SAMPLING OF SURFACES AND LEARNING FUNCTIONS IN HIGH DIMENSIONS.

Area of Science:

• Machine Learning
• Data Representation
• Signal Processing

Background:

• Efficient data representation in high-dimensional spaces is crucial for machine learning.
• Non-linear data structures pose challenges for traditional methods.
• Modeling data as points on a smooth surface offers a new perspective.

Purpose of the Study:

• To develop a novel method for representing non-linear data in high-dimensional spaces.
• To introduce algorithms for surface recovery and multidimensional function learning.
• To demonstrate the utility of the proposed model in practical applications.

Main Methods:

• Modeling data as points on a smooth surface, specifically the zero level-set of a bandlimited function.
• Developing a non-linear lifting technique to map surface points to a low-dimensional subspace.
• Introducing algorithms for surface recovery from sparse samples and learning bandlimited functions from data.

Main Results:

• A novel representation of data in high-dimensional spaces using bandlimited functions.
• Development of algorithms for surface recovery and multidimensional function learning.
• Demonstrated applicability in image denoising tasks.

Conclusions:

• The proposed surface model and associated algorithms offer an effective way to handle non-linear data structures.
• This approach facilitates efficient data representation and learning in machine learning.
• The method shows promise for practical applications such as image denoising.