As it is known, nature cannot be always represented by functions having an exact solution or by a system of equations having a mathematical solution. In the exact sciences a model may be constructed for simplifying the relationships and helping the mathematical inspection to be carried out in order to achieve a mathematical representation of the relationships between data or a correlation among data appearing as not correlated, and a mathematical tool for evaluating the degree or level of the correlation of the data. Furthermore the models may consist in recognizing and constructing images or structures and provide for a graphic representation.
Further to the problem of having tools for apparati having artificial intelligence, in order to better understand, classify and evaluate the physical or chemical world and nature, it has to be noted that artificial intelligence is not limited to the analysis and inspection of nature only relatively to exact scientific or technical problems or structures but must be also confronted with social problems which are far most difficult to be represented by the mathematical tools or by exact computable functions. In this case, the apparatus is confronted with individuals having a specific behavior and acting on their own mind or by means of reactions to instincts, which actions cannot be described by mathematical models because there is no mathematical model and also because there is no clear and univocal rule defining the relations between events whichever kind they are and the behavior.
Human beings have the capacity of analyzing environmental stimuli and deciding to carry out an action as a response to said stimuli also when apparently the stimuli have no relationship among them or are not correlated. This process is carried out sometimes in a non-conscious way giving rise to logically non-predictable actions if considering the known relationship of the stimuli if one ever exists. Nevertheless the action is often correct or approximately correct or leads to a certain successful effect. Such kind of behavior which we can define as intuition or the like seems not to have any logical basis or seems not to be caused by a logical thought.
Since artificial intelligence is based on computational machines there is the need of instruments which may help these machines to analyze or transform information data in such a way as to be simply handled and used by the machine and in such a way as to allow the machine to recognize and/or generate relationship functions which are easier to handle from the mathematical or computational point of view without distorting or leaving information and giving thus the opportunity to simulate at least at a certain degree the “intuitive” behavior of the human intelligence.
Records of a database may be represented as points in a space, the position of the points being determined by variables values which describes the records of the database. In principle the representation may also be reversed in the sense that the variables are represented as points in a space, while the position of each variable is defined by the records. This projection brings certain advantages. As a first technical advantage, certain relationship may be discovered which were hidden in the n-dimensional space of the information data being not intelligible either by human beings nor by machines, since the relative position of the records and/or of the variables in the space where the records or the variables are represented by points is a measure of their similarity or difference. A second technical advantage is that the simplifying of the information data helps in transforming the data in data which may subjected to a computational evaluation and thus to help the machine to analyze the data to determine an appropriate response to the data and to carrying out its computational job in a more rapid and simple way. One might not forget that for mathematical or computational problems there might be theoretically a solution, which cannot be computed in practice.
The solution of a mapping problem allowing to reduce a three dimensional space for the data in a two dimensional space without losing or distorting the information represented by the data has also a great relevance if one considers for example a machine, which collects image data from the environment and which has to generate an image recognizing the objects or at least discriminating certain objects between objects constituting obstacles and objects which do not constitute obstacles and also between objects that might constitute obstacles at a later time. In this case a machine which has the possibility of reducing information about physical objects placed in a three dimensional space and which have a three dimensional extension in a two dimensional map would allow to dramatically simplify the machine construction and to dramatically reduce the computational burden.
The above described technical advantages are present already if one considers non humanoid machines having artificial intelligence. Considering for example humanoid machines such like humanoid robots, the advantages become more important since such a machine has a large number of sensors and a very high computing and evaluation burden is sent to the processing units.
The algorithm to which the present invention relates has not only relevance for artificial intelligence, but can also help human intelligence in inspecting and analyzing the relationships between information data belonging to a n-dimensional space, where n is bigger then 3 by projecting the data onto a two or three dimensional space. This is a representation which can be understood by human intelligence having its senses constructed to sense a three dimensional or two dimensional space. Thus a representation of data in this space can help human intelligence to understand and find out relationships which could be not be recognized in a four or more dimensional space.
A known algorithm for projecting data from a n-dimensional space into a less dimensional space, and particularly onto a three or two dimensional space, uses a predetermined characteristic projection function for computing the position of each point in the projection space. An example for such kind of projection algorithm is the so called Principal Component Analysis, briefly PCA which is described in H. Hotelling “Analysis of a Complex of Statistical Variables into Principal Components” J. Educ. Psychol., 24:498-520, 1933. This algorithm provides the steps of defining N factors and N new variables which are orthogonal. Using this base of new variables a reorganisation of the data is carried out by attempting to put as much information as possible in the first factors under the constraint of linearity. The mapping consist in rewriting the observations/variables using the computed factors and in plotting each one on a two dimensional map using as coordinates the computed factors F1/F2, F3/F4 and so on.
This kind of projection algorithm working only on the base of linear projections determines that some information will be lost during the projection. In order to understand this situation consider a normal projection from a three dimensional space onto a two dimensional space. In a linear projections two points having a certain distance along one of the three dimensions might appear very near if the two dimensional projection space is perpendicular to the third dimension along which the two points are spaced apart. In a very simplified manner this situation takes place using a PCA algorithm. The result of the known technique is that, in the less dimensional space where the information data has been projected, the data relationships is distorted in a dramatic way and the distortion can go so far as to cancel or abnormally enhance relationships between data.