Topological Autoencoders++ Teaser

Pipeline description

This example illustrates the Topological Autoencoders++ dimensionality reduction technique on two toy examples and one real-world example. Topolocial Autoencoders used autoencoders with a topological regularization term that constrains the preservation of the 0-dimensional persistent homology. Topological Autoencoders++ proposes a regularization term and efficient algorithms to additionally constrain the preservation of 1-dimensional persistent homology.

Three examples are provided as CSV ASCII files (one line per point, one column per dimension). These are two toy point clouds in 3D and one high-dimensional image dataset, which Topological Autoencoders++ projects down to 2D. Each of these point clouds features one or several significant 1-dimensional persistent homology class(es), that Topological Autoencoders++ aims to preserve when projecting down to 2D.

The dataset twistedEllipse.csv consists in an ellipse embedded in 3D which has been twisted along its long axis (top left view in the above screenshot). It features one significant persistent pair.

The dataset K4.csv contains points noisily sampled next to the edges of a regular tetrahedron embedded in 3D. It features three significant persistent pairs. For each of them, a persistent generator is computed (one color per generator in the above screenshot, top center view).

The dataset coil20.csv contains 20 object classes, each seen from 72 different angles. The subset of this dataset containing only the pictures of the first object (a rubber duck, data is shown in the top right table in the above screenshot) features one significant persistent pair, that corresponds to the fact these views from different angles live on a manifold homeomorphic to a circle.

For each of these datasets, Topological Autoencoders++ computes a 2D projection (corresponding bottom views in the above screenshot).

ParaView

To reproduce the above screenshot, go to your ttk-data directory and enter the following command:

paraview states/topoAEppTeaser.pvsm

Python code

#!/usr/bin/env python

from paraview.simple import *

twistedEllipsecsv = CSVReader(FileName=["twistedEllipse.csv"])
tTKDimensionReduction1 = TTKDimensionReduction(Input=twistedEllipsecsv)
tTKDimensionReduction1.InputColumns = ["x", "y", "z"]
tTKDimensionReduction1.Method = "Autoencoder"
tTKDimensionReduction1.Lossfunction = "Asymmetric Cascade Autoencoder (TopoAE++)"
tTKDimensionReduction1.Numberofepochs = 1000
tTKDimensionReduction1.Hiddenlayers = "32 32"
tTKDimensionReduction1.Enforcedeterminism = 1
tTKDimensionReduction1.Randomseed = 5
SaveData("twistedEllipse_topoAE++.csv", tTKDimensionReduction1)

k4csv = CSVReader(FileName=["K4.csv"])
tTKDimensionReduction2 = TTKDimensionReduction(Input=k4csv)
tTKDimensionReduction2.InputColumns = ["x", "y", "z"]
tTKDimensionReduction2.Method = "Autoencoder"
tTKDimensionReduction2.Lossfunction = "Asymmetric Cascade Autoencoder (TopoAE++)"
tTKDimensionReduction1.Numberofepochs = 1000
tTKDimensionReduction2.Hiddenlayers = "128 32"
tTKDimensionReduction2.Enforcedeterminism = 1
tTKDimensionReduction2.Randomseed = 100
SaveData("K4_topoAE++.csv", tTKDimensionReduction2)

coil20csv = CSVReader(FileName=["coil20.csv"])
thresholdTable = ThresholdTable(Input=coil20csv)
thresholdTable.Column = ["ROWS", "1025"]
thresholdTable.MinValue = 1.0
thresholdTable.MaxValue = 1.0
tTKDimensionReduction3 = TTKDimensionReduction(Input=thresholdTable)
tTKDimensionReduction3.InputColumns = sorted([str(x) for x in range(1, 1025)])
tTKDimensionReduction3.Method = "Autoencoder"
tTKDimensionReduction3.Lossfunction = "Asymmetric Cascade Autoencoder (TopoAE++)"
tTKDimensionReduction3.Numberofepochs = 1000
tTKDimensionReduction3.Hiddenlayers = "128 32"
tTKDimensionReduction3.Enforcedeterminism = 1
tTKDimensionReduction3.Randomseed = 0
SaveData("coil20-1_topoAE++.csv", tTKDimensionReduction3)

To run the above Python script, go to your ttk-data directory and enter the following command:

pvpython python/topoAEppTeaser.py

Inputs

• twistedEllipse.csv: a 3D point cloud in CSV ASCII format (one line per point, one column per dimension), which exhibits 1 significant persistent cycle.
• K4.csv: a 3D point cloud in CSV ASCII format (one line per point, one column per dimension), which exhibits 3 significant persistent cycles.
• coil20.csv: a high-dimensional point cloud in CSV ASCII format (one line per point, one column per dimension), of which the first class of images exhibits 1 significant persistent cycle.

Outputs

• twistedEllipse_topoAE++.csv: the output 2D projection of the twistedEllipse.csv dataset in CSV ASCII format (one line per point, one column per dimension).
• K4_topoAE++.csv: the output 2D projection of the K4.csv dataset in CSV ASCII form at (one line per point, one column per dimension).
• coil20-1_topoAE++csv: the output 2D projection of the first class of the coil20.csv dataset in CSV ASCII form at (one line per point, one column per dimension).

C++/Python API

DimensionReduction