The calculated principal components are called eigengalaxies and span the lower dimensional image space. The eigengalaxies are shown below for the 2D case,they are ordered by their explained variance ratio (EVR), which is the fraction of the total variance explained by each eigengalaxy.
Figure 2: Left: Cumulative explained variance ratio (EVR) for up to 400 eigengalaxies: Achieving the same explained variance requires significantly more eigengalaxies in 3D compared to 2D. To surpass an 90% EVR, ∼ 60 (215) eigengalaxies are needed in 2D (3D).
We define the reconstruction error (RE) as the fractional difference in pixel values between the PCA representation,
where we sum over all pixel values.
Figure 3: Reconstruction error (RE) for fixed-dimensionality reduction on 60 (215) eigengalaxies in 2D (3D). The dashedline represents the 90% quantile. Impressively, 90% of all images exhibit a RE below 0.022 (0.027).
Figure 4: 90th percentile of RE as a function of the number of eigengalaxies. Reconstruction is a strong function of eigengalaxies, and already 15 (60) eigengalaxies lead to a RE better than 5% in 2D (3D).
We calculate the Nearest Neighbors in Euclidean distance in the lower dimensional Image space. The Nearest Neighbors are shown below for the 2D case, they are ordered by their distance to the query galaxy.
Figure 5: Nearest Neighbors in 2D for an elliptical (left) and a spiral (right) galaxy. The query galaxy is shown in the top left corner.
We show the eigengalaxies that contribute most to the reconstruction of the query galaxy. The eigengalaxies are ordered by their contribution to the reconstruction.