LaTeX Documentation

Comprehensive mathematical notes and formal documentation

About the Documentation

This section contains our formal LaTeX documentation, including detailed mathematical notes, proofs, and theoretical foundations for manifold learning. All documents are compiled weekly and maintain rigorous academic standards.

The documentation follows the structure of our study group meetings and provides comprehensive coverage of topics from foundational mathematics to advanced manifold learning algorithms.

Explore the individual member contributions below to dive into specialised topics we're studying together.

Member Notes

Personal study dossiers prepared by members to deepen and share understanding beyond the main syllabus.

Member Note

Repeated Eigendirection Problem (REP)

Oğuzhan Recep Akkol examines repeated eigendirections, their impact on embeddings, and ways to mitigate artefacts.

Oğuzhan Recep Akkol PDF · 11 pages
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Laplace–Beltrami and Graph Laplacians

Oğuzhan Recep Akkol links physical intuition, discrete graphs, and eigenmaps to show how graph Laplacians converge to their Laplace–Beltrami counterparts.

Oğuzhan Recep Akkol PDF · 6 pages
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Local Linear Approximations

Oğuzhan Recep Akkol reviews the manifold assumption, random projections, and local PCA to motivate neighbourhood-based dimensionality reduction.

Oğuzhan Recep Akkol PDF · 5 pages
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Local Tangent Space Alignment (LTSA)

Oğuzhan Recep Akkol outlines the LTSA pipeline, from tangent patch construction to aligning affine charts into a coherent embedding.

Oğuzhan Recep Akkol PDF · 4 pages
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Section 5.3 – Estimating the Intrinsic Dimension

Oğuzhan Recep Akkol summarises Section 5.3, highlighting intrinsic-dimension estimators and how bias analyses guide parameter choices.

Oğuzhan Recep Akkol PDF
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Principal Curves and d-Manifolds

Naki Eren Şengezer revisits principal curves and intrinsic dimensionality, connecting geometric intuition with manifold learning objectives.

Naki Eren Şengezer PDF · 3 pages
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Riemannian Metrics in Context

An exploration by Naki Eren Şengezer on framing Riemannian metrics for data-driven manifolds, with annotated derivations and references.

Naki Eren Şengezer PDF · 6 pages
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Statistical Basis of Sampling Biases

Naki Eren Şengezer formalises Section 5 intuitions with statistical tools that explain sampling-density bias and its interaction with graph construction.

Naki Eren Şengezer PDF · 6 pages
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Embedding Distortions

Guest summaries chart the typical distortion patterns that appear in spectral embeddings and explain how to diagnose them in practice.

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Estimating the Laplace–Beltrami Operator

The guest extends our Laplace–Beltrami conversations with derivations, convergence notes, and an emphasis on estimator stability.

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Section 2 Overview

A concise overview from the guest summarising Section 2 of the paper, highlighting the structural assumptions and notation we rely on later.

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Phase 2

ML & Singular Learning Theory

Guest explores the intersection of machine learning and singular learning theory, discussing algebraic geometry applications and ML's role in solving mathematical problems.

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Phase 2

Singular Learning & Manifolds

Guest captures the neuroalgebraic geometry discussion, dataset diagnostics, and the shift toward manifold-inspired explainability ideas.

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Phase 2

Week 17 Strategy Notes

Guest summarises the Sixty-Four curves workshop, the cubic replication plan, singularity diagnostics, and the determinantal-to-NN assignments.

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Phase 1 TP/TD Archive

Structured Travaux Pratiques / Travaux Dirigés materials we prepared while wrapping Phase 1.

Phase 1 wrap-up checkpoint: Use the TP/TD packet to test yourself before jumping into Singular Learning Theory. New exercises will appear here as soon as they’re published.
TP/TD

Manifold Learning TP/TD #1

A curated set of exercises and discussion prompts summarising the key takeaways from Phase 1.

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