Sal­im Bouzeb­da is a tenured uni­ver­si­ty pro­fes­sor and Direc­tor of UTC-LMAC (Applied Maths). He describes the dif­fer­ent math­e­mat­i­cal tools used in arti­fi­cial intel­li­gence (AI) and their role.

Math­e­mat­ics lies At the heart of AI tech­nolo­gies. The first math­e­mat­i­cal mod­el of neur­al net­works was devel­oped in the 1940s. How­ev­er, it was in 1956 that the term AI was used first. Since then, var­i­ous AI tech­nolo­gies have been devel­oped over the years. The explo­sion of Big Data, in par­tic­u­lar since 2010, has changed the game with so-called “gen­er­a­tive” AI, which relies on com­plex algo­rithms capa­ble of pro­cess­ing large amounts of data to mim­ic real­world sit­u­a­tions and behaviours.

These algo­rithms require spe­cif­ic math­e­mat­i­cal tools, depend­ing on the AI mod­els devel­oped and their fields of appli­ca­tion. First of all, there’s lin­ear alge­bra. “This is an essen­tial branch for cal­cu­la­tions per­formed by neur­al net­works. Input data, con­nec­tion points between neu­rons and trans­for­ma­tions car­ried out in the net­work lay­ers are gen­er­al­ly rep­re­sent­ed in either matrix or vec­tor form. These tools are used, for exam­ple, in image recog­ni­tion, where each pix­el is rep­re­sent­ed by a num­ber and the final image is then sym­bol­ized by a matrix or a vec­tor”, describes Sal­im Bouzebda.

AI tools also call on oth­er math­e­mat­i­cal tools, includ­ing dif­fer­en­tial cal­cu­lus, ran­dom mod­els based on prob­a­bil­i­ty and sta­tis­tics, graph the­o­ry and search algo­rithms, infor­ma­tion the­o­ry and data com­pres­sion. The for­mer enables the para­me­ters of AI mod­els to be adjust­ed, par­tic­u­lar­ly in super­vised learn­ing, and thus mod­els can be opti­mized. “In this case, we know the input data and we know the results obtained. Once the oper­a­tion has been repeat­ed many times on sev­er­al exper­i­ments with large amounts of his­tor­i­cal data, we try to min­i­mize the mod­el using a cost func­tion that will enable us to reduce, as far as pos­si­ble, the dis­tance between real­i­ty and what we are esti­mat­ing”, he points out.

The sec­ond are used in sit­u­a­tions of uncer­tain­ty. “These mod­els enable us to mea­sure the uncer­tain­ty asso­ci­at­ed with deci­sions made by AI sys­tems. Thus, if you have a huge num­ber of para­me­ters to man­age, you’ll have a real prob­lem inter­pret­ing and clas­si­fy­ing data. To get around this prob­lem while retain­ing as much infor­ma­tion as pos­si­ble, we per­form an encod­ing process that we project into a new space of small­er dimen­sions, and there­fore make things more rea­son­able to study. Once we’ve clas­si­fied the ini­tial data, we decode it. This enables us to obtain a new sig­nal very sim­i­lar to the orig­i­nal one, but with the cor­re­spond­ing class,” he explains.

As for graphs, they apply in par­tic­u­lar to rela­tion­ships between objects. “Using the data avail­able, we need to find a cer­tain graph com­pat­i­ble with that data. In this case, the nodes rep­re­sent the “indi­vid­u­als” and the edges the “rela­tion­ships” between them. This method was applied dur­ing the Covid-19 pan­dem­ic to trace pos­si­ble con­t­a­m­i­na­tion but is also used in a large num­ber of con­sumer appli­ca­tions: social net­works, mobil­i­ty or, more anec­do­tal­ly, dat­ing sites…”, assures Sal­im Bouzebda.

Final­ly, infor­ma­tion the­o­ry and data com­pres­sion. “In par­tic­u­lar, it is used to trans­mit and store data at the right lev­el of qual­i­ty. Gen­er­al­ly speak­ing, data has a cer­tain size and, if stored as such, will not only con­sume a lot of mem­o­ry but will also be more dif­fi­cult to retrieve. To avoid this pit­fall, data is com­pressed to a min­i­mum size, while retain­ing almost all the orig­i­nal infor­ma­tion. This method is used in com­put­er vision tools in par­tic­u­lar,” he concludes.

MSD