https://arshadhs.github.io/tags/exam-revision/Recent content in Exam Revision on Arshad SiddiquiHugoen-ushttps://arshadhs.github.io/docs/ai/maths/mfml-exam-revision-index/Mon, 01 Jan 0001 00:00:00 +0000https://arshadhs.github.io/docs/ai/maths/mfml-exam-revision-index/<h1 id="mfml-exam-revision-index"> MFML Exam Revision Index <a class="anchor" href="#mfml-exam-revision-index">#</a> </h1> <p>This is a practical revision index for the uploaded Mathematical Foundations for Machine Learning material.</p> <h2 id="exam-split"> Exam split <a class="anchor" href="#exam-split">#</a> </h2> <table> <thead> <tr> <th>Exam</th> <th>Coverage</th> <th>Main files</th> </tr> </thead> <tbody> <tr> <td>Mid-Semester</td> <td>Weeks/Sessions 1-8</td> <td>Lecture 1 to Lecture 8, Webinar 1, Webinar 2</td> </tr> <tr> <td>Comprehensive</td> <td>Sessions 1-16</td> <td>Lecture 1 to Lecture 15, webinars, and any missing Lecture 16/kernel material</td> </tr> </tbody> </table> <h2 id="high-priority-concept-checklist"> High-priority concept checklist <a class="anchor" href="#high-priority-concept-checklist">#</a> </h2> <h3 id="linear-systems-and-matrices"> Linear systems and matrices <a class="anchor" href="#linear-systems-and-matrices">#</a> </h3> <ul> <li>Convert equations into matrix form <code>Ax = b</code></li> <li>Understand solution types: no solution, unique solution, infinite solutions</li> <li>Identify pivot and free variables</li> <li>Understand row operations, REF/RREF, rank, nullity</li> <li>Know matrix inverse and transpose properties</li> </ul> <h3 id="vector-spaces"> Vector spaces <a class="anchor" href="#vector-spaces">#</a> </h3> <ul> <li>Definition of vector space and subspace</li> <li>Closure under addition and scalar multiplication</li> <li>Span, linear combination, linear independence</li> <li>Basis, dimension, rank</li> <li>Column space, row space, nullspace</li> </ul> <h3 id="analytic-geometry"> Analytic geometry <a class="anchor" href="#analytic-geometry">#</a> </h3> <ul> <li>Norm properties</li> <li>Manhattan norm and Euclidean norm</li> <li>Inner product definition</li> <li>Symmetric positive-definite matrices</li> <li>Distance, angle, orthogonality</li> <li>Orthonormal basis and Gram-Schmidt</li> </ul> <h3 id="matrix-decompositions"> Matrix decompositions <a class="anchor" href="#matrix-decompositions">#</a> </h3> <ul> <li>Determinant and trace</li> <li>Cofactor expansion</li> <li>Row operation effect on determinant</li> <li>Eigenvalue equation <code>Av = λv</code></li> <li>Characteristic equation <code>det(A - λI) = 0</code></li> <li>Diagonalisation <code>A = PDP^{-1}</code></li> <li>Spectral theorem for symmetric matrices</li> <li>Cholesky decomposition</li> <li>SVD <code>A = UΣV^T</code></li> <li>Low-rank approximation</li> </ul> <h3 id="vector-calculus"> Vector calculus <a class="anchor" href="#vector-calculus">#</a> </h3> <ul> <li>Derivative from first principles</li> <li>Partial derivatives</li> <li>Gradient as direction of steepest ascent</li> <li>Gradient of vector-valued functions</li> <li>Matrix-gradient identities</li> <li>Chain rule</li> <li>Backpropagation and automatic differentiation</li> </ul> <h3 id="taylor-series-and-hessian"> Taylor series and Hessian <a class="anchor" href="#taylor-series-and-hessian">#</a> </h3> <ul> <li>Taylor polynomial</li> <li>Taylor series and Maclaurin series</li> <li>Remainder term</li> <li>Taylor series in two variables</li> <li>Hessian matrix</li> <li>First derivative and second derivative tests</li> <li>Maxima, minima and saddle points</li> </ul> <h3 id="gradient-descent-and-optimisation"> Gradient descent and optimisation <a class="anchor" href="#gradient-descent-and-optimisation">#</a> </h3> <ul> <li>Negative gradient direction</li> <li>Learning rate/step size</li> <li>Line search</li> <li>Convergence and local minima</li> <li>Constrained vs unconstrained optimisation</li> <li>Lagrange multipliers</li> <li>Convex optimisation</li> <li>SGD and optimisation in ML</li> <li>Feature preprocessing and scaling</li> <li>Overfitting in optimisation examples</li> </ul> <h3 id="nonlinear-optimisation-algorithms"> Nonlinear optimisation algorithms <a class="anchor" href="#nonlinear-optimisation-algorithms">#</a> </h3> <ul> <li>Difficult surfaces: cliffs, valleys, flat regions</li> <li>Curvature and why first-order methods can struggle</li> <li>Momentum update and intuition</li> <li>AdaGrad</li> <li>RMSProp</li> <li>Adam</li> <li>Learning rate decay</li> </ul> <h3 id="pca"> PCA <a class="anchor" href="#pca">#</a> </h3> <ul> <li>Dimensionality reduction problem</li> <li>Centred data and covariance matrix</li> <li>Maximum variance view</li> <li>Projection/reconstruction view</li> <li>Principal components as eigenvectors of covariance matrix</li> <li>SVD relation to PCA</li> <li>Low-rank approximation and Eckart-Young theorem</li> <li>PCA in high dimensions</li> <li>Practical PCA steps</li> </ul> <h3 id="svm"> SVM <a class="anchor" href="#svm">#</a> </h3> <ul> <li>Linear classifiers</li> <li>Margin and support vectors</li> <li>Hard-margin SVM primal formulation</li> <li>Lagrangian for SVM</li> <li>KKT conditions</li> <li>Primal vs dual perspective</li> <li>Role of inner products</li> <li>Kernel trick</li> <li>Hinge loss</li> <li>Soft-margin SVM</li> </ul> <h2 id="suggested-revision-order"> Suggested revision order <a class="anchor" href="#suggested-revision-order">#</a> </h2> <h3 id="phase-1-foundations"> Phase 1: Foundations <a class="anchor" href="#phase-1-foundations">#</a> </h3> <ol> <li>Lecture 1</li> <li>Lecture 2</li> <li>Lecture 3</li> <li>Webinar 1 problems related to REF, nullspace, column space and subspaces</li> </ol> <h3 id="phase-2-matrix-decompositions"> Phase 2: Matrix decompositions <a class="anchor" href="#phase-2-matrix-decompositions">#</a> </h3> <ol> <li>Lecture 4</li> <li>Lecture 5</li> <li>Webinar 1 and Webinar 2 eigenvalue/eigendecomposition problems</li> </ol> <h3 id="phase-3-calculus-and-optimisation-foundations"> Phase 3: Calculus and optimisation foundations <a class="anchor" href="#phase-3-calculus-and-optimisation-foundations">#</a> </h3> <ol> <li>Lecture 6</li> <li>Lecture 7</li> <li>Lecture 8</li> <li>Webinar 2 maxima/minima and Hessian problems</li> </ol> <h3 id="phase-4-optimisation-for-ml"> Phase 4: Optimisation for ML <a class="anchor" href="#phase-4-optimisation-for-ml">#</a> </h3> <ol> <li>Lecture 9</li> <li>Lecture 10</li> <li>Lecture 11</li> <li>Webinar 3 gradient-descent step-size problems</li> </ol> <h3 id="phase-5-pca-and-svm"> Phase 5: PCA and SVM <a class="anchor" href="#phase-5-pca-and-svm">#</a> </h3> <ol> <li>Lecture 12</li> <li>Lecture 13</li> <li>Lecture 14</li> <li>Lecture 15</li> <li>Webinar 4 / SVM problems</li> </ol> <h2 id="what-to-ask-me-next"> What to ask me next <a class="anchor" href="#what-to-ask-me-next">#</a> </h2> <p>Use these prompts when generating Hugo pages:</p>