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We are inundated with huge volumes of data from all different domains, including scientific, medical, social media, and educational data. Analysis of such data is a crucial requirement. With the increasing amount of data, it is important to have approaches to extract simple and meaningful representations of complex data. The above methods work on the same assumption that the data lies close to a small dimensional manifold despite having a large environmental dimension and look for the lowest dimensional manifold that best characterizes the data.
Multiple learning methods are used in representation learning, where high-dimensional data is transformed into lower-dimensional space while keeping crucial data features intact. Although the multiple hypothesis works for most types of data, it does not work well on data with singularities. Singularities are the regions where the multiple assumption breaks down and may contain important information. These regions violate the uniformity or regularity properties of a variety.
Researchers have proposed a topological framework called TARDIS (Topological Algorithm for Robust Singularity Discovery) to address the challenge of identifying and characterizing singularities in data. This unsupervised representation learning framework detects singular regions in point cloud data and has been designed to be independent of the geometric or stochastic properties of the data, requiring only a notion of the intrinsic dimension of the neighborhoods. Its objective is to address two key aspects: quantifying the local intrinsic dimension and evaluating the variety of a point on multiple scales.
The authors have mentioned that local intrinsic dimension quantization measures the effective dimensionality of the neighborhood of a data point. The framework has achieved this by using topological methods, particularly persistent homology, which is a mathematical tool used to study the shape and structure of data at different scales. It estimates the intrinsic dimension of the neighborhood of a point by applying persistent homology, which gives information about the local geometric complexity. This local intrinsic dimension measures the degree to which the data point is multiple and indicates whether it conforms to the low-dimensional multiple assumption or behaves differently.
The Euclidean Score, which evaluates the variety of a point on different scales, quantifies the deviation of a point from Euclidean behavior, revealing the existence of singularities or non-manifold structures. The framework captures differences in the multiplicity of a point by accounting for Euclidicity at various scales, allowing for the detection of singularities and understanding of local geometric complexity.
The team has provided theoretical assurances about the quality of approximation of this framework for certain classes of spaces, including manifolds. They have performed experiments on a variety of data sets, from high-dimensional image collections to spaces with known singularities, to validate their theory. These findings showed how well the approach identifies and processes non-multiple slices in the data, shedding light on the limitations of the multiple hypothesis and exposing important data hidden in singular regions.
In conclusion, this approach effectively challenges the multiplicity assumption and is efficient in detecting singularities that are the points that violate the multiplicity assumption.
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Tanya Malhotra is a final year student at the University of Petroleum and Power Studies, Dehradun, studying BTech in Computer Engineering with a specialization in Artificial Intelligence and Machine Learning.
She is a data science enthusiast with good analytical and critical thinking, along with a keen interest in acquiring new skills, leading groups, and managing work in an organized manner.
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