This document provides an overview of structured probabilistic models for deep learning. It discusses using graphs to describe model structure, sampling from graphical models, and the advantages of structured modeling over unstructured approaches. Key topics covered include directed and undirected models, separation properties, converting between graph representations, learning model structure, and using latent variables. The document serves as lecture slides outlining concepts in structured probabilistic modeling.
Waltz algorithm in artificial intelligenceMinakshi Atre
The document discusses the Waltz algorithm for constraint satisfaction problems. It presents the algorithm in three parts. Part 1 discusses constraints in search and knowledge representation, and how constraint propagation allows reaching a global solution using local search. It provides an example of line labeling in computer vision. Part 2 discusses how constraints can reduce complexity in perceptual tasks like line drawings. It explains Waltz's labeling scheme and valid junction configurations. Part 3 works through an example of applying Waltz labeling to a pyramid drawing, showing how constraints successively eliminate possible labelings until a unique solution is reached.
The boundary element method is used to simulate the flow of an emulsion drop through a converging channel. The drop flow is governed by the Stokes equations since the Reynolds number is low. Integral equations are derived relating the velocity and stress fields on the drop surface and channel boundaries. The equations account for the viscosity ratio, capillary number, and drop shape. Nodes and quadratic elements are used to discretize the boundaries and solve the integral equations numerically. The results show the effect of parameters on the flow rate-pressure relation and drop shape dynamics.
You can use this presentation to introduce students in how to write linear equations given the slope and the y-intercept. This is the first case in writing linear equations.
1) The document compares different simplified models for simulating rivulet flow down a slowly varying substrate to a direct numerical simulation using computational fluid dynamics (CFD).
2) Three simplified models are considered: constant contact angle and varying width, constant width and varying contact angle, and varying both contact angle and width.
3) The CFD simulation uses the lubrication approximation to directly solve the Navier-Stokes equations, while the simplified models make additional assumptions to obtain analytical solutions.
4) Comparison of the results from the different methods show that the model allowing both contact angle and width to vary provides the closest agreement to the CFD simulation.
1) The document discusses the tangent line problem, which involves finding the slope of the tangent line to a curve at a point. This led to the development of the concept of the derivative.
2) The derivative of a function f(x) is defined as the limit of the difference quotient and represents the instantaneous rate of change of the function.
3) A function is differentiable if its derivative exists, and differentiability requires a function to be continuous but continuity does not guarantee differentiability. Specifically, a function can be continuous at a point where its graph involves a sharp turn but not differentiable there.
The document discusses various graph theory topics including isomorphism, cut sets, labeled graphs, and Hamiltonian circuits. It defines isomorphism as two graphs being structurally identical with a one-to-one correspondence between their vertices and edges. Cut sets are edges whose removal would disconnect a connected graph. Labeled graphs assign labels or weights to their vertices and/or edges. A Hamiltonian circuit is a closed walk that visits each vertex exactly once.
Application of Vertex Colorings with Some Interesting Graphsijtsrd
Firstly, basic concepts of graph and vertex colorings are introduced. Then, some interesting graphs with vertex colorings are presented. A vertex coloring of graph G is an assignment of colors to the vertices of G. And then by using proper vertex coloring, some interesting graphs are described. By using some applications of vertex colorings, two problems is presented interestingly. The vertex coloring is the starting point of graph coloring. The chromatic number for some interesting graphs and some results are studied. Ei Ei Moe "Application of Vertex Colorings with Some Interesting Graphs" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29263.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/29263/application-of-vertex-colorings-with-some-interesting-graphs/ei-ei-moe
Waltz algorithm in artificial intelligenceMinakshi Atre
The document discusses the Waltz algorithm for constraint satisfaction problems. It presents the algorithm in three parts. Part 1 discusses constraints in search and knowledge representation, and how constraint propagation allows reaching a global solution using local search. It provides an example of line labeling in computer vision. Part 2 discusses how constraints can reduce complexity in perceptual tasks like line drawings. It explains Waltz's labeling scheme and valid junction configurations. Part 3 works through an example of applying Waltz labeling to a pyramid drawing, showing how constraints successively eliminate possible labelings until a unique solution is reached.
The boundary element method is used to simulate the flow of an emulsion drop through a converging channel. The drop flow is governed by the Stokes equations since the Reynolds number is low. Integral equations are derived relating the velocity and stress fields on the drop surface and channel boundaries. The equations account for the viscosity ratio, capillary number, and drop shape. Nodes and quadratic elements are used to discretize the boundaries and solve the integral equations numerically. The results show the effect of parameters on the flow rate-pressure relation and drop shape dynamics.
You can use this presentation to introduce students in how to write linear equations given the slope and the y-intercept. This is the first case in writing linear equations.
1) The document compares different simplified models for simulating rivulet flow down a slowly varying substrate to a direct numerical simulation using computational fluid dynamics (CFD).
2) Three simplified models are considered: constant contact angle and varying width, constant width and varying contact angle, and varying both contact angle and width.
3) The CFD simulation uses the lubrication approximation to directly solve the Navier-Stokes equations, while the simplified models make additional assumptions to obtain analytical solutions.
4) Comparison of the results from the different methods show that the model allowing both contact angle and width to vary provides the closest agreement to the CFD simulation.
1) The document discusses the tangent line problem, which involves finding the slope of the tangent line to a curve at a point. This led to the development of the concept of the derivative.
2) The derivative of a function f(x) is defined as the limit of the difference quotient and represents the instantaneous rate of change of the function.
3) A function is differentiable if its derivative exists, and differentiability requires a function to be continuous but continuity does not guarantee differentiability. Specifically, a function can be continuous at a point where its graph involves a sharp turn but not differentiable there.
The document discusses various graph theory topics including isomorphism, cut sets, labeled graphs, and Hamiltonian circuits. It defines isomorphism as two graphs being structurally identical with a one-to-one correspondence between their vertices and edges. Cut sets are edges whose removal would disconnect a connected graph. Labeled graphs assign labels or weights to their vertices and/or edges. A Hamiltonian circuit is a closed walk that visits each vertex exactly once.
Application of Vertex Colorings with Some Interesting Graphsijtsrd
Firstly, basic concepts of graph and vertex colorings are introduced. Then, some interesting graphs with vertex colorings are presented. A vertex coloring of graph G is an assignment of colors to the vertices of G. And then by using proper vertex coloring, some interesting graphs are described. By using some applications of vertex colorings, two problems is presented interestingly. The vertex coloring is the starting point of graph coloring. The chromatic number for some interesting graphs and some results are studied. Ei Ei Moe "Application of Vertex Colorings with Some Interesting Graphs" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29263.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/29263/application-of-vertex-colorings-with-some-interesting-graphs/ei-ei-moe
Monte Carlo methods can be used to estimate sums and integrals by approximating them as expectations under a probability distribution. Samples are drawn from the distribution and the average of the function evaluated at each sample is calculated. This provides an unbiased estimate with variance that decreases as more samples are taken. Importance sampling improves upon this by drawing samples from a different distribution that puts more weight on important areas, which can reduce variance. Markov chain Monte Carlo methods like Gibbs sampling are used to draw samples from distributions that cannot be directly sampled, like those represented by undirected graphs, by iteratively updating variables conditioned on others.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
1. Unsupervised pretraining of deep neural networks (DNNs) on multiple datasets usually provides no benefit and sometimes harms performance compared to DNNs trained on individual datasets.
2. Representation learning aims to learn representations of input data that make a task easier by separating explanatory factors of variations. For example, mixture models can discover separate classes in data and distributed representations can divide the input space into uniquely identifiable regions.
3. Generative adversarial networks (GANs) can learn vector spaces that support semantic operations like representing a woman with glasses by combining vectors for concepts of gender and wearing glasses.
IRJET- On the Generalization of Lami’s TheoremIRJET Journal
1) The document generalizes Lami's Theorem, which relates the magnitudes of three coplanar, concurrent and non-collinear forces in static equilibrium, to systems with any odd number of forces.
2) It shows that for cyclic polygons with an odd number of sides, the sine rule relating side lengths and opposite interior angles takes the same form as Lami's Theorem.
3) The paper also describes how to transform n-gons into vector diagrams and vice versa, and derives a condition for when a vector diagram can represent a cyclic polygon.
Map Coloring and Some of Its Applications MD SHAH ALAM
This is a research paper which I have conducted at the final year of undergrad study and got 4.00/4.00. It is mainly related to graph theory and has many applications in practical life.
The Power of Graphs in Immersive Communicationstonizza82
This document discusses graph signal processing and its applications in immersive communications. It begins with an introduction to graphs and how they can represent network-structured data. It then discusses how machine learning can be applied to graph-structured data through tasks like graph classification, node classification, and graph clustering. The document outlines challenges with 360-degree video streaming like delivering large volumes of data under low-delay constraints. It proposes that graph signal processing approaches may help address these challenges by accounting for both the data and relationships in the network.
In this chapter we have to Learn Graph Terminologies, Types of Graph, Representation of Graph, Traversal of Graph BFS and DFS. Its Application Use Advantages and Disadvantages
This project developed algorithms to convert between square and hexagonal identifications of surfaces. Python was used to program the data transformations and transfer of curves between representations. The algorithms compare vertex values and track boundaries to determine properties like genus from the Euler characteristic. Examples demonstrate converting a square model into a hexagonal one and vice versa, and transferring curves between representations remains a focus of further work.
Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Ro...James Smith
Using ideas developed in detail in http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016, this document solves one of the special cases of the famous Problem of Apollonius. A new Appendix presents alternative solutions.
See also:
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial
FREQUENT SUBGRAPH MINING ALGORITHMS - A SURVEY AND FRAMEWORK FOR CLASSIFICATIONcscpconf
Data mining algorithms are facing the challenge to deal with an increasing number of complex
objects. Graph is a natural data structure used for modeling complex objects. Frequent subgraph
mining is another active research topic in data mining . A graph is a general model to represent
data and has been used in many domains like cheminformatics and bioinformatics. Mining
patterns from graph databases is challenging since graph related operations, such as subgraph
testing, generally have higher time complexity than the corresponding operations on itemsets,
sequences, and trees. Many frequent subgraph Mining algorithms have been proposed. SPIN,
SUBDUE, g_Span, FFSM, GREW are a few to mention. In this paper we present a detailed
survey on frequent subgraph mining algorithms, which are used for knowledge discovery in
complex objects and also propose a frame work for classification of these algorithms. The
purpose is to help user to apply the techniques in a task specific manner in various application domains and to pave wave for further research.
Introduction to Graphs
Topics:
Definition: Graph
Related Definitions
Applications
Teaching material for the course of "Tecniche di Programmazione" at Politecnico di Torino in year 2012/2013. More information: http://bit.ly/tecn-progr
This document contains several multi-part calculus problems involving estimating areas under curves using Riemann sums with rectangles. The problems ask the student to:
- Estimate areas under graphs using left, right, and midpoint Riemann sums with varying numbers of rectangles
- Interpret Riemann sums as approximations of definite integrals involving the area under a curve
- Evaluate definite integrals using properties such as interpreting them in terms of areas under curves
The document describes a graph-based method for finding all candidate keys in a relational database scheme. It first presents how to construct a functional dependency graph (FDG) from the functional dependencies in a relational scheme. It then describes a two-phase process to transform the FDG into a candidate graph (GC) that contains only the candidate nodes. The first phase involves augmenting the FDG by adding new edges and nodes. The second phase reduces the augmented graph by deleting nodes and edges according to certain rules. From the resulting GC, all candidate keys for the relational scheme can be identified as either individual candidate nodes or parts of cycles in the graph. Theorems are also presented regarding properties of the candidate graph and
This document discusses graphs and their representation in Java. It begins with basic graph terminology like vertices, edges, directed/undirected graphs. It then discusses modeling graphs in Java using interfaces and data structures like arrays and lists to store vertices and edges. Specific implementations are shown to represent sample graphs. The document is intended to introduce basic graph concepts and their modeling in Java.
Analysis of the Boston Housing Data from the 1970 censusShuai Yuan
This document analyzes the Boston housing data from 1970 using R. It examines the relationships between variables using scatterplots and correlation. Various regression models are tested to analyze properties of the data. Model selection methods like forward selection, backward selection, and information criteria are used to identify the best fitting model. The selected model is then used to compute statistics like SSPE on a subset of the data.
A graph is a non-linear data structure consisting of nodes and edges where the nodes are connected via edges. There are different ways to represent graphs including using an adjacency matrix or adjacency lists. Common graph terminology includes vertices, edges, degree, and traversal algorithms like depth-first search (DFS) and breadth-first search (BFS) which are used to search graphs. DFS uses a stack and explores nodes as deep as possible before backtracking while BFS uses a queue and explores all neighbor nodes at the present depth before moving deeper.
This document discusses graphs and spanning trees. A graph is a data structure containing vertices and edges connecting the vertices. A spanning tree of a graph is a subgraph that connects all vertices without cycles. Spanning trees can be found using graph search algorithms like breadth-first search or depth-first search. Spanning trees have properties like containing no cycles, having the minimum number of edges to connect all vertices, and being uniquely defined up to the choice of starting vertex for the search algorithm.
for sbi so Ds c c++ unix rdbms sql cn osalisha230390
This document contains 35 questions related to data structures and algorithms. It covers topics like data structures used in different areas like databases, networks and hierarchies. Other topics covered include trees, graphs, sorting, hashing and file structures. Sample problems are given related to these topics to test understanding.
This document defines and provides examples of congruent angles, congruent segments, and congruent triangles in geometry. It explains that angles are congruent if they have the same measure, segments are congruent if they have the same length, and triangles are congruent if corresponding angles and sides are congruent. The document provides exercises asking the reader to identify corresponding parts of congruent triangles without diagrams as well as included angles and sides within triangles.
1. The research question investigates the relationship between the force applied to one side of a cantilever beam and the maximum acceleration reached by the free end.
2. Materials used include an acrylic plastic cantilever beam, accelerometer, string, and force meter.
3. Experiments are conducted to pull the cantilever with varying forces and measure the corresponding maximum accelerations using the accelerometer.
4. Results are analyzed to understand the elastic properties of the cantilever beam based on the relationship between applied force and acceleration. Potential and kinetic energy concepts are also explored.
The document summarizes various applications of deep learning that have progressed rapidly since the book was written. It discusses applications in areas such as computer vision, natural language processing, speech recognition and synthesis, robotics, healthcare, and autonomous vehicles. Specific examples mentioned include neural machine translation models from Google, AlphaGo from DeepMind, and autonomous vehicle research from Waymo. It notes that many applications now rely on techniques such as attention mechanisms, generative models, reinforcement learning, and model compression that have developed significantly in recent years.
This document provides an overview of sequence modeling using recurrent and recursive neural networks. It begins with an introduction to classical dynamical systems and unfolding computational graphs for recurrent networks. It then discusses different types of recurrent networks, including those with recurrence through the hidden and output states. Bidirectional and encoder-decoder sequence-to-sequence architectures are also covered. Finally, the document discusses issues like exploding gradients and presents solutions like LSTMs and gradient clipping. Recursive networks and networks with explicit memory components are also introduced.
Monte Carlo methods can be used to estimate sums and integrals by approximating them as expectations under a probability distribution. Samples are drawn from the distribution and the average of the function evaluated at each sample is calculated. This provides an unbiased estimate with variance that decreases as more samples are taken. Importance sampling improves upon this by drawing samples from a different distribution that puts more weight on important areas, which can reduce variance. Markov chain Monte Carlo methods like Gibbs sampling are used to draw samples from distributions that cannot be directly sampled, like those represented by undirected graphs, by iteratively updating variables conditioned on others.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
1. Unsupervised pretraining of deep neural networks (DNNs) on multiple datasets usually provides no benefit and sometimes harms performance compared to DNNs trained on individual datasets.
2. Representation learning aims to learn representations of input data that make a task easier by separating explanatory factors of variations. For example, mixture models can discover separate classes in data and distributed representations can divide the input space into uniquely identifiable regions.
3. Generative adversarial networks (GANs) can learn vector spaces that support semantic operations like representing a woman with glasses by combining vectors for concepts of gender and wearing glasses.
IRJET- On the Generalization of Lami’s TheoremIRJET Journal
1) The document generalizes Lami's Theorem, which relates the magnitudes of three coplanar, concurrent and non-collinear forces in static equilibrium, to systems with any odd number of forces.
2) It shows that for cyclic polygons with an odd number of sides, the sine rule relating side lengths and opposite interior angles takes the same form as Lami's Theorem.
3) The paper also describes how to transform n-gons into vector diagrams and vice versa, and derives a condition for when a vector diagram can represent a cyclic polygon.
Map Coloring and Some of Its Applications MD SHAH ALAM
This is a research paper which I have conducted at the final year of undergrad study and got 4.00/4.00. It is mainly related to graph theory and has many applications in practical life.
The Power of Graphs in Immersive Communicationstonizza82
This document discusses graph signal processing and its applications in immersive communications. It begins with an introduction to graphs and how they can represent network-structured data. It then discusses how machine learning can be applied to graph-structured data through tasks like graph classification, node classification, and graph clustering. The document outlines challenges with 360-degree video streaming like delivering large volumes of data under low-delay constraints. It proposes that graph signal processing approaches may help address these challenges by accounting for both the data and relationships in the network.
In this chapter we have to Learn Graph Terminologies, Types of Graph, Representation of Graph, Traversal of Graph BFS and DFS. Its Application Use Advantages and Disadvantages
This project developed algorithms to convert between square and hexagonal identifications of surfaces. Python was used to program the data transformations and transfer of curves between representations. The algorithms compare vertex values and track boundaries to determine properties like genus from the Euler characteristic. Examples demonstrate converting a square model into a hexagonal one and vice versa, and transferring curves between representations remains a focus of further work.
Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Ro...James Smith
Using ideas developed in detail in http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016, this document solves one of the special cases of the famous Problem of Apollonius. A new Appendix presents alternative solutions.
See also:
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial
FREQUENT SUBGRAPH MINING ALGORITHMS - A SURVEY AND FRAMEWORK FOR CLASSIFICATIONcscpconf
Data mining algorithms are facing the challenge to deal with an increasing number of complex
objects. Graph is a natural data structure used for modeling complex objects. Frequent subgraph
mining is another active research topic in data mining . A graph is a general model to represent
data and has been used in many domains like cheminformatics and bioinformatics. Mining
patterns from graph databases is challenging since graph related operations, such as subgraph
testing, generally have higher time complexity than the corresponding operations on itemsets,
sequences, and trees. Many frequent subgraph Mining algorithms have been proposed. SPIN,
SUBDUE, g_Span, FFSM, GREW are a few to mention. In this paper we present a detailed
survey on frequent subgraph mining algorithms, which are used for knowledge discovery in
complex objects and also propose a frame work for classification of these algorithms. The
purpose is to help user to apply the techniques in a task specific manner in various application domains and to pave wave for further research.
Introduction to Graphs
Topics:
Definition: Graph
Related Definitions
Applications
Teaching material for the course of "Tecniche di Programmazione" at Politecnico di Torino in year 2012/2013. More information: http://bit.ly/tecn-progr
This document contains several multi-part calculus problems involving estimating areas under curves using Riemann sums with rectangles. The problems ask the student to:
- Estimate areas under graphs using left, right, and midpoint Riemann sums with varying numbers of rectangles
- Interpret Riemann sums as approximations of definite integrals involving the area under a curve
- Evaluate definite integrals using properties such as interpreting them in terms of areas under curves
The document describes a graph-based method for finding all candidate keys in a relational database scheme. It first presents how to construct a functional dependency graph (FDG) from the functional dependencies in a relational scheme. It then describes a two-phase process to transform the FDG into a candidate graph (GC) that contains only the candidate nodes. The first phase involves augmenting the FDG by adding new edges and nodes. The second phase reduces the augmented graph by deleting nodes and edges according to certain rules. From the resulting GC, all candidate keys for the relational scheme can be identified as either individual candidate nodes or parts of cycles in the graph. Theorems are also presented regarding properties of the candidate graph and
This document discusses graphs and their representation in Java. It begins with basic graph terminology like vertices, edges, directed/undirected graphs. It then discusses modeling graphs in Java using interfaces and data structures like arrays and lists to store vertices and edges. Specific implementations are shown to represent sample graphs. The document is intended to introduce basic graph concepts and their modeling in Java.
Analysis of the Boston Housing Data from the 1970 censusShuai Yuan
This document analyzes the Boston housing data from 1970 using R. It examines the relationships between variables using scatterplots and correlation. Various regression models are tested to analyze properties of the data. Model selection methods like forward selection, backward selection, and information criteria are used to identify the best fitting model. The selected model is then used to compute statistics like SSPE on a subset of the data.
A graph is a non-linear data structure consisting of nodes and edges where the nodes are connected via edges. There are different ways to represent graphs including using an adjacency matrix or adjacency lists. Common graph terminology includes vertices, edges, degree, and traversal algorithms like depth-first search (DFS) and breadth-first search (BFS) which are used to search graphs. DFS uses a stack and explores nodes as deep as possible before backtracking while BFS uses a queue and explores all neighbor nodes at the present depth before moving deeper.
This document discusses graphs and spanning trees. A graph is a data structure containing vertices and edges connecting the vertices. A spanning tree of a graph is a subgraph that connects all vertices without cycles. Spanning trees can be found using graph search algorithms like breadth-first search or depth-first search. Spanning trees have properties like containing no cycles, having the minimum number of edges to connect all vertices, and being uniquely defined up to the choice of starting vertex for the search algorithm.
for sbi so Ds c c++ unix rdbms sql cn osalisha230390
This document contains 35 questions related to data structures and algorithms. It covers topics like data structures used in different areas like databases, networks and hierarchies. Other topics covered include trees, graphs, sorting, hashing and file structures. Sample problems are given related to these topics to test understanding.
This document defines and provides examples of congruent angles, congruent segments, and congruent triangles in geometry. It explains that angles are congruent if they have the same measure, segments are congruent if they have the same length, and triangles are congruent if corresponding angles and sides are congruent. The document provides exercises asking the reader to identify corresponding parts of congruent triangles without diagrams as well as included angles and sides within triangles.
1. The research question investigates the relationship between the force applied to one side of a cantilever beam and the maximum acceleration reached by the free end.
2. Materials used include an acrylic plastic cantilever beam, accelerometer, string, and force meter.
3. Experiments are conducted to pull the cantilever with varying forces and measure the corresponding maximum accelerations using the accelerometer.
4. Results are analyzed to understand the elastic properties of the cantilever beam based on the relationship between applied force and acceleration. Potential and kinetic energy concepts are also explored.
The document summarizes various applications of deep learning that have progressed rapidly since the book was written. It discusses applications in areas such as computer vision, natural language processing, speech recognition and synthesis, robotics, healthcare, and autonomous vehicles. Specific examples mentioned include neural machine translation models from Google, AlphaGo from DeepMind, and autonomous vehicle research from Waymo. It notes that many applications now rely on techniques such as attention mechanisms, generative models, reinforcement learning, and model compression that have developed significantly in recent years.
This document provides an overview of sequence modeling using recurrent and recursive neural networks. It begins with an introduction to classical dynamical systems and unfolding computational graphs for recurrent networks. It then discusses different types of recurrent networks, including those with recurrence through the hidden and output states. Bidirectional and encoder-decoder sequence-to-sequence architectures are also covered. Finally, the document discusses issues like exploding gradients and presents solutions like LSTMs and gradient clipping. Recursive networks and networks with explicit memory components are also introduced.
1. Autoencoders are neural networks that are trained to reconstruct their input. They have an internal representation or "code" layer that compresses the input into a lower-dimensional form.
2. There are different types of regularized autoencoders that can learn meaningful representations, including sparse, denoising, contractive, and stochastic autoencoders. Denoising autoencoders in particular are trained to reconstruct clean inputs from corrupted versions, which can learn the manifold of the data.
3. Contractive autoencoders explicitly regularize the hidden layer activations to resist small changes to the input, encouraging a smooth hidden representation that captures the manifold structure of the data.
This document discusses adversarial machine learning attacks. It begins by reminding the reader about evasion attacks, which aim to find model weaknesses at test time, and poisoning attacks, which compromise the training process. It then discusses evasion attacks in more detail, including the use of adversarial examples and real-world attacks. The document outlines different types of evasion attacks and formulations. It also discusses adversarial training as a potential mitigation technique, as well as universal adversarial perturbations. The document presents experimental results showing that perlin noise attacks can outperform other attacks and that adversarial training is not fully effective. It concludes by emphasizing the need to understand vulnerabilities in machine learning systems and develop effective defenses and testing methodologies.
This document discusses adversarial machine learning and data poisoning attacks. It notes that machine learning systems can be compromised through evasion and poisoning attacks. Poisoning attacks aim to degrade the performance of a machine learning system by compromising the training data. Optimal poisoning attacks can be modeled as bi-level optimization problems and efficiently computed using back-gradient optimization, allowing poisoning points to be generated at scale. Poisoning attacks may be transferable across different machine learning algorithms. The document explores different types of poisoning attacks and defenses like anomaly detection and label sanitization, noting that defenses must account for detectability constraints to defend against more sophisticated attacks.
This document discusses confronting the partition function in probabilistic models. It explains that many probabilistic models are defined by an unnormalized probability distribution that is normalized by a partition function. The partition function is difficult to compute, making the gradient of the log-likelihood challenging. Basic learning algorithms for undirected models involve generating model samples to estimate the negative phase of the gradient. Estimating the partition function is also important for evaluating trained models.
This document discusses approximate inference techniques for probabilistic models. It begins with an introduction to variational inference and how it can be used to approximate intractable distributions. It then discusses applying variational inference to mixture of Gaussian models and exponential family distributions. Finally, it briefly introduces expectation propagation as another approximate inference method before concluding with a summary.
1) Linear factor models represent observed data vectors as a linear combination of latent factors plus noise. They include probabilistic principal component analysis (PCA) and factor analysis.
2) Independent component analysis learns components that are closer to statistically independent than the raw features, and can separate signals like voices or EEG signals.
3) Sparse coding finds a sparse representation of data by solving an optimization problem that minimizes a factor's value and reconstruction error, producing sparse weights.
Boudoir photography, a genre that captures intimate and sensual images of individuals, has experienced significant transformation over the years, particularly in New York City (NYC). Known for its diversity and vibrant arts scene, NYC has been a hub for the evolution of various art forms, including boudoir photography. This article delves into the historical background, cultural significance, technological advancements, and the contemporary landscape of boudoir photography in NYC.
Hadj Ounis's most notable work is his sculpture titled "Metamorphosis." This piece showcases Ounis's mastery of form and texture, as he seamlessly combines metal and wood to create a dynamic and visually striking composition. The juxtaposition of the two materials creates a sense of tension and harmony, inviting viewers to contemplate the relationship between nature and industry.
Fashionista Chic Couture Maze & Coloring Adventures is a coloring and activity book filled with many maze games and coloring activities designed to delight and engage young fashion enthusiasts. Each page offers a unique blend of fashion-themed mazes and stylish illustrations to color, inspiring creativity and problem-solving skills in children.
This document announces the winners of the 2024 Youth Poster Contest organized by MATFORCE. It lists the grand prize and age category winners for grades K-6, 7-12, and individual age groups from 5 years old to 18 years old.
storyboard: Victor and Verlin discussing about top hat
16_graphical_models.pdf
1. Structured Probabilistic
Models for Deep Learning
Lecture slides for Chapter 16 of Deep Learning
www.deeplearningbook.org
Ian Goodfellow
2016-10-04
2. (Goodfellow 2017)
Roadmap
• Challenges of Unstructured Modeling
• Using Graphs to Describe Model Structure
• Sampling from Graphical Models
• Advantages of Structured Modeling
• Structure Learning and Latent Variables
• Inference and Approximate Inference
• The Deep Learning Approach to Structured Probabilistic Modeling
3. (Goodfellow 2017)
Tasks for Generative Models
• Density estimation
• Denoising
• Sample generation
• Missing value imputation
• Conditional sample generation
• Conditional density estimation
4. (Goodfellow 2017)
Samples from a BEGAN
(Berthelot et al, 2017)
Images are 128 pixels wide, 128 pixels tall
R, G, and B pixel at each location.
5. (Goodfellow 2017)
Cost of Tabular Approach
g P(x) by storing
quires kn param
not feasible for s
Number of values per variable
Number of variables
For BEGAN faces: 256
For BEGAN faces:
128 ⇥ 128 = 16384
There are roughly ten to the power of forty thousand
times more points in the discretized domain of the BEGAN
face model than there are atoms in the universe.
6. (Goodfellow 2017)
Tabular Approach is Infeasible
• Memory: cannot store that many parameters
• Runtime: inference and sampling are both slow
• Statistical efficiency: extremely high number of
parameters requires extremely high number of
training examples
7. (Goodfellow 2017)
Roadmap
• Challenges of Unstructured Modeling
• Using Graphs to Describe Model Structure
• Sampling from Graphical Models
• Advantages of Structured Modeling
• Structure Learning and Latent Variables
• Inference and Approximate Inference
• The Deep Learning Approach to Structured Probabilistic Modeling
8. (Goodfellow 2017)
Insight of Model Structure
• Most variables influence each other
• Most variables do not influence each other directly
• Describe influence with a graph
• Edges represent direct influence
• Paths represent indirect influence
• Computational and statistical savings come from omissions
of edges
9. (Goodfellow 2017)
Directed Models
ED PROBABILISTIC MODELS FOR DEEP LEA
t0
t0 t1
t1 t2
t2
Alice Bob Carol
phical model depicting the relay race example. A
shing time t1, because Bob does not get to start
arol only gets to start running after Bob finis
fluences Carol’s finishing time t2.
Figure 16.2
elay race example from section 16.1, suppose we name
Bob’s finishing time t1, and Carol’s finishing time t2.
mate of t1 depends on t0. Our estimate of t2 depends
rectly on t0. We can draw this relationship in a directed
d in figure 16.2.
raphical model defined on variables x is defined by a
hose vertices are the random variables in the model, and
al probability distributions p(xi | PaG(xi)), where
of xi in G. The probability distribution over x is given
p(x) = ⇧ip(xi | PaG(xi)). (16.1)
le, this means that, using the graph drawn in figure 16.2,
, t1, t2) = p(t0)p(t1 | t0)p(t2 | t1). (16.2)
e t0, Bob’s finishing time t1, and Carol’s finishing time t2.
ur estimate of t1 depends on t0. Our estimate of t2 depends
y indirectly on t0. We can draw this relationship in a directed
strated in figure 16.2.
ted graphical model defined on variables x is defined by a
h G whose vertices are the random variables in the model, and
itional probability distributions p(xi | PaG(xi)), where
rents of xi in G. The probability distribution over x is given
p(x) = ⇧ip(xi | PaG(xi)). (16.1)
xample, this means that, using the graph drawn in figure 16.2,
p(t0, t1, t2) = p(t0)p(t1 | t0)p(t2 | t1). (16.2)
time seeing a structured probabilistic model in action. We
t of using it, to observe how structured modeling has many
Directed models work best when influence
clearly flows in one direction
10. (Goodfellow 2017)
Undirected Models
RUCTURED PROBABILISTIC MODELS FOR DEE
hr
hr hy
hy hc
hc
undirected graph representing how your roomma
r work colleague’s health hc affect each other. You
other with a cold, and you and your work colleagu
Undirected models work best when influence
has no clear direction or is best modeled as
flowing in both directions
Do you have a cold?
Does your
roommate have a
cold?
Does your work
colleague have a
cold?
11. (Goodfellow 2017)
Undirected Models
ph G. For each clique C in the graph,3 a factor (C)
al) measures the affinity of the variables in that clique
ssible joint states. The factors are constrained to be
efine an unnormalized probability distribution
p̃(x) = ⇧C2G (C). (16.3)
bility distribution is efficient to work with so long as
encodes the idea that states with higher affinity are
in a Bayesian network, there is little structure to the
here is nothing to guarantee that multiplying them
obability distribution. See figure 16.4 for an example
mation from an undirected graph.
unction
ability distribution is guaranteed to be nonnegative
teed to sum or integrate to 1. To obtain a valid
must use the corresponding normalized probability
p(x) =
1
Z
p̃(x), (16.4)
esults in the probability distribution summing or
Z =
Z
p̃(x)dx. (16.5)
unction
ability distribution is guaranteed to be nonnegative
teed to sum or integrate to 1. To obtain a valid
must use the corresponding normalized probability
p(x) =
1
Z
p̃(x), (16.4)
esults in the probability distribution summing or
Z =
Z
p̃(x)dx. (16.5)
Unnormalized probability
Partition function
12. (Goodfellow 2017)
Separation
R 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARN
a s b a s b
(a) (b)
6: (a) The path between random variable a and random variable b t
cause s is not observed. This means that a and b are not separated.
in, to indicate that it is observed. Because the only path between
and that path is inactive, we can conclude that a and b are separat
Separation and D-Separation
When s is not observed,
influence can flow from a
to b and vice versa through s.
When s is observed,
it blocks the flow of
influence between a
and b: they are
separated
13. (Goodfellow 2017)
Separation example
HAPTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARN
a
b c
d
igure 16.7: An example of reading separation properties from an undirected gr
is shaded to indicate that it is observed. Because observing b blocks the only
to c, we say that a and c are separated from each other given b. The observ
so blocks one path between a and d, but there is a second, active path betw
herefore, a and d are not separated given b.
The nodes a and c are separated
One path between a and d is still active,
though the other path is blocked, so these
two nodes are not separated.
14. (Goodfellow 2017)
d-separation
The flow of influence is more complicated for directed models
The path between a and b is active for all of these graphs:
CHAPTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
a s b
a s b a s b
(a) (b)
a
s
b a s b
c
(c) (d)
CHAPTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
a s b
a s b a s b
(a) (b)
a
s
b a s b
c
(c) (d)
Figure 16.8: All the kinds of active paths of length two that can exist between random
variables a and b. (a) Any path with arrows proceeding directly from a to b or vice versa.
This kind of path becomes blocked if s is observed. We have already seen this kind of
path in the relay race example. (b) Variables a and b are connected by a common cause s.
CHAPTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
a s b
a s b a s b
(a) (b)
a
s
b a s b
c
CHAPTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
a s b
a s b a s b
(a) (b)
a
s
b a s b
c
(c) (d)
Figure 16.8: All the kinds of active paths of length two that can exist between random
15. (Goodfellow 2017)
d-separation example
a b
c
d e
graph, we can read out several d-separation properties. Examples
parated given the empty set
parated given c
parated given c
some variables are no longer d-separated when we observe some
d
Figure 16.9: From this graph, we can read out sever
include:
• a and b are d-separated given the empty set
• a and e are d-separated given c
• d and e are d-separated given c
We can also see that some variables are no longer
variables:
• a and b are not d-separated given c
• a and b are not d-separated given d
c
d
Figure 16.9: From this graph, we can read ou
include:
• a and b are d-separated given the empt
• a and e are d-separated given c
• d and e are d-separated given c
We can also see that some variables are no
variables:
• a and b are not d-separated given c
• a and b are not d-separated given d
Observing variables can activate paths!
16. (Goodfellow 2017)
A complete graph can represent
any probability distribution
R 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARN
10: Examples of complete graphs, which can describe any probability d
how examples with four random variables. (Left) The complete undirec
directed case, the complete graph is unique. (Right) A complete direc
ected case, there is not a unique complete graph. We choose an orde
and draw an arc from each variable to every variable that comes afte
The benefits of graphical models come from omitting edges
17. (Goodfellow 2017)
Converting between graphs
• Any specific probability distribution can be
represented by either an undirected or a directed
graph
• Some probability distributions have conditional
independences that one kind of graph fails to imply
(the distribution is simpler than the graph
describes; need to know the conditional probability
distributions to see the independences)
18. (Goodfellow 2017)
Converting directed to
undirected
APTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
h1
h1 h2
h2 h3
h3
v1
v1 v2
v2 v3
v3
a b
c
a
c
b
h1
h1 h2
h2 h3
h3
v1
v1 v2
v2 v3
v3
a b
c
a
c
b
ure 16.11: Examples of converting directed models (top row) to undirected models
Must add an edge between
unconnected coparents
19. (Goodfellow 2017)
Converting undirected to
directed
ER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
a b
d c
a b
d c
a b
d c
16.12: Converting an undirected model to a directed model. (Left) This und
cannot be converted to a directed model because it has a loop of length fo
ds. Specifically, the undirected model encodes two different independenc
cted model can capture simultaneously: a?c | {b, d} and b?d | {a, c}. (C
vert the undirected model to a directed model, we must triangulate the
uring that all loops of greater than length three have a chord. To do so,
No loops of
length
greater than
three allowed!
Add edges to
triangulate
long loops
Assign
directions to
edges. No
directed cycles
allowed.
20. (Goodfellow 2017)
Factor graphs are less
ambiguous
CHAPTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
a b
c
a b
c
f1
f1
a b
c
f1
f1
f2
f2
f3
f3
Figure 16.13: An example of how a factor graph can resolve ambiguity in the interpretation
of undirected networks. (Left) An undirected network with a clique involving three
variables: a, b and c. (Center) A factor graph corresponding to the same undirected
model. This factor graph has one factor over all three variables. (Right) Another valid
actor graph for the same undirected model. This factor graph has three factors, each
over only two variables. Representation, inference, and learning are all asymptotically
heaper in this factor graph than in the factor graph depicted in the center, even though
Undirected graph: is
this three pairwise
potentials or one
potential over three
variables?
Factor graphs
disambiguate by
placing each potential
in the graph
21. (Goodfellow 2017)
Roadmap
• Challenges of Unstructured Modeling
• Using Graphs to Describe Model Structure
• Sampling from Graphical Models
• Advantages of Structured Modeling
• Structure Learning and Latent Variables
• Inference and Approximate Inference
• The Deep Learning Approach to Structured Probabilistic Modeling
22. (Goodfellow 2017)
Sampling from directed models
• Easy and fast to draw fair samples from the whole
model
• Ancestral sampling: pass through the graph in
topological order. Sample each node given its
parents.
• Harder to sample some nodes given other nodes,
unless the observed nodes are at the start of the
topology
23. (Goodfellow 2017)
Sampling from undirected
models
• Usually requires Markov chains
• Usually cannot be done exactly
• Usually requires multiple iterations even to
approximate
• Described in Chapter 17
24. (Goodfellow 2017)
Roadmap
• Challenges of Unstructured Modeling
• Using Graphs to Describe Model Structure
• Sampling from Graphical Models
• Advantages of Structured Modeling
• Structure Learning and Latent Variables
• Inference and Approximate Inference
• The Deep Learning Approach to Structured Probabilistic Modeling
25. (Goodfellow 2017)
Tabular Case
• Assume each node has a tabular distribution given its parents
• Memory, sampling, inference are now exponential in number of
variables in factor with largest scope
• For many interesting models, this is very small
• e.g., RBMs: all factor scopes are size 2 or 1
• Previously, these costs were exponential in total number of nodes
• Statistically, much easier to estimate this manageable number of
parameters
26. (Goodfellow 2017)
Roadmap
• Challenges of Unstructured Modeling
• Using Graphs to Describe Model Structure
• Sampling from Graphical Models
• Advantages of Structured Modeling
• Structure Learning and Latent Variables
• Inference and Approximate Inference
• The Deep Learning Approach to Structured Probabilistic Modeling
27. (Goodfellow 2017)
Learning about dependencies
• Suppose we have thousands of variables
• Maybe gene expression data
• Some interact
• Some do not
• We do not know which ahead of time
28. (Goodfellow 2017)
Structure learning strategy
• Try out several graphs
• See which graph does best job of some criterion
• Fitting training set with small model complexity
• Fitting validation set
• Iterative search, propose new graphs similar to best
graph so far (remove edge / add edge / flip edge)
29. (Goodfellow 2017)
Latent variable strategy
• Use one graph structure
• Many latent variables
• Dense connections of latent variables to observed variables
• Parameters learn that each latent variable interacts
strongly with only a small subset of observed variables
• Trainable just with gradient descent; no discrete search
over graphs
30. (Goodfellow 2017)
Roadmap
• Challenges of Unstructured Modeling
• Using Graphs to Describe Model Structure
• Sampling from Graphical Models
• Advantages of Structured Modeling
• Structure Learning and Latent Variables
• Inference and Approximate Inference
• The Deep Learning Approach to Structured Probabilistic Modeling
31. (Goodfellow 2017)
Inference and Approximate
Inference
• Inferring marginal distribution over some nodes or
conditional distribution of some nodes given other nodes
is #P hard
• NP-hardness describes decision problems. #P-
hardness describes counting problems, e.g., how many
solutions are there to a problem where finding one
solution is NP-hard
• We usually rely on approximate inference, described in
chapter 19
32. (Goodfellow 2017)
Roadmap
• Challenges of Unstructured Modeling
• Using Graphs to Describe Model Structure
• Sampling from Graphical Models
• Advantages of Structured Modeling
• Structure Learning and Latent Variables
• Inference and Approximate Inference
• The Deep Learning Approach to Structured Probabilistic Modeling
33. (Goodfellow 2017)
Deep Learning Stylistic
Tendencies
• Nodes organized into layers
• High amount of connectivity between layers
• Examples: RBMs, DBMs, GANs, VAEs
CHAPTER 16. STRUCTURED PROBABILISTIC MODELS FOR DEEP LEARNING
h1
h1 h2
h2 h3
h3
v1
v1 v2
v2 v3
v3
h4
h4
Figure 16.14: An RBM drawn as a Markov network.