Operatorski račun nad programskimi prostori

SAJOVIC, ŽIGA (author), Robič, Borut (mentor)

V delu razvijemo algebraični jezik, ki predstavlja formalni račun za globoko učenje, in je hkrati model, v katerem je programe mogoče tako implementirati kot tudi preučevati. V ta namen razvijemo abstraktni računski model avtomatsko odvedljivih programov. V njem so programi elementi t. i. programskih prostorov. Programe gledamo kot preslikave končno-dimenzionalnega vektorskega prostora vase, imenovanega navidezni pomnilniški prostor. Navidezni pomnilniški prostor je algebra programov, torej algebraična podatkovna struktura (s katero je mogoče računati). Elementi navideznega pomnilniškega prostora pa omogočajo razvoj programov v neskončne tenzorske vrste. Na programskih prostorih definiramo operator odvajanja, s pomočjo njegovih potenc pa implementiramo posplošen operator premika in operator kompozicije programov. Tako konstruiran algebraični jezik je poln model globokega učenja. Omogoča takšen način izražanja programov, da že njihov zapis nudi teoretični vpogled vanje.

• Language: Slovenian
• Keywords: operatorska algebra, tenzorska algebra, nevronske mreže, globoko učenje, avtomatsko odvajanje
• Work type: Bachelor thesis/paper
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2017
• PID: 20.500.12556/RUL-95108
• Publication date in RUL: 14.09.2017

SAJOVIC, ŽIGA, 2017, Operatorski račun nad programskimi prostori [online]. Bachelor’s thesis. [Accessed 4 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=95108

Secondary language

• Language: English
• Title: Operational calculus on program spaces

Abstract

In this work we develop an algebraic language that represents a formal calculus for deep learning and is, at the same time a model which enables implementations and investigations of programs. To this purpose, we develop an abstract computational model of automatically differentiable programs. In the model, programs are elements of op. cit. programming spaces. Programs are viewed as maps from a finite-dimensional vector space to itself op. cit. virtual memory space. Virtual memory space is an algebra of programs, an algebraic data structure (one can calculate with). The elements of the virtual memory space give the expansion of a program into an infinite tensor series. We define a differential operator on programming spaces and, using its powers, implement the general shift operator and the operator of program composition. The algebraic language constructed in this way is a complete model of deep learning. It enables us to express programs in such a way, that their properties may be derived from their source codes.

• Keywords: operator algebra, tensor algebra, neural networks, deep learning, automatic differentiation