Extractable patterns

Though a term support vector machines sounds pretty fancy, the basic idea behind them is pretty easy. For simplicity, let us assume that we have a binary classification problem with a dataset $\mathcal{D} = \{(x^{(i)},y^{(i)}\}$ where $x^{(i)}\in \mathbb{R}^n$ and $y^{(i)}\in \pm1$. Our aim is to find somehow 'optimal' hyperplane, high-dimensional version of plane, that separates positive and negative datapoints sufficiently well. To understand how do we find that optimal hyperplane let us transform our setting to $n$-dimensional Euclidean space.\\ Here we can define an operation of dot product between any two vectors, $\cdot:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$. This dot product is defined as follows: \[ x\cdot y = \sum_{i=1}^n x_iy_i \] where $x = (x_1,x_2,\ldots,x_n)$ and $y=(y_1,y_2,\ldots,y_n)$. Let us fix some vector $w$ and look at the dot product of the type: \[ w\cdot x, \text{ where } \Vert x \Vert =1 \] One might ask when is such product is m...

In this series of posts, I wish to introduce a notion of automatic groups, i.e. groups which has an automatic presentation. To do so, we need to revise notions of regularity and that of automatic relation. Let us start with a notion of regularity. Regular Languages The concept of regularity for formal languages can be approached from several sides. We are going to define regulaity using finite state machines. We start with a finite set $\Sigma$ called an \emph{alphabet}. By concatenating elements of this alphabet, we can form \emph{words} or \emph{strings}. For example, if $\Sigma = \{0,1\}$, then possible words are $01$ and $00111$. Let us denote a collection of all possible words in the given alphabet $\Sigma$ as $\Sigma^*$. We refer to subsets of $\Sigma^*$ as \emph{formal languages} or \emph{languages} for short. There are might be different kinds of languages, in fact $\vert 2^{\mathbb{N}}\vert \}$ many, some of them easy to describe, while others extremely complicated...

In the previous post, we were left with the construction of graph $\Gamma$ from group $G$ with a promise to transform the graph into a metric space. A metric space consists of vertices together with edges. Each edge is assumed to have a length of $1$.  Let's denote the given space $X$. As for metric, we first define a distance between two vertices. A distance between two vertices $h,k$ corresponds to the length of the shortest path between them. Formally: \[ d(g,h) = \min\{n\mid g^{-1}h = s_1s_2\ldots s_n \},\quad s_i\in S \] As for points on the edges, the distance defined similarly with consideration that each point should first travel to one of its two neighboring vertices. With this metric, $X$ or $Cay(G,S)$ becomes a metric space and referred to as Cayley graph of $G$ with respect to $S$. We can say few things about $X$ immediately. Firstly, $X$ is a proper metric space, i.e. its closed balls are compact. Secondly, $X$ is a geodesic metric space, i.e. there is a short...

From earliest of times, people have been using tools which simplified and enriched their lives. At first humans used natural tool such as stones to manufacture necessary items, and animals as a source of food, transportation and clothing. Then people learned to design more sophisticated tools such as engines, planes and cars. So the tools were progressing from the simplest to more advanced by the means on human intellect. In twentieth century people invented computers which could perform millions of  operations in a fractions of second, this allowed people to scale things up in an unprecedented levels. Currently, people are trying to use these computers to produce the intelligence, a mighty feat if it is to be accomplished. But the new tool has a potential of altering our very lifestyle. If earlier tools were intended to quicken thing up, the tool of artificial intelligence has a potential of turning things upside down. For this reason, I believe, we need to be aware of general de...

A Cayley graph can be considered as an attempt to geometrize or visualize groups . Algebraically, a group is just a bunch of letters some of them having an inverse script. By visualizing groups, we may observe some patterns in their structure. So let's see how can we construct Cayley graphs. Take an arbitrary group $G$ with a set of generators $S\subset G$. In other words, every element of $G$ can be written as: \begin{equation} g = s_1^{\epsilon_1}s_2^{\epsilon_2}\ldots s_n^{\epsilon_n},\quad \epsilon_i\in\{\pm1 \} \end{equation} As for $S$, we assume that $S^{-1} = S$, i.e. $s\in S$ implies $s^{-1}\in S$, and $1\not\in S$. In fact these assumptions are not necessary, but tend to make life a bit easier. Let us construct a Cayley graph $\Gamma=(V,E)$ as follows: $V=G$ $(g,h)\in E$ if $gs = h$ for some $s\in S$, or equivalently $g^{-1}h\in S$ We can observe few things straight away about $\Gamma$. It is undirected as $gs = h$ happens when $g = hs^{-1}$. Any two vert...

In this post I would like to relate some of the reasons why I decided to study automatic groups near the end of my PhD term. I have always been fascinated about patterns occurring in universe. I marvel at the objects of ever-increasing complexities populating our universe. Probably, that is the reason why I chose to study mathematics. For me mathematics is a study of patterns. But these patterns are not limited to our physical universe. Most of the time, they live in some abstract universes. In those universes, patterns live in different continents which can interpreted as fields like Algebra, Analysis, Geometry, Topology, Mathematical Logic and much more. Some patterns occur at different continents and travel back and forth along some bridges. One of the most famous among such bridges is Langlands program. Pondering on such bridges gives me a deep sense of wonder. In my third year, I mainly focused on a notion of algorithmic randomness. A randomness occurs when some patterns are no...

Hi, my name is Birzhan Moldagaliyev. In this blog I hope to share my journey in search for patterns in the universe. I believe that Mathematics is a study of general patterns taking place in abstract spaces. My current research involves studying geometries of Cayley graphs corresponding to automatic and Cayley-automatic groups. I intend to share my research progress in this blog. As for patterns found in the real world, I believe that machine learning is one of the most potent tools to harness such patterns. So I hope to learn some Machine Learning and share my learning experience here. I hope that you, reader, will find some of the things written here useful. Feel free to comment anything here. Thanks, Birzhan.