Geometric Methods in Data Preprocessing: Enhancing Your Data Through Spatial Thinking

When working with complex datasets, traditional data preprocessing in machine learning methods sometimes fall short of revealing the deeper patterns hidden in your data.

You might have clean, complete data with solid quality pipelines, but still struggle to extract meaningful insights that drive better model performance.

This is where geometric methods come in. By thinking of your data as points, shapes, or spaces, much like a map or blueprint, you can spot patterns and connections that standard methods might overlook. Drawing on the clarity of spatial reasoning, we'll explore geometric approaches that can reshape your preprocessing workflow and help you get insights in ways you might not expect.

What Are Geometric Methods in Data Preprocessing?

Geometric methods apply concepts from geometry and topology to understand and transform your data. Think of your dataset as points in a multidimensional space, where each feature represents a different dimension (See Fig 1). Geometric preprocessing helps you analyze relationships between these points based on their positions, distances, and the shapes they form.

Fig 1: Geometric Feature Engineering

Geometric feature engineering creates new, informative features based on spatial properties of your data. Here are some examples:

• Angles between data points or vectors
• Distances between points or to reference landmarks
• Density of points in different regions
• Shape metrics like convex hull area or perimeter
• Centroids and distances to them
• Isolation measures for outlier detection

Unlike traditional preprocessing that handles each feature independently, geometric methods consider how data points relate to each other in space. This perspective often reveals insights about your data that might remain hidden when using conventional approaches.

Let's say you're working with a dataset from a retail store that tracks customer purchases. Each customer is represented by two features: total spending (in dollars) and number of visits per year. You want to preprocess this data to understand customer behavior better, maybe to identify loyal shoppers or detect unusual patterns before feeding it into a clustering model.

Here's how you can use geometric methods to create new features based on spatial relationships.

Imagine you have 1,000 customers, and each is a point in a 2D space where:

• The x-axis is their total spending (e.g., $100 to $5,000).
• The y-axis is their number of visits (e.g., 1 to 50 visits).

Instead of just using these raw numbers, you apply geometric feature engineering to capture how customers relate to each other in this "spending-visits" space.

Suppose the average spending across all customers is $1,200, and the average number of visits is 15. This gives you a "center" point at ($1,200, 15).

For each customer, measure their distance to this center. For instance, Customer A spends $2,000 and visits 10 times. Using simple distance math (like the Pythagorean theorem in 2D), their distance is roughly 800 units (dollars and visits combined).

So, in this case, customers far from the center might be big spenders, rare visitors, or both (potentially VIPs or outliers) worth studying.

Why Add Geometric Thinking to Your Preprocessing Toolkit?

Now that you've seen how geometric methods can transform your data into a spatial map, showing patterns through distances and shapes, let's explore why this approach is great for your preprocessing workflow:

1. Capture complex relationships that aren't obvious in tabular formats.
2. Reduce dimensionality while preserving the meaningful structure of your data.
3. Identify outliers based on their spatial positions relative to other points.
4. Transform non-linear data into more ML-friendly representations.
5. Handle imbalanced datasets by understanding their geometric distribution.

Essential Geometric Techniques You Can Use

Building on the spatial perspective of viewing your data as points and shapes, you can enhance your pipeline with techniques that capture meaningful patterns and relationships. Here are some practical geometric methods you can easily incorporate:

1. Distance-Based Transformations

Distance calculations are the foundation of many geometric methods. By computing how far apart data points are from each other, you can:

• Group similar items using clustering algorithms
• Identify anomalies that lie far from most points
• Create new features based on distances to landmark points

The most common distance metrics include Euclidean (straight-line), Manhattan (city-block), and Mahalanobis (accounts for correlations).

Example: In a fraud detection system, you can calculate the Mahalanobis distance between a new transaction and the centroid of a user's normal transaction patterns. Transactions with distances beyond a threshold get flagged for review. This allows you to identify subtle fraud patterns that simple rule-based systems might miss.

2. Manifold Learning

Manifold learning helps you understand the intrinsic structure of high-dimensional data by projecting it onto a lower-dimensional space while preserving important relationships (See Fig 2).

Fig 2: Manifold Learning

Popular manifold learning techniques include:

• t-SNE (t-Distributed Stochastic Neighbor Embedding): Excellent for visualization by emphasizing local similarities.
• UMAP (Uniform Manifold Approximation and Projection): Faster than t-SNE and better preserves global structure.
• LLE (Locally Linear Embedding): Preserves local neighborhoods of points.

Let's go through an example to understand this better. When analyzing thousands of product reviews with hundreds of text features, UMAP can project this high-dimensional data onto a 2D map where similar reviews cluster together. This visualization helps you identify distinct customer sentiment groups and discover nuanced opinion patterns that simple positive/negative categorization would miss.

3. Topological Data Analysis (TDA)

TDA examines the "shape" of your data across multiple scales. It helps you understand persistent features that remain stable despite noise or variations in your dataset (See Fig 3).

Fig 3: Persistent Homology

The core technique in TDA is persistent homology (See Fig 4), which tracks how topological features (like connected components, loops, and voids) appear and disappear as you analyze data at different resolutions.

Fig 4: Persistent Diagram

Example: In healthcare, TDA helps analyze complex patient data to identify disease subtypes. For instance, when applied to diabetes patient data, TDA might reveal distinct clusters and connectivity patterns that correspond to different disease progression paths to help doctors develop more personalized treatment approaches for each subtype.

4. Geometric Feature Engineering

You can create new features based on geometric properties:

• Angles between data points or features
• Volumes of simplices formed by points
• Curvature of manifolds where data lies
• Density of points in different regions

Example: In a retail location analysis, you can create a "competitive pressure" feature by calculating the density of competitor stores within different radii of your locations. This geometric feature often predicts store performance better than simple counts, as it captures the spatial distribution of competition more accurately.

Real-World Applications

Now that you understand what geometric methods are, let's look at some of their real-world applications:

Customer Segmentation

When analyzing customer behavior data, traditional clustering might miss subtle patterns. You can project customer profiles onto a 2D or 3D space where natural groupings become visible by applying manifold learning techniques. These groups often represent market segments with distinct behaviors that standard approaches might lump together.

Medical Image Analysis

In healthcare, topological data analysis helps examine the structure of medical images. For example, when analyzing mammograms, TDA can help you identify persistent features that correspond to potentially cancerous tissue. These features might be missed by looking only at pixel-level information.

Financial Fraud Detection

Distance-based anomaly detection helps identify fraudulent transactions by measuring how far they deviate from normal patterns in multi-dimensional feature space. This geometric approach spots suspicious activities that might look normal when examining individual features in isolation.

How And When You Can Start With Geometric Methods

Having explored how geometric methods can reveal patterns by treating data as points and shapes in a spatial landscape, you're now ready to apply these concepts to your own preprocessing tasks.

Here's a simple way to begin:

1. Visualize your data geometrically using dimensionality reduction techniques like PCA, t-SNE, or UMAP.
2. Examine distance distributions between points to understand the geometric structure.
3. Try a simple distance-based approach such as k-nearest neighbors for imputation or anomaly detection.
4. Experiment with manifold learning to transform your data while preserving important relationships.
5. Create geometric features based on distances, angles, or local densities.

For example, in an e-commerce dataset with features like purchase frequency, average order value, and product category diversity, you can apply UMAP to project the data into a 2D plot. This visualization might reveal clusters of customers, such as frequent low-spenders versus occasional high-spenders, to help you identify market segments before clustering.

But when would it be ideal to use geometric methods for your data processing in the first place?

• When your data has complex, nonlinear relationships
• When traditional feature engineering doesn't capture important patterns
• When you need to reduce dimensions while preserving structure
• When working with naturally geometric data (images, spatial information, network data)
• When dealing with imbalanced datasets where minority classes form distinct regions

Conclusion

You'll have noticed how the geometric methods we went through add a powerful dimension to your data preprocessing toolkit. You gain insights that table-focused approaches might miss by thinking about your data spatially. These techniques help you transform complex, high-dimensional data into more manageable representations that machine learning models can process effectively.

That said, as you build your next machine learning project, you'll need to consider whether a geometric perspective might help you better understand and prepare your data. The spatial relationships between your data points often contain valuable information waiting to be discovered!