In the last post, we have looked at the Deutsch-Jozsa algorithm that is considered to be the first example of a quantum algorithm that is structurally more efficient than any classical algorithm can probably be. However, the problem solved by the algorithm is rather special. This does, of course, raise the question whether a similar … Continue reading Grover’s algorithm – unstructured search with a quantum computer

# Category: Mathematics

# Qubits and Hilbert spaces

When you use your favorite search engine to search for information on quantum computing, the first term that will most likely jump at you is the qubit. In this post, I will try to explain what this is and how it is related to the usual framework of quantum mechanics. Please be aware that this … Continue reading Qubits and Hilbert spaces

# The EM algorithm and Gaussian mixture models – part II

In this post, I will discuss the general form of the EM algorithm to obtain a maximum likelihood estimator for a model with latent variables. First, let us describe our model. We suppose that we are given some joint distribution of a random variable X (the observed variables) and and random variable Z (the latent … Continue reading The EM algorithm and Gaussian mixture models – part II

# The EM algorithm and Gaussian mixture models – part I

In the last few posts on machine learning, we have looked in detail at restricted Boltzmann machines. RBMs are a prime example for unsupervised learning - they learn a given distribution and are able to extract features from a data set, without the need to label the data upfront. However, there are of course many … Continue reading The EM algorithm and Gaussian mixture models – part I

# Why you need statistics to understand neuronal networks

When I tried to learn about neuronal networks first, I did what probably most of us would do - I started to look for tutorials, blogs etc. on the web and was surprised by the vast amount of resources that I found. Almost every blog or webpage about neuronal networks has a section on training … Continue reading Why you need statistics to understand neuronal networks

# The Metropolis-Hastings algorithm

In this post, we will investigate the Metropolis-Hastings algorithm, which is still one of the most popular algorithms in the field of Markov chain Monte Carlo methods, even though its first appearence (see [1]) happened in 1953, more than 60 years in the past. It does for instance appear on the CiSe top ten list … Continue reading The Metropolis-Hastings algorithm

# Recurrent and ergodic Markov chains

Today, we will look in more detail into convergence of Markov chains - what does it actually mean and how can we tell, given the transition matrix of a Markov chain on a finite state space, whether it actually converges. So suppose that we are given a Markov chain on a finite state space, with … Continue reading Recurrent and ergodic Markov chains