t-SNE and UMAP are powerful tools for visualizing high-dimensional data in lower dimensions. These non-linear techniques preserve local structure, making them great for revealing hidden patterns and relationships that linear methods like PCA might miss.

Understanding how to apply and tune t-SNE and UMAP is crucial for effective data visualization. By adjusting key parameters like perplexity and n_neighbors, you can balance local and global structure preservation, tailoring the output to your specific dataset and analysis goals.

Non-linear Dimensionality Reduction

Overview of t-SNE and UMAP

• t-SNE (t-Distributed Stochastic Neighbor Embedding) and UMAP (Uniform Manifold Approximation and Projection) are non-linear dimensionality reduction techniques used for visualizing high-dimensional data in lower-dimensional spaces (typically 2D or 3D)
• Both t-SNE and UMAP aim to preserve the local structure of the high-dimensional data in the low-dimensional representation
  • Similar data points in the original space should remain close together in the reduced space
  • Dissimilar data points should be further apart in the reduced space

Key Concepts and Algorithms

• t-SNE converts the high-dimensional Euclidean distances between data points into conditional probabilities that represent similarities
  • Minimizes the Kullback-Leibler divergence between the joint probabilities of the low-dimensional embedding and the high-dimensional data
  • The t-distribution is used to compute the similarity between two points in the low-dimensional space, allowing for a higher probability of dissimilar points being further apart
• UMAP constructs a weighted k-neighbor graph in the high-dimensional space and then optimizes a low-dimensional graph to be as structurally similar as possible
  • Optimization is based on cross-entropy between the two graphs
  • Assumes that the data lies on a locally connected Riemannian manifold and uses a fuzzy topological structure to approximate the manifold
• Both t-SNE and UMAP have a non-convex optimization objective
  • The resulting low-dimensional embeddings can vary across different runs
  • Embeddings are sensitive to the initial random state

t-SNE vs UMAP vs PCA

Linearity and Non-linearity

• Principal Component Analysis (PCA) is a linear dimensionality reduction technique, while t-SNE and UMAP are non-linear techniques
  • PCA finds a new set of orthogonal axes (principal components) that maximize the variance of the projected data
  • Data is transformed linearly onto these axes in PCA
  • t-SNE and UMAP do not rely on linear transformations and can capture more complex, non-linear relationships in the data

Global vs Local Structure Preservation

• PCA preserves the global structure of the data
  • Low-dimensional representation maintains the relative distances between far apart points in the original space
• t-SNE and UMAP focus on preserving the local structure
  • Often at the expense of the global structure
  • Prioritize maintaining the relationships between nearby points in the original space

Deterministic vs Stochastic Results

• PCA is deterministic and has a unique solution for a given dataset
• t-SNE and UMAP are stochastic and can produce different results across runs due to their non-convex optimization

Suitable Data Characteristics and Use Cases

• PCA is better suited for datasets with linear relationships and Gaussian-distributed data
• t-SNE and UMAP are more appropriate for non-linear relationships and complex data distributions
• t-SNE and UMAP are primarily used for visualization purposes
  • They do not provide a direct mapping from the high-dimensional space to the low-dimensional space
  • Difficult to embed new, unseen data points
• PCA can be used for both visualization and as a pre-processing step for other machine learning tasks

Applying t-SNE and UMAP

Input Data and Preprocessing

• The input to t-SNE and UMAP is typically a high-dimensional feature matrix
  • Each row represents a data point
  • Each column represents a feature or dimension
• Before applying t-SNE or UMAP, it is essential to preprocess the data by scaling the features to a consistent range
  • Use standardization or min-max scaling to ensure that the distance calculations are not dominated by features with larger magnitudes

Output and Visualization

• The output of t-SNE and UMAP is a low-dimensional embedding of the data points, usually in 2D or 3D
  • Visualize using scatter plots or other visualization techniques
• Experiment with different hyperparameter settings to find the best representation of the data
  • Perplexity for t-SNE
  • n_neighbors and min_dist for UMAP

Applicability to Various Data Types

• t-SNE and UMAP can be applied to various types of high-dimensional data
  • Images
  • Text embeddings
  • Gene expression data
• Gain insights into the underlying structure and relationships between data points

Comparison with Other Techniques

• Compare the results of t-SNE and UMAP with other dimensionality reduction techniques (PCA)
  • Assess the quality and interpretability of the low-dimensional representations
  • Evaluate the preservation of important patterns and structures in the data

Tuning t-SNE and UMAP Hyperparameters

t-SNE Hyperparameters

• Perplexity balances the attention between local and global aspects of the data
  • Higher values (30-50) result in more global structure
  • Lower values (5-10) emphasize local structure
• Learning_rate determines the speed of the optimization process
  • Higher values lead to faster convergence but potentially less stable results

UMAP Hyperparameters

• n_neighbors controls the trade-off between local and global structure
  • Higher values capture more global structure
  • Lower values focus on local neighborhoods
• min_dist determines the minimum distance between points in the low-dimensional space, affecting the compactness of the clusters
  • Smaller values lead to tighter clusters
  • Larger values produce more dispersed clusters
• n_components specifies the number of dimensions in the low-dimensional embedding (typically set to 2 or 3 for visualization purposes)

Hyperparameter Tuning Strategies

• Use a grid search or random search approach to tune the hyperparameters
  • Evaluate the quality of the visualizations based on domain knowledge and visual inspection
• Optimal hyperparameter settings may vary depending on the characteristics of the dataset
  • Size
  • Dimensionality
  • Presence of noise or outliers
• Assess the stability and reproducibility of the visualizations
  • Run the algorithms multiple times with different random seeds
  • Compare the results

Computational Considerations

• Consider the computational complexity of t-SNE and UMAP when tuning hyperparameters
  • Larger datasets and higher perplexity or n_neighbors values can significantly increase the runtime of the algorithms
  • Balance the quality of the visualizations with the computational resources available