Geometric Deep Learning 06.03.2023|Using Machine Learning to Approximate Functions

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The text discusses the use of machine learning algorithms to approximate a given function based on a finite set of data points. The approach involves reducing the space of possible functions to a manageable class, such as Lipschitz continuous functions, and using symmetries to further constrain the class. This is done by making assumptions about the structure of the space from which the data points were obtained. The goal is to find a function that best approximates the given data points, and this is achieved by selecting a family of functions and using a loss function to measure the quality of the approximation. The algorithm is trained by selecting the best function within the family to approximate the data points, and then tested on new data to evaluate its performance.

The text also mentions theoretical results that show certain classes of functions, such as those defined by a two-layer perceptron, can approximate any continuous function in a given space. However, the limitations of computational resources and available data must be taken into account when implementing these algorithms. The text also discusses the curse of dimensionality, which refers to the exponential increase in the amount of data required to accurately reconstruct a function in high-dimensional spaces. To address this issue, the text suggests using a measure of complexity to select the simplest function within a family that satisfies the desired constraints. The text also briefly mentions Bayesian approaches to selecting functions based on prior probabilities.

Additionally, the text introduces the concept of symmetries in the context of signal processing and how they can be used to further constrain the class of functions. The text also discusses the use of vector spaces to represent signals and how this can be used to define the domain of a signal. In addition, the text introduces the concept of groups and their properties, such as closure under multiplication and the use of symmetries to constrain the group. The text also discusses the connection between group actions on vectors and the inverse action on the group itself. Finally, the text relates these concepts to the use of algorithms in image processing, where the desired function should be invariant to certain transformations of the image. The text also discusses the concept of automorphisms, which are functions that preserve the structure of a given set, such as a group or a topological space. These functions can be used to identify isomorphic structures and to study the properties of groups and topological spaces.

Видео Geometric Deep Learning 06.03.2023|Using Machine Learning to Approximate Functions канала Data Lounge