Dimensionality reduction techniques simplify complex data by reducing the number of variables while preserving essential information. Methods like PCA, SVD, and t-SNE help visualize high-dimensional data, enhance model performance, and uncover hidden patterns, crucial in data science and linear algebra.
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Principal Component Analysis (PCA)
- Reduces dimensionality by transforming data to a new set of variables (principal components) that capture the most variance.
- Utilizes eigenvalue decomposition of the covariance matrix to identify the directions of maximum variance.
- Helps in visualizing high-dimensional data and improving model performance by eliminating noise.
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Singular Value Decomposition (SVD)
- Factorizes a matrix into three components: U (left singular vectors), Σ (singular values), and V (right singular vectors).
- Useful for dimensionality reduction, noise reduction, and data compression.
- Forms the basis for other techniques like PCA and Latent Semantic Analysis (LSA).
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Linear Discriminant Analysis (LDA)
- A supervised technique that finds a linear combination of features that best separates two or more classes.
- Maximizes the ratio of between-class variance to within-class variance.
- Often used for classification tasks and dimensionality reduction in labeled datasets.
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t-Distributed Stochastic Neighbor Embedding (t-SNE)
- A non-linear technique primarily used for visualizing high-dimensional data in two or three dimensions.
- Preserves local structure while revealing global structure through a probabilistic approach.
- Effective for clustering and understanding complex datasets, especially in exploratory data analysis.
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Autoencoders
- Neural network architectures designed to learn efficient representations of data through unsupervised learning.
- Consists of an encoder that compresses the input and a decoder that reconstructs it, minimizing reconstruction error.
- Useful for dimensionality reduction, denoising, and feature learning.
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Truncated SVD (LSA)
- A variant of SVD that retains only the top k singular values and corresponding vectors, reducing dimensionality.
- Commonly used in Latent Semantic Analysis for text data to uncover latent structures.
- Helps in improving computational efficiency and reducing noise in data.
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Independent Component Analysis (ICA)
- A computational technique to separate a multivariate signal into additive, independent components.
- Assumes that the observed data is a mixture of non-Gaussian signals and aims to recover the original sources.
- Widely used in fields like signal processing and neuroimaging.
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Non-negative Matrix Factorization (NMF)
- Decomposes a matrix into two non-negative matrices, allowing for parts-based representation.
- Useful for extracting interpretable features from data, especially in image and text analysis.
- Enforces non-negativity constraints, making it suitable for applications where negative values are not meaningful.
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Multidimensional Scaling (MDS)
- A technique for visualizing the level of similarity of individual cases of a dataset in a low-dimensional space.
- Preserves the distances between points in high-dimensional space as closely as possible in lower dimensions.
- Useful for exploratory data analysis and understanding relationships between data points.
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Isomap
- An extension of MDS that incorporates geodesic distances on a manifold, preserving global structure.
- Constructs a neighborhood graph and computes shortest paths to maintain the intrinsic geometry of the data.
- Effective for non-linear dimensionality reduction in complex datasets.
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Locally Linear Embedding (LLE)
- A non-linear dimensionality reduction technique that preserves local relationships between data points.
- Constructs a neighborhood graph and reconstructs each point as a linear combination of its neighbors.
- Useful for uncovering the underlying manifold structure of high-dimensional data.
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Factor Analysis
- A statistical method used to identify underlying relationships between variables by modeling observed variables as linear combinations of potential factors.
- Helps in data reduction and identifying latent constructs in datasets.
- Commonly used in psychology, social sciences, and market research.
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Random Projections
- A technique that reduces dimensionality by projecting data onto a randomly generated lower-dimensional subspace.
- Based on the Johnson-Lindenstrauss lemma, which guarantees that distances are approximately preserved.
- Efficient and simple, making it suitable for large datasets.
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Kernel PCA
- An extension of PCA that uses kernel methods to perform non-linear dimensionality reduction.
- Maps data into a higher-dimensional space using a kernel function, allowing for the capture of complex structures.
- Useful for datasets where linear separability is not achievable.
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Uniform Manifold Approximation and Projection (UMAP)
- A non-linear dimensionality reduction technique that preserves both local and global structure of data.
- Utilizes concepts from topology and manifold theory to create a low-dimensional representation.
- Effective for visualizing complex datasets and maintaining meaningful relationships between data points.