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# Topology-enhanced machine learning model (Top-ML) for anticancer peptide
prediction

Sequence analysis

Joshua Zhi En Tan1 JunJie Wee1,2,∗ Xue Gong1,∗ and Kelin Xia1,∗ ${}^{\text{\sf
1}}$Division of Mathematical Sciences, School of Physical and Mathematical
Sciences, Nanyang Technological University, Singapore 637371.
${}^{\text{\sf 2}}$Michigan State University, Department of Mathematics, East
Lansing, MI 48824, USA.

(2020)

###### Abstract

Recently, therapeutic peptides have demonstrated great promise for cancer
treatment. To explore powerful anticancer peptides, artificial intelligence
(AI)-based approaches have been developed to systematically screen potential
candidates. However, the lack of efficient featurization of peptides has
become a bottleneck for these machine-learning models. In this paper, we
propose a topology-enhanced machine learning model (Top-ML) for anticancer
peptide prediction. Our Top-ML employs peptide topological features derived
from its sequence "connection" information characterized by vector and
spectral descriptors. Our Top-ML model has been validated on two widely used
AntiCP 2.0 benchmark datasets and has achieved state-of-the-art performance.
Our results highlight the potential of leveraging novel topology-based
featurization to accelerate the identification of anticancer peptides.
Availability: The Top-ML code can be found in https://github.com/XueGong-
git/TopPep-ML. The datasets for ACPs can be found in
https://webs.iiitd.edu.in/raghava/anticp2/download.php.
Contact<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

## Introduction

Despite the extensive research into the prevention and diagnosis of cancer
over the past few decades, cancer remains one of the leading causes of death
worldwide. Statistics from the World Health Organization [1] show that cancer
accounted for an estimated 10 million deaths in 2020, representing nearly one
in six deaths. Traditional approaches to cancer treatment often come with
significant limitations, such as systemic toxicity and drug resistance.
Traditional cancer treatment relies on radiotherapy and chemotherapy; however,
both methods have downsides, as they target both normal and cancerous cells.
Furthermore, highly adaptable cancer cells can develop drug resistance,
further reducing the efficacy of these treatments.

In recent years, there has been a growing interest in Anticancer Peptides
(ACPs) as a novel alternative treatment for cancer. ACPs are a specific type
of antimicrobial peptides (AMP) typically consisting of less than 50 amino
acids. They differ from other AMPs by their cationic nature and low molecular
weight [2]. Peptide therapy offers significant advantages over traditional
cancer treatments due to its high specificity, low toxicity, superior membrane
penetration ability, and easier chemical modifications. However, the
experiment-based discovery and design of ACPs are extremely costly, time-
consuming, and labor-intensive, which leads to poor scalability. To address
these challenges, researchers are increasingly turning to machine learning for
the screening and design of ACPs.

Featurization or feature engineering is key to the performance of machine
learning models for ACPs design and discovery. The ACPs identification task
was formulated as a binary classification machine learning problem in these
studies [3, 4, 5, 6, 7, 8]. In these methods, features are constructed from
the peptides and then used to train ML models. Traditional molecular
descriptors are primarily designed based on molecular structure properties,
such as amino acid composition, dipeptide composition, and terminus
composition to establish strong structure-function relationships [5]. In
addition, other classes of physicochemical properties, including
hydrophobicity and amphipathicity, have been utilized as feature encoding
schemes [9]. The effectiveness of machine learning techniques heavily depends
on molecular featurization and representations [10, 11, 12].

Recently, topological data analysis (TDA) [13, 14], which is a model deeply
rooted in algebraic topology, has demonstrated its great power in molecular
representation and featurization [15]. In particular, TDA-based machine
learning models have achieved great successes in various steps of drug design,
including protein-ligand binding affinity prediction [15], protein stability
change upon mutation prediction [16], toxicity prediction [17], solvation free
energy prediction [18], partition coefficient and aqueous solubility [19],
binding pocket detection [20], protein mutation analysis [21], and drug
discovery [22]. In the D3R Grand Challenge, a worldwide competition for
computational models in drug design, TDA-based models have consistently
delivered state-of-the-art results [23]. Further, TDA-based models have also
achieved great successes in material data analysis [24, 25]. The key idea of
these models is to characterize the structural "connection" information with
topological tools.

Here, we propose a topology-enhanced machine learning model (Top-ML) for ACPs
prediction. The key idea is to use topological features to characterize the
‘‘connection’’ pattern between amino acids within peptide sequences. More
specifically, we employ vector features and spectral features derived from
amino acid sequences to describe these connection properties, i.e., how amino
acids, amino acid pairs, and amino acid groups relate to each other. Our Top-
ML considers four types of peptide features, namely Magnus vector, natural
vector, terminal composition feature, and spectral feature. By combining all
four types of features with the Extra Trees classifier, we demonstrate that
our proposed Top-ML can achieve state-of-the-art results in the widely-used
benchmark datasets.

## Results

### Peptide vector representations

##### Natural Vector

Natural vector has been widely used in the construction of phylogenetic trees
[26, 27, 28]. Here we use natural vector representation for peptide sequences.
The natural vector was first conceived for the analysis of genomes and has
been previously applied to DNA sequences [29]. Consider a peptide sequence
$\mathcal{S}=x_{1}x_{2}\cdots x_{n}$ of length $n$, where $x_{i}$ lies in the
set of 20 unique amino acids $\\{A,C,D,\cdots,Y\\}$. For each peptide sequence
$\mathcal{S}$ and amino acid $R$, we compute three statistical metrics:
$n_{R}$, representing the number of times $R$ appears in $\mathcal{S}$;
$\mu_{R}$, the average position of $R$; and $D_{2}^{R}$, the variation of
position. The natural vector concatenates the three statistics calculated for
each unique amino acid, forming a vector of length 60, i.e., ($n_{A}$,
$\mu_{A}$, $D_{2}^{A}$, … $n_{R}$, $\mu_{R}$, $D_{2}^{R}$, … $n_{Y}$,
$\mu_{Y}$, and $D_{2}^{Y}$). The natural vector encodes the presence and
location of amino acids which correlate with anticancer properties. For
instance, a previous study found that A, F, H, K, L, and W are more prevalent
in ACPs, whereas C, G, R, and S are more common in non-ACPs [5].

##### Magnus Vector

In combinatorial group theory, Wilhelm Magnus studied representations of free
groups by non-commutative power series. For a free group $F$ with basis
$x_{1},\cdots,x_{n}$ and a power series ring $\prod$ in indeterminates
$\xi_{1},\cdots,\xi_{n}$, Magnus proved that the map $\mu:\xi_{i}\mapsto
1+\xi_{i}$ defines an isomorphism from $F$ into the multiplicative group
$\prod^{\times}$ of units in $\prod$ [30]. Magnus’s work laid the foundation
for the development of Magnus representation, which has been recently applied
to the analysis of DNA and RNA sequences in genome analysis [31].

Peptide sequence-based Magnus representation and Magnus vector can be
developed and used for the characterization of peptide properties. The Magnus
representation of a peptide sequence $\mathcal{S}$, denoted by
$\rho(\mathcal{S})$, is defined as the product
$\rho(\mathcal{S})=\prod_{i=1}^{n}(1+x_{i})$ in the non-commutative polynomial
algebra $R\langle A,C,D,\cdots,Y\rangle$, where $R$ is a commutative ring. In
this paper, we use $R=\mathbb{Z}$. For example, the peptide sequence
$\mathcal{S}=ACD$ will be mapped to
$\rho(\mathcal{S})=(1+A)(1+C)(1+D)=1+A+C+D+AC+AD+CD+ACD$.

In implementation, the Magnus vector of a peptide sequence $S$, denoted as
$v(S)$, is obtained by the following steps. First, we arrange the set of
possible words over the amino acids $\\{A,C,D,\cdots,W,Y\\}$ of length less
than or equal to $n$, first by ascending order of length and then by
lexicographic order. With respect to the above arrangement, assign each term
present in $\rho(S)$ with a constant coefficient $c\in R$, and 0 for each term
not present in $\rho(S)$. In our model, we use $c=1$ and only consider
subsequences with a maximum length of 2. As each peptide sequence can contain
a maximum of 20 unique amino acids [5], there are a maximum of
$\sum_{i=1}^{2}20^{i}=420$ possible subsequences of up to length 2, i.e.,
$A,C,D,\cdots,W,Y,AA,AC,AD,\cdots,YW,YY$. Note that due to the non-
commutativity of the variables, if $S^{\prime}=ADC$, we observe that
$\rho(S^{\prime})\neq\rho(S)$ and $v(S^{\prime})\neq v(S)$. From our numerical
experiments detailed in the Results section, we found that the Magnus vector
of length 420 are sufficient to achieve state-of-the-art results in anticancer
peptide classification.

Figure 1: Illustration of the transformation of (a) the peptide sequence
‘‘ACDE’’ into (b) a sequence-based Laplacian $\mathbf{L}_{0}$ and (c) its
corresponding symmetric normalized Laplacian matrix $\tilde{\mathbf{L}}_{0}$
based on the mean index positions of the amino acids. The largest off-diagonal
entry in (b) $\mathbf{L}_{0}$ in terms of absolute value corresponds to the
amino acid pair (D, E) since D and E have the highest mean index positions.
After normalization, the diagonal entries in (c) $\tilde{\mathbf{L}}_{0}$
become ones, and the off-diagonal entries take values between -1 and 0.

In our implementation, we consider the Magnus vectors for non-overlapping
$k$-mers for each peptide sequence. A $k$-mer represents a segment of $k$
consecutive amino acids within a given peptide sequence. We compute the Magnus
vector for each $k$-mer, selected using non-overlapping sliding windows of
size $k$ to avoid redundancy and reduce computational cost. The sliding window
begins at the first index, and we truncate the tail if the remaining sequence
is shorter than $k$. Therefore, the number of $k$-mers for a peptide sequence
of length $n$ is the largest integer less than or equal to $\frac{n}{k}$. We
then form a mean Magnus vector $m(S)$ by averaging the Magnus vectors across
all $k$-mers, capturing both $k$-mer frequency information and internal
subsequence structure. We experiment with sliding window sizes $k=4,5,6,7$ and
find that $k=5$ yields good prediction performance for AntiCP 2.0 Datasets, as
detailed in the Supporting Information Table 3. This result aligns with the
observation of minimal classification errors for the Baltimore and genus
classification labels reported in [32]. Therefore, we selected $k=5$ for our
final Top-ML model.

##### Terminal Composition Features

Previous research into peptide representation has found that terminal residues
provide vital information about the biological and physiological functions of
peptides [33]. The terminus composition provides local sequence information in
contrast to the global sequential information when the full length of the
peptide is used for featurization. We compute the Magnus and natural vectors
based on the terminus compositions of peptides (i.e., N5, C5, N5C5, $\cdots$,
C15, N15C15). For example, extracting the first 5 residues from the N-terminus
is termed N5. Our experiments show that the C15 and N15C15 terminus provided
the best prediction accuracy for AntiCP 2.0 Datasets A and B respectively, as
shown in Table 5 in the Supplementary Information.

### Peptide spectral representation

In this section, we introduce a peptide spectral representation, which is
inspired by the spectral theory of combinatorial Laplacian matrices.
Simplicial complexes offer an effective representation for capturing the
underlying structural information and its interactions. We refer the reader to
the Method section for more information regarding simplicial complexes and
their combinatorial Laplacian matrices. Using simplicial complexes as a
molecular structural representation has generated tremendous success for
combinatorial Laplacian matrices in applications such as drug design and
discovery [34]. However, unlike molecular structures, which inherently possess
3D structures, peptide data is sequential and cannot be directly represented
by various topological objects such as graphs and simplicial complexes.
Therefore, we need to first construct simplicial complexes from peptide
sequences.

In order to construct a peptide spectral representation, we first define a
sequence-based boundary matrix. For any peptide sequence $S$ of length $n$, we
denote the set of possible subsequences of length $k$ as $S_{k}$, where $1\leq
k\leq n$. Then the sequence-based boundary matrix $\mathbf{B}_{k}$ ($1\leq
k\leq n$) is an $|S_{k}|\times|S_{k+1}|$ matrix constructed from the
relationships between subsequences $w^{k}\in S_{k}$ and $w^{k+1}\in S_{k+1}$.
For a peptide sequence $S$ of length $n$, its $k$-th sequence-based boundary
matrix $\mathbf{B}_{k}$ is defined as follows:

$\mathbf{B}_{k}(i,j)=\begin{cases}f(w_{i}^{k}),&\text{if }w_{i}^{k}\preceq
w_{j}^{k+1},\\\ 0,&\text{if }w_{i}^{k}\not\preceq w_{j}^{k+1},\end{cases}$ (1)

where $f(w)$ is a real-valued function, $w_{i}^{k}\preceq w_{j}^{k+1}$ means
that the $w_{i}^{k}$ is a subsequence of $w_{j}^{k+1}$. Likewise,
$w_{i}^{k}\not\preceq w_{j}^{k+1}$ means that $w_{i}^{k}$ is not a subsequence
of $w_{j}^{k+1}$. We are considering two definitions for $f(w)$. The first
defines $f(w)$ as the number of occurrences of the subsequence $w$ within the
sequence $S$. For instance, in the sequence ‘‘FFSFS", the subsequence ‘‘FS"
occurs twice. Another definition of $f(w)$ is the mean index position of the
first amino acid in $w$ across all its occurrences within sequence $S$. For
example, in ‘‘FFSFS", the index positions of ‘‘FS" are 2 and 4, resulting in a
mean index position of 3. The choice of $f(w)$ for the final model is the mean
index position based on the model performance as presented in the
Supplementary Information Table 2.

Similar to the combinatorial Laplacian defined on simplicial complexes, we
compute our sequence-based Laplacian matrices $\mathbf{L}_{k}$ using Equation
(3) in the Method section. For $\mathbf{L}_{k}$ to be positive semi-definite,
we update the diagonals of $\mathbf{L}_{k}$ using the sum of its off-diagonals
and then assign all off-diagonal values a negative sign. This ensures the sum
of every row in $\mathbf{L}_{k}$ is zero. We then convert $\mathbf{L}_{k}$ to
its symmetric normalized version $\tilde{\mathbf{L}}_{k}$ as defined in
Equation (4).

Let’s consider $\mathbf{B}_{1}$ as an example. Given that all peptides consist
of a maximum of 20 unique amino acids [5], the rows in $\mathbf{B}_{1}$
represent these 20 amino acids (A, C, D, …, Y), while the columns represent
the 400 pairs of amino acids (AA, AC, AD, …, YY). Therefore, $\mathbf{B}_{1}$
is a $20\times 400$ matrix, $\mathbf{L}_{0}$ is a $20\times 20$ matrix, and
$\mathbf{L}_{1}$ is a $400\times 400$ matrix. This resulting vector containing
eigenvalues is used as input features for downstream machine learning models.
$\mathbf{L}_{0}$ produces a feature vector of size 20 while $\mathbf{L}_{1}$
produces a feature vector of size 400. The two feature vectors are
concatenated to form a feature vector of size 420.

### Topology-enhanced Machine Learning (Top-ML)

In this section, we present our topology-enhanced machine learning model (Top-
ML) and its results for anticancer peptide classification. The natural vector,
Magnus vector, terminus composition features, and peptide spectral
representations are combined to form the input features for our Top-ML model.
Figure 2 shows the general pipeline of our Top-ML model.

We train and assess the performance of our Top-ML model using the AntiCP 2.0
benchmark datasets [5]. The AntiCP 2.0 datasets comprise Dataset A and Dataset
B. Dataset A consists of 970 experimentally validated ACPs and 970 randomly
selected peptides, assumed to be non-ACPs. Dataset B contains 861
experimentally validated ACPs and 861 non-ACPs, which are antimicrobial
peptides (AMPs). AntiCP 2.0 (Datasets A and B) was partitioned into training
and test sets using an 80/20 split. We assess the effectiveness of our Top-ML
model by employing a range of performance metrics, which include accuracy,
sensitivity (true positive rate), specificity (true negative rate), and the
Matthews correlation coefficient (MCC). The model selection process is
detailed in the Supplementary Information. The final Top-ML model for AntiCP
2.0 is an Extra Trees classifier with 400 trees, each using a random subset of
features equal to the square root of the total number of features to determine
the optimal split. In this finalized model, peptide spectral representations
are derived from the mean index position, and Magnus vectors are computed
using a window size of 5. The terminus used to build terminal composition
features is C15 for AntiCP 2.0 (Dataset A) and N15C15 for AntiCP 2.0 (Dataset
B). The final Top-ML model has been further optimized by finding the optimal
probability threshold that returns the best accuracy. All reported
performances of the Top-ML model are medians over 100 repeated tests.

Figure 2: Pipeline of topology-enhanced machine learning model (Top-ML) for
anticancer peptide prediction. (A) The peptide sequences from AntiCP 2.0
(Dataset A or B). (B) The peptide spectral features are obtained from the
sequence-based Laplacian matrices. (C) The Magnus vectors, natural vectors,
and terminal composition features are generated from peptide sequences. (D)
The combined features serve as inputs for the classification of anticancer
peptides (ACPs) and non-ACPs.

Table 1 summarizes the results of Top-ML and existing state-of-the-art machine
learning models on AntiCP 2.0 (Datasets A and B). It can be observed that on
AntiCP 2.0 (Dataset A), our Top-ML model achieves one of the highest
accuracies and MCC scores, outperforming several existing state-of-the-art
machine learning models. This is noteworthy because these models were built on
deep learning algorithms while only tree-based classifiers have been used for
Top-ML. Compared to AntiCP 2.0 [5], which serves as a key benchmark model for
comparison, our Top-ML model demonstrates an improvement of 1.3% in accuracy
and 3.1% in specificity. On AntiCP 2.0 (Dataset B), the accuracy of Top-ML
stands at 78.89%, which is higher than all other models compared, except for
iACP-FSCM (82.50% accuracy) and ME-ACP(79.2%). When compared with the
benchmark model AntiCP 2.0, there is an improvement of 3.5% in accuracy and
8.5% in specificity. Although iACP-FSCM and ME-ACP report higher accuracy, our
Top-ML model can obtain a more balanced sensitivity and specificity score.

Additionally, our results confirm the observations made in [5] that the
accuracy of results on AntiCP 2.0 (Dataset A) is higher than that on AntiCP
2.0 (Dataset B). This may be attributed to the fact that ACPs exhibit greater
compositional and structural similarities to AMPs, which are the negative
class in AntiCP 2.0 (Dataset B), than to random peptides, which serve as the
negative class in AntiCP 2.0 (Dataset A). As a result, it is observed in our
numerical experiments for model selection (see Supporting Information) that
models trained on AntiCP 2.0 (Dataset A) demonstrate better performance,
irrespective of the chosen model or feature encoding technique.

Table 1: Performance comparison of Top-ML with existing state-of-the-art
machine learning models on AntiCP 2.0 (Datasets A and B).  AntiCP 2.0 (Dataset
A)
---
Model | Acc | Sen | Spc | MCC
ME-ACP [3] | 93.30 | 91.70 | 94.80 | 0.87
Top-ML | 93.30 | 91.49 | 94.85 | 0.87
iACP-DRLF [4] | 93.00 | 89.60 | 96.40 | 0.86
AntiCP 2.0 [5] | 92.01 | 92.27 | 91.75 | 0.84
iACP-FSCM [6] | 88.90 | 87.60 | 90.20 | 0.78
AntiMF [7] | 91.03 | 88.24 | 93.78 | 0.83
ACP-MHCNN [8] | 90.00 | 86.50 | 93.30 | 0.80
AntiCP [35] | 89.95 | 89.69 | 90.20 | 0.80
ACP-DL [36] | 88.10 | 86.00 | 90.20 | 0.76
GRCI-Net [37] | 87.60 | 87.00 | 88.10 | 0.75
ACPred [38] | 85.31 | 87.11 | 83.51 | 0.71
ACPred-Fuse [39] | 78.87 | 64.43 | 93.30 | 0.60
iACP [40] | 77.58 | 78.35 | 76.80 | 0.55
PEPred-Suite [41] | 57.47 | 40.21 | 74.74 | 0.16
AntiCP 2.0 (Dataset B)
Models | Acc | Sen | Spc | MCC
iACP-FSCM [6] | 82.50 | 72.60 | 90.30 | 0.65
ME-ACP [3] | 79.20 | 74.90 | 81.18 | 0.58
Top-ML | 78.89 | 76.90 | 81.96 | 0.63
iACP-DRLF [4] | 77.50 | 80.70 | 74.30 | 0.55
AntiMF [7] | 75.94 | 65.34 | 86.63 | 0.53
AntiCP 2.0 [5] | 75.43 | 77.46 | 73.41 | 0.51
ACP-DL [36] | 74.60 | 74.90 | 74.30 | 0.49
GRCI-Net [37] | 74.60 | 75.40 | 73.70 | 0.49
ACP-MHCNN [8] | 74.00 | 73.70 | 74.30 | 0.48
ACPred-Fuse [39] | 68.90 | 69.19 | 68.60 | 0.38
iACP [40] | 55.10 | 77.91 | 32.16 | 0.11
PEPred-Suite [41] | 53.49 | 33.14 | 73.84 | 0.08
ACPred [38] | 53.47 | 85.55 | 21.39 | 0.09
AntiCP [35] | 50.58 | 100.00 | 01.16 | 0.07

## Discussion

In this paper, we propose a topology-enhanced machine learning model (Top-ML)
combining peptide spectral features with Magnus and natural vector-based
peptide features. We demonstrate that our Top-ML model outperforms or achieves
the same level of accuracy as several existing state-of-the-art machine
learning models for anticancer peptide classification. Our results highlight
the potential of mathematics-assisted peptide-based features to improve the
performance of anticancer peptide identification.

In spite of the overall increase in predictive accuracy, there are several
limitations associated with this study. Additionally, since this study focuses
only on the topological featurization of peptides, it has not considered
physicochemical properties such as hydrophobicity and helicity, which have
been shown to also provide valuable signals in encoding a peptide. These
different classes of feature encoding methods can be combined in future
research.

## Method

### Combinatorial Laplacian

Our sequence-based Laplacian construction is heavily inspired by combinatorial
Laplacian matrices constructed from simplicial complexes. Here, we provide the
background information on combinatorial Laplacian matrices. A simplicial
complex $K$ is made of simplexes. A $p$-simplex is a geometric object formed
by $p+1$ affinely independent points, resulting in a convex hull. For example,
a $0$-simplex is a node, a $1$-simplex is an edge, a $2$-simplex is a triangle
(with a solid inside region), and a $3$-simplex is a solid tetrahedron.
Mathematically, a $p$-simplex
$\sigma^{p}=\\{v_{0},v_{1},v_{2},\cdots,v_{p}\\}$ can be written as follows:

$\sigma^{p}=\bigg{\\{}\lambda_{0}v_{0}+\lambda_{1}v_{1}+\cdots+\lambda_{p}v_{p}\bigg{|}\sum_{i=0}^{p}\lambda_{i}=0;\forall
i,0\leq\lambda_{i}\leq 1\bigg{\\}}.$

In order to construct combinatorial Laplacian matrices from a simplicial
complex $K$, the boundary matrices are first computed by obtaining the
orientation and relevant faces for each simplex in $K$. Here,
$\sigma_{i}^{k-1}\subset\sigma_{j}^{k}$ indicates that $\sigma_{i}^{k-1}$ is a
face of $\sigma_{j}^{k}$. In this case, $\sigma_{j}^{k}$ is called a co-face
of $\sigma_{i}^{k-1}$. Otherwise, we denote
$\sigma_{i}^{k-1}\not\subset\sigma_{j}^{k}$. Additionally,
$\sigma_{i}^{k-1}\sim\sigma_{j}^{k}$ indicates that the two simplices are
similarly oriented and $\sigma_{i}^{k-1}\not\sim\sigma_{j}^{k}$ means two
simplices are dissimilarly oriented. For an oriented simplicial complex, its
$k$-th boundary matrix $\mathbf{B}_{k}$ can be defined as follows:

$\mathbf{B}_{k}(i,j)=\begin{cases}1,&\text{if
}\sigma_{i}^{k-1}\subset\sigma_{j}^{k}\text{ and
}\sigma_{i}^{k-1}\sim\sigma_{j}^{k}\\\ -1,&\text{if
}\sigma_{i}^{k-1}\subset\sigma_{j}^{k}\text{ and
}\sigma_{i}^{k-1}\nsim\sigma_{j}^{k}\\\ 0,&\text{if
}\sigma_{i}^{k-1}\not\subset\sigma_{j}^{k}.\end{cases}$ (2)

Laplacian matrices encapsulate the topological and geometrical information of
the simplicial complex $K$. Once the boundary matrices are obtained, the
$k$-th combinatorial Laplacian matrices can be computed as follows:

$\mathbf{L}_{k}=\begin{cases}\mathbf{B}_{1}\mathbf{B}_{1}^{T},&\text{if
}k=0\\\
\mathbf{B}_{k}^{T}\mathbf{B}_{k}+\mathbf{B}_{k+1}\mathbf{B}_{k+1}^{T},&\text{if
}k\geq 1.\end{cases}$ (3)

For a simple illustration, refer to Figure 3, which displays an oriented
simplicial complex $K$ along with its corresponding boundary and Laplacian
matrices. For the combinatorial Laplacian matrices $\mathbf{L}_{0}$ and
$\mathbf{L}_{1}$, every diagonal entry corresponds to the degree of a
$k$-simplex $\sigma$ (i.e., the total number of faces and co-faces of
$\sigma$). The symmetric normalized Laplacian is defined as follows:

$\tilde{\mathbf{L}}_{k}=\mathbf{D}^{\frac{1}{2}}_{k}\mathbf{L}_{k}\mathbf{D}_{k}^{-\frac{1}{2}},$
(4)

where $\mathbf{D}_{k}$ is a $k$-th diagonal matrix with diagonal entries from
$\mathbf{L}_{k}$.

Figure 3: Illustration of boundary and combinatorial Laplacian matrices for an
oriented simplicial complex.

### Model Selection

Three classifiers were considered to train our Top-ML features: Random Forest
(RF), Extra Trees (Etrees), and Gradient Boosting Trees (GBT). The set of
features tested consists of the natural vector, mean and sum Magnus vector ,
while the $L_{0}$ and $L_{1}$ peptide spectral features have been aggregated
using mean and sum. The set of features was then augmented with additional
feature classes generated using the terminus composition representation.

The hyperparameters for both the Random Forest and Extra Trees classifiers
have been configured as follows during experimentation: a total of 400 trees
are used, with each tree considering a subset of features equal to the square
root of the total number of features. The splitting process is evaluated using
the Gini impurity. For the Gradient Boosting classifier, the hyperparameters
are configured as follows: 400 boosting stages, a learning rate of 0.01, a
maximum depth of 8, and a subsample ratio of 0.8. Furthermore, each
combination of feature inputs has been systematically studied. The results
shown for each model are the medians over 100 iterations. The threshold-
dependent metrics, including accuracy, specificity, sensitivity, and the
Matthews correlation coefficient, are generated at the 0.5 probability
threshold.

#### Peptide Spectral Features

In the following experiment, we systematically tested two definitions of
$f(w)$ in Equation (1) for constructing Laplacian matrices: the number of
occurrences of amino acid $w$ (frequency), and its mean index position. We
compared the performance of two construction methods using three types of
classifiers: Random Forest (RF), Extra Trees (ETrees), and Gradient Boosting
Trees (GBT). Models are evaluated on the benchmark dataset AntiCP 2.0, using
peptide spectral features derived from $L_{0}$ and $L_{1}$ as inputs.

As shown in Table 2, on AntiCP 2.0 (Dataset A), when controlling for the
classifier, the mean index position definition achieves higher accuracy and
AUROC compared to the frequency. GBT with mean index position-based features
has the highest accuracy of 80.16%. On AntiCP 2.0 (Dataset B), the mean
position-based feature outperforms the frequency-based feature when using
Etrees, whereas the frequency-based spectral feature performs better with GBT
and RF. The highest accuracy of 71.99% is achieved using Random Forest when
peptide spectral features are constructed from the frequency. For the final
model, we selected the mean index position feature because it outperforms the
frequency feature in more settings. Consequently, we chose Etrees as the
classifier due to its higher accuracy with mean index position-based features.

AntiCP 2.0 (Dataset A)
---
Spectral Feature | Classifier | Acc | Sen | Spc | MCC | AUROC
Mean Position | Etrees | 79.64 | 76.80 | 82.47 | 0.59 | 0.80
RF | 79.64 | 77.32 | 81.96 | 0.59 | 0.80
GBT | 80.16 | 78.35 | 81.96 | 0.60 | 0.80
Frequency | Etrees | 79.38 | 77.84 | 80.93 | 0.59 | 0.79
RF | 78.35 | 77.06 | 79.90 | 0.57 | 0.78
GBT | 78.87 | 77.84 | 79.90 | 0.58 | 0.79
AntiCP 2.0 (Dataset B)
Spectral Feature | Classifier | Acc | Sen | Spc | MCC | AUROC
Mean Position | Etrees | 70.67 | 67.25 | 74.12 | 0.41 | 0.71
RF | 70.38 | 66.67 | 74.12 | 0.41 | 0.70
GBT | 68.62 | 64.91 | 72.35 | 0.37 | 0.69
Frequency | Etrees | 70.38 | 63.74 | 76.47 | 0.41 | 0.70
RF | 71.99 | 63.74 | 80.00 | 0.45 | 0.72
GBT | 70.67 | 63.16 | 78.24 | 0.42 | 0.71
Table 2: The performance of models using peptide spectral features on AntiCP
2.0 (Datasets A and B)

#### Sequence-based Features

##### Magnus Vector

In this experiment, detailed in Table 3, we focus on the Extra Trees
classifier and evaluate its performance using the mean Magnus vector across
various window sizes. A window size of 4 yields the highest accuracy on AntiCP
2.0 (Dataset A), whereas a window size of 5 achieves the highest accuracy on
AntiCP 2.0 (Dataset B). We selected a window size of 5 for our final Top-ML
model because it achieves the highest AUROC for both AntiCP 2.0 Datasets A and
B, the highest accuracy of 76.54% on AntiCP 2.0 Dataset B, and the second-
highest accuracy of 92.17% on AntiCP 2.0 Dataset A.

AntiCP 2.0 (Dataset A)
---
Window Size | Classifier | Acc | Sen | Spc | MCC | AUROC
4 | Etrees | 92.53 | 88.56 | 95.77 | 0.85 | 0.92
5 | Etrees | 92.17 | 88.66 | 95.36 | 0.84 | 0.92
6 | Etrees | 91.69 | 90.36 | 93.08 | 0.83 | 0.92
7 | Etrees | 91.80 | 89.64 | 94.05 | 0.84 | 0.92
AntiCP 2.0 (Dataset B)
Window Size | Classifier | Acc | Sen | Spc | MCC | AUROC
4 | Etrees | 75.95 | 74.85 | 77.06 | 0.52 | 0.76
5 | Etrees | 76.54 | 74.27 | 78.82 | 0.53 | 0.77
6 | Etrees | 75.52 | 72.78 | 78.24 | 0.51 | 0.76
7 | Etrees | 75.67 | 72.78 | 78.57 | 0.52 | 0.76
Table 3: The performance of models using Magnus vector representation on
AntiCP 2.0 (Datasets A and B)

##### Natural Vector

In this section, we tested three classifiers, i.e., RF, ETrees, and GBT, with
the natural vector as input features. The Gradient Boosting Trees classifier
provides the best accuracy of 92.53% on AntiCP 2.0 (Dataset A) (Table 4). On
AntiCP 2.0 (Dataset B), Extra Trees achieves the highest accuracy of 74.78%.
For both Magnus representation and natural representation, the specificity
metric generally trends higher than sensitivity across different classifiers.

Table 4: Performance of models using natural vector-based features on AntiCP
2.0 (Datasets A and B) AntiCP 2.0 (Dataset A)
---
Features | Classifier | Acc | Sen | Spc | MCC | AUROC
Natural Vector | Etrees | 92.40 | 88.14 | 96.91 | 0.85 | 0.92
Natural Vector | RF | 92.27 | 89.69 | 94.85 | 0.85 | 0.92
Natural Vector | GBT | 92.53 | 90.21 | 94.85 | 0.85 | 0.93
AntiCP 2.0 (Dataset B)
Features | Classifier | Acc | Sen | Spc | MCC | AUROC
Natural Vector | Etrees | 74.78 | 73.10 | 76.47 | 0.50 | 0.75
Natural Vector | RF | 74.19 | 73.10 | 75.29 | 0.48 | 0.74
Natural Vector | GBT | 74.19 | 74.27 | 74.12 | 0.48 | 0.74

##### Terminal Composition Features

To test which terminus to use, we generate the natural vector and mean Magnus
vector based on the terminus instead of the entire peptide sequence and test
the performance using the Extra Trees classifier. For AntiCP 2.0 (Dataset A)
(Table 5), the C15-based natural and Magnus vectors outperformed all other
terminal composition encoding with an accuracy score of 92.78%. On AntiCP 2.0
(Dataset B) shown in Table 5, the N15C15 terminus composition feature provided
the best results with an accuracy of 77.13%.

Considering the different feature encodings individually using the Extra Trees
classifier, it has been observed that the Magnus representation and natural
vector representation yield better results than the terminal composition
feature and the peptide spectral representation. The natural vector and Magnus
vector have similar accuracy on AntiCP 2.0 (Dataset A), while the Magnus
representation performs better on AntiCP 2.0 (Dataset B).

Table 5: The performance of models using Terminal composition features on
AntiCP 2.0 (Datasets A and B) AntiCP 2.0 (Dataset A)
---
Features | Classifier | Acc | Sen | Spc | MCC | AUROC
N5 | Etrees | 85.83 | 84.02 | 87.63 | 0.72 | 0.86
C5 | Etrees | 84.02 | 79.38 | 88.66 | 0.68 | 0.84
N5C5 | Etrees | 89.05 | 85.57 | 92.78 | 0.78 | 0.89
N10 | Etrees | 89.43 | 85.05 | 93.81 | 0.79 | 0.89
C10 | Etrees | 87.1 | 79.90 | 94.33 | 0.75 | 0.87
N10C10 | Etrees | 90.46 | 85.05 | 95.88 | 0.81 | 0.91
N15 | Etrees | 89.95 | 82.99 | 96.91 | 0.81 | 0.90
C15 | Etrees | 92.78 | 89.69 | 95.88 | 0.86 | 0.93
N15C15 | Etrees | 91.50 | 86.60 | 95.88 | 0.83 | 0.92
AntiCP 2.0 (Dataset B)
Features | Classifier | Acc | Sen | Spc | MCC | AUROC
N5 | Etrees | 75.37 | 75.44 | 75.29 | 0.51 | 0.75
C5 | Etrees | 67.30 | 67.25 | 67.65 | 0.35 | 0.67
N5C5 | Etrees | 75.07 | 71.93 | 78.24 | 0.50 | 0.75
N10 | Etrees | 75.95 | 74.27 | 78.24 | 0.52 | 0.76
C10 | Etrees | 73.61 | 70.18 | 77.06 | 0.47 | 0.74
N10C10 | Etrees | 75.66 | 74.27 | 77.06 | 0.51 | 0.76
N15 | Etrees | 75.66 | 73.10 | 78.24 | 0.51 | 0.76
C15 | Etrees | 76.69 | 74.85 | 78.82 | 0.53 | 0.77
N15C15 | Etrees | 77.13 | 74.85 | 79.41 | 0.54 | 0.77

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## Acknowledgments

J.T., J.W., and K.X. acknowledge the funding support from the Singapore
Ministry of Education Academic Research Fund (MOE-T2EP20220-0010 and
MOE-T2EP20221-0003). X.G. was supported by Nanyang Technological University
under the Presidential Postdoctoral Fellowship grant 023545-00001.

## Code and Data Availability

The Top-ML code can be found in https://github.com/XueGong-git/TopPep-ML. The
datasets for ACPs can be found in
https://webs.iiitd.edu.in/raghava/anticp2/download.php.

## Author contributions

K.X. designed the experiment. J.T., J.W., and X.G. conducted the experiment
and analyzed the data. J.T. wrote the initial draft of the manuscript. K.X.,
J.W., and X.G. subsequently revised the manuscript. All authors reviewed and
contributed to the final manuscript.


# The Ricci decomposition of the inertia tensor for a rigid body in arbitrary
spatial dimensions

Edward Parker<EMAIL_ADDRESS>
###### Abstract

The rotations of rigid bodies in Euclidean space are characterized by their
instantaneous angular velocity and angular momentum. In an arbitrary number of
spatial dimensions, these quantities are represented by bivectors
(antisymmetric rank-2 tensors), and they are related by a rank-4 inertia
tensor. Remarkably, this inertia tensor belongs to a well-studied class of
_algebraic curvature tensors_ that have the same index symmetries as the
Riemann curvature tensor used in general relativity. Any algebraic curvature
tensor can be decomposed into irreducible representations of the orthogonal
group via the _Ricci decomposition_. We calculate the Ricci decomposition of
the inertia tensor for a rigid body in any number of dimensions, and we find
that (unlike for the Riemann curvature tensor) its Weyl tensor is always zero,
so the inertia tensor is completely characterized by its (rank-2) Ricci
contraction. So unlike in general relativity, the Weyl tensor does not cause
any qualitatively new phenomenology for rigid-body dynamics in $n\geq 4$
dimensions.

##  I Introduction

A first course in classical mechanics usually begins by discussing the
rotation of rigid plane figures in two dimensions. In this context, students
are taught the familiar equations for the 2D rigid-body rotation about a fixed
point

$L=I^{(2D)}\omega,\quad\tau=\frac{dL}{dt}=I^{(2D)}\alpha,\quad
T=\frac{1}{2}I^{(2D)}\omega^{2},$ (1)

where $L$ is the body’s angular momentum, $I^{(2D)}=\int dm\,r^{2}$ is its
moment of inertia about the point of rotation ($dm=\sigma(\bm{r})\,d^{2}r$
with $\sigma(\bm{r})$ the area mass density), $\omega$ is its angular
velocity, $\tau$ is the net external torque $\sum r\,F_{\theta}$ applied to
it, $t$ is time, $\alpha$ is the body’s angular acceleration, and $T$ is its
rotational kinetic energy. In the 2D context, $I^{(2D)}$ and $T$ are
considered to be nonnegative scalars and $L$, $\omega$, $\tau$, and $\alpha$
to be signed scalars whose signs represent a counterclockwise or clockwise
orientation. Students are sometimes vaguely told that similar formulas often
work for 3D rotation about axes with high symmetry and given somewhat
mysterious formulas for the moments of inertia about various axes for various
3D shapes.

A later course will usually cover the rotation of rigid bodies in 3D more
systematically. Students learn that in 3D, equations (1) generalize to

$\bm{L}=I^{(3D)}\bm{\omega},\ \bm{\tau}=\frac{d\bm{L}}{dt},\
T=\frac{1}{2}\bm{\omega}\cdot
I^{(3D)}\cdot\bm{\omega}:=\frac{1}{2}\omega_{i}I^{(3D)}_{ij}\omega_{j}$ (2)

(in the inertial “laboratory” frame of reference) [1]. The kinetic energy $T$
remains a scalar quantity, but $\bm{L}$, $\bm{\omega}$, $\bm{\tau}$, and
$\bm{\alpha}$ are now considered to be _vector_ quantities. Now
$\bm{L}=\int\bm{r}\times dm\,\bm{v}(\bm{r})$ (where now
$dm=\rho(\bm{r})d^{3}r$ with $\rho(\bm{r})$ the volume mass density),
$\bm{\omega}$ is oriented along the axis of rotation with a magnitude equal to
the angular speed, $\bm{\tau}=\int\bm{r}\times d\bm{F}_{\text{ext}}(\bm{r})$,
and $\bm{\alpha}=d\bm{\omega}/dt$. (More precisely, they are all pseudovector
quantities that do not change orientation under a parity inversion.) The
scalar moment of inertia $I^{(2D)}$ generalizes to a symmetric rank-2 inertia
_tensor_ , or (more prosaically but concretely) a symmetric $3\times 3$ matrix
$I^{(3D)}$. The components of this tensor are determined by the rigid body’s
mass distribution:

$I^{(3D)}_{ij}=\int
dm\mathopen{}\mathclose{{}\left(r^{2}\delta_{ij}-r_{i}r_{j}}\right).$ (3)

Since the tensor is represented by a real symmetric matrix, it can always be
diagonalized. Its eigenvectors are referred to as the rigid body’s _principal
axes_ and the corresponding eigenvalues are the _principal moments of inertia_
about those axes. Only for rotations about the principal axes do the vector
equations (2) simplify to the scalar equations (1).

The tensor $I^{(3D)}$ is no longer necessarily constant in the laboratory
frame as its orientation changes, so the easiest course of action is often to
shift to a non-inertial “body” frame of reference in which $I^{(3D)}$ is
constant, even at the expense of the additional complications from working in
a non-inertial reference frame. Any student who has studied rigid-body motion
in 3D can testify that the relatively simple equations (2) and (3) can already
lead to very complicated and unintuitive rotational dynamics.

But three dimensions are not the end of the story. What about an arbitrary
number of dimensions $n$? Rigid-body rotation in higher than three dimensions
is admittedly less realistic than $n=2$ or $n=3$, but considering the fully
general case is still an interesting thought exercise that will yield
unexpected connections to the study of general relativity, which _is_
naturally formulated in greater than three dimensions.

Although this article will eventually use some advanced tools developed to
study general relativity, most of it should be accessible to someone with a
solid understanding of advanced undergraduate classical mechanics. We leave
some of the more technical mathematical details in the footnotes.

##  II Mathematical preliminaries

(This section can be skipped by those who are less concerned with the
mathematical details.)

We define a _rigid body_ to be a set of points whose relative distances remain
constant. An extended body can only remain rigid if the causal influence of an
external force on one point is instantly transmitted to all other points, so
rigid bodies cannot exist in a relativistic setting. Moreover, we will require
that the body is free to perform unconstrained rotation, which requires that
space be flat. We will therefore work in the Euclidean space $\mathbb{R}^{n}$
endowed with the usual flat and positive-definite Euclidean inner product
(except where indicated otherwise).111In this article, the notation
$\mathbb{R}^{n}$ always refers to the full Euclidean inner product space, not
just the vector space. We will always work in Cartesian coordinates, in which
the metric indices are given by the Kronecker delta $\delta_{ij}$. We will use
the Einstein summation convention that repeated indices are summed from 1 to
$n$, and we will not distinguish between raised and lowered tensor indices. We
will denote the (constant) Euclidean metric tensor by $\delta$ and will only
occasionally use $g$ to denote the metric tensor _field_ for an arbitrary
(potentially curved) manifold.

###  II.1 The exterior algebra

Several ideas from the _exterior algebra_ will be very useful [2]. If $k$ is a
natural number, then a _$k$ -vector_ or _multivector_ is an element of
$\Lambda^{k}(V)$, the $k$th exterior power of a vector field $V$.222We use the
terminology convention that “multivectors” must be homogeneous with fixed $k$.
The terms “multivector”, “bivector”, etc. are often associated with the
somewhat obscure formalism of _geometric algebra_ , but this article does not
use any concepts from geometric algebra – just the simpler and much more
standard exterior algebra of totally antisymmetric tensors. Some physicists
familiar with general relativity might be more used to referring to totally
antisymmetric tensors as “differential forms”. But strictly speaking,
differential forms are smooth multivector _fields_ that are functions of a
spacetime manifold. The bivectors discussed in this article are not local
fields but correspond to individual extended objects, so they are just fixed
bivectors and _not_ differential forms. At a “physicist’s level of rigor”,
$\Lambda^{k}(V)$ is the space of totally antisymmetric rank-$k$ tensors over
$V$. A $p$-vector $A$ and a $q$-vector $B$ can be combined together into a
$(p+q)$-vector using the _wedge product_

$(A\wedge
B)_{\mu_{1},\dots,\mu_{p+q}}=\frac{(p+q)!}{p!\,q!}A_{[\mu_{1}\dots\mu_{p}}B_{\mu_{p+1}\dots\mu_{p+q}]},$

where $[\,]$ around tensor indices denotes total antisymmetrization.333There
are two different normalization conventions for the wedge product in common
use. In this article, we use the “geometer’s convention” that is standard in
physics rather than the “algebraist’s convention”. See [3] for a detailed
discussion of the pros and cons of each convention. A multivector is _simple_
(or _decomposable_ or a _$k$ -blade_) if it can be expressed as a wedge
product $v_{1}\wedge\dots\wedge v_{k}$ of $k$ rank-1 vectors $v_{i}$.

$\Lambda^{k}(\mathbb{R}^{n})$ is a real vector space of dimension
$\binom{n}{k}$, since a natural basis is the set of simple wedge products
$e_{i_{1}}\wedge\dots\wedge e_{i_{k}}$ of $k$ unit vectors within an
orthonormal basis for $\mathbb{R}^{n}$. Moreover, we can use the inner product
on $\mathbb{R}^{n}$ to map any multivector $A\in\Lambda^{k}(V)$ to its _Hodge
dual_ multivector $\star A\in\Lambda^{n-k}$. For Euclidean space, the Hodge
dual of a multivector is just proportional to its contraction with the totally
antisymmetric Levi-Civita tensor $\epsilon_{\mu_{1}\dots\mu_{n}}$:

$(\star
A)_{\mu_{1}\dots\mu_{n-k}}=\frac{1}{k!}\epsilon_{\mu_{1}\dots\mu_{n-k}\nu_{1}\dots\nu_{k}}A_{\nu_{1}\dots\nu_{k}}.$

(The formula is more complicated for more general manifolds.) For a Reimannian
(i.e. positive-definite) metric, the double Hodge star of a $k$-vector
$A\in\Lambda^{k}(\mathbb{R}^{n})$ is $\star\star A=(-1)^{k(n-k)}A$. The vector
space $\Lambda^{k}(\mathbb{R}^{n})$ inherits its own inner product from the
Euclidean inner product. The inner product between simple $k$-vectors is given
by

$\langle v_{1}\wedge\dots\wedge v_{k},w_{1}\wedge\dots\wedge
w_{k}\rangle:=\det M,\ \ M_{ij}:=\langle v_{i},w_{j}\rangle,$

and it extends to general multivectors by linearity. As a special case, the
norm-squared of a simple $k$-vector $v_{1}\wedge\dots\wedge v_{k}$ equals the
_Gram determinant_ of the matrix with entries $\langle
v_{i},v_{j}\rangle$.444We will use the normalization convention that when
calculating the inner product of $k$-vectors, the combinatorial factor $1/k!$
goes into the index contraction rather than into the antisymmetric tensors
that represent the orthonormal basis vectors
$\hat{e}_{i_{1}}\wedge\dots\wedge\hat{e}_{i_{k}}$ for
$\Lambda^{k}(\mathbb{R}^{n})$. That is, the tensors that represent
$\hat{e}_{i_{1}}\wedge\dots\wedge\hat{e}_{i_{k}}$ have elements $1$, $0$, and
$-1$ for all $k$, while the inner product on the exterior algebra
$\Lambda^{k}(\mathbb{R}^{n})$ is given by $\langle
A,B\rangle_{\Lambda^{k}(\mathbb{R}^{n})}=\frac{1}{k!}A_{I}B_{I}$ (where $I$
denotes the multi-index $(i_{1},\dots,i_{k})$) instead of by the usual inner
product $A_{I}B_{I}$ on the tensor algebra.

###  II.2 Rotations in arbitrary dimensions

By definition, a rotation $R$ of $n$-dimensional Euclidean space preserves
angles and distances between points, and more generally it preserves the
Euclidean inner product between vectors. It is also straightforward to show
that a rotation must be a linear transformation on vectors. Therefore, for any
vectors $v$ and $u$, $\delta(v,u)\equiv\delta(Rv,Ru)$. In matrix language,
this becomes

$(Rv)^{T}\delta(Ru)=v^{T}R^{T}\delta Ru=v^{T}\delta u,$

where $\delta$ represents the $n\times n$ identity matrix. Since this equation
must hold for all vectors $u$ and $v$, we must have that $R^{T}\delta
R=\delta$. If we only consider proper rotations, which are connected to the
identity operator, then the set of proper rotation operators form the Lie
group $\mathrm{SO}(n)$.

Angular velocity and angular momentum correspond to infinitesimal rotations,
which are elements of the Lie algebra $\mathfrak{so}(n)$. To see what the
elements $A$ of (the fundamental representation of) $\mathfrak{so}(n)$ look
like, we can Taylor expand the rotation operator in equation $R^{T}R=\delta$
to first order in the rotation angle $\theta$: letting $R=\delta+\theta
A+o(\theta^{2})$ gives

$\displaystyle\mathopen{}\mathclose{{}\left(\delta+\theta
A^{T}+o(\theta^{2})}\right)$
$\displaystyle\mathopen{}\mathclose{{}\left(\delta+\theta
A+o(\theta^{2})}\right)$
$\displaystyle=\delta+\theta\mathopen{}\mathclose{{}\left(A+A^{T}}\right)+o(\theta^{2})$
$\displaystyle=\delta,$

so $A+A^{T}=0$ and A must be an $n\times n$ antisymmetric matrix. An
infinitesimal rotation generator $A\in\mathfrak{so}(n)$ can be mapped to a
rotation $R\in\mathrm{SO}(n)$ through a non-infinitesimal angle $\theta$ by
the _exponential map_ $R=\exp(\theta A)$.555In the context of quantum
mechanics, physicists usually use the convention that generators are Hermitian
operators and the exponential map is given by $A\to\exp(-iAt/\hbar)$, where
$t$ is a continuous real parameter like time, distance, or angle. In this
article, it will be easier to stick to real numbers and use the phase
convention more common among mathematicians. For matrix representations like
the one that we are implicitly considering, the exponential map is just the
ordinary matrix exponential.

At our level of rigor, antisymmetric matrices are _bivectors_ in
$\Lambda^{2}(\mathbb{R}^{n})$. So in general dimensions, an infinitesimal
rotation is not represented by a (pseudo-)vector but by a bivector [4].666The
Lie algebra $\mathfrak{so}(n)$ is isomorphic to $\Lambda^{2}(\mathbb{R}^{n})$
as a vector space, but instead of the wedge product it has a _Lie bracket_
given by the matrix commutator. We will not need this Lie bracket in this
article.

More concretely, if $x_{i}$ and $x_{j}$ are orthonormal vectors in
$\mathbb{R}^{n}$, then an infinitesimal rotation in the $i$-$j$ plane
(oriented so that $x_{i}$ rotates into $x_{j}$) is generated by the simple
bivector $x_{i}\wedge x_{j}$. More generally, the magnitude of a simple
bivector gives the (infinitesimal) angle of rotation, and its sign (or
equivalently, the ordering of the two vectors being wedged together) gives the
orientation of the rotation.777When we say that rotations are “infinitesimal”,
we mean that they are small enough that we can neglect any small non-
commutative composition effects and add them together without keeping track of
ordering. The higher-order non-commutative effects are captured by the Lie
bracket structure mentioned in a previous footnote. A _simple_ bivector
generates a rotation in a single plane that leaves all orthogonal directions
unchanged. (In $n>3$ dimensions, we cannot describe this as a rotation about a
single 1D axis, because there are multiple directions that are all orthogonal
to the plane and to each other.) All bivectors over $\mathbb{R}^{n}$ are
simple if $n\leq 3$, so all rotations occur in a single plane. But not all
bivectors are simple if $n\geq 4$; instead, any bivector in
$\Lambda^{2}(\mathbb{R}^{n})$ can be decomposed into a sum of at most
$\mathopen{}\mathclose{{}\left\lfloor\frac{n}{2}}\right\rfloor$ orthogonal
simple bivectors (where $\lfloor\,\rfloor$ denotes the floor function). This
decomposition is generically unique, unless multiple simple bivectors have the
same magnitude [4].888In matrix language, a real antisymmetric $n\times n$
matrix $A$ represents a _simple_ bivector iff there exists a nonzero vector
$v\in\mathbb{R}^{n}$ such that $A_{[ij}v_{k]}\equiv 0$. This is true for all
real antisymmetric $n\times n$ matrices $A$ if $n\leq 3$, but only for some
such matrices if $n\geq 4$. Geometrically, it means that the vector $v$ lies
in the unique plane in $\mathbb{R}^{n}$ spanned by the simple bivector $A$.
Therefore, not all rotations of higher-dimensional Euclidean space occur in a
single plane; a general rotation of Euclidean space in $n$ dimensions is
generated by orthogonal planes rotating simultaneously (generically at
different speeds).

In three dimensions, the usual (pseudo-)vector representations of angular
velocity, angular acceleration, angular momentum, and torque are all derived
from the fundamental infinitesimal pseudovector rotation $d\bm{\theta}$. These
pseudovector representations are the Hodge duals of the corresponding
bivectors. (The fact that they transform as pseudovectors is a clue that a
bivector description is more fundamental, because unlike pseudovectors,
bivectors transform in the natural way under parity inversion [4].) The
bivector representations of all of these quantities are defined in any
dimension, but the pseudovector representations only make sense for $n=3$. As
expected, for $n=2$ the bivector space is one-dimensional, representing the
single scalar degree of freedom for plane rotations. For $n=3$, the bivector
space is three-dimensional, corresponding to the usual axis-magnitude
representation of a 3D rotation. But for $n=4$, the bivector space is _six_
-dimensional – more than the four degrees of freedom that we might expect
based on our 3D intuition.

##  III The inertia tensor in arbitrary dimensions

In three dimensions, the inertia tensor is a linear map that maps an angular
velocity (pseudo-)vector to an angular momentum (pseudo-)vector. But in
arbitary dimensions, angular velocity and angular momentum are represented by
bivectors, not vectors. The inertia tensor therefore generalizes to a rank-4
tensor $I:\Lambda^{2}(\mathbb{R}^{n})\to\Lambda^{2}(\mathbb{R}^{n})$ that
linearly maps bivectors to bivectors. In terms of indices, this becomes999The
factor of $1/2$ in (4) is not necessary. We simply include it to match the
standard normalization for the 3D rank-2 tensor, and to parallel our
conventions for the inner product on $\Lambda^{k}(\mathbb{R}^{n})$ and the
Hodge star operator that we normalize a contraction of $k$ totally
antisymmetric tensor indices by $1/k!$. But the tensor contraction in (4) does
not represent an inner product on $\Lambda^{k}(\mathbb{R}^{n})$, so we could
also consistently absorb the factor of $1/2$ into the normalization of $I$.

$L_{ij}=\frac{1}{2}I_{ijkl}\omega_{kl}.$ (4)

More abstractly, the inertia tensor can still be thought of as a linear
operator on a real inner product space – but the inner product space is no
longer the $n$-dimensional physical Euclidean space, but the
$\binom{n}{2}=\frac{1}{2}n(n-1)$-dimensional inner product space
$\Lambda^{2}(\mathbb{R}^{n})$ of bivectors on $\mathbb{R}^{n}$. This linear
operator will turn out to be self-adjoint, just like in the 3D case.

After all this setup, it is actually very simple to derive the inertia tensor.
In arbitrary dimensions, we do not have a cross product, so the angular
momentum generalizes to a bivector wedge product of the vectors $\bm{r}$ and
$dm\,\bm{v}$:

$L=\int\bm{r}\wedge(dm\,\bm{v}).$

($dm$ now represents the arbitrary-dimensional volume form
$\rho(\bm{r})\,d^{n}r$.) With our choice of sign conventions, the 3D rigid-
rotation formula $\bm{v}=\bm{\omega}\times\bm{r}$ (which ultimately derives
from $d\bm{r}=d\bm{\theta}\times\bm{r}$) generalizes to
$v_{j}=r_{k}\omega_{kj}$. So

$\displaystyle L_{ij}$ $\displaystyle=\int
dm\mathopen{}\mathclose{{}\left(2r_{[i}v_{j]}}\right)=\int
dm\mathopen{}\mathclose{{}\left(2r_{[i|}r_{k}\omega_{k|j]}}\right)$ (5)
$\displaystyle=\int
dm\mathopen{}\mathclose{{}\left(2r_{[i|}r_{k}\delta_{|j]l}\omega_{kl}}\right)$

This equation would seem to suggest that $I_{ijkl}=\int
dm\mathopen{}\mathclose{{}\left(4r_{[i}\delta_{j]l}r_{k}}\right)$. Strictly
speaking, this formula is correct in the sense that it returns the correct
value of $L$, but it contains unphysical degrees of freedom. The inertia
tensor inputs a bivector angular momentum $\omega$ that is always
antisymmetric, so any part of $I_{ijkl}$ that is symmetric in $k$ and $l$ will
vanish by symmetry when contracted with $\omega_{kl}$, and only the part that
is antisymmetric in $k$ and $l$ will affect the output $L$. We therefore
explicitly antisymmetrize $I$ on $k$ and $l$ to more clearly show which are
the true degrees of freedom that affect the angular momentum:

$\displaystyle I_{ijkl}$ $\displaystyle=\int
dm\,\mathopen{}\mathclose{{}\left(-4r_{[i}\delta_{j][k}r_{l]}}\right)$ (6)
$\displaystyle=\int
dm\,\mathopen{}\mathclose{{}\left(-r_{i}\delta_{jk}r_{l}+r_{i}\delta_{jl}r_{k}+r_{j}\delta_{ik}r_{l}-r_{j}\delta_{il}r_{k}}\right).$

In three dimensions, (4) can be reformulated in terms of pseudovector
quantities as

$\bm{L}=\star
L=\star\mathopen{}\mathclose{{}\left(\frac{1}{2}I\omega}\right)=\star\mathopen{}\mathclose{{}\left(\frac{1}{2}I(\star\bm{\omega})}\right),$

or in index notation,

$L_{p}=\frac{1}{4}\epsilon_{pij}I_{ijkl}\epsilon_{klq}\omega_{q}.$

Comparing with the first equation in (2), we see that

$I^{(3D)}_{pq}=\frac{1}{4}\epsilon_{pij}I_{ijkl}\epsilon_{klq}=\int
dm\mathopen{}\mathclose{{}\left(r^{2}\delta_{pq}-r_{p}r_{q}}\right),$

which agrees with (3).

The inertia tensor satisfies several index symmetries:

$\displaystyle I_{ijkl}$ $\displaystyle=-I_{jikl}=-I_{ijlk}$ (7a)
$\displaystyle I_{ijkl}$ $\displaystyle=I_{klij}$ (7b) $\displaystyle
I_{ijkl}$ $\displaystyle+I_{iklj}+I_{iljk}=0.$ (7c)

Remarkably, these are the exact same symmetries satisfied by the Riemann
curvature tensor in general relativity [5]. But there is one important
structural difference between the inertia tensor and the Riemann curvature
tensor: the inertia tensor is a single fixed tensor, while the Riemann
curvature tensor is a tensor _field_ defined over a spacetime
manifold.101010There is a subtle point here. The integrated tensor $I$ is
indeed just a single tensor with no spatial dependence. But, as mentioned
above, the differential $dm$ is technically a true differential volume _form_
on $\mathbb{R}^{n}$ (although the full machinery of differential forms is
somewhat overkill for integrating over Euclidean space). The integrand in
parentheses in (6) explicitly depends on the position $\bm{r}$ and so is
obviously a tensor _field_ that varies over space. Taken together, the full
differential form being integrated in (6) is a _tensor-valued volume form_
[6]. Tensor-valued differential forms cannot be integrated over generic curved
spaces, because there is no natural way to parallel-transport the tensor at
each point in the manifold to the same base point so that the tensors can be
added together within the same vector space. But vector-valued differential
forms defined on a flat manifold _can_ be integrated, because the vector
spaces at each point are naturally isomorphic. The last section of this
article exploits this parallel by using tools from general relativity to study
the inertia tensor for rigid bodies in Euclidean spacetime.

##  IV The Ricci decomposition of the inertia tensor

###  IV.1 Algebraic curvature tensors

Any rank-4 tensor that satisfies the index symmetries (7), including the
inertia tensor $I$ given by (6), is referred to as an _algebraic curvature
tensor_ by analogy with the Riemann curvature tensor [7].

Identity (7c) is known as the _first_ or _algebraic Bianchi identity_.111111In
keeping with Stigler’s law of eponymy, the algebraic Bianchi identity was
discovered by Ricci. It follows from identities (7a) and (7b) if $n=2$ or $3$,
but is an independent condition if $n\geq 4$ [7]. Identity (7a) simply means
that an algebraic curvature tensor can be thought of as a linear operator on
$\Lambda^{2}(\mathbb{R}^{n})$. Identity (7b) means that this operator is self-
adjoint, just as the rank-2 inertia tensor is in 3D. Therefore, there always
exists a complete orthonormal basis of $\binom{n}{2}$ eigenbivectors
$\omega^{(i)}$ with eigenvalues $I^{(i)}$ (the principal moments of inertia)
such that if the rigid body is rotating with angular velocity $\omega^{(i)}$,
then its angular momentum $L=I^{(i)}\omega^{(i)}$. These eigenbivectors are
the generalizations of the principal axes in 3D (but in higher dimensions,
they may not be simple and so may not correspond to rotations within a single
plane). In general relativity, this approach (thinking of the Riemann
curvature tensor as a self-adjoint operator on $\Lambda^{2}(M)$ and
considering its eigendecomposition) leads to the _Petrov classification_ of
spacetimes [8].

###  IV.2 The Kulkarni-Nomizu product

It will be convenient to introduce the bilinear _Kulkarni-Nomizu product_ of
symmetric rank-2 tensors [7]. If $A$ and $B$ are symmetric rank-2 tensors,
then their Kulkarni-Nomizu product $A\owedge B$ is a rank-4 tensor defined by

$(A\owedge B)_{ijkl}:=A_{ik}B_{jl}-A_{il}B_{jk}-A_{jk}B_{il}+A_{jl}B_{ik}.$

Any Kulkarni-Nomizu product is an algebraic curvature tensor that satisfies
(7) (although the converse is not true).121212There is a _formal_ similarity
between the commutation relations $[J^{\mu\nu},J^{\rho\sigma}]$ for the Lie
algebra $\mathfrak{so}(3,1)$ (the generators of the $(3+1)D$ Lorentz group)
and the formal Kulkarni-Nomizu product $-i\,\eta\owedge J$, where $\eta$ is
the flat Minkowski metric with signature $(-,+,+,+)$ (and we use the standard
physicists’ convention for the normalization of the generators, rather than
the mathematicians’ convention used in the main text) [9]. But the latter
expression is not actually a true Kulkarni-Nomizu product, because the tensor
$J$ is antisymmetric rather than symmetric. The commutator is therefore not an
algebraic curvature tensor; it is antisymmetric rather than symmetric under
the simultaneous exchange $(\mu\leftrightarrow\rho,\
\nu\leftrightarrow\sigma)$, and requirement (7b) is violated. The Kulkarni-
Nomizu product is symmetric: $A\owedge B\equiv B\owedge A$.

Note that

$(A\owedge A)_{ijkl}=4A_{i[k}A_{l]j}.$

For an arbitrary two-dimensional curved surface, the Riemann curvature tensor
field equals $\frac{1}{4}R\,g\owedge g$, where $R$ is the Ricci scalar field
(twice the Gaussian curvature) and $g$ is the metric tensor field. All _space
forms_ – Riemannian manifolds of any dimension with constant _sectional
curvature_ – also have a Riemann curvature tensor field equal to
$\frac{1}{4}R\,g\owedge g$, although in this case the Ricci scalar field $R$
is constant over the manifold.

Also note that if $v$ is a rank-1 vector, then

$(A\owedge(v\otimes v))_{ijkl}=-4v_{[i}A_{j][k}v_{l]}.$

Equation (6) therefore simplifies to the compact expression

$\boxed{I=\delta\owedge\int dm\,(\bm{r}\otimes\bm{r}).}$ (8)

###  IV.3 The Ricci decomposition

If $R$ is either an algebraic curvature tensor on $\mathbb{R}^{n}$ or an
algebraic curvature tensor _field_ on an arbitrary $n$-dimensional manifold
with metric tensor $g$, then $R$ has only one independent single trace: the
rank-2 symmetric _Ricci-contracted_ tensor (or tensor field)[7]131313If we
consider the linear map from the space of symmetric rank-2 tensors $A$ to the
space of algebraic curvature tensors that is given by $A\to g\owedge A$ (where
$g$ is an arbitrary inner product on $\mathbb{R}^{n}$), then this map turns
out to be exactly the transpose of the Ricci contraction.

$R^{(2)}_{jl}:=R_{ijil}.$

The other five single traces equal $\pm R_{jl}$ or 0. The only independent
double trace of $R$ is the Ricci-contracted scalar (or scalar field)

$R^{(0)}:=R^{(2)}_{ii}=R_{ijij}.$

From these, we can form the trace-free Ricci-contracted tensor (or tensor
field)

$\hat{R}^{(2)}:=R^{(2)}-\frac{1}{n}R^{(0)}g.$

$R$ can be naturally decomposed into a (direct) sum of three irreducible
representations of the orthogonal group $\mathrm{O}(n)$:

$R=S+E+C.$ (9)

This decomposition is known as the _Ricci decomposition_ [7]. Here $S$, $E$,
and $C$ are themselves algebraic curvature tensors (or tensor fields) given by

$\displaystyle S$ $\displaystyle:=\frac{R^{(0)}}{2n(n-1)}g\owedge g$
$\displaystyle E$ $\displaystyle:=\frac{1}{n-2}\hat{R}^{(2)}\owedge g$
$\displaystyle C$ $\displaystyle:=R-S-E.$

For $n=2$, only the $S$ term is well defined, so the decomposition is trivial.
For $n=3$, the $C$ term vanishes identically. For $n\geq 4$, all three terms
are generically nonzero.

Equation (9) is mathematically trivial by the definition of $C$, but the Ricci
decomposition is useful because each term lies in a different irreducible
representation of the orthogonal group. Loosely speaking, the $S$ term
contains the doubly-contracted degree of freedom in the algebraic curvature
tensor that transforms under rotations as a scalar, the $E$ term contains the
singly-contracted degrees of freedom that transform under rotations as a
traceless symmetric rank-2 tensor, and the $C$ term contains the uncontracted
degrees of freedom that transform under rotations as a rank-4 tensor.

In general relativity, the $S+E$ terms contain the same information as the
Ricci tensor field, which reflects spacetime’s local response to matter, while
$C$ is the Weyl tensor field, which is totally traceless and contains the
gravitational degrees of freedom that propagate through vacuum. We can
simplify the former sum to

$S+E=A\owedge g,$ (10)

where the symmetric rank-2 tensor

$\displaystyle A:=$ $\displaystyle\
\frac{1}{n-2}\hat{R}^{(2)}+\frac{1}{2n(n-1)}R^{(0)}g$ $\displaystyle=$
$\displaystyle\
\frac{1}{n-2}\mathopen{}\mathclose{{}\left(R^{(2)}-\frac{1}{2(n-1)}R^{(0)}g}\right)$

is known as the _Schouten tensor_ for the algebraic curvature tensor
[10].141414Some sources normalize the Schouten tensor to be twice this
expression or to have the opposite sign. The Schouten tensor and the Ricci
tensor are very closely related, and either can be easily derived from the
other; they are essentially just rescaled and trace-adjusted versions of each
other.

For the fixed inertia tensor (6), we have (shifting notation from $R$ to $I$)

$\displaystyle I^{(2)}$ $\displaystyle=\int
dm\mathopen{}\mathclose{{}\left((n-2)\bm{r}\otimes\bm{r}+r^{2}\delta}\right)$
$\displaystyle I^{(0)}$ $\displaystyle=2(n-1)\int
dm\mathopen{}\mathclose{{}\left(r^{2}}\right)$ $\displaystyle\hat{I}^{(2)}$
$\displaystyle=(n-2)\int
dm\mathopen{}\mathclose{{}\left(\bm{r}\otimes\bm{r}-\frac{1}{n}r^{2}\delta}\right)$
$\displaystyle A$ $\displaystyle=\int dm\,(\bm{r}\otimes\bm{r}).$ (11)

Combining equations (10), (11), and (8), we see that $I=S+E$ and so $C=0$.
Unlike for the Riemann curvature tensor in general relativity, the Weyl
component of the inertia tensor for rigid-body rotation vanishes identically
in _all_ dimensions.

The inertia tensor $I$ is fully characterized by its Schouten tensor (11) via
equation (8). For $n\geq 3$ dimensions, the inertia tensor only has the
$\frac{1}{2}n(n+1)$ independent degrees of freedom of its Schouten tensor. For
$n\geq 4$, this is less than the $\frac{1}{12}n^{2}(n^{2}-1)$ degrees of
freedom in a generic algebraic curvature tensor.

Moreover, the fact that the Weyl part of $I$ vanishes follows directly from
equation (8) by rotational symmetry. $I$ depends only on a rank-2 symmetric
tensor. So the degrees of freedom of $I$ must all transform under rotations as
rank-2 symmetric tensors, which means that they must lie in the $S\oplus E$
representation of the orthogonal group. This implies that $I$ can only have
nonzero $S$ and $E$ components in the Ricci decomposition, and its Weyl
component $C$ must vanish. We therefore could have concluded that $C=0$
directly from equation (8) without explicitly working out any of the traces of
$I$ or the Schouten tensor (although in this case, doing so is not difficult).

Equation (8) implies a high redundancy in the components of the inertia tensor
$I$ that simply reflects the fact that much of the “work” that $I$ is doing is
simply matching up indices correctly in the tensor contraction. If an
algebraic curvature tensor on $\mathbb{R}^{n}$ has vanishing Weyl component –
or equivalently, if it can be expressed in the form $A\owedge\delta$ – then
its action (4) simply maps the bivector $\omega$ to twice the antisymmetric
part of the matrix product $A\omega$. So in order to calculate the angular
momentum $L$ corresponding to an explicit angular velocity bivector $\omega$,
the most efficient course of action is often to entirely skip calculating $I$
and to directly use equation (5).

In three dimensions, the standard 3D inertia tensor (3) can be expressed in
terms of the Schouten tensor (11) by

$I^{(3D)}=\operatorname{Tr}(A)\delta-A.$

The phenomenology of general relativity changes qualitatively between $n\leq
3$ and $n\geq 4$ spacetime dimensions. If $n\leq 3$, then the Weyl tensor
field vanishes (or is undefined), so no gravitational degrees of freedom can
propagate locally through a vacuum. But if $n\geq 4$, then the possibility of
a nontrivial Weyl tensor field enables much richer phenomenology, such as
gravitational waves. We have shown that in arbitrary dimensions, the inertia
tensor for rigid-body rotation is an algebraic curvature tensor that shares
many mathematical similarities with the Riemann curvature tensor. From this
parallel alone, we might have guessed (by analogy with general relativity)
that the possibility of a nonzero Weyl tensor in the inerta tensor
qualitatively changes the phenomenology of rigid-body dynamics in $n\geq 4$
dimensions. But we have shown that this guess is not true, because there is a
crucial difference between the Riemann and the inertia tensors: the latter
cannot contain a Weyl tensor even in $n\geq 4$ dimensions.151515Of course,
there is a much more obvious difference between the Riemann and the inertia
tensors: the former is a tensor _field_ that varies over a spacetime manifold,
while the latter is just a fixed tensor. So the analogy is only rough, and
this guess may not have been very plausible in the first place. Moreover, we
are not claiming that there are no qualitative differences between rigid-body
dynamics in $n\leq 3$ and $n\geq 4$ dimensions. There are — most notably, the
fact that in $n\geq 4$ dimensions there exist non-simple bivectors, which
generate proper rotations that are not confined within a single plane. (In
four dimensions, these are sometimes called _double rotations_.) We are only
making the narrower claim that – in contrast with general relativity – _Weyl
tensors_ do not cause qualitatively new phenomenology in higher dimensions.

###### Acknowledgements.

The author thanks Brayden Ware and Gavin Hartnett for helpful discussions.

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