Dimensionality reduction is a process used in data analysis and machine learning to reduce the number of random variables under consideration, by obtaining a set of principal variables. This technique helps in mitigating the curse of dimensionality, improving model performance, and visualizing high-dimensional data in a more comprehensible way.
Manifold Learning is a type of unsupervised learning that seeks to uncover the low-dimensional structure embedded within high-dimensional data by assuming that the data lies on a manifold. It is particularly useful for dimensionality reduction and visualization, helping to preserve the intrinsic geometry of the data while reducing computational complexity.
High-dimensional geometry studies the properties and behaviors of geometric objects in spaces with more than three dimensions, often revealing counterintuitive phenomena not present in lower dimensions. This field is crucial for understanding complex systems in data science, machine learning, and theoretical physics, where high-dimensional spaces frequently occur.
Monte Carlo Integration is a numerical method that uses random sampling to approximate the value of definite integrals, particularly useful when dealing with high-dimensional spaces or complex domains. It relies on the law of large numbers to converge to an accurate estimate as the number of samples increases, making it a powerful tool in fields like finance, physics, and engineering.
Locality-Sensitive Hashing (LSH) is a technique used to approximate nearest neighbor searches in high-dimensional spaces by hashing input items so that similar items map to the same 'buckets' with high probability. This method significantly reduces the dimensionality of data, enabling efficient similarity searches in large datasets while sacrificing some precision for speed.
Sampling-based algorithms are a class of algorithms used primarily in robotics and computer graphics for solving high-dimensional path planning problems. They work by randomly sampling the search space and constructing a graph or tree that approximates the solution space, making them particularly effective for complex, non-linear environments where traditional methods struggle.
The Probabilistic Roadmap (PRM) method is a sampling-based algorithm used in robotics and computational geometry to find paths for moving objects in complex environments by constructing a network of feasible paths. It efficiently navigates through high-dimensional spaces by randomly sampling the configuration space and connecting these samples to form a roadmap that captures the connectivity of the space.