High-dimensional data

As explained in Math and Architectures of Deep Learning

A dataset whose instances are represented with many dimensions or features, making it suitable for dimensionality-reduction methods.

High-dimensional data represents each input instance as a feature vector with many dimensions. In a data matrix, rows represent instances and columns represent feature dimensions; geometrically, each row is a point in feature space. section 2.13 section 4.0

Machine learning searches these spaces for patterns. Points from a class may occupy a relatively small subregion rather than being spread uniformly, creating structure that linear algebra can analyze. section 2.13 section 4.0

High dimensionality matters because it supports both pattern discovery and simplification:

• Treat each feature vector as a point whose coordinates specify an input instance in feature space. section 2.13
• Transform the feature space when some axes show little variation, reducing dimensions while preserving broader patterns. section 2.13
• Use PCA to project multidimensional data onto its first principal axes when storage or representation must be reduced. section 4.4.5

Compression through principal components

Given a large multidimensional dataset and a fixed byte budget, discard components associated with smaller principal values and store projections onto the first few principal axes. This reduces the number of components while limiting information loss. section 4.4.5

Figure 2.13 During rotation, points on the axis of rotation do not change position.

Use the rotation analogy to visualize how a transformation can leave an axis unchanged, motivating eigenvectors as tools for analyzing structured feature spaces. section 2.13

How Math and Architectures of Deep Learning covers it

How deep it goes

Defines feature-space geometry and matrix organization, then connects low-variation axes to reduction without deriving a complete reduction algorithm.

Examples

Uses class regions in feature space, a rotation-axis analogy, and PCA projections for compression.

What you can do after

Apply PCA to project multidimensional data onto fewer principal axes for compact representation.

How it connects

Links feature vectors, feature spaces, matrices, eigenvalues, eigenvectors, dimensionality reduction, PCA, and data compression.

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Also explained in

• 

  liveBookmanning.com
  Advanced Algorithms and Data StructuresMultidimensional data consists of points or records described by multiple independent coordinates or dimensions.
• 

  liveBookmanning.com
  Outlier Detection in PythonA high-dimensional dataset is a dataset whose observations are represented by a large number of feature dimensions.

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Related concepts

• sparsityproduces · Sparsity is the property of data, a representation, or a structure in which relatively few possible entries, relationships, or components are present, nonzero, or active.
• outlier detectionused by · Outlier detection is the process of identifying observations that substantially deviate from an expected or prevailing pattern in data.
• k-d treecontrasts with · A k-d tree is a binary space-partitioning tree that organizes points in multidimensional space by recursively comparing coordinate dimensions.
• Gaussian processused by · A Gaussian process is a probability distribution over functions in which every finite collection of function values has a joint Gaussian distribution.

This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.