Overview

TRAILED provides topological representation learning methods for EHR data, built on the differentiable Euler Characteristic Transform (ECT). It bridges topological data analysis with clinical machine learning.

Note

TRAILED is under active development. The current release provides the foundational ECT implementation. Healthcare-specific extensions are in progress.

The Problem

Longitudinal EHR data and synthetic data generation face two persistent challenges:

Mode Collapse

Rare but clinically significant phenotypes — pediatric rare diseases, underrepresented demographics — are often absent from synthetic datasets. Models trained on such data fail on these populations. Standard statistical metrics cannot detect these coverage gaps because they rely on pairwise comparisons that miss higher-order structure.

Pathological Interpolation

Generative models produce synthetic patient trajectories that pass through biologically implausible states: impossible lab value transitions, contradictory comorbidity sequences, or clinically incoherent progressions. These failures degrade downstream model reliability and create safety risks.

Why Topology?

Traditional fidelity metrics are coordinate-dependent and local — they compare distributions point-by-point but cannot capture the global structure of patient trajectory spaces. Topological methods provide:

Coordinate-Free Representations

ECT descriptors encode shape without relying on specific coordinate systems, making them robust to embedding choices.

Higher-Order Structure

Topology captures connectivity, holes, and voids — the “shape” of data distributions that pairwise statistics miss.

Differentiability

TRAILED’s implementation supports gradients, enabling topological objectives as training-time regularizers.

What is ECT?

The Euler Characteristic Transform is a topological descriptor that captures shape information through directional filtrations:

  1. Direction sampling: Choose directions on the unit sphere

  2. Filtration: Sweep a hyperplane along each direction

  3. Euler characteristic: Count connected components, holes, and voids at each level

  4. Vectorization: Concatenate all curves into a fixed-length descriptor

        graph LR
   subgraph "Input"
      A["Patient Embeddings"]
   end

   subgraph "Direction Sampling"
      B1["θ₁"]
      B2["θ₂"]
      B3["θₙ"]
   end

   subgraph "Filtration"
      C1["EC curve 1"]
      C2["EC curve 2"]
      C3["EC curve n"]
   end

   subgraph "Output"
      D["Topological Descriptor"]
   end

   A --> B1
   A --> B2
   A --> B3
   B1 --> C1
   B2 --> C2
   B3 --> C3
   C1 --> D
   C2 --> D
   C3 --> D

   style A fill:#f9f9f9,stroke:#999
   style D fill:#DFF0D8,stroke:#3C763D,stroke-width:2px

ECT descriptors have powerful properties:

Injectivity

ECT can distinguish between almost all shapes — it’s injective on a dense subset of shapes.

Stability

Small perturbations to input data produce small changes in descriptors.

Differentiability

TRAILED’s implementation supports gradients for end-to-end learning.

Roadmap

TRAILED is being developed in phases:

Current: ECT Foundation

Fast ECT computation with NumPy, sklearn, and tabular (pandas/polars) integrations. This is the building block for healthcare-specific methods. For PyTorch use cases, see the upstream aidos-lab/dect package.

Planned: Density-Aware Descriptors

Extensions that fuse topological structure with local density information, addressing limitations of standard ECT for statistical inference.

Planned: Patient Manifold

Methods for constructing and analyzing patient manifolds from longitudinal EHR embeddings, characterizing viable pathways vs. impossible states.

Planned: Fidelity Metrics

Topological fidelity scores for synthetic data validation, designed to correlate with downstream clinical task utility.

Architecture

TRAILED has a layered design:

        graph TB
   subgraph "Core"
      A["Direction sampling"]
      B["Filtration computation"]
      C["EC curve calculation"]
   end

   subgraph "Adapters"
      D["NumPy interface"]
      E["pandas interface"]
   end

   subgraph "Plugins (optional)"
      F["sklearn transformer"]
   end

   A --> B
   B --> C
   C --> D
   C --> E
   D --> F

   style A fill:#D9EDF7,stroke:#31708F,stroke-width:2px
   style B fill:#D9EDF7,stroke:#31708F,stroke-width:2px
   style C fill:#D9EDF7,stroke:#31708F,stroke-width:2px

Key Components

Direction Sampling

TRAILED supports multiple sampling strategies:

  • Uniform: Random directions on the sphere

  • Stratified: Evenly distributed directions

  • Custom: User-defined direction sets

Resolution Control

The resolution parameter controls filtration granularity:

  • Higher resolution = more detail, larger descriptors

  • Lower resolution = faster computation, coarser features

Framework Integration

Framework

Installation

NumPy/pandas

pip install trailed (included)

sklearn

pip install trailed[sklearn]

pandas/polars

pip install trailed[dataframe]

All

pip install trailed[all]

PyTorch

pip install dect (upstream package)

Use Cases

Synthetic Data Validation

Compare topological structure of real and synthetic EHR cohorts to detect mode collapse and coverage gaps.

Training Regularization

Use differentiable ECT as a loss term to steer generative models away from pathological solutions.

Patient Trajectory Analysis

Characterize clinical pathways and identify anomalous trajectories in topological latent space.

Representation Learning

Extract topological features from longitudinal health records for downstream prediction tasks.

Next Steps