11.3 Other Dimensionality Reduction Methods
3 min read•august 7, 2024
Dimensionality reduction techniques go beyond PCA, offering diverse methods to simplify complex datasets. From non-linear approaches like t-SNE and UMAP to linear methods like LDA and ICA, these tools tackle different data challenges.
Neural network-based techniques like provide powerful alternatives for reducing dimensions. These methods, along with matrix factorization and manifold learning, expand our toolkit for handling high-dimensional data effectively.
Manifold Learning Techniques
t-SNE and UMAP: Non-linear Dimensionality Reduction
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Stop stretching your T-SNE, UMAP, and networks! – Reticular View original
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Top images from around the web for t-SNE and UMAP: Non-linear Dimensionality Reduction
Frontiers | A Comparison for Dimensionality Reduction Methods of Single-Cell RNA-seq Data View original
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t-SNE in Python [single cell RNA-seq example and hyperparameter optimization] - Renesh Bedre View original
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Stop stretching your T-SNE, UMAP, and networks! – Reticular View original
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Frontiers | A Comparison for Dimensionality Reduction Methods of Single-Cell RNA-seq Data View original
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- t-SNE () is a non-linear dimensionality reduction technique
- Preserves local structure of high-dimensional data in low-dimensional space
- Converts high-dimensional Euclidean distances between data points into conditional probabilities representing similarities
- Minimizes the divergence between joint probabilities in high-dimensional and low-dimensional space using
- UMAP () is another non-linear dimensionality reduction method
- Constructs a high-dimensional graph representation of the data and optimizes a low-dimensional graph to be as structurally similar as possible
- Assumes the data is uniformly distributed on a Riemannian manifold and tries to learn the manifold's local metric
- Faster than t-SNE and better preserves global structure (clusters at different scales)
Multidimensional Scaling (MDS)
- MDS is a technique used for visualizing the level of similarity of individual cases in a dataset
- Aims to find a low-dimensional representation of the data where the distances between points are preserved as well as possible
- Classical MDS: Uses eigenvector decomposition to preserve pairwise distances exactly in the low-dimensional space
- Non-metric MDS: Preserves the rank order of the pairwise distances (used for ordinal data)
- measures the discrepancy between the distances in the low-dimensional space and the original dissimilarities
- Applications include visualizing the relationships between objects (cities on a map) or individuals (based on survey responses)
Linear Dimensionality Reduction Methods
Supervised and Unsupervised Linear Methods
- (LDA) is a supervised dimensionality reduction method
- Finds a linear combination of features that best separates the classes
- Projects the data onto a lower-dimensional space while maximizing the separation between classes
- Assumes the data is normally distributed and the classes have equal covariance matrices
- (ICA) is an unsupervised method for separating a multivariate signal into additive subcomponents
- Assumes the subcomponents are non-Gaussian and statistically independent
- Finds a linear transformation that minimizes the statistical dependence between the components
- Applications include blind source separation (cocktail party problem) and
Matrix Factorization Techniques
- (NMF) is a dimensionality reduction technique that factorizes a non-negative matrix into two non-negative matrices
- Finds a low-rank approximation of the original matrix: , where , , and are non-negative
- Interpretable parts-based representation: Each column of represents a basis vector, and each column of represents the coefficients
- Applications include image processing (facial recognition), text mining (topic modeling), and recommender systems
Neural Network-based Dimensionality Reduction
Autoencoders
- Autoencoders are neural networks trained to reconstruct their input data
- Consist of an encoder that maps the input to a lower-dimensional latent space and a decoder that reconstructs the input from the latent representation
- Bottleneck layer in the middle has a lower dimensionality than the input, forcing the network to learn a compressed representation
- Types of autoencoders:
- Undercomplete autoencoders: Latent space has lower dimensionality than the input, used for dimensionality reduction
- Regularized autoencoders: Add regularization terms to the loss function to learn more robust representations (sparse, contractive, or denoising autoencoders)
- Variational autoencoders (VAEs): Latent space is constrained to follow a prior distribution (usually Gaussian), enabling generation of new samples
Key Terms to Review (25)
Autoencoders: Autoencoders are a type of artificial neural network used for unsupervised learning that aim to compress input data into a lower-dimensional representation and then reconstruct the original data from this compressed form. They are particularly useful in dimensionality reduction tasks because they learn to capture important features while discarding noise and irrelevant information, making them valuable tools for data preprocessing and feature extraction.
Bias-variance tradeoff: The bias-variance tradeoff is a fundamental concept in statistics and machine learning that describes the balance between two types of error in predictive models: bias, which refers to the error due to overly simplistic assumptions in the learning algorithm, and variance, which refers to the error due to excessive complexity in the model. Understanding this tradeoff is crucial when creating models that aim for good predictive performance while avoiding overfitting or underfitting.
Clustering: Clustering is a method used in data analysis that groups similar data points together based on their features, allowing for the discovery of patterns and structures within a dataset. It helps in reducing the complexity of data by summarizing it into clusters, which can make it easier to visualize and interpret. This technique is particularly useful in dimensionality reduction methods, where large datasets can be simplified while retaining essential information.
Data Visualization: Data visualization is the graphical representation of information and data, using visual elements like charts, graphs, and maps to convey complex information in a clear and understandable manner. This practice is essential for making sense of large datasets and is deeply intertwined with the processes of data analysis, as it allows for better insights and communication of findings to various audiences.
Eigenvalues: Eigenvalues are scalar values that indicate how much a corresponding eigenvector is stretched or compressed during a linear transformation represented by a matrix. They play a critical role in various statistical methods, as they help to understand the variance captured by components in dimensionality reduction techniques, the relationships between variables, and the overall structure of the data being analyzed.
Eigenvectors: Eigenvectors are special vectors in linear algebra that, when transformed by a linear transformation represented by a matrix, change only in scale and not in direction. They play a crucial role in understanding data structures, simplifying complex datasets, and are essential in techniques for dimensionality reduction.
Feature Extraction: Feature extraction is the process of transforming raw data into a set of measurable characteristics or features that can be used in machine learning models and statistical analysis. By selecting and refining these features, you can enhance the model's performance and interpretability, making it easier to understand relationships within the data. This process plays a crucial role in simplifying data and reducing its dimensionality while retaining the most relevant information.
Gradient descent: Gradient descent is an optimization algorithm used to minimize a function by iteratively moving towards the steepest descent as defined by the negative of the gradient. This method is essential in machine learning and statistical methods for efficiently finding the minimum of a cost function, particularly in the context of dimensionality reduction techniques where reducing complexity while preserving variance is crucial.
Hierarchical Clustering: Hierarchical clustering is a method of cluster analysis that seeks to build a hierarchy of clusters, creating a tree-like structure called a dendrogram. This technique is useful for identifying patterns and relationships in data by grouping similar objects based on their features, which can help in recognizing outliers and understanding data distributions. It can be applied in various domains, offering insights into the data structure without requiring a predefined number of clusters.
Independent Component Analysis: Independent Component Analysis (ICA) is a computational method used to separate a multivariate signal into additive, independent components. It is particularly useful in situations where signals are mixed together and we want to retrieve the original sources without prior knowledge of the mixing process. This technique plays a significant role in dimensionality reduction, helping to enhance the interpretability of data by uncovering hidden factors or sources that contribute to the observed signals.
K-means clustering: K-means clustering is an unsupervised learning algorithm used to partition a dataset into K distinct clusters based on feature similarities. The algorithm aims to minimize the variance within each cluster while maximizing the variance between clusters, making it a powerful tool for dimensionality reduction and data analysis.
Linear Discriminant Analysis: Linear Discriminant Analysis (LDA) is a statistical technique used for classifying data by finding a linear combination of features that separates two or more classes of objects. This method not only helps in classification tasks but also aids in understanding the underlying structure of the data, making it useful for both discrimination and dimensionality reduction.
Model complexity: Model complexity refers to the degree of sophistication or intricacy in a statistical model, often defined by the number of parameters or the structure of the model. A more complex model may fit the training data better but can lead to issues like overfitting, where the model captures noise rather than the underlying pattern. Balancing model complexity is crucial as it impacts performance, interpretability, and generalizability.
Multidimensional Scaling: Multidimensional scaling (MDS) is a statistical technique used for visualizing the level of similarity or dissimilarity of data points in a multi-dimensional space. By converting complex data into a lower-dimensional representation, MDS helps uncover relationships between items, making it easier to analyze and interpret patterns within the data. It’s particularly valuable in exploring data sets where dimensions can be abstract or difficult to interpret directly.
Noise reduction: Noise reduction refers to the techniques and methods used to minimize or eliminate irrelevant or extraneous data, which can obscure meaningful patterns in datasets. This process is crucial for enhancing the quality of data analysis, as it helps in focusing on the most significant signals and improves model performance by reducing overfitting and improving interpretability.
Non-negative Matrix Factorization: Non-negative Matrix Factorization (NMF) is a dimensionality reduction technique that factorizes a non-negative matrix into two lower-dimensional non-negative matrices, often used for data representation and interpretation. By ensuring that the components are non-negative, NMF allows for a more interpretable representation of data, particularly in applications like image processing, text mining, and bioinformatics.
Principal Component Analysis: Principal Component Analysis (PCA) is a statistical technique used to reduce the dimensionality of a dataset while preserving as much variance as possible. It helps in identifying patterns, simplifying data analysis, and visualizing complex datasets by transforming correlated variables into a set of uncorrelated variables called principal components. This method is crucial for various applications, such as exploratory data analysis, model fitting, handling multicollinearity, and facilitating factor analysis.
Reconstruction error: Reconstruction error is a metric that quantifies how well a dimensionality reduction technique reconstructs the original data from its lower-dimensional representation. It reflects the difference between the original data points and their corresponding approximations in the reduced space, making it a crucial measure of accuracy in various dimensionality reduction methods, such as Principal Component Analysis (PCA) and autoencoders.
Scikit-learn: Scikit-learn is an open-source machine learning library for Python that provides simple and efficient tools for data mining and data analysis. It offers a wide range of algorithms for classification, regression, clustering, and dimensionality reduction, making it an essential toolkit for anyone working in data science and machine learning.
Silhouette score: The silhouette score is a metric used to evaluate the quality of clustering in data science. It provides a way to measure how similar an object is to its own cluster compared to other clusters, with a higher silhouette score indicating better-defined and separated clusters. This score helps in assessing the effectiveness of clustering algorithms like K-means, hierarchical, and density-based clustering, as well as understanding the impact of dimensionality reduction methods on clustering results.
Stress Function: A stress function is a mathematical tool used in the field of applied mathematics and engineering to describe the distribution of stress in a material under load. It provides a way to simplify complex stress analysis problems by representing stress as a function of other variables, thus making it easier to study how materials respond to external forces.
T-distributed stochastic neighbor embedding: t-distributed stochastic neighbor embedding (t-SNE) is a dimensionality reduction technique that is particularly effective for visualizing high-dimensional data in a lower-dimensional space, typically two or three dimensions. It works by converting the similarities between data points into probabilities and then minimizing the divergence between these probabilities in high-dimensional and low-dimensional spaces. This method preserves local structures, making it useful for clustering and revealing patterns in complex datasets.
Tensorflow: TensorFlow is an open-source machine learning framework developed by Google, designed to facilitate the development and training of deep learning models. It provides a comprehensive ecosystem for building, training, and deploying machine learning applications, making it easier for developers to work with complex data sets and perform numerical computations efficiently. With its flexible architecture, TensorFlow supports various programming languages and platforms, enabling seamless integration into different workflows.
Uniform Manifold Approximation and Projection: Uniform Manifold Approximation and Projection (UMAP) is a nonlinear dimensionality reduction technique that helps visualize high-dimensional data by projecting it into a lower-dimensional space while preserving its topological structure. UMAP emphasizes preserving the local and global structure of the data, making it particularly effective for visualizing complex datasets in various fields, from biology to machine learning.
Variance Explained: Variance explained refers to the proportion of the total variance in a dataset that is accounted for by a particular model or method. It is a crucial concept in statistical analysis, particularly in understanding how well a model captures the underlying structure of the data, allowing for dimensionality reduction techniques to effectively summarize information while minimizing loss.