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[Phase 2 Parity] Implement Modern Dimensionality Reduction Algorithms
Problem: AiDotNet has decomposition methods but lacks high-level dimensionality reduction. Missing: t-SNE (CRITICAL), UMAP (HIGH), PCA wrapper (HIGH), LDA (HIGH), ICA (MEDIUM). Existing: SVD and Eigen decomposition. Goal: Build high-level API on existing foundation.
Issue #383: Implement Modern Dimensionality Reduction Algorithms
Junior Developer Implementation Guide
For: Developers new to dimensionality reduction and manifold learning Difficulty: Advanced Estimated Time: 40-50 hours Prerequisites: Linear algebra, understanding of SVD/PCA basics, optimization fundamentals
Understanding Dimensionality Reduction
For Beginners: Dimensionality reduction is like creating a map from a globe. A globe is 3D, but we represent it on a 2D map. We lose some information (distances get distorted), but we gain the ability to see the whole world at once. Similarly, dimensionality reduction takes high-dimensional data (hundreds or thousands of features) and creates a 2D or 3D representation that preserves the important structure.
Why Build Dimensionality Reduction?
vs Using Raw High-Dimensional Data:
- ✅ Visualization (can't plot 100-dimensional data, but can plot 2D/3D)
- ✅ Faster computation (algorithms run faster in lower dimensions)
- ✅ Reduces noise (eliminates less important features)
- ✅ Overcomes curse of dimensionality (ML algorithms work better in lower dimensions)
- ❌ Loses some information (trade-off between compression and accuracy)
Real-World Use Cases:
- Data Visualization: Visualize high-dimensional datasets (gene expression, customer features, document embeddings)
- Feature Engineering: Create compact feature representations for downstream ML
- Anomaly Detection: Detect outliers in compressed space
- Data Compression: Reduce storage/transmission costs
- Exploratory Analysis: Discover hidden patterns and clusters
Key Concepts
Linear vs Nonlinear Methods
Linear Methods (PCA, LDA):
- Assume data lies on a linear subspace (flat hyperplane)
- Fast, deterministic, interpretable
- Work well for linearly separable data
- Example: Principal Component Analysis (PCA)
Nonlinear Methods (t-SNE, UMAP):
- Discover curved manifolds (nonlinear structures)
- Can capture complex patterns linear methods miss
- Slower, may require tuning
- Example: t-SNE for visualization
t-SNE (t-Distributed Stochastic Neighbor Embedding)
How it Works:
- Compute pairwise similarities in high-dimensional space (Gaussian kernel)
- Compute pairwise similarities in low-dimensional space (t-distribution)
- Minimize difference between high-D and low-D similarities using gradient descent
- Result: Points close in high-D space are close in low-D, preserving local structure
Beginner Analogy: Imagine 1000 people scattered across a country. You want to create a 2D map where friends are drawn close together, but you don't care about exact distances - just that friends end up near each other. t-SNE does this for data points.
Key Insight: t-SNE prioritizes preserving local structure (neighborhoods) over global structure (distances between far points). This makes it excellent for visualization but not for measuring actual distances.
Algorithm Complexity: O(n² log n) with tree-based approximations, O(n²) naive
Best For:
- Visualization of high-dimensional data
- Discovering clusters and patterns
- Biological data (gene expression, single-cell RNA-seq)
- When local relationships matter more than global distances
Limitations:
- Nondeterministic (different runs give different results)
- Doesn't preserve global structure
- Can't transform new data (must rerun entire algorithm)
- Slow for large datasets (> 10,000 points)
- Sensitive to perplexity parameter
Key Parameter: Perplexity (default: 30)
- Controls neighborhood size (roughly number of nearest neighbors to preserve)
- Low perplexity (5-15): Focuses on very local structure, may create false clusters
- High perplexity (30-50): Balances local and global structure
- Rule of thumb: perplexity = sqrt(n) or 30-50 for most datasets
UMAP (Uniform Manifold Approximation and Projection)
How it Works:
- Construct fuzzy topological graph in high-D space (k-nearest neighbors)
- Optimize low-D graph to match high-D graph structure
- Uses Riemannian geometry and algebraic topology (sounds fancy, works amazing)
- Result: Preserves both local and global structure better than t-SNE
Beginner Analogy: Think of your data as cities on a curved planet surface. UMAP tries to flatten the surface (like making a map) while keeping both nearby cities close AND the overall continent shapes recognizable. t-SNE only focuses on keeping nearby cities close.
Key Advantages over t-SNE:
- Faster: Scales to millions of points
- Preserves global structure: Distances between clusters are meaningful
- Deterministic: Same seed = same result
- Supports transform: Can embed new points without retraining
- Better for downstream ML: UMAP features work better for classification/clustering
Algorithm Complexity: O(n^1.14) - subquadratic, much faster than t-SNE
Best For:
- Large datasets (100,000+ points)
- When you need to transform new data
- Preserving global structure (cluster relationships)
- Downstream machine learning tasks
- General-purpose dimensionality reduction
Key Parameters:
-
n_neighbors (default: 15): Size of local neighborhood, like t-SNE's perplexity
- Small (5-10): Focus on fine-grained local structure
- Large (50-100): Focus on global structure
-
min_dist (default: 0.1): How tightly to pack points in embedding
- Small (0.0-0.1): Dense, tightly-packed clusters
- Large (0.5-0.99): Looser, more spread-out embedding
- metric (default: "euclidean"): Distance metric to use
PCA (Principal Component Analysis) Wrapper
Why a Wrapper When We Have SVD? AiDotNet has SVD (Singular Value Decomposition), which is the mathematical foundation of PCA. But PCA adds:
- Automatic centering (subtracting mean)
- Explained variance ratio (how much information each component captures)
- Component selection (choose top K components automatically)
- Standardization options
- High-level API matching scikit-learn
PCA Math (simplified):
- Center data: X_centered = X - mean(X)
- Compute covariance: C = (X_centered^T × X_centered) / n
- Eigen-decomposition: C = V × Λ × V^T
- Principal components = eigenvectors with largest eigenvalues
- Transform: X_reduced = X_centered × V_k (keep top k components)
Beginner Analogy: Imagine a swarm of fireflies in 3D space. PCA finds the "main directions" the swarm spreads out in. The first direction captures the most spread, the second captures the next most, etc. You can then describe each firefly's position using just these main directions instead of x,y,z coordinates.
When to Use PCA vs t-SNE/UMAP:
- PCA: Fast, interpretable components, preserves distances, reversible
- t-SNE/UMAP: Better visualization, finds nonlinear patterns, not reversible
Implementation Overview
src/DimensionalityReduction/
├── TSNE.cs [NEW - AC 1.1]
├── UMAP.cs [NEW - AC 1.2]
├── PCA.cs [NEW - AC 1.3]
└── Base/
└── DimensionalityReductionBase.cs [NEW - base class]
src/Interfaces/
└── IDimensionalityReduction.cs [NEW - AC 1.0]
tests/UnitTests/DimensionalityReduction/
├── TSNETests.cs [NEW - AC 2.1]
├── UMAPTests.cs [NEW - AC 2.2]
└── PCATests.cs [NEW - AC 2.3]
Phase 1: Core Dimensionality Reduction Algorithms
AC 1.0: Create IDimensionalityReduction Interface (3 points)
File: C:\Users\cheat\source\repos\AiDotNet\src\Interfaces\IDimensionalityReduction.cs
namespace AiDotNet.Interfaces;
/// <summary>
/// Defines the contract for dimensionality reduction algorithms.
/// </summary>
/// <typeparam name="T">The numeric data type used for calculations (e.g., float, double).</typeparam>
/// <remarks>
/// <para>
/// Dimensionality reduction transforms high-dimensional data into a lower-dimensional space
/// while preserving important structure (distances, neighborhoods, or variance).
/// </para>
/// <para><b>For Beginners:</b> Simplifying complex data for visualization and analysis.
///
/// Think of dimensionality reduction as creating a simplified version of your data:
/// - Original: 1000-dimensional vectors (too complex to visualize or understand)
/// - Reduced: 2D or 3D points (can plot and visually inspect)
///
/// The goal is to keep the important information while discarding noise and redundancy.
///
/// <b>Common use cases:</b>
/// - Visualize high-dimensional data (plot gene expression, customer features, etc.)
/// - Speed up machine learning (fewer features = faster training)
/// - Denoise data (low-dimensional representation filters out noise)
/// - Feature engineering (create compact representations)
///
/// <b>Types of algorithms:</b>
/// - Linear (PCA): Assumes data lies on flat hyperplanes
/// - Nonlinear (t-SNE, UMAP): Discovers curved manifolds and complex patterns
/// </para>
/// </remarks>
public interface IDimensionalityReduction<T>
{
/// <summary>
/// Fits the dimensionality reduction model to the provided data.
/// </summary>
/// <param name="data">
/// A matrix where each row is a data point and each column is a feature.
/// For example, a 1000×100 matrix represents 1000 samples with 100 features each.
/// </param>
/// <remarks>
/// <para>
/// This method learns the mapping from high-dimensional to low-dimensional space.
/// The specific learning process varies by algorithm:
/// - PCA: Computes principal components via eigendecomposition
/// - t-SNE: Optimizes embedding via gradient descent
/// - UMAP: Constructs fuzzy topological graphs and optimizes layout
/// </para>
/// <para><b>For Beginners:</b> Learning the transformation.
///
/// Fit is where the algorithm analyzes your high-dimensional data and learns
/// how to compress it. After fitting:
/// - The model has discovered the important patterns
/// - You can use Transform to apply the compression
///
/// Think of it like learning to summarize:
/// - Fit: Read many articles and learn what's important vs fluff
/// - Transform: Apply that knowledge to create concise summaries
///
/// For PCA: Fit finds the principal components (main directions of variation)
/// For t-SNE/UMAP: Fit creates the actual low-dimensional embedding
/// </para>
/// </remarks>
void Fit(Matrix<T> data);
/// <summary>
/// Transforms high-dimensional data into the learned low-dimensional space.
/// </summary>
/// <param name="data">
/// A matrix where each row is a data point to be transformed.
/// Must have the same number of columns as the training data.
/// </param>
/// <returns>
/// A matrix with the same number of rows but reduced number of columns (dimensions).
/// For example, 100×1000 data transformed to 100×2 (2D embedding).
/// </returns>
/// <remarks>
/// <para>
/// This method applies the learned transformation to new or existing data.
/// Note: Not all algorithms support transforming new data:
/// - PCA: Fully supports transform (linear projection)
/// - UMAP: Supports transform with trained model
/// - t-SNE: Does NOT support transform (must refit with new data included)
/// </para>
/// <para><b>For Beginners:</b> Applying the compression.
///
/// After fitting, Transform compresses high-dimensional data to low-dimensional.
///
/// For PCA:
/// - Transform = matrix multiplication (X_new × principal_components)
/// - Fast and deterministic
/// - Can transform any new data with same features
///
/// For UMAP:
/// - Transform = find position in learned embedding
/// - Requires fitted model
/// - Can transform new data (unlike t-SNE)
///
/// For t-SNE:
/// - Transform is NOT supported
/// - Must include new data in Fit call
/// - This is a fundamental limitation of t-SNE
///
/// <b>Example:</b>
/// ```csharp
/// var pca = new PCA<double>(nComponents: 2);
/// pca.Fit(trainingData); // Learn from 1000×100 data
/// var reduced = pca.Transform(newData); // Apply to 500×100 data → 500×2
/// ```
/// </para>
/// </remarks>
Matrix<T> Transform(Matrix<T> data);
/// <summary>
/// Fits the model and immediately transforms the same data (convenience method).
/// </summary>
/// <param name="data">
/// A matrix where each row is a data point.
/// </param>
/// <returns>
/// The low-dimensional embedding of the input data.
/// </returns>
/// <remarks>
/// <para>
/// This is equivalent to calling Fit(data) followed by Transform(data).
/// It's the most common usage pattern for dimensionality reduction.
/// </para>
/// <para><b>For Beginners:</b> One-step compression.
///
/// FitTransform is a shortcut that learns and applies the compression in one call.
///
/// Instead of:
/// ```csharp
/// pca.Fit(data);
/// var reduced = pca.Transform(data);
/// ```
///
/// You can do:
/// ```csharp
/// var reduced = pca.FitTransform(data);
/// ```
///
/// This is what you'll use most often:
/// - For visualization: Reduce data to 2D/3D for plotting
/// - For analysis: Compress before clustering/classification
/// - For exploration: See what patterns emerge in low-D
/// </para>
/// </remarks>
Matrix<T> FitTransform(Matrix<T> data);
/// <summary>
/// Gets the number of dimensions in the reduced space.
/// </summary>
/// <remarks>
/// <para>
/// This is typically 2 or 3 for visualization, but can be higher (e.g., 10-50)
/// for feature engineering or denoising applications.
/// </para>
/// <para><b>For Beginners:</b> How many dimensions in the output?
///
/// - 2D: For 2D scatter plots (most common for visualization)
/// - 3D: For interactive 3D visualizations
/// - 10-50: For downstream machine learning (compact features)
/// - 100+: For denoising (keep most info, remove noise)
/// </para>
/// </remarks>
int NComponents { get; }
}
Key Design Decisions:
- Fit/Transform pattern: Matches scikit-learn API for familiarity
- FitTransform: Convenience method for common usage
- Generic T: Supports float/double for different precision needs
- NComponents: Configurable output dimensionality
AC 1.1: Implement t-SNE (18 points)
File: C:\Users\cheat\source\repos\AiDotNet\src\DimensionalityReduction\TSNE.cs
using AiDotNet.Interfaces;
namespace AiDotNet.DimensionalityReduction;
/// <summary>
/// Implements t-SNE (t-Distributed Stochastic Neighbor Embedding) for nonlinear dimensionality reduction.
/// </summary>
/// <typeparam name="T">The numeric data type used for calculations (e.g., float, double).</typeparam>
/// <remarks>
/// <para>
/// t-SNE (van der Maaten & Hinton, 2008) is a nonlinear technique particularly well-suited for
/// visualizing high-dimensional data. It works by:
///
/// 1. Computing pairwise similarities in high-dimensional space using Gaussian kernel
/// 2. Computing pairwise similarities in low-dimensional space using Student-t distribution
/// 3. Minimizing KL divergence between high-D and low-D distributions via gradient descent
///
/// The use of heavy-tailed t-distribution in low-D space helps avoid crowding (all points
/// squashing into center) and creates well-separated clusters.
/// </para>
/// <para>
/// <b>Mathematical Formulation:</b>
///
/// High-dimensional similarity (conditional probability):
/// p(j|i) = exp(-||xi - xj||² / 2σi²) / Σk≠i exp(-||xi - xk||² / 2σi²)
///
/// Symmetrized joint probability:
/// pij = (p(j|i) + p(i|j)) / 2n
///
/// Low-dimensional similarity (Student-t with df=1):
/// qij = (1 + ||yi - yj||²)^(-1) / Σk≠l (1 + ||yk - yl||²)^(-1)
///
/// Objective (KL divergence):
/// Cost = Σi Σj pij × log(pij / qij)
///
/// Gradient:
/// dC/dyi = 4 × Σj (pij - qij) × (yi - yj) × (1 + ||yi - yj||²)^(-1)
/// </para>
/// <para><b>For Beginners:</b> Creating beautiful 2D visualizations of complex data.
///
/// t-SNE is the go-to algorithm when you want to visualize high-dimensional data.
/// It's the reason you see those beautiful cluster plots in papers and presentations.
///
/// <b>How to think about t-SNE:</b>
/// Imagine you have 1000 points in 1000-dimensional space (impossible to visualize).
/// t-SNE creates a 2D "map" where:
/// - Points that were close neighbors in 1000-D are drawn close in 2D
/// - Points that were far apart can be far or close (less emphasis on global distances)
///
/// <b>When to use t-SNE:</b>
/// - Visualizing high-dimensional data (images, gene expression, word embeddings)
/// - Discovering clusters and patterns
/// - Exploratory data analysis
/// - Creating figures for papers/presentations
///
/// <b>When NOT to use t-SNE:</b>
/// - When you need to transform new data (t-SNE can't do this, use UMAP)
/// - When distances between far points matter (t-SNE only preserves local structure)
/// - For downstream ML tasks (clusters may be artifacts, use UMAP or PCA)
/// - Large datasets (> 10,000 points without approximations)
///
/// <b>Key parameter: perplexity</b>
/// - Think of it as "how many neighbors should each point have?"
/// - Low (5-15): Very local structure, may create false small clusters
/// - Medium (30-50): Balanced (RECOMMENDED for most data)
/// - High (100+): More global structure, may merge distinct clusters
/// - Rule of thumb: 5 < perplexity < 50, typically 30
///
/// <b>Common mistakes:</b>
/// 1. Over-interpreting cluster sizes (t-SNE doesn't preserve cluster sizes)
/// 2. Measuring distances between far clusters (they're not meaningful)
/// 3. Running only once (try multiple random seeds, pick best result)
/// 4. Using default parameters without tuning (try different perplexity values)
///
/// <b>Default values from research:</b>
/// - nComponents=2: Standard for visualization (van der Maaten & Hinton, 2008)
/// - perplexity=30: Empirically validated default (scikit-learn, original paper)
/// - learningRate=200: Adaptive learning rate (modern implementations)
/// - nIter=1000: Sufficient for convergence on most datasets
/// </para>
/// </remarks>
public class TSNE<T> : IDimensionalityReduction<T>
{
private static readonly INumericOperations<T> NumOps = MathHelper.GetNumericOperations<T>();
private readonly int _nComponents;
private readonly double _perplexity;
private readonly double _learningRate;
private readonly int _nIter;
private readonly int _randomSeed;
private Matrix<T>? _embedding;
/// <summary>
/// Initializes a new instance of t-SNE.
/// </summary>
/// <param name="nComponents">
/// Number of dimensions in the embedding space.
/// Default: 2 (for 2D visualization).
/// Use 3 for 3D interactive visualizations, or 1 for 1D ordering.
/// </param>
/// <param name="perplexity">
/// The perplexity parameter, which balances local vs global structure.
/// Default: 30.0 (scikit-learn default, works well for most datasets).
///
/// Perplexity can be thought of as the target number of nearest neighbors.
/// - Low (5-15): Focus on very local structure, risk of false clusters
/// - Medium (20-50): Balanced structure (RECOMMENDED)
/// - High (50-100): More global structure, may merge clusters
///
/// Rule of thumb: perplexity should be smaller than number of points.
/// For datasets with < 100 points, use perplexity = 5-15.
/// For datasets with > 1000 points, try perplexity = 30-50.
/// </param>
/// <param name="learningRate">
/// The learning rate for gradient descent.
/// Default: 200.0 (scikit-learn default, works well for most data).
///
/// - Too low (< 10): Slow convergence, may get stuck
/// - Just right (100-1000): Good convergence
/// - Too high (> 1000): Unstable, poor results
///
/// Adaptive rule: learningRate = max(n / early_exaggeration / 4, 50)
/// where early_exaggeration = 12 (compression factor for first iterations).
/// </param>
/// <param name="nIter">
/// Number of gradient descent iterations.
/// Default: 1000 (sufficient for most datasets).
///
/// - Small datasets (< 1000 points): 250-500 iterations
/// - Medium datasets (1000-10000): 1000 iterations
/// - Large datasets (> 10000): 1000-5000 iterations
///
/// Monitor KL divergence - it should decrease and plateau.
/// If still decreasing at nIter, increase iterations.
/// </param>
/// <param name="randomSeed">
/// Random seed for reproducible initialization.
/// Default: 42 (makes results reproducible).
///
/// Note: t-SNE is stochastic - different seeds give different (but similar) results.
/// Best practice: Run with multiple seeds and choose best (lowest KL divergence).
/// </param>
/// <exception cref="ArgumentException">
/// Thrown when parameters are out of valid ranges.
/// </exception>
public TSNE(
int nComponents = 2,
double perplexity = 30.0,
double learningRate = 200.0,
int nIter = 1000,
int randomSeed = 42)
{
if (nComponents < 1)
{
throw new ArgumentException("Number of components must be at least 1.", nameof(nComponents));
}
if (perplexity < 1.0)
{
throw new ArgumentException("Perplexity must be at least 1.0.", nameof(perplexity));
}
if (learningRate <= 0)
{
throw new ArgumentException("Learning rate must be positive.", nameof(learningRate));
}
if (nIter < 1)
{
throw new ArgumentException("Number of iterations must be at least 1.", nameof(nIter));
}
_nComponents = nComponents;
_perplexity = perplexity;
_learningRate = learningRate;
_nIter = nIter;
_randomSeed = randomSeed;
}
/// <inheritdoc/>
public int NComponents => _nComponents;
/// <summary>
/// Gets the final embedding after fitting.
/// </summary>
/// <remarks>
/// <para><b>For Beginners:</b> The compressed data.
///
/// After calling Fit or FitTransform, this contains the low-dimensional coordinates.
/// Each row corresponds to one input data point.
///
/// For visualization:
/// ```csharp
/// var tsne = new TSNE<double>(nComponents: 2);
/// tsne.Fit(highDimData);
/// var embedding = tsne.Embedding; // n × 2 matrix
///
/// // Plot as scatter plot:
/// for (int i = 0; i < embedding.Rows; i++)
/// {
/// double x = embedding[i, 0];
/// double y = embedding[i, 1];
/// plot.AddPoint(x, y, labels[i]);
/// }
/// ```
/// </para>
/// </remarks>
public Matrix<T>? Embedding => _embedding;
/// <inheritdoc/>
public void Fit(Matrix<T> data)
{
if (data.Rows < 2)
{
throw new ArgumentException("Data must have at least 2 points.", nameof(data));
}
if (_perplexity >= data.Rows)
{
throw new ArgumentException(
$"Perplexity ({_perplexity}) must be less than number of points ({data.Rows}).",
nameof(data));
}
int n = data.Rows;
var random = new Random(_randomSeed);
// Step 1: Compute pairwise affinities in high-dimensional space
var P = ComputeAffinities(data);
// Step 2: Initialize low-dimensional embedding randomly
_embedding = InitializeEmbedding(n, random);
// Step 3: Optimize embedding using gradient descent
OptimizeEmbedding(P, _embedding, random);
}
/// <inheritdoc/>
public Matrix<T> Transform(Matrix<T> data)
{
throw new NotSupportedException(
"t-SNE does not support transforming new data. " +
"You must include all data points in the initial Fit call. " +
"For out-of-sample embedding, consider using UMAP instead.");
}
/// <inheritdoc/>
public Matrix<T> FitTransform(Matrix<T> data)
{
Fit(data);
return _embedding!;
}
/// <summary>
/// Computes pairwise affinities (similarities) in high-dimensional space.
/// </summary>
/// <remarks>
/// <para>
/// This implements the Gaussian kernel with adaptive bandwidth (σi) chosen
/// to achieve target perplexity for each point.
///
/// For each point i, we find σi such that the perplexity of the distribution
/// p(j|i) equals the target perplexity. Perplexity is defined as:
///
/// Perplexity(Pi) = 2^(H(Pi))
///
/// where H(Pi) is the Shannon entropy: H(Pi) = -Σj p(j|i) × log2(p(j|i))
///
/// This adaptive bandwidth ensures each point has approximately the same
/// number of "effective neighbors" regardless of local density.
/// </para>
/// </remarks>
private Matrix<double> ComputeAffinities(Matrix<T> data)
{
int n = data.Rows;
var P = new Matrix<double>(n, n);
// Compute pairwise distances
var distances = new Matrix<double>(n, n);
for (int i = 0; i < n; i++)
{
for (int j = i + 1; j < n; j++)
{
double dist = Convert.ToDouble(EuclideanDistance(data.GetRow(i), data.GetRow(j)));
distances[i, j] = dist * dist; // Squared distance for Gaussian kernel
distances[j, i] = distances[i, j];
}
}
// For each point, find σi to achieve target perplexity
double targetEntropy = Math.Log(_perplexity, 2);
for (int i = 0; i < n; i++)
{
// Binary search for sigma
double betaMin = 0.0;
double betaMax = double.MaxValue;
double beta = 1.0; // beta = 1 / (2 * sigma^2)
double tolerance = 1e-5;
int maxIterations = 50;
for (int iter = 0; iter < maxIterations; iter++)
{
// Compute probabilities for current beta
double sum = 0.0;
var probs = new double[n];
for (int j = 0; j < n; j++)
{
if (i == j) continue;
probs[j] = Math.Exp(-distances[i, j] * beta);
sum += probs[j];
}
// Normalize
for (int j = 0; j < n; j++)
{
probs[j] /= sum + 1e-10;
}
// Compute entropy
double entropy = 0.0;
for (int j = 0; j < n; j++)
{
if (probs[j] > 1e-10)
{
entropy -= probs[j] * Math.Log(probs[j], 2);
}
}
// Check convergence
double diff = entropy - targetEntropy;
if (Math.Abs(diff) < tolerance)
{
// Store probabilities
for (int j = 0; j < n; j++)
{
P[i, j] = probs[j];
}
break;
}
// Adjust beta
if (diff > 0)
{
betaMin = beta;
beta = (betaMax == double.MaxValue) ? beta * 2 : (beta + betaMax) / 2;
}
else
{
betaMax = beta;
beta = (beta + betaMin) / 2;
}
}
}
// Symmetrize: pij = (p(j|i) + p(i|j)) / 2n
var Psym = new Matrix<double>(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
Psym[i, j] = (P[i, j] + P[j, i]) / (2.0 * n);
}
}
return Psym;
}
/// <summary>
/// Initializes low-dimensional embedding randomly.
/// </summary>
private Matrix<T> InitializeEmbedding(int n, Random random)
{
var Y = new Matrix<T>(n, _nComponents);
// Initialize with small random values (Gaussian noise)
for (int i = 0; i < n; i++)
{
for (int j = 0; j < _nComponents; j++)
{
// Box-Muller transform for Gaussian random values
double u1 = random.NextDouble();
double u2 = random.NextDouble();
double gaussianValue = Math.Sqrt(-2.0 * Math.Log(u1)) * Math.Cos(2.0 * Math.PI * u2);
Y[i, j] = NumOps.FromDouble(gaussianValue * 1e-4); // Scale to small values
}
}
return Y;
}
/// <summary>
/// Optimizes the embedding using gradient descent.
/// </summary>
private void OptimizeEmbedding(Matrix<double> P, Matrix<T> Y, Random random)
{
int n = Y.Rows;
// Momentum for gradient descent
var gains = new Matrix<double>(n, _nComponents);
var velocity = new Matrix<double>(n, _nComponents);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < _nComponents; j++)
{
gains[i, j] = 1.0;
}
}
double momentum = 0.5; // Initial momentum
double finalMomentum = 0.8; // Momentum after early exaggeration
double minGain = 0.01;
int earlyExaggerationIter = 250; // First 250 iterations use exaggerated P
double earlyExaggeration = 12.0; // Multiply P by this during early iterations
for (int iter = 0; iter < _nIter; iter++)
{
// Switch to final momentum after early exaggeration
if (iter == earlyExaggerationIter)
{
momentum = finalMomentum;
}
// Compute Q matrix (low-dimensional affinities) and gradient
var (Q, grad) = ComputeGradient(Y, P, iter < earlyExaggerationIter ? earlyExaggeration : 1.0);
// Update embedding with adaptive learning rate and momentum
for (int i = 0; i < n; i++)
{
for (int d = 0; d < _nComponents; d++)
{
// Adaptive gains: increase if gradient direction consistent, decrease otherwise
double gradValue = grad[i, d];
double velValue = velocity[i, d];
if (Math.Sign(gradValue) != Math.Sign(velValue))
{
gains[i, d] += 0.2;
}
else
{
gains[i, d] *= 0.8;
}
gains[i, d] = Math.Max(gains[i, d], minGain);
// Update velocity with momentum
velocity[i, d] = momentum * velValue - _learningRate * gains[i, d] * gradValue;
// Update position
double currentY = Convert.ToDouble(Y[i, d]);
Y[i, d] = NumOps.FromDouble(currentY + velocity[i, d]);
}
}
// Center embedding (remove mean)
for (int d = 0; d < _nComponents; d++)
{
double mean = 0.0;
for (int i = 0; i < n; i++)
{
mean += Convert.ToDouble(Y[i, d]);
}
mean /= n;
for (int i = 0; i < n; i++)
{
Y[i, d] = NumOps.FromDouble(Convert.ToDouble(Y[i, d]) - mean);
}
}
}
}
/// <summary>
/// Computes Q matrix and gradient for current embedding.
/// </summary>
private (Matrix<double> Q, Matrix<double> grad) ComputeGradient(
Matrix<T> Y,
Matrix<double> P,
double exaggeration)
{
int n = Y.Rows;
var Q = new Matrix<double>(n, n);
var grad = new Matrix<double>(n, _nComponents);
// Compute Q matrix (Student-t kernel)
double sum = 0.0;
for (int i = 0; i < n; i++)
{
for (int j = i + 1; j < n; j++)
{
double distSq = 0.0;
for (int d = 0; d < _nComponents; d++)
{
double diff = Convert.ToDouble(Y[i, d]) - Convert.ToDouble(Y[j, d]);
distSq += diff * diff;
}
double qval = 1.0 / (1.0 + distSq);
Q[i, j] = qval;
Q[j, i] = qval;
sum += 2.0 * qval;
}
}
// Normalize Q
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
Q[i, j] /= (sum + 1e-10);
}
}
// Compute gradient
for (int i = 0; i < n; i++)
{
for (int d = 0; d < _nComponents; d++)
{
double gradSum = 0.0;
for (int j = 0; j < n; j++)
{
if (i == j) continue;
double pij = P[i, j] * exaggeration;
double qij = Q[i, j];
double diff = Convert.ToDouble(Y[i, d]) - Convert.ToDouble(Y[j, d]);
double multiplier = (pij - qij) * (1.0 / (1.0 + SquaredDistance(Y, i, j)));
gradSum += 4.0 * multiplier * diff;
}
grad[i, d] = gradSum;
}
}
return (Q, grad);
}
/// <summary>
/// Computes squared Euclidean distance between two embedding points.
/// </summary>
private double SquaredDistance(Matrix<T> Y, int i, int j)
{
double sum = 0.0;
for (int d = 0; d < _nComponents; d++)
{
double diff = Convert.ToDouble(Y[i, d]) - Convert.ToDouble(Y[j, d]);
sum += diff * diff;
}
return sum;
}
/// <summary>
/// Computes Euclidean distance between two high-dimensional points.
/// </summary>
private T EuclideanDistance(Vector<T> a, Vector<T> b)
{
T sum = NumOps.Zero;
for (int i = 0; i < a.Length; i++)
{
T diff = NumOps.Subtract(a[i], b[i]);
sum = NumOps.Add(sum, NumOps.Multiply(diff, diff));
}
return NumOps.Sqrt(sum);
}
}
Key Implementation Details:
- Adaptive bandwidth: σi chosen to achieve target perplexity for each point
- Early exaggeration: Multiply P by 12 for first 250 iterations (prevents crowding)
- Adaptive learning rate: Gains increase/decrease based on gradient direction consistency
- Momentum: Helps escape local minima
- Student-t kernel: Heavy-tailed distribution prevents crowding in low-D
- Time complexity: O(n²) per iteration, O(n² × iterations) total
Performance Note: For large datasets (> 10,000 points), implement Barnes-Hut approximation (O(n log n) per iteration) or use FFT-accelerated t-SNE (O(n) per iteration).
AC 1.2: Implement UMAP (18 points)
File: C:\Users\cheat\source\repos\AiDotNet\src\DimensionalityReduction\UMAP.cs
Due to UMAP's complexity (fuzzy simplicial sets, Riemannian geometry), I'll provide a simplified version focusing on the core algorithm. A production implementation would require additional optimizations.
using AiDotNet.Interfaces;
namespace AiDotNet.DimensionalityReduction;
/// <summary>
/// Implements UMAP (Uniform Manifold Approximation and Projection) for dimensionality reduction.
/// </summary>
/// <typeparam name="T">The numeric data type used for calculations (e.g., float, double).</typeparam>
/// <remarks>
/// <para>
/// UMAP (McInnes et al., 2018) is a state-of-the-art technique for both visualization and
/// general dimensionality reduction. It builds on mathematical foundations from Riemannian
/// geometry and algebraic topology to:
///
/// 1. Construct a fuzzy topological representation of high-dimensional data
/// 2. Optimize a low-dimensional graph to match the high-dimensional structure
/// 3. Preserve both local and global structure better than t-SNE
///
/// <b>Key advantages over t-SNE:</b>
/// - Faster (O(n^1.14) vs O(n²))
/// - Preserves global structure (distances between clusters are meaningful)
/// - Supports transform (can embed new points)
/// - Better for downstream ML (UMAP features work well for classification)
/// - More tunableUMAP has clearer parameter interpretations
/// </para>
/// <para><b>For Beginners:</b> The modern standard for dimensionality reduction.
///
/// UMAP is like t-SNE's smarter, faster cousin. It creates beautiful visualizations
/// while also being practical for machine learning pipelines.
///
/// <b>When to use UMAP:</b>
/// - Almost any dimensionality reduction task (it's very general-purpose)
/// - Large datasets (100,000+ points scale well)
/// - When you need to embed new data points later
/// - When global structure matters (cluster relationships)
/// - Downstream ML (UMAP features work great for classification/clustering)
///
/// <b>When to use t-SNE instead:</b>
/// - Historical comparisons (many papers used t-SNE)
/// - Extremely fine-grained local structure (t-SNE may be slightly better)
/// - Very small datasets (< 100 points, either works)
///
/// <b>Key parameters:</b>
///
/// <b>n_neighbors</b> (like t-SNE's perplexity):
/// - Controls local vs global balance
/// - Low (5-10): Focus on fine detail, many small clusters
/// - Medium (15-30): Balanced (RECOMMENDED)
/// - High (50-100): Focus on broad structure, fewer clusters
///
/// <b>min_dist</b> (how tightly to pack points):
/// - Low (0.0-0.1): Dense clusters, clear separation
/// - Medium (0.1-0.5): Balanced
/// - High (0.5-0.99): Loose, spread-out embedding
///
/// <b>metric</b> (distance function):
/// - "euclidean": Standard choice for continuous data
/// - "manhattan": For sparse data or counts
/// - "cosine": For normalized vectors (word embeddings)
/// - "hamming": For binary data
///
/// <b>Default values from research:</b>
/// - nComponents=2: Standard for visualization (McInnes et al., 2018)
/// - nNeighbors=15: Empirically validated (original paper)
/// - minDist=0.1: Good separation without over-compression
/// - metric="euclidean": Most common use case
/// </para>
/// </remarks>
public class UMAP<T> : IDimensionalityReduction<T>
{
private static readonly INumericOperations<T> NumOps = MathHelper.GetNumericOperations<T>();
private readonly int _nComponents;
private readonly int _nNeighbors;
private readonly double _minDist;
private readonly string _metric;
private readonly int _nEpochs;
private readonly int _randomSeed;
private Matrix<T>? _embedding;
private Matrix<T>? _trainingData;
/// <summary>
/// Initializes a new instance of UMAP.
/// </summary>
/// <param name="nComponents">
/// Number of dimensions in the embedding space.
/// Default: 2 (for visualization).
/// </param>
/// <param name="nNeighbors">
/// Size of local neighborhood (number of neighbors to consider for each point).
/// Default: 15 (empirically validated, works for most datasets).
///
/// - Low (5-10): Emphasizes fine local structure
/// - Medium (15-30): Balanced local/global (RECOMMENDED)
/// - High (50-100): Emphasizes broader global structure
///
/// This is similar to t-SNE's perplexity but more interpretable.
/// </param>
/// <param name="minDist">
/// Minimum distance between points in the embedding.
/// Default: 0.1 (allows some spread while maintaining cluster structure).
///
/// - Low (0.0-0.1): Tightly packed, well-separated clusters
/// - Medium (0.1-0.5): Balanced
/// - High (0.5-0.99): Loose, topological structure emphasized
/// </param>
/// <param name="metric">
/// Distance metric to use in high-dimensional space.
/// Default: "euclidean" (standard for continuous features).
///
/// Options: "euclidean", "manhattan", "cosine", "hamming".
/// </param>
/// <param name="nEpochs">
/// Number of training epochs (optimization iterations).
/// Default: 200 (sufficient for most datasets).
///
/// - Small datasets (< 1000): 100-200
/// - Medium datasets (1000-10000): 200-500
/// - Large datasets (> 10000): 200 (scales automatically)
/// </param>
/// <param name="randomSeed">
/// Random seed for reproducible results.
/// Default: 42.
/// </param>
public UMAP(
int nComponents = 2,
int nNeighbors = 15,
double minDist = 0.1,
string metric = "euclidean",
int nEpochs = 200,
int randomSeed = 42)
{
if (nComponents < 1)
{
throw new ArgumentException("Number of components must be at least 1.", nameof(nComponents));
}
if (nNeighbors < 2)
{
throw new ArgumentException("Number of neighbors must be at least 2.", nameof(nNeighbors));
}
if (minDist < 0 || minDist >= 1.0)
{
throw new ArgumentException("Min distance must be in [0, 1).", nameof(minDist));
}
if (metric != "euclidean" && metric != "manhattan" && metric != "cosine" && metric != "hamming")
{
throw new ArgumentException(
"Metric must be 'euclidean', 'manhattan', 'cosine', or 'hamming'.",
nameof(metric));
}
if (nEpochs < 1)
{
throw new ArgumentException("Number of epochs must be at least 1.", nameof(nEpochs));
}
_nComponents = nComponents;
_nNeighbors = nNeighbors;
_minDist = minDist;
_metric = metric;
_nEpochs = nEpochs;
_randomSeed = randomSeed;
}
/// <inheritdoc/>
public int NComponents => _nComponents;
/// <inheritdoc/>
public void Fit(Matrix<T> data)
{
_trainingData = data;
// Step 1: Compute k-nearest neighbors graph
var knnGraph = ComputeKNNGraph(data);
// Step 2: Compute fuzzy simplicial set (high-D graph)
var graph = ComputeFuzzyGraph(data, knnGraph);
// Step 3: Initialize embedding
_embedding = InitializeEmbedding(data.Rows, new Random(_randomSeed));
// Step 4: Optimize embedding
OptimizeEmbedding(graph, _embedding);
}
/// <inheritdoc/>
public Matrix<T> Transform(Matrix<T> data)
{
if (_embedding == null || _trainingData == null)
{
throw new InvalidOperationException("Model must be fitted before calling Transform.");
}
// Simplified transform: find nearest neighbors in training data and interpolate
var transformed = new Matrix<T>(data.Rows, _nComponents);
for (int i = 0; i < data.Rows; i++)
{
// Find k nearest neighbors in training data
var neighbors = FindKNearestNeighbors(_trainingData, data.GetRow(i), _nNeighbors);
// Average their embeddings (weighted by inverse distance)
var weights = new double[_nNeighbors];
double weightSum = 0.0;
for (int k = 0; k < _nNeighbors; k++)
{
double dist = Convert.ToDouble(CalculateDistance(data.GetRow(i), _trainingData.GetRow(neighbors[k])));
weights[k] = 1.0 / (dist + 1e-10);
weightSum += weights[k];
}
for (int d = 0; d < _nComponents; d++)
{
double sum = 0.0;
for (int k = 0; k < _nNeighbors; k++)
{
sum += Convert.ToDouble(_embedding[neighbors[k], d]) * weights[k];
}
transformed[i, d] = NumOps.FromDouble(sum / weightSum);
}
}
return transformed;
}
/// <inheritdoc/>
public Matrix<T> FitTransform(Matrix<T> data)
{
Fit(data);
return _embedding!;
}
/// <summary>
/// Computes k-nearest neighbors graph.
/// </summary>
private List<int>[] ComputeKNNGraph(Matrix<T> data)
{
int n = data.Rows;
var knnGraph = new List<int>[n];
for (int i = 0; i < n; i++)
{
knnGraph[i] = FindKNearestNeighbors(data, data.GetRow(i), _nNeighbors + 1)
.Where(idx => idx != i)
.Take(_nNeighbors)
.ToList();
}
return knnGraph;
}
/// <summary>
/// Finds k nearest neighbors for a given point.
/// </summary>
private List<int> FindKNearestNeighbors(Matrix<T> data, Vector<T> point, int k)
{
var distances = new List<(int index, double distance)>();
for (int i = 0; i < data.Rows; i++)
{
double dist = Convert.ToDouble(CalculateDistance(point, data.GetRow(i)));
distances.Add((i, dist));
}
return distances.OrderBy(x => x.distance)
.Take(k)
.Select(x => x.index)
.ToList();
}
/// <summary>
/// Computes fuzzy simplicial set (weighted graph).
/// </summary>
private Matrix<double> ComputeFuzzyGraph(Matrix<T> data, List<int>[] knnGraph)
{
int n = data.Rows;
var graph = new Matrix<double>(n, n);
// For each point, compute fuzzy membership strengths
for (int i = 0; i < n; i++)
{
var neighbors = knnGraph[i];
var distances = neighbors.Select(j =>
Convert.ToDouble(CalculateDistance(data.GetRow(i), data.GetRow(j)))).ToList();
// Find rho (distance to nearest neighbor)
double rho = distances.Min();
// Find sigma (normalization parameter)
double sigma = ComputeSigma(distances, rho);
// Compute memberships
for (int k = 0; k < neighbors.Count; k++)
{
int j = neighbors[k];
double dist = distances[k];
// Fuzzy membership strength
double membership = Math.Exp(-(Math.Max(0, dist - rho) / sigma));
graph[i, j] = membership;
}
}
// Symmetrize using probabilistic OR: a + b - a*b
var symGraph = new Matrix<double>(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
symGraph[i, j] = graph[i, j] + graph[j, i] - graph[i, j] * graph[j, i];
}
}
return symGraph;
}
/// <summary>
/// Computes sigma parameter for fuzzy membership.
/// </summary>
private double ComputeSigma(List<double> distances, double rho)
{
// Binary search to find sigma such that sum of memberships ≈ log2(k)
double target = Math.Log2(_nNeighbors);
double sigmaMin = 1e-10;
double sigmaMax = 1000.0;
double sigma = 1.0;
for (int iter = 0; iter < 64; iter++)
{
double sum = 0.0;
foreach (double dist in distances)
{
sum += Math.Exp(-(Math.Max(0, dist - rho) / sigma));
}
double logSum = Math.Log2(sum + 1e-10);
if (Math.Abs(logSum - target) < 1e-5)
{
break;
}
if (logSum > target)
{
sigmaMax = sigma;
sigma = (sigma + sigmaMin) / 2;
}
else
{
sigmaMin = sigma;
sigma = (sigma + sigmaMax) / 2;
}
}
return sigma;
}
/// <summary>
/// Initializes embedding with spectral initialization (PCA-like).
/// </summary>
private Matrix<T> InitializeEmbedding(int n, Random random)
{
// Simplified: random initialization (production would use spectral initialization)
var Y = new Matrix<T>(n, _nComponents);
for (int i = 0; i < n; i++)
{
for (int d = 0; d < _nComponents; d++)
{
Y[i, d] = NumOps.FromDouble(random.NextDouble() * 20 - 10);
}
}
return Y;
}
/// <summary>
/// Optimizes embedding using stochastic gradient descent.
/// </summary>
private void OptimizeEmbedding(Matrix<double> graph, Matrix<T> Y)
{
int n = Y.Rows;
double learningRate = 1.0;
// Collect all edges (i, j) where graph[i,j] > 0
var edges = new List<(int i, int j, double weight)>();
for (int i = 0; i < n; i++)
{
for (int j = i + 1; j < n; j++)
{
if (graph[i, j] > 0)
{
edges.Add((i, j, graph[i, j]));
}
}
}
var random = new Random(_randomSeed);
for (int epoch = 0; epoch < _nEpochs; epoch++)
{
// Shuffle edges
edges = edges.OrderBy(x => random.Next()).ToList();
// Decay learning rate
double alpha = learningRate * (1.0 - epoch / (double)_nEpochs);
foreach (var (i, j, weight) in edges)
{
// Compute low-dimensional distance
double distSq = 0.0;
for (int d = 0; d < _nComponents; d++)
{
double diff = Convert.ToDouble(Y[i, d]) - Convert.ToDouble(Y[j, d]);
distSq += diff * diff;
}
// Compute gradient for attractive force
double attractive = -2.0 * weight / (1.0 + distSq);
// Compute gradient for repulsive force
double repulsive = 2.0 * (1.0 - weight) / ((1.0 + distSq) * (1.0 + distSq));
double gradient = attractive + repulsive;
// Update positions
for (int d = 0; d < _nComponents; d++)
{
double diff = Convert.ToDouble(Y[i, d]) - Convert.ToDouble(Y[j, d]);
double update = alpha * gradient * diff;
Y[i, d] = NumOps.FromDouble(Convert.ToDouble(Y[i, d]) + update);
Y[j, d] = NumOps.FromDouble(Convert.ToDouble(Y[j, d]) - update);
}
}
}
}
/// <summary>
/// Calculates distance between two points based on the selected metric.
/// </summary>
private T CalculateDistance(Vector<T> a, Vector<T> b)
{
return _metric switch
{
"euclidean" => EuclideanDistance(a, b),
"manhattan" => ManhattanDistance(a, b),
"cosine" => CosineDistance(a, b),
"hamming" => HammingDistance(a, b),
_ => EuclideanDistance(a, b)
};
}
private T EuclideanDistance(Vector<T> a, Vector<T> b)
{
T sum = NumOps.Zero;
for (int i = 0; i < a.Length; i++)
{
T diff = NumOps.Subtract(a[i], b[i]);
sum = NumOps.Add(sum, NumOps.Multiply(diff, diff));
}
return NumOps.Sqrt(sum);
}
private T ManhattanDistance(Vector<T> a, Vector<T> b)
{
T sum = NumOps.Zero;
for (int i = 0; i < a.Length; i++)
{
T diff = NumOps.Abs(NumOps.Subtract(a[i], b[i]));
sum = NumOps.Add(sum, diff);
}
return sum;
}
private T CosineDistance(Vector<T> a, Vector<T> b)
{
T dotProduct = NumOps.Zero;
T normA = NumOps.Zero;
T normB = NumOps.Zero;
for (int i = 0; i < a.Length; i++)
{
dotProduct = NumOps.Add(dotProduct, NumOps.Multiply(a[i], b[i]));
normA = NumOps.Add(normA, NumOps.Multiply(a[i], a[i]));
normB = NumOps.Add(normB, NumOps.Multiply(b[i], b[i]));
}
T cosine = NumOps.Divide(dotProduct, NumOps.Multiply(NumOps.Sqrt(normA), NumOps.Sqrt(normB)));
return NumOps.Subtract(NumOps.One, cosine); // Convert similarity to distance
}
private T HammingDistance(Vector<T> a, Vector<T> b)
{
int count = 0;
for (int i = 0; i < a.Length; i++)
{
if (!NumOps.AreEqual(a[i], b[i]))
{
count++;
}
}
return NumOps.FromDouble(count);
}
}
Key Implementation Details:
- Fuzzy simplicial sets: Graph representation with adaptive bandwidth
- Probabilistic symmetrization: Combines directed similarities
- Stochastic gradient descent: Efficient optimization
- Multiple distance metrics: Euclidean, Manhattan, Cosine, Hamming
- Transform support: Can embed new points (unlike t-SNE)
- Time complexity: O(n^1.14) - much faster than t-SNE's O(n²)
Production Note: This is a simplified implementation. Production UMAP would include:
- Spectral initialization (instead of random)
- Nearest neighbor indexing (Annoy, NNDescent)
- Negative sampling for repulsive forces
- Parallel/GPU acceleration
Common Pitfalls
-
Not standardizing features before t-SNE/UMAP:
- Pitfall: Features on different scales dominate distance calculations
- Solution: Always standardize (mean=0, std=1) before dimensionality reduction
-
Over-interpreting t-SNE visualizations:
- Pitfall: Assuming cluster sizes/distances are meaningful
- Solution: t-SNE preserves local structure only; use UMAP if global structure matters
-
Using default perplexity/n_neighbors without tuning:
- Pitfall: Missing important structure at different scales
- Solution: Try multiple values and compare results
-
Expecting t-SNE to be reproducible:
- Pitfall: Different runs give different results (stochastic optimization)
- Solution: Use fixed random seed or run multiple times and pick best (lowest KL divergence)
-
Using t-SNE embeddings for downstream ML:
- Pitfall: t-SNE distorts global structure, bad for classification/clustering
- Solution: Use UMAP or PCA for feature engineering
Conclusion
You've built modern dimensionality reduction with t-SNE, UMAP, and PCA:
What You Built:
- t-SNE: Beautiful visualizations, preserves local structure
- UMAP: Fast, preserves global structure, supports transform
- PCA: Fast linear baseline, interpretable components
Impact:
- Enables visualization of complex high-dimensional data
- Provides features for downstream ML tasks
- Completes unsupervised learning toolkit
Key Takeaways:
- UMAP is usually the best default choice (fast, accurate, versatile)
- Use t-SNE for publication-quality visualizations of small datasets
- Use PCA for baseline comparisons and when interpretability matters
- Always standardize features before dimensionality reduction
You've mastered dimensionality reduction - the art of seeing patterns in complex data!