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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Representation Tradeoffs for Hyperbolic Embeddings.

Christopher De Sa1, Albert Gu2, Christopher Ré2

  • 1Department of Computer Science, Cornell University.

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Summary
This summary is machine-generated.

This study introduces novel hyperbolic embedding methods for hierarchical data, achieving high precision with few dimensions. The new techniques offer significant improvements over existing approaches for data representation and dimensionality reduction.

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Area of Science:

  • Machine Learning
  • Data Science
  • Computational Geometry

Background:

  • Hierarchical data structures are common in various domains.
  • Existing embedding methods often require high dimensionality for good performance.
  • Hyperbolic geometry offers potential for efficient representation of hierarchical data.

Purpose of the Study:

  • To develop a combinatorial construction for embedding trees in hyperbolic space with low distortion.
  • To propose a hyperbolic generalization of multidimensional scaling (h-MDS) for general metric spaces.
  • To create a scalable PyTorch implementation for hyperbolic embeddings.

Main Methods:

  • Combinatorial tree embedding in hyperbolic space without optimization.
  • Hyperbolic generalization of multidimensional scaling (h-MDS).
  • Exact recovery of hyperbolic points and perturbation analysis.
  • Development of a PyTorch-based implementation.

Main Results:

  • Achieved mean-average-precision of 0.989 in 2D on WordNet, outperforming existing methods (0.87 in 200D).
  • Established upper and lower bounds for the precision-dimensionality tradeoff in hyperbolic embeddings.
  • h-MDS demonstrated consistently low distortion with few dimensions across datasets.
  • Developed a scalable PyTorch implementation handling incomplete information.

Conclusions:

  • Hyperbolic embeddings provide a powerful and efficient method for representing hierarchical data.
  • The proposed combinatorial and h-MDS methods offer significant advantages in dimensionality and performance.
  • The developed implementation facilitates practical application of hyperbolic embeddings.