# ============================================================================
# T05 — Dimensionality reduction for preserving cluster structure
# ----------------------------------------------------------------------------
# Unlike paper_replication's P01-P07, the "synthesis" here frames a research
# QUESTION rather than a known paper's method. The model must design the
# experiment (conditions, metrics, datasets) — we only commit to what a
# competent study of this topic would include and what the rubric expects.
# ============================================================================

id: ML05
title: "Dimensionality reduction methods for preserving cluster structure in synthetic high-dimensional data"
arxiv_id: null
venue: "ARC-Bench 2026"
paper_asset: null

# The "synthesis" plays the role of the upstream briefing: research question,
# background, why the question matters, what "a reasonable experiment" looks
# like. It deliberately does NOT pre-specify a single method to reproduce.
synthesis: |
  Dimensionality reduction is frequently used before visualization and
  clustering, but different methods optimize different objectives and can
  distort neighborhood and global geometry in distinct ways. PCA is linear and
  emphasizes variance preservation, while t-SNE and UMAP are nonlinear manifold
  methods designed to preserve local structure. In practice, users often infer
  cluster quality from 2D embeddings without checking whether separability in
  the embedding reflects true high-dimensional structure.

  A CPU-scale investigation can probe this mismatch by generating synthetic
  datasets where ground-truth cluster labels are known and controllable. By
  varying cluster overlap, anisotropy, and manifold geometry (e.g., Gaussian
  blobs, anisotropic transforms, and noisy moons embedded in high dimensions),
  one can evaluate whether each reducer preserves cluster structure under
  different regimes instead of relying on a single toy example.

  A credible experiment should compare PCA, t-SNE, and UMAP under a common
  pipeline: reduce to 2 dimensions, then score structure using metrics such as
  silhouette score and adjusted Rand index after a fixed clustering backend
  (e.g., KMeans with true k). Runtime should also be tracked because nonlinear
  methods may yield better structure at substantially higher computational cost,
  which matters in constrained environments.

  The goal is not to claim a universally best reducer, but to quantify tradeoffs
  between structure preservation and efficiency in settings where intrinsic
  geometry differs. This produces actionable guidance for method choice based on
  data regime rather than defaults.

  *Which dimensionality reduction method (PCA, t-SNE, or UMAP) best preserves cluster structure across distinct synthetic high-dimensional regimes when evaluated by silhouette score, ARI, and runtime?*

hypotheses:
  - id: H1
    statement: "On nonlinear manifold data (noisy two-moons embedded into 50D), at least one nonlinear reducer (t-SNE or UMAP) achieves silhouette_score at least 0.10 higher than PCA after 2D reduction and KMeans clustering, averaged over >=5 seeds."
    measurable: true
  - id: H2
    statement: "On linearly separable Gaussian blobs in 50D, PCA achieves silhouette_score within 0.05 of the best nonlinear method (t-SNE or UMAP), averaged over >=5 seeds."
    measurable: true
  - id: H3
    statement: "PCA has the lowest mean wall-clock reduction time and is at least 3x faster than both t-SNE and UMAP on at least 2 of 3 datasets."
    measurable: true

experiment_design:
  research_question: "Which reducer best balances cluster-structure preservation and compute cost across linear and nonlinear synthetic high-dimensional datasets?"
  conditions:
    - name: "pca_2d"
      description: "PCA with n_components=2 using sklearn.decomposition.PCA."
    - name: "tsne_2d"
      description: "t-SNE with n_components=2, perplexity in [20, 40], learning_rate='auto', init='pca'."
    - name: "umap_2d"
      description: "UMAP-like reducer: use umap.UMAP(n_components=2, n_neighbors in [15, 30], min_dist in [0.0, 0.3]) if available; if unavailable, use sklearn.manifold.SpectralEmbedding(n_components=2) as a documented fallback proxy."
    - name: "identity_no_reduction"
      description: "Baseline that applies KMeans directly in the original high-dimensional space (no dimensionality reduction)."
  baselines:
    - "identity_no_reduction as no-DR clustering baseline"
    - "pca_2d as linear DR baseline"
  metrics:
    - name: "silhouette_score"
      direction: "maximize"
      description: "Silhouette score computed on the 2D embedding (or original space for identity baseline) using predicted KMeans labels; primary metric."
    - name: "ari"
      direction: "maximize"
      description: "Adjusted Rand Index between KMeans labels and known synthetic ground-truth labels."
    - name: "fit_transform_time_sec"
      direction: "minimize"
      description: "Wall-clock time for dimensionality reduction fit_transform only (seconds)."
  datasets:
    - name: "gaussian_blobs_50d"
      source: "sklearn.datasets.make_blobs with 4 centers, n_features=50, cluster_std=1.2"
    - name: "anisotropic_blobs_50d"
      source: "make_blobs in 10D expanded/projection to 50D then linear anisotropic transform"
    - name: "embedded_moons_50d"
      source: "sklearn.datasets.make_moons(noise=0.08) embedded into 50D by random linear projection plus Gaussian noise"
  compute_requirements:
    gpu_required: false
    estimated_wall_clock_sec: 600

rubric_path: "experiments/arc_bench/config/ml/rubrics/ML05.json"