Boltzmann machines and deep learning
Contents
Plans for the week of April 7-11, 2025
Energy models
Probability model
Marginal and conditional probabilities
Change of notation
Optimization problem
Further simplifications
Optimizing the logarithm instead
Expression for the gradients
The derivative of the partition function
Explicit expression for the derivative
Final expression
Introducing the energy model
More compact notation
Anticipating results to be derived
Boltzmann Machines, marginal and conditional probabilities
Generative and discriminative models
Basics of the Boltzmann machine
More about the basics
Difficult to train
The network layers
Goal of hidden layer
The parameters
Joint distribution
Network Elements, the energy function
Defining different types of RBMs
Gaussian-binary RBM
Cost function
Optimization / Training
Gradients
Simplifications
Positive and negative phases
Gradient examples
Kullback-Leibler relative entropy
Kullback-Leibler divergence
Maximizing log-likelihood
More on the partition function
Setting up for gradient descent calculations
Difference of moments
More observations
Adding hyperparameters
Mathematical details
Binary-binary (BB) RBMs
Marginal probability
Marginal Probability Density Function for the visible units
Marginal probability for hidden units
Conditional Probability Density Functions
On and off probabilities
Conditional probability for visible units
Gaussian-Binary Restricted Boltzmann Machines
Joint Probability Density Function
Partition function
Marginal Probability Density Functions
Then the visible units
Conditional Probability Density Functions
Hidden units
Visible units
Comments
Neural Quantum States
Model for the trial wave function
Allowing for complex valued functions
Squared wave function
Cost function
And the gradient
Additional details
Final equation
Code example
Project 2, VMC for fermions: Efficient calculation of Slater determinants
Matrix elements of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
Efficient calculation of Slater determinants
The gradient and the Laplacian
How to compute the derivates of the Slater determinant
The Slater determinant
Rewriting the Slater determinant
Splitting the Slater determinant
Spin up and spin down parts
Factorization
Number of operations
Counting the number of FLOPS
Computation of ratios
Scaling properties
How to get the determinant
Expectation value of the kinetic energy
Second derivative of the Jastrow factor
Functional form
Second derivative of the Jastrow factor
Gradient and Laplacian
The gradient for the determinant
Jastrow gradient in quantum force
Metropolis Hastings part
Proof for updating algorithm for Slater determinant
Proof for updating algorithm for Slater determinant
Proof for updating algorithm for Slater determinant
Proof for updating algorithm for Slater determinant
Proof for updating algorithm for Slater determinant
Proof for updating algorithm for Slater determinant
Proof for updating algorithm for Slater determinant
Plans for the week of April 7-11, 2025
Neural Networks and Boltzmann Machines, examples of generative deep learning
Discussions of project 2
Video of lecture TBA
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