7.3 t-SNE and UMAP

t-SNE and UMAP are powerful tools for visualizing complex quantum data. They reduce high-dimensional datasets to 2D or 3D, revealing hidden patterns and clusters that linear methods like PCA might miss.

These techniques help us understand quantum systems better by showing relationships between data points. t-SNE focuses on local structure, while UMAP balances local and global, offering different insights into quantum datasets.

t-SNE and UMAP for Dimensionality Reduction

Principles and Advantages

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  t-SNE in Python [single cell RNA-seq example and hyperparameter optimization] - Renesh Bedre View original

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• t-SNE (t-Distributed Stochastic Neighbor Embedding) preserves the local structure of high-dimensional data in a low-dimensional space (typically 2D or 3D) for visualization purposes
• t-SNE minimizes the divergence between the probability distributions of pairwise similarities in the high-dimensional and low-dimensional spaces using a Student's t-distribution to compute the similarities in the low-dimensional space
• UMAP (Uniform Manifold Approximation and Projection) preserves both the local and global structure of the high-dimensional data in the low-dimensional representation
• UMAP constructs a weighted k-neighbor graph in the high-dimensional space and optimizes a low-dimensional graph to have a similar topological structure using a cross-entropy loss function and stochastic gradient descent

Benefits for Quantum Machine Learning

• t-SNE and UMAP are advantageous for visualizing and exploring complex, high-dimensional datasets encountered in quantum machine learning where linear dimensionality reduction techniques (PCA) may fail to capture nonlinear relationships
• t-SNE and UMAP reveal hidden patterns, clusters, and structures in the data facilitating the understanding and interpretation of high-dimensional quantum datasets
• The low-dimensional representations generated by t-SNE and UMAP provide valuable insights into the structure and patterns present in high-dimensional quantum datasets
• Clusters in the low-dimensional representation suggest the presence of distinct groups or states in the quantum data corresponding to different quantum system configurations, measurement outcomes, or physical properties

Implementing t-SNE and UMAP on Quantum Data

t-SNE Implementation Steps

1. Compute the pairwise similarities between data points in the high-dimensional space using a Gaussian kernel
2. Compute the pairwise similarities between data points in the low-dimensional space using a Student's t-distribution
3. Minimize the Kullback-Leibler (KL) divergence between the high-dimensional and low-dimensional similarity distributions using gradient descent
4. Visualize the resulting low-dimensional representations (2D or 3D) using scatter plots where each point represents a quantum data sample and the proximity of points indicates their similarity in the original high-dimensional space

UMAP Implementation Steps

1. Construct a weighted k-neighbor graph in the high-dimensional space where the edge weights represent the similarities between data points
2. Compute a low-dimensional graph that preserves the topological structure of the high-dimensional graph by minimizing the cross-entropy between the two graphs using stochastic gradient descent
3. Apply UMAP to high-dimensional quantum datasets (measurements from quantum sensors, quantum state vectors, or features extracted from quantum circuits)
4. Use the visualizations to identify clusters, outliers, or patterns in the quantum data guiding further analysis and interpretation

t-SNE vs UMAP for Quantum Data

Optimization Objectives and Algorithms

• t-SNE and UMAP have different optimization objectives and algorithms leading to different low-dimensional representations and performance characteristics when applied to quantum data
• t-SNE focuses more on preserving the local structure of the data often resulting in visually distinct clusters but may not always preserve the global structure as effectively as UMAP
• UMAP balances the preservation of both local and global structure often resulting in more continuous and connected low-dimensional representations compared to t-SNE

Performance and Hyperparameters

• UMAP is generally faster than t-SNE, especially for larger datasets, due to its more efficient optimization algorithm and the use of approximate nearest neighbor search
• UMAP has fewer hyperparameters to tune compared to t-SNE making it more user-friendly and less sensitive to parameter choices (main hyperparameters: number of neighbors and minimum distance between points in the low-dimensional space)
• Both t-SNE and UMAP can be sensitive to the choice of hyperparameters and it is essential to experiment with different settings to obtain meaningful and stable low-dimensional representations of quantum data
• The performance and suitability of t-SNE and UMAP for a given quantum dataset may depend on factors (size of the dataset, intrinsic dimensionality of the data, presence of noise, desired balance between local and global structure preservation)

Interpreting Low-Dimensional Representations

Insights from Low-Dimensional Representations

• The proximity of data points in the low-dimensional space indicates their similarity in the original high-dimensional space (points close together likely share similar characteristics or belong to the same cluster)
• The overall shape and topology of the low-dimensional representation can reveal the presence of manifolds, trajectories, or transitions in the quantum data indicative of underlying physical processes or quantum system dynamics
• Outliers or isolated points in the low-dimensional representation may correspond to rare or anomalous quantum events, measurement errors, or data points significantly different from the majority of the dataset

Validation and Caution

• Coloring the data points in the low-dimensional representation according to known labels or properties allows visual assessment of the relationship between the discovered structure and available metadata facilitating the interpretation of the quantum data in terms of physical or computational concepts
• The interpretation of the low-dimensional representations should be done cautiously considering the limitations and potential distortions introduced by the dimensionality reduction techniques
• It is essential to validate the findings using domain knowledge and complementary analysis methods to ensure accurate interpretation of the low-dimensional representations generated by t-SNE and UMAP for quantum data analysis