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AI and Geometry

Columbia University_021124C
[Columbia University - Kevin Dale]

- Overview

Artificial intelligence (AI) can understand geometry concepts and can be trained to recognize and interpret shapes, angles, and measurements. AI algorithms can learn geometric relationships directly from data, which can help with statistical analysis. 

AI has made significant progress in solving math problems, but geometry problems can be challenging because they rely on both text and diagrams. For example, symbolic characters like "\triangleABC" can connect the text description to the corresponding diagram. 

AI has been used to find the properties of atomic pieces of geometry, which can help accelerate mathematical discoveries. Moreover, geometric methods have enabled the development of algorithms that can discern patterns in high-dimensional spaces that are not tractable for humans to visualize. 

Algorithms powered by AI are capable of learning these geometric relationships directly from data, further augmenting the statistical analysis.

AI and geometry are related in a few ways: 

  • Machine vision, solid modeling, and robot planning: Algorithms for these areas are largely based on geometric concepts.
  • Machine learning: Machine learning (ML) can be used to find the properties of atomic pieces of geometry, which can help accelerate mathematical discoveries.
  • Geometrical models: One example of a geometrical model in ML is the Support Vector Machine (SVM). SVM is a supervised learning algorithm that separates data points by finding the optimal hyperplane.
  • Geometric deep learning: This is an umbrella term for approaches that consider a broad class of ML problems from the perspectives of symmetry and invariance.
  • Generative AI: This is a subset of ML that relies on mathematical models to generate new data instances that resemble a given set of data.


Please refer to the following for details:


- Elementary Geometry

Geometry is the branch of mathematics that studies the properties of space, such as distance, shape, size, and the relative position of figures. Geometry, like arithmetic, is one of the oldest branches of mathematics. Mathematicians who work in the field of geometry are called geometers. Until the 19th century, geometry was devoted almost entirely to Euclidean geometry, which included basic concepts such as points, lines, surfaces, distances, angles, surfaces, and curves. 

Geometry was originally developed to simulate the physical world, and it has applications in almost all fields of science, as well as in art, architecture, and other graphics-related activities. Geometry also has applications in apparently unrelated areas of mathematics. For example, the method of algebraic geometry was the basis for Wiles' proof of Fermat's Last Theorem, a problem formulated in elementary arithmetic that had remained unsolved for centuries. 

Since the late 19th century, the scope of geometry has greatly expanded, and the field has been split into many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also Called combinatorial geometry) geometry) etc. - or about neglected properties of Euclidean spaces - projective geometry which considers only point alignments and not distances and parallelisms, affine geometry which omits the notions of angles and distances, omits Continuity of finite geometries, etc. 

The expansion of the scope of geometry led to a change in the meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its space provided by Euclidean geometry Model; Currently, a geometric space, or space for short, is a mathematical structure on which some geometry is defined.


- Relations, Geometry, and AI 

Geometry is the branch of mathematics that studies the size, shape, position, angles and dimensions of things. Artificial intelligence (AI) can use geometric models to analyze data, sort, group and predict. For example, nearest neighbor methods are geometric models used in classification and regression problems.

In their provocative paper, Peter Battaglia and colleagues argue that in order for AI to achieve the capabilities of human intelligence, it must be able to compute using highly structured data types, such as those processed by humans. 

This means that AI must be able to process information about the relationships between objects that is efficiently encoded as a graph, a collection of nodes and the edges between those nodes, and the attributes attached to the nodes and edges.

A simple example is an image, where consider a graph shaped like a square grid, where pixel values are attributes attached to specific nodes in the grid. The authors go on to describe the framework they developed for computing with graph-structured data, and in particular how to build neural networks for such computing.


- Graph Computing for Big Data

The problem recognizes that most data is "sensory" in nature, i.e. it is obtained by measuring numerical values and recording their values as vector coordinates in a high-dimensional space, and therefore is not equipped with any explicit underlying graphical structure. 

It follows that in order to achieve human-like intelligence when studying such data, it is necessary to construct graphs or relational structures from sensory data, and performing such construction is a key element of human intelligence. 

The goal is to describe a method to perform this task precisely and to describe how to construct a neural network directly from the output of such a method. We can summarize the approach as follows. 

  • The key element is the set of features or sensor geometry or measurements that capture sensory data.
  • The second key element is covering feature sets by subsetting. Covers are usually chosen to relate well to the geometry in an appropriate sense. 
  • Geometry and correlation overlays are used together to build graph structures based on the characteristics that define the data. 
  • They are also used to build similar pooling structures used in convolutional neural networks.


Geometric Models

Graph technology is used in artificial intelligence (AI) to show how data is connected. Graphs can represent relationships between entities, such as people, devices, and locations. Graph AI uses neural networks to provide insights when relationships between entities are important. 

Geometric deep learning (DL) is a subfield of ML that analyzes non-Euclidean data, such as graphs and manifolds. The goal of geometric DL is to generalize neural networks to non-Euclidean structured data. 

Geometric models use geometric concepts to solve problems, such as classification, regression, and clustering. These models are based on the idea that data points in a high-dimensional space can be represented by a lower-dimensional subspace, known as a manifold. 

Geometric models can be used in a variety of ML applications, including: Data analysis, Sorting, Grouping, Prediction.  The nearest neighbor approach is one example of a geometric model.


- Geometric Models in Machine Learning

(Salem, Massachusetts - Harvard Taiwan Student Association)

A geometric model in ML is a class of models that describe and process data using geometrical ideas and methods. When working with structured data or data that naturally has a spatial or relational nature, these models are especially helpful. 

A geometric model in machine learning is a mathematical model that uses geometry to explain the properties and connections of a system or element. Geometric models can be used to represent information in a way that makes it easier to analyze its features and connections. 

Geometric models are based on the idea that data points in a high-dimensional space can be represented by a lower-dimensional subspace. These models can be used to solve problems like classification, regression, and clustering. 

Geometric models can also be used in computer vision to solve visual tasks. These models define similarity by considering the geometry of the instance space. 

Geometric models can describe features as points in two dimensions (x- and y-axis) or a three-dimensional space (x, y, and z). For example, temperature as a function of time can be modeled in two axes. 

Geometric models can be used to describe regions of the data which correspond to man-made constructions. 


- Geometric Deep Learning 

Geometric deep learning (DL) is a broad term for approaches that consider ML problems from the perspectives of symmetry and invariance. It allows us to analyze data in its native form, instead of converting it into a lower dimensional space. This allows us to take advantage of data with inherent relationships, connections, and shared properties.

Geometric DL is useful because it can generalize to a wide range of problems, even those involving complex and structured data. This makes it an essential area of research for many fields where traditional deep learning methods may not be effective. 

Geometric DL has been applied to macromolecular structure (i.e., molecular graph) in structure-based drug design. Structure-based drug design identifies appropriate ligands by utilizing the three-dimensional geometric information of macromolecules such as proteins or nucleic acids. 


[More to come ...]

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