Orthogonal projections of hypercubes

TL;DR

This work demonstrates that principal component analysis can be used to produce reproducible, interpretable two-dimensional projections of high-dimensional hypercubes representing binary state spaces. By interpreting PC loadings as contribution vectors, a biplot framework emerges that connects PCA to orthogonal projections of hypercubes, enabling analytic insight into which vertices dominate the projection and how vertex overlaps distort pairwise similarities. The authors analytically and numerically show that PC1 tracks weighted, low-energy states (e.g., magnetization-like order parameters) while higher PCs encode additional structure, with inner-product error providing a quantitative measure of projection quality. They apply this framework to finite artificial kagome spin-ice systems, revealing polarized state distributions, correlated spin-flip pathways, and energy landscapes, and discuss extensions to mean-field Hopfield-type models. Overall, PCA-based hypercube projections offer a principled, interpretable tool for visualizing high-dimensional binary dynamics and their transitions, with practical utility for exploring complex many-body systems.

Abstract

Projections of hypercubes have been applied to visualize high-dimensional binary state spaces in various scientific fields. Conventional methods for projecting hypercubes, however, face practical difficulties. Manual methods require nontrivial adjustments of the projection basis, while optimization-based algorithms limit the interpretability and reproducibility of the resulting plots. These limitations motivate us to explore theoretically analyzable projection algorithms such as principal component analysis (PCA). Here, we investigate the mathematical properties of PCA-projected hypercubes. Our numerical and analytical results show that PCA effectively captures polarized distributions within the hypercubic state space. This property enables the assessment of the asymptotic distribution of projected vertices and error bounds, which characterize the performance of PCA in the projected space. We demonstrate the application of PCA to visualize the hypercubic energy landscapes of Ising spin systems, specifically finite artificial spin-ice systems, including those with geometric frustration. By adding projected hypercubic edges, these visualizations reveal pathways of correlated spin flips. We confirm that the time-integrated probability flux exhibits patterns consistent with the pathways identified in the projected hypercubic energy landscapes. Using the mean-field model, we show that dominant state transition pathways tend to emerge around the periphery of the projected hypercubes. Our work provides a better understanding of how PCA discovers hidden patterns in high-dimensional binary data.

Figures (14)

• Figure 1: A gallery of hypercubes and Hamiltonian (directed) paths on them. (a) A cube, or three-dimensional hypercube, with three-dimensional coordinates of vertices. (b) A tesseract, or four-dimensional hypercube, with four-dimensional coordinates of vertices. (c) A Hamiltonian (directed) path on a cube. (d) A Hamiltonian (directed) path on a tesseract. In (c) and (d), the arrow indicates the direction of the path. One obtains a Hamiltonian path by converting the decimals to Gray code Gray1953Gardner1975 and following them in ascending order. See Tables \ref{['tab:decimal-binary-gray-3d']} and \ref{['tab:decimal-binary-gray-4d']} for Gray codes used to visualize Hamiltonian paths on three-dimensional and four-dimensional hypercubes.
• Figure 2: A gallery of orthogonal projections of hypercubes. Colored arrows represent the contribution vectors corresponding to the original dimensions. The boxes on the bottom right indicate the correspondence between the colors and the original dimensions. (a) An isometric projection of a cube. Notice that $101^\top$ and $010^\top$ are overlapped. (b) An isometric projection of a tesseract. (c) A Hamming projection of a cube. (d) A Hamming projection of a tesseract. Notice that $1001^\top$ and $0110^\top$ are overlapped. In Hamming projections (c) and (d), the contribution vector of each dimension has the same horizontal contribution. (e) A fractal projection of a six-dimensional hypercube. (f) A fractal projection of an eight-dimensional hypercube. In fractal projections (e) and (f), the contribution vectors of the first half (left half) of the code are ten times longer than the rest. More projections of hypercubes are available in the Supplemental Material sm.
• Figure 3: Exchanging the labels of vertices by reversing the direction of the contribution vectors of corresponding digits. The boxes on the bottom right indicate the correspondence between the colors and the dimensions. (a) By reversing the contribution vector for the second digit, one can exchange the cubic labels of pairs of vertices, $000^\top$ and $010^\top$, $100^\top$ and $110^\top$, $001^\top$ and $011^\top$, and $101^\top$ and $111^\top$. Compare this with Fig. \ref{['fig:isometric-hamming-fractal']}(a). (b) Obtaining different labels for a tesseract by swapping two contribution vectors. The first and fourth unit vectors are reversed, cf. Fig. \ref{['fig:isometric-hamming-fractal']}(b).
• Figure 4: Orthogonal projections of four-dimensional hypercubic vertices using PCA. (a) A projection of a four-dimensional hypercube where vertices are weighted randomly by uniform random numbers in $\left[0, z\right)$. Red filled circles are the vertices and lines are the edges of the hypercube. The magnitude of weight is proportional to the area of the vertex. Arrows are biplot vectors originating from $----^\top$ and the boxes on the bottom right indicate the correspondence between the colors of the arrows and the original dimensions. (b) Fraction of explained variance by each PC of (a). (c) PC1 loading, and (d) PC2 loading of random weighted hypercubic vertices of (a). (e) Hamiltonian path on a four-dimensional hypercube in (a). For a different realization of random weight, see the Supplemental Material sm. (f)--(j) Same as (a)--(e) but with bipolar distribution. Two of the vertices, $----^\top$ and $++++^\top$, are two-times more weighted than the others.
• Figure 5: Orthogonal projections of four-dimensional hypercubic vertices using PCA. $----^\top$ and $++++^\top$ are the most weighted vertices with weight $3z$, $--++^\top$ and $++--^\top$ are the second most weighted vertices with weight $2z$, $-+-+^\top$ and $+-+-^\top$ are the third most weighted vertices with weight $z$, and the rest of them are weighted randomly by uniform random numbers in $\left[0, z\right)$. (a) Projection of the hypercube by PC1 and PC2. Red filled circles are the vertices and lines are the edges of the hypercube. The magnitude of weight is proportional to the area of the vertices. Arrows are biplot vectors originating from $----^\top$ and the boxes on the bottom right indicate the correspondence between the colors of the arrows and the original dimensions. Dashed arrows correspond to the projection of original high-dimensional vectors, $-+++^\top$ and $++++^\top$. The original coordinate is indicated around the lower right of the arrowhead as an array of filled $\blacksquare$ (indicates $+$) or empty $\square$ (indicates $-$) boxes. Horizontal and vertical dashed lines crossing the origin are for visual aid. (b) Same as (a) but by PC1 and PC3. (c) Fraction of explained variance by each PC. (d) PC1 loading, (e) PC2 loading, and (f) PC3 loading.