaly detection using deep one class classifier홍배 김
The document discusses anomaly detection techniques using deep one-class classifiers and generative adversarial networks (GANs). It proposes using an autoencoder to extract features from normal images, training a GAN on those features to model the distribution, and using a one-class support vector machine (SVM) to determine if new images are within the normal distribution. The method detects and localizes anomalies by generating a binary mask for abnormal regions. It also discusses Gaussian mixture models and the expectation-maximization algorithm for modeling multiple distributions in data.
Data science involves extracting insights from large volumes of data. It is an interdisciplinary field that uses techniques from statistics, machine learning, and other domains. The document provides examples of classification algorithms like k-nearest neighbors, naive Bayes, and perceptrons that are commonly used in data science to build models for tasks like spam filtering or sentiment analysis. It also discusses clustering, frequent pattern mining, and other machine learning concepts.
Lecture 8: Decision Trees & k-Nearest NeighborsMarina Santini
This document provides an overview of two machine learning algorithms: decision trees and k-nearest neighbors (k-NN). Decision trees use a hierarchical structure to classify instances based on the values of their features, splitting the data at each node based on tests of individual features. k-NN classifies new instances based on the majority class of its k nearest neighbors in the training data, where distance between instances is measured using a metric like overlap. The document discusses key aspects of both algorithms like decision criteria, parameters, and properties.
1. The document discusses various machine learning classification algorithms including neural networks, support vector machines, logistic regression, and radial basis function networks.
2. It provides examples of using straight lines and complex boundaries to classify data with neural networks. Maximum margin hyperplanes are used for support vector machine classification.
3. Logistic regression is described as useful for binary classification problems by using a sigmoid function and cross entropy loss. Radial basis function networks can perform nonlinear classification with a kernel trick.
- Unsupervised learning aims to find hidden patterns in unlabeled data. Expectation-maximization and k-means clustering are common unsupervised learning algorithms.
- Principal component analysis performs dimension reduction by projecting data onto dimensions that maximize variance. Independent component analysis finds underlying generating sources in data.
- This document provides an overview of various unsupervised learning techniques including expectation-maximization, k-means clustering, hierarchical clustering, principal component analysis, and independent component analysis. Formulas and algorithms for each technique are defined.
The document discusses various clustering algorithms and concepts:
1) K-means clustering groups data by minimizing distances between points and cluster centers, but it is sensitive to initialization and may find local optima.
2) K-medians clustering is similar but uses point medians instead of means as cluster representatives.
3) K-center clustering aims to minimize maximum distances between points and clusters, and can be approximated with a farthest-first traversal algorithm.
This document discusses machine learning techniques including k-means clustering, expectation maximization (EM), and Gaussian mixture models (GMM). It begins by introducing unsupervised learning problems and k-means clustering. It then describes EM as a general algorithm for maximum likelihood estimation and density estimation. Finally, it discusses using GMM with EM to model data distributions and for classification tasks.
One of the central tasks in computational mathematics and statistics is to accurately approximate unknown target functions. This is typically done with the help of data — samples of the unknown functions. The emergence of Big Data presents both opportunities and challenges. On one hand, big data introduces more information about the unknowns and, in principle, allows us to create more accurate models. On the other hand, data storage and processing become highly challenging. In this talk, we present a set of sequential algorithms for function approximation in high dimensions with large data sets. The algorithms are of iterative nature and involve only vector operations. They use one data sample at each step and can handle dynamic/stream data. We present both the numerical algorithms, which are easy to implement, as well as rigorous analysis for their theoretical foundation.
This document provides an overview of dimensionality reduction techniques, specifically principal component analysis (PCA). It begins with acknowledging dimensionality reduction aims to choose a lower-dimensional set of features to improve classification accuracy. Feature extraction and feature selection are introduced as two common dimensionality reduction methods. PCA is then explained in detail, including how it seeks a new set of basis vectors that maximizes retained variance from the original data. Key mathematical steps of PCA are outlined, such as computing the covariance matrix and its eigenvectors/eigenvalues to determine the principal components.
This document provides an overview of support vector machines (SVMs) for machine learning. It explains that SVMs find the optimal separating hyperplane that maximizes the margin between examples of separate classes. This is achieved by formulating SVM training as a convex optimization problem that can be solved efficiently. The document discusses how SVMs can handle non-linear decision boundaries using the "kernel trick" to implicitly map examples to higher-dimensional feature spaces without explicitly performing the mapping.
The document discusses clustering, mixture models, and the EM algorithm. It provides an overview of k-means clustering and Gaussian mixture models (GMM). K-means aims to partition observations into K clusters while minimizing within-cluster variance. GMM represents data as a weighted sum of Gaussian distributions. The EM algorithm is introduced for training GMM through maximum likelihood. It iteratively performs E-steps to estimate posterior distribution of latent variables, and M-steps to update model parameters, converging to a local optimum.
This document provides a summary of supervised learning techniques including linear regression, logistic regression, support vector machines, naive Bayes classification, and decision trees. It defines key concepts such as hypothesis, loss functions, cost functions, and gradient descent. It also covers generative models like Gaussian discriminant analysis, and ensemble methods such as random forests and boosting. Finally, it discusses learning theory concepts such as the VC dimension, PAC learning, and generalization error bounds.
The document discusses machine learning concepts including supervised and unsupervised learning algorithms like clustering, dimensionality reduction, and classification. It also covers parallel computing strategies for machine learning like partitioning problems across distributed memory systems.
The document discusses machine learning concepts including supervised and unsupervised learning algorithms like clustering, dimensionality reduction, and classification. It also covers parallel computing strategies for machine learning like partitioning problems across distributed memory architectures.
The document discusses machine learning concepts including supervised and unsupervised learning algorithms like clustering, dimensionality reduction, and classification. It also covers parallel computing strategies for machine learning like partitioning problems across systems.
The document discusses machine learning concepts including supervised and unsupervised learning algorithms like clustering, dimensionality reduction, and classification. It also covers parallel computing strategies for machine learning like partitioning problems across distributed memory systems.
The document discusses machine learning concepts including supervised and unsupervised learning algorithms like clustering, dimensionality reduction, and classification. It also covers parallel computing strategies for machine learning like partitioning problems across distributed memory systems.
The document discusses machine learning concepts including supervised and unsupervised learning algorithms like clustering, dimensionality reduction, and classification. It also covers parallel computing strategies for machine learning like partitioning problems across distributed memory architectures.
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as an important issue for both cloud providers and users. To
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1. cloud environment is dynamically changing; 2. cloud
environment consists of many heterogeneous physical and virtual
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This document outlines a course on mining heterogeneous information networks. The course will cover phrase mining and topic modeling from large text corpora, entity extraction and typing through relational graph construction and propagation, as well as mining and constructing heterogeneous information networks.
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This document discusses influenceability estimation in social networks. It describes the independent cascade model of influence diffusion, where each node has an independent probability of influencing its neighbors. The problem is to estimate the expected number of nodes reachable from a given seed node. The document presents the naive Monte Carlo (NMC) approach, which samples possible graphs and averages the number of reachable nodes over the samples. While NMC provides an unbiased estimator, it has high variance. The document aims to reduce the variance to improve estimation accuracy.
Calton pu experimental methods on performance in cloud and accuracy in big da...jins0618
Experimental Methods on Performance in Clouds provides an overview of cloud computing and benchmarks for measuring cloud performance. It discusses:
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This document discusses challenges and opportunities in parallel graph processing for big data. It describes how graphs are ubiquitous but processing large graphs at scale is difficult due to their huge size, complex correlations between data entities, and skewed distributions. Current computation models have problems with ghost vertices, too much interaction between partitions, and lack of support for iterative graph algorithms. New frameworks are needed to handle these graphs in a scalable way with low memory usage and balanced computation and communication.
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Wang ke mining revenue-maximizing bundling configurationjins0618
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The document summarizes an entity extraction and typing framework proposed by the author. The framework constructs a heterogeneous graph connecting entity mentions, surface names, and relation phrases extracted from documents. It then performs joint type propagation and relation phrase clustering on the graph to infer types for entity mentions. Evaluation on news, tweets and reviews shows the framework outperforms existing methods in recognizing new types and domains without extensive feature engineering or human supervision. It obtains improvements by modeling each mention individually and addressing data sparsity through relation phrase clustering.
Strategy 3 for topical phrase mining first performs phrase mining on a corpus to extract candidate phrases, then applies topic modeling with the phrases as constraints. This approach generates coherent topics where words within a phrase share a topic label. Strategy 3 outputs high-quality topics and phrases faster than Strategies 1 and 2, and the topics and phrases have better coherence, quality, and intrusion scores. However, Strategy 3's topic inference relies on the accuracy of the initial phrase mining.
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2. Remind:Two``-MainProblemsinML
• Two-mainproblemsinML:
– Regression: Linear Regression, Neural net...
– Classification: Decision Tree, kNN, Bayessian Classifier...
• Today, we will learn:
– K-means: a trivial unsupervised classification algorithm.
– Expectation Maximization: a general algorithm for density estimation.
∗ We will see how to use EM in general cases and in specific case of GMM.
– GMM: a tool for modelling Data-in-the-Wild (density estimator)
∗ We also learn how to use GMM in a Bayessian Classifier
3. Contents
• Unsupervised Learning
• K-means clustering
• Expectation Maximization (E.M.)
• Gaussian mixtures as a Density Estimator
– Gaussian mixtures
– EM for mixtures
• Gaussian mixtures for classification
4. Unsupervised Learning
•
– Label of each sample is included in the training set
Sample Label
x1 y1
... ...
xn yk
• Unsupervised learning:
– Traning set contains the samples only
Sample Label
x1
...
xn
Supervised learning techniques:
6. What is unsupervised learning useful for?
• Collecting and labeling a large training set can be very expensive.
• Be able to find features which are helpful for categorization.
• Gain insight into the natural structure of the data.
7. Contents
• Unsupervised Learning
• K-means clustering
• Expectation Maximization (E.M.)
• Gaussian mixtures as a Density Estimator
– Gaussian mixtures
– EM for mixtures
• Gaussian mixtures for classification
8. K-means clustering
• Clustering algorithms aim to find
groups of “similar” data points among
the input data.
• K-means is an effective algorithm to ex-
tract a given number of clusters from a
training set.
• Once done, the cluster locations can
be used to classify data into distinct
classes. −10 0 10 20 30 40 50
0
10
20
30
40
50
60
9. K-means clustering
• Given:
– The dataset: {xn}N
n=1 = {x1, x2, ..., xN}
– Number of clusters: K (K < N)
• Goal: find a partition S = {Sk}K
k=1 so that it minimizes the objective function
J =
N∑
n=1
K∑
k=1
rnk ∥ xn − µk ∥2
(1)
where rnk = 1 if xn is assigned to cluster Sk, and rnj = 0 for j ̸= k.
i.e. Find values for the {rnk} and the {µk} to minimize (1).
10. K-means clustering
J =
N∑
n=1
K∑
k=1
rnk ∥ xn − µk ∥2
• Select some initial values for the µk.
• Expectation: keep the µk fixed, minimize J respect to rnk.
• Maximization: keep the rnk fixed, minimize J respect to the µk.
• Loop until no change in the partitions (or maximum number of interations is
exceeded).
11. K-means clustering
J =
N∑
n=1
K∑
k=1
rnk ∥ xn − µk ∥2
• Expectation: J is linear function of rnk
rnk =
1 if k = arg minj ∥ xn − µj ∥2
0 otherwise
• Maximization: setting the derivative of J with respect to µk to zero, gives:
µk =
∑
n rnkxn
∑
n rnk
Convergence of K-means: assured [why?], but may lead to local minimum of J
[8]
12. K-means clustering: How to understand?
J =
N∑
n=1
K∑
k=1
rnk ∥ xn − µk ∥2
• Expectation: minimize J respect to rnk
– For each xn, find the “closest” cluster mean µk and put xn into cluster Sk.
• Maximization: minimize J respect to µk
– For each cluster Sk, re-estimate the cluster mean µk to be the average value
of all samples in Sk.
• Loop until no change in the partitions (or maximum number of interations is
exceeded).
24. Expectation Maximization
• A general-purpose algorithm for MLE in a wide range of situations.
• First formally stated by Dempster, Laird and Rubin in 1977 [1]
• An excellent way of doing our unsupervised learning problem, as we will see
– EM is also used widely in other domains.
25. EM: a solution for MLE
• Given a statistical model with:
– a set X of observed data,
– a set Z of unobserved latent data,
– a vector of unknown parameters θ,
– a likelihood function L (θ; X, Z) = p (X, Z | θ)
• Roughly speaking, the aim of MLE is to determine θ = arg maxθ L (θ; X, Z)
– We known the old trick: partial derivatives of the log likelihood...
– But it is not always tractable [e.g.]
– Other solutions are available.
26. EM: General Case
L (θ; X, Z) = p (X, Z | θ)
• EM is just an iterative procedure for finding the MLE
• Expectation step: keep the current estimate θ(t)
fixed, calculate the expected
value of the log likelihood function
Q
(
θ | θ(t)
)
= E [log L (θ; X, Z)] = E [log p (X, Z | θ)]
• Maximization step: Find the parameter that maximizes this quantity
θ(t+1)
= arg max
θ
Q
(
θ | θ(t)
)
27. EM: Motivation
• If we know the value of the parameters θ, we can find the value of latent variables
Z by maximizing the log likelihood over all possible values of Z
– Searching on the value space of Z.
• If we know Z, we can find an estimate of θ
– Typically by grouping the observed data points according to the value of asso-
ciated latent variable,
– then averaging the values (or some functions of the values) of the points in
each group.
To understand this motivation, let’s take K-means as a trivial example...
28. EM: informal description
Both θ and Z are unknown, EM is an iterative algorithm:
1. Initialize the parameters θ to some random values.
2. Compute the best values of Z given these parameter values.
3. Use the just-computed values of Z to find better estimates for θ.
4. Iterate until convergence.
29. EM Convergence
• E.M. Convergence: Yes
– After each iteration, p (X, Z | θ) must increase or remain [NOT OBVIOUS]
– But it can not exceed 1 [OBVIOUS]
– Hence it must converge [OBVIOUS]
• Bad news: E.M. converges to local optimum.
– Whether the algorithm converges to the global optimum depends on the ini-
tialization.
• Let’s take K-means as an example, again...
• Details can be found in [9].
30. Contents
• Unsupervised Learning
• K-means clustering
• Expectation Maximization (E.M.)
• Gaussian mixtures as a Density Estimator
– Gaussian mixtures
– EM for mixtures
• Gaussian mixtures for classification
–
31. Remind: Bayes Classifier
0 10 20 30 40 50 60 70 80
−10
0
10
20
30
40
50
60
70
p (y = i | x) =
p (x | y = i) p (y = i)
p (x)
32. Remind: Bayes Classifier
0 10 20 30 40 50 60 70 80
−10
0
10
20
30
40
50
60
70
In case of Gaussian Bayes Classifier:
p (y = i | x) =
1
(2π)
d/2
∥Σi∥1/2
exp
[
−1
2 (x − µi)T
Σi (x − µi)
]
pi
p (x)
How can we deal with the denominator p (x)?
33. Remind: The Single Gaussian Distribution
• Multivariate Gaussian
N (x; µ, Σ) =
1
(2π)
d/2
∥ Σ ∥1/2
exp
−
1
2
(x − µ)T
Σ−1
(x − µ)
• For maximum likelihood
0 =
∂ ln N (x1, x2, ..., xN; µ, Σ)
∂µ
• and the solution is
µML =
1
N
N∑
i=1
xi
ΣML =
1
N
N∑
i=1
(xi − µML)T
(xi − µML)
34. The GMM assumption
• There are k components: {ci}k
i=1
• Component ci has an associated mean
vector µi
•
•
µ1
µ2
µ3
35. The GMM assumption
• There are k components: {ci}k
i=1
• Component ci has an associated mean
vector µi
• Each component generates data from a
Gaussian with mean µi and covariance
matrix Σi
• Each sample is generated according to
the following guidelines:
µ1
µ2
µ3
36. The GMM assumption
• There are k components: {ci}k
i=1
• Component ci has an associated mean
vector µi
• Each component generates data from a
Gaussian with mean µi and covariance
matrix Σi
• Each sample is generated according to
the following guidelines:
– Randomly select component ci
with probability P (ci) = wi, s.t.
∑k
i=1 wi = 1
µ2
37. The GMM assumption
• There are k components: {ci}k
i=1
• Component ci has an associated mean
vector µi
• Each component generates data from a
Gaussian with mean µi and covariance
matrix Σi
• Each sample is generated according to
the following guidelines:
– Randomly select component ci with
probability P (ci) = wi, s.t.
∑k
i=1 wi = 1
– Sample ~ N (µi, Σi)
µ2
x
38. Probability density function of GMM
“Linear combination” of Gaussians:
f (x) =
k∑
i=1
wiN (x; µi, Σi) , where
k∑
i=1
wi = 1
0 50 100 150 200 250
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
w1N µ1, σ2
1 w2N µ2, σ2
2
w3N µ3, σ2
3
f (x)
(a) The pdf of an 1D GMM with 3 components. (b) The pdf of an 2D GMM with 3 components.
Figure 2: Probability density function of some GMMs.
39. GMM: Problem definition
f (x) =
k∑
i=1
wiN (x; µi, Σi) , where
k∑
i=1
wi = 1
Given a training set, how to model these data point using GMM?
• Given:
– The trainning set: {xi}N
i=1
– Number of clusters: k
• Goal: model this data using a mixture of Gaussians
– Weights: w1, w2, ..., wk
– Means and covariances: µ1, µ2, ..., µk; Σ1, Σ2, ..., Σk
40. Computing likelihoods in unsupervised case
f (x) =
k∑
i=1
wiN (x; µi, Σi) , where
k∑
i=1
wi = 1
• Given a mixture of Gaussians, denoted by G. For any x, we can define the
likelihood:
P (x | G) = P (x | w1, µ1, Σ1, ..., wk, µk, Σk)
=
k∑
i=1
P (x | ci) P (ci)
=
k∑
i=1
wiN (x; µi, Σi)
• So we can define likelihood for the whole training set [Why?]
P (x1, x2, ..., xN | G) =
N∏
i=1
P (xi | G)
=
N∏
i=1
k∑
j=1
wjN (xi; µj, Σj)
41. Estimating GMM parameters
• We known this: Maximum Likelihood Estimation
ln P (X | G) =
N∑
i=1
ln
k∑
j=1
wjN (xi; µj, Σj)
– For the max likelihood:
0 =
∂ ln P (X | G)
∂µj
– This leads to non-linear non-analytically-solvable equations!
• Use gradient descent
– Slow but doable
• A much cuter and recently popular method...
42. E.M. for GMM
• Remember:
– We have the training set {xi}N
i=1, the number of components k.
– Assume we know p (c1) = w1, p (c2) = w2, ..., p (ck) = wk
– We don’t know µ1, µ2, ..., µk
The likelihood:
p (data | µ1, µ2, ..., µk) = p (x1, x2, ..., xN | µ1, µ2, ..., µk)
=
N∏
i=1
p (xi | µ1, µ2, ..., µk)
=
N∏
i=1
k∑
j=1
p (xi | wj, µ1, µ2, ..., µk) p (cj)
=
N∏
i=1
k∑
j=1
K exp
−
1
2σ2
(
xi − µj
)
2
wi
43. E.M. for GMM
• For Max. Likelihood, we know ∂
∂µi
log p (data | µ1, µ2, ..., µk) = 0
• Some wild algebra turns this into: For Maximum Likelihood, for each j:
µj =
N∑
i=1
p (cj | xi, µ1, µ2, ..., µk) xi
N∑
i=1
p (cj | xi, µ1, µ2, ..., µk)
This is N non-linear equations of µj’s.
• So:
– If, for each xi, we know p (cj | xi, µ1, µ2, ..., µk), then we could easily compute
µj,
– If we know each µj, we could compute p (cj | xi, µ1, µ2, ..., µk) for each xi
and cj.
44. E.M. for GMM
• E.M. is coming: on the t’th iteration, let our estimates be
λt = {µ1 (t) , µ2 (t) , ..., µk (t)}
• E-step: compute the expected classes of all data points for each class
p (cj | xi, λt) =
p (xi | cj, λt) p (cj | λt)
p (xi | λt)
=
p
(
xi | cj, µj (t) , σjI
)
p (cj)
k∑
m=1
p (xi | cm, µm (t) , σmI) p (cm)
• M-step: compute µ given our data’s class membership distributions
µj (t + 1) =
N∑
i=1
p (cj | xi, λt) xi
N∑
i=1
p (cj | xi, λt)
45. E.M. for General GMM: E-step
• On the t’th iteration, let our estimates be
λt = {µ1 (t) , µ2 (t) , ..., µk (t) , Σ1 (t) , Σ2 (t) , ..., Σk (t) , w1 (t) , w2 (t) , ..., wk (t)}
• E-step: compute the expected classes of all data points for each class
τij (t) ≡ p (cj | xi, λt) =
p (xi | cj, λt) p (cj | λt)
p (xi | λt)
=
p
(
xi | cj, µj (t) , Σj (t)
)
wj (t)
k∑
m=1
p (xi | cm, µm (t) , Σj (t)) wm (t)
46. E.M. for General GMM: M-step
• M-step: compute µ given our data’s class membership distributions
wj (t + 1) =
N∑
i=1
p (cj | xi, λt)
N
µj (t + 1) =
N∑
i=1
p (cj | xi, λt) xi
N∑
i=1
p (cj | xi, λt)
=
1
N
N∑
i=1
τij (t) =
1
Nwj (t + 1)
N∑
i=1
τij (t) xi
Σj (t + 1) =
N∑
i=1
p (cj | xi, λt)
[
xi − µj (t + 1)
] [
xi − µj (t + 1)
]
T
N∑
i=1
p (cj | xi, λt)
=
1
Nwj (t + 1)
N∑
i=1
τij (t)
[
xi − µj (t + 1)
] [
xi − µj (t + 1)
]
T
47. E.M. for General GMM: Initialization
• wj = 1/k, j = 1, 2, ..., k
• Each µj is set to a randomly selected point
– Or use K-means for this initialization.
• Each Σj is computed using the equation in previous slide...
56. Local optimum solution
• E.M. is guaranteed to find the local optimal solution by monotonically increasing
the log-likelihood
• Whether it converges to the global optimal solution depends on the initialization
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8
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14
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18
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5
10
15
57. GMM: Selecting the number of components
• We can run the E.M. algorithm with different numbers of components.
– Need a criteria for selecting the “best” number of components
−10 −5 0 5 10 15
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5
10
15
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0
2
4
6
8
10
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58. Contents
• Unsupervised Learning
• K-means clustering
• Expectation Maximization (E.M.)
– Regularized EM
– Model Selection
• Gaussian mixtures as a Density Estimator
– Gaussian mixtures
– EM for mixtures
• Gaussian mixtures for classification
59. Gaussian mixtures for classification
p (y = i | x) =
p (x | y = i) p (y = i)
p (x)
• To build a Bayesian classifier based on GMM, we can use GMM to model data in
each class
– So each class is modeled by one k-component GMM.
• For example:
Class 0: p (y = 0) , p (x | θ0), (a 3-component mixture)
Class 1: p (y = 1) , p (x | θ1), (a 3-component mixture)
Class 2: p (y = 2) , p (x | θ2), (a 3-component mixture)
...
60. GMM for Classification
• As previous, each class is modeled by a k-component GMM.
• A new test sample x is classified according to
c = arg max
i
p (y = i) p (x | θi)
where
p (x | θi) =
k∑
i=1
wiN (x; µi, Σi)
• Simple, quick (and is actually used!)
63. References
[1] N. Laird A. Dempster and D. Rubin. Maximum likelihood from incomplete data
via the em algorithm. Journal of the Royal Statistical Society. Series B (Method-
ological), 39(1):pp. 1–38., 1977.
[2] David Arthur and Sergei Vassilvitskii. k-means ++ : The Advantages of Careful
Seeding. In Proceedings of the eighteenth annual ACM-SIAM symposium on
Discrete algorithms, volume 8, pages 1027–1035, 2007.
[3] N. Gumerov C. Yang, R. Duraiswami and L. Davis. Improved fast gauss transform
and efficient kernel density estimation. In IEEE International Conference on
Computer Vision, pages pages 464–471, 2003.
[4] Charles Elkan. Using the Triangle Inequality to Accelerate k-Means. In Proceed-
64. ings of the Twentieth International Conference on Machine Learning (ICML),
2003.
[5] Keshu Zhang Haifeng Li and Tao Jiang. The regularized em algorithm. In
Proceedings of the 20th National Conference on Artificial Intelligence, pages
pages 807 – 812, Pittsburgh, PA, 2005.
[6] Greg Hamerly and Charles Elkan. Learning the k in k-means. In In Neural
Information Processing Systems. MIT Press, 2003.
[7] Tapas Kanungo, David M Mount, Nathan S Netanyahu, Christine D Piatko, Ruth
Silverman, and Angela Y Wu. An efficient k-means clustering algorithm: anal-
ysis and implementation. IEEE Transactions on Pattern Analysis and Machine
Intelligence, 24(7):881–892, July 2002.
[8] J MacQueen. Some methods for classification and analysis of multivariate obser-
vations. In Proceedings of 5th Berkeley Symposium on Mathematical Statistics
and Probability, volume 233, pages 281–297. University of California Press, 1967.
65. [9] C.F. Wu. On the convergence properties of the em algorithm. The Annals of
Statistics, 11:95–103, 1983.