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...