Dimensionality Reduction and Model Families Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

    Dimensionality Reduction and Model Families notes for GATE DA: 43 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

    Chapter Roadmap: Dimensionality Reduction and Model Families

    Chapter Roadmap

    Dimensionality Reduction and Model Families

    1. Principal Component Analysis

    Geometry of variance, covariance matrices, eigen-decomposition, and strict orthogonality of principal axes.

    2. Generative & Discriminative Models

    Classifying algorithms by probabilistic assumptions. Joint vs conditional probabilities.

    Machine Learning › Supervised Learning › Dimensionality Reduction

    Principal Component Analysis and Orthogonal Components

    Principal Component Analysis & Orthogonal Components

    Finding the orthogonal axes of maximum variance in high-dimensional data.

    What you will learn

    • Geometric intuition behind dimensionality reduction
    • Constructing and interpreting the Covariance Matrix
    • Eigen-decomposition and finding the Principal Axes
    • The strict orthogonality of Principal Components
    • Calculating variance explained for component selection
    Dimensionality Reduction › PCA & Orthogonal Components

    The Goal of Dimensionality Reduction

    The Goal of Dimensionality Reduction

    The Curse of Dimensionality

    As features increase, space volume grows exponentially, making data sparse. This leads to slower training, high overfitting risk, and visualization difficulty.

    How PCA Solves This

    PCA creates new, compressed features called Principal Components (PCs). Each PC is a linear combination of original features.

    1st PC: Captures maximum possible variance.
    2nd PC: Captures next highest variance, constrained to be orthogonal to the 1st.

    The Covariance Matrix

    The Covariance Matrix

    Quantifying how original features vary and co-vary.

    Diagonal ()
    Variance of the -th feature.
    Off-Diagonal ()
    Covariance between -th and -th features.
    Crucial Step: You must standardize features (zero mean, unit variance) before computing . Otherwise, features with larger scales will artificially dominate.

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    Dimensionality Reduction and Model Families Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

    Dimensionality Reduction and Model Families notes for GATE DA: 43 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice ques

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