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Managing signal sampling rates is essential in digital signal processing to maintain signal integrity. A decimated signal, characterized by a reduced frequency range due to its lower sampling rate, can be upsampled by inserting zeros between each sample. This upsampling process expands the original spectrum and introduces repeated spectral replicas at intervals dictated by the new Nyquist frequency. To refine this zero-inserted sequence, it is passed through a lowpass filter with a cutoff...
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When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Intrinsic dimension estimation for locally undersampled data.

Vittorio Erba1, Marco Gherardi2, Pietro Rotondo3

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This study introduces a novel intrinsic dimension estimator to overcome limitations in current methods, particularly with undersampled data. The new approach enhances accuracy for complex datasets, improving dimensionality estimation.

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Area of Science:

  • Machine Learning
  • Data Science
  • Manifold Learning

Background:

  • Intrinsic dimension estimation is crucial for understanding complex datasets.
  • Existing methods fail with locally undersampled data, a challenge known as the curse of dimensionality.
  • Accurate dimensionality is vital for tasks like invariant object recognition.

Purpose of the Study:

  • To develop a robust intrinsic dimension estimator that addresses the curse of dimensionality.
  • To improve the reliability of intrinsic dimension estimation for locally undersampled and highly curved manifolds.
  • To introduce a multiscale approach for identifying multiple dimensionalities within a dataset.

Main Methods:

  • Leveraging properties of the tangent space of a manifold.
  • Extending the correlation integral estimator to handle extreme undersampling.
  • Developing a multiscale generalization of the intrinsic dimension estimation algorithm.

Main Results:

  • The new estimator demonstrates improved reliability on locally undersampled datasets.
  • The multiscale approach accurately estimates intrinsic dimensions of extremely curved manifolds.
  • The method successfully handles manifolds generated from global transformations of high-contrast images.

Conclusions:

  • The proposed intrinsic dimension estimator offers a significant advancement over existing techniques.
  • This method enhances the accuracy and robustness of dimensionality estimation for challenging datasets.
  • The findings have implications for invariant object recognition and manifold learning applications.