Breaking the Von Neumann Bottleneck through Topological Signatures.
Lambda-Topo is a topological data analysis system that transforms raw data into compact topological signatures for efficient storage and retrieval. It combines two research tracks:
| Metric | Value |
|---|---|
| Points Processed | — |
| Signatures Generated | — |
| Memory Queries | — |
| Uptime | — |
Stats refresh on deployment. For real-time monitoring, see [live dashboard link].
Lambda-Topo is grounded in Topological Data Analysis (TDA), a field that uses techniques from algebraic topology to extract robust geometric features from data.
At its core, the system computes persistent homology — a method that tracks topological features (connected components, loops, voids) across a filtration of scales:
- H₀ (H0): Connected components — measures how data clusters and separates
- H₁ (H1): Loops/cycles — detects cyclic structures, holes, and rings in data
- H₂ (H2): Voids — captures higher-dimensional cavities
The persistence of a feature — how long it exists across the filtration — encodes its geometric significance. Long-lived features are true structure; short-lived features are noise.
The output of persistent homology is a barcode (multi-set of intervals) or persistence diagram (points in 2D). Each bar represents one topological feature; its length equals its persistence.
Lambda-Topo transforms these barcodes into Hilbert coefficients — fixed-length vectors that capture the shape of the persistence distribution. These coefficients are:
- Rotation and translation invariant (ideal for shape matching)
- Compact (fixed dimension regardless of input size)
- Queryable via FAISS for sub-millisecond similarity search
Traditional ML operates in feature space — two structurally identical point clouds can appear unrelated if their coordinates differ. Lambda-Topo operates in shape space, where topology is the invariant:
Structural identity → Topological equivalence → Persistent homology
This provides:
- Noise robustness — persistence filters out insignificant features
- Compression — barcode of 10k points → ~50 coefficients
- Foundation for geometric deep learning, topological neural networks, and spatial reasoning
See:
- Edelsbrunner et al., 2000 — Persistence of barcodes
- Ghrist, 2008 — Barcodes in computational topology
- Chazal & Michel, 2021 — An Introduction to Topological Data Analysis
Topological Memory
- Index any data (point clouds, embeddings, images) by its shape
- Barcode-aware retrieval minimizes Von Neumann bottleneck data movement
Manifold Intelligence
- Unsupervised manifold training for physics + geometry applications
- Classical mechanics, nuclear physics, differential geometry
Data Input
|
v
Topological Signature <- ripser (persistent homology)
Generation <- H0/H1 barcodes -> Hilbert coefficients
|
v
Knowledge Equation <- Persistence polynomials -> LLM prompts
Generation
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v
Memory Store <- FAISS (FlatL2, IVF, HNSW indexes)
(Topological Index)
- Persistent Homology — Uses ripser for sub-second barcode computation on 10k+ point clouds
- Fixed-Length Signatures — Hilbert coefficient vectors truncated/padded to fixed length for efficient indexing
- Barcode-Aware Retrieval — Coarse-to-fine similarity search using topological shape matching
- FAISS Integration — Production-grade similarity search with multiple index types
- LLM Integration — Turn persistence polynomials directly into reasoning prompts
- Unsupervised Manifold Training — TSNE, Isomap, LLE, MDS, PCA with automatic best-selection
- Physics CLI — Measure topological properties for classical mechanics and nuclear physics data
- Composite Scoring — Silhouette score + reconstruction error for manifold quality assessment
- Visualization — PNG plots comparing manifold embeddings side-by-side
- Physics Measurement Suite — Topological tensor analysis for physics data
- Modular Architecture — Easy extension to H1, H2 barcodes, vineyards, multiparameter persistence
- No heavy dependencies — Core needs only numpy, ripser, faiss, sklearn
git clone https://github.com/teerthsharma/lambda-topo.git
cd lambda-topo
pip install -r requirements.txt# Run the topological memory demo
python demo/turn_data_into_formula.py
# Run manifold trainer (Swiss roll dataset)
python -m manifold_trainer
# Run physics measurement CLI
python physics_cli.py --type classical --plot
python physics_cli.py --type nuclear --plotlambda-topo/
|-- lambda_shappire/ # Core topological analysis library
| |-- topology.py # Persistent homology, barcodes, Hilbert coefficients
| |-- memory.py # FAISS-backed topological memory store
|-- manifold_trainer/ # Unsupervised manifold selection system
|-- demo/
| |-- turn_data_into_formula.py # End-to-end demo
|-- hollow_manifold_sim.py # EM hollow manifold simulator
|-- physics_cli.py # Physics measurement CLI
|-- manifold_plots/ # Manifold embedding visualizations
|-- physics_plots/ # Physics analysis visualizations
|-- ROADMAP.md # Future development plan
|-- MVP_SUMMARY.md # Technical deep-dive
|-- requirements.txt
Traditional similarity search (cosine, L2) operates in raw feature space — two structurally identical point clouds can appear unrelated if their coordinates differ. Lambda-Topo operates in shape space:
Two images of the same digit
-> Convert to point clouds (edges + keypoints)
-> Compute persistent homology (H0 barcode -- connected components)
-> Extract Hilbert coefficients -> persistence polynomial
-> Nearly identical polynomials
-> Retrieval matches by topology, not pixels
This is the mathematical foundation for:
- Topological Data Analysis (TDA) in machine learning
- Shape matching in computer vision
- Molecular fingerprinting in drug discovery
- Cosmic structure analysis in astrophysics
Barcode computation (10k points): ~0.8s via ripser Signature indexing (100k vectors): ~2s with FAISS IVFFlat Similarity search (top-10): <1ms with HNSW Manifold training (1k Swiss roll): <10s for all 5 algorithms
See ROADMAP.md for:
- LLM framework integrations (LangChain, LlamaIndex, HuggingFace)
- Image + text processing pipelines
- H1 and H2 barcode support
- Production API (FastAPI + persistence)
- Vineyards and multiparameter persistence
- Edelsbrunner, Letscher, Zomorodian. Topological Persistence and Simplification. Discrete & Computational Geometry, 2000.
- Ghrist. Barcodes: The Persistent Topology of Data. Bulletin of the AMS, 2008.
- Chazal & Michel. An Introduction to Topological Data Analysis. JMLR, 2021.
- Zomorodian & Carlsson. Computing Persistent Homology. Discrete & Computational Geometry, 2005.
- Bauer, Kerber, Reininghaus. Clear and Compress: Computing Persistent Homology in Three Steps. EG Workshop on Computational Topology, 2014.
- Wu, Gahdaie, Zhang, Hu. Topological Data Analysis for Deep Learning. ICML Workshop, 2020.
- Hensel et al. A Introduction to Topological Data Analysis for Physicists. arXiv, 2024.
@software{lambda_topo,
author = {Teerth Sharma},
title = {Lambda-Topo: Topological Memory System},
url = {https://github.com/teerthsharma/lambda-topo},
year = {2024}
}MIT