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How to reduce dimension with PCA and random projections?

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Combining data-oblivious sketching with data-aware principal component analysis (PCA) offers efficient dimension reduction. This study analyzes "sketch and solve" methods, finding they provide accurate results even with weak signals and non-negligible projection dimensions.

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Area of Science:

  • Data Science
  • Machine Learning
  • Statistical Analysis

Background:

  • Increasing data size and complexity necessitate effective dimension reduction techniques.
  • Data-oblivious methods (e.g., random projections) offer speed, while data-aware methods (e.g., PCA) offer adaptivity.
  • Combining these approaches can leverage the strengths of both.

Purpose of the Study:

  • To investigate "sketch and solve" methods that combine random projection (sketching) with principal component analysis (PCA).
  • To analyze the performance of various sketching techniques in a signal-plus-noise data model.
  • To provide asymptotically exact results applicable to weak signals and non-negligible projection dimensions.

Main Methods:

  • Evaluation of popular sketching methods including random iid projections, random sampling, subsampled Hadamard transform, and CountSketch.
  • Analysis within a general "signal-plus-noise" (spiked) data model.
  • Study of performance under varying signal strengths and covariance structures.

Main Results:

  • Signal strength reduction under projection depends intricately on data structure and sketching method.
  • Orthogonal projections demonstrate slightly higher accuracy.
  • Randomization has minimal negative impact due to the concentration of measure phenomenon.
  • CountSketch performance can be enhanced through normalization.

Conclusions:

  • The "sketch and solve" approach effectively combines the benefits of sketching and PCA for dimension reduction.
  • The findings offer accurate theoretical guarantees for statistical learning and data analysis, particularly in challenging scenarios.
  • Simulations and empirical data analysis confirm the high accuracy of the proposed methods.