Stable Coorbit Embeddings of Orbifold Quotients Analysis

Core Concepts

Coorbit filter banks unify max filter banks and finite coorbit filter banks, providing injective and locally Lipschitz mappings in the quotient metric.

Abstract

The content discusses stable coorbit embeddings of orbifold quotients, focusing on constructing G-invariant real-valued functions called coorbit filter banks. It unifies previous notions of max filter banks and finite coorbit filter banks. The article establishes properties such as injectivity and local lower Lipschitzness in the quotient metric. It explores the construction of coorbit filter banks for all compact groups G ≤ O(d) and addresses theoretical questions regarding their bi-Lipschitz bounds. The paper delves into the geometric analysis of coorbit maps, emphasizing principal points, semialgebraicity, avoidance notions, and group components realization. Additionally, it covers preliminary concepts on max filtering and continuity of coorbit maps.

Construction and Basic Properties of Coorbit Maps:

• Coorbit maps are invariant, symmetric, and semialgebraic.
• Component coorbit maps interact with group actions.
• Semialgebraicity ensures continuity in embedding orbit spaces.

Realizing Group Components:

• Separation scale determines shrinkage or expansion of realizing group components.
• Small perturbations can only shrink realizing group components.

Continuity of Coorbit Maps:

• Local Lipschitz continuity ensures stability under small perturbations.
• Global Lipschitz continuity guarantees smooth transitions between points.

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Key Insights Distilled From

Stable Coorbit Embeddings of Orbifold Quotients

by Yousef Qaddu... at arxiv.org 03-22-2024

https://arxiv.org/pdf/2403.14042.pdf

Deeper Inquiries

How do coorbit filter banks compare to other types of filters in signal processing

Coorbit filter banks are a generalization of max filter banks and finite coorbit filter banks in signal processing. They provide a unified framework for dealing with ambiguity in data representations caused by group symmetries. Unlike traditional filters that operate on individual data points, coorbit filter banks consider entire orbits of data points under the action of a group of linear isometries. This approach allows for capturing the underlying structure and relationships within the data more effectively.

What implications do the results have for machine learning algorithms dealing with ambiguous data representations

The results obtained from stable coorbit embeddings have significant implications for machine learning algorithms dealing with ambiguous data representations. By augmenting training sets with entire group orbits of each datapoint, these algorithms can address ambiguities stemming from subgroup symmetries in the dataset. This approach helps reduce sample complexity and computational expenses by providing a more structured representation that preserves essential information while minimizing distortion to the original metric space.

How can the concept of avoiding zero vectors be applied to other mathematical models or real-world scenarios

The concept of avoiding zero vectors, as explored in mathematical models like coorbit filtering, can be applied to various real-world scenarios where distinguishing between different classes or categories is crucial. For example:

In anomaly detection: Avoiding zero vectors can help identify abnormal patterns or outliers in datasets by focusing on features that deviate significantly from normal behavior.
In image recognition: Zero vectors could represent noise or irrelevant information, so avoiding them ensures that only meaningful features are considered during classification tasks.
In financial analysis: Zero vectors might indicate missing or incomplete data points, so avoiding them can improve risk assessment models and decision-making processes based on accurate information.