Asymptotically Stable Quaternion-valued Hopfield-structured Neural Network with Periodic Projection-based Supervised Learning Rules
Authors: Tianwei Wang
Xinhui Ma
Wei Pang
Venue: NeurIPS 2025 poster ()
Keywords: Quaternion-valued Neural Network
Supervised Learning
Asymptotic Stability
Neural Dynamical System
Hopfield Neural Network
Periodic Projection Method
Machine Learning Theory
Neural Control
Primary area: general_machine_learning
TL;DR: A pathbreaking model based on neural differential equations with rigorous mathematical foundation

Abstract: Motivated by the geometric advantages of quaternions in representing rotations and postures, we propose a quaternion-valued supervised learning Hopfield-structured neural network (QSHNN) with a fully connected structure inspired by the classic Hopfield neural network (HNN). Starting from a continuous-time dynamical model of HNNs, we extend the formulation to the quaternionic domain and establish the existence and uniqueness of fixed points with asymptotic stability. For the learning rules, we introduce a periodic projection strategy that modifies standard gradient descent by periodically projecting each $4\times 4$ block of the weight matrix onto the closest quaternionic structure in the least-squares sense. This approach preserves both convergence and quaternionic consistency throughout training. Benefiting from this rigorous mathematical foundation, the experimental model implementation achieves high accuracy, fast convergence, and strong reliability across randomly generated target sets. Moreover, the evolution trajectories of the QSHNN exhibit well-bounded curvature, i.e., sufficient smoothness, which is crucial for applications such as control systems or path planning modules in robotic arms, where joint postures are parameterized by quaternion neurons. Beyond these application scenarios, the proposed model offers a practical implementation framework and a general mathematical methodology for designing neural networks under hypercomplex or non-commutative algebraic structures.

Asymptotically Stable Quaternion-valued Hopffeld-structured Neural Networks with Periodic Projection-based Supervised Learning Rules
Tianwei Wang University of Edinburgh T. Wang-1100sms 1100sms. ed. ac. uk
Xinhui Ma University of Hull xinhui mw@hulll.a.......mm.. ac..llacc..... .u
Wei Pang Heriot-Watt University pang@hu ac. uk
Abstract
Motivated by the geometric advantages of quaternions in representing rotations and postures, we propose quaternion-valuee supervised learning Hopfield-structurr neural network (QSHNN) with fully connected structure inspired by the classic Hopfield neural network (HNN). Starting from continuous-time dynamical model of HNNs, we extend the formulation to the quaternionic domain and establish the existence and uniqueness of fixed points with asymptotic stability. For the learning rules, we introduce a periodic projection strategy that modifies standard gradient descent by periodically projecting each 4 block of the weight matrix onto the closest quaternionic structure in the least-squares sense. This approach preserves both convergence and quaternionic consistency throughout training. Benefiting from this rigorous mathematical foundation, the experimental model implementation achieves high accuracy, fast convergence, and strong reliability across randomly generated target sets. Moreover, the evolution trajectories of the QSHNN exhibit well-bounded curvature, i.e., sufficient smoothness, which is crucial for applications such control systems path planning modules in robotic arms, where joint postures are parameterized by quaternion neurons. Beyond these application scenarios, the proposed model offers practical implementation framework and general mathematical methodology for designing neural networks under hypercomplex non-commutative algebraic structures.
Introduction
Hopfield neural network (HNN) could be recognized as the precursor of many fundamental models 27 33 10]. It has a symmetric network topology and recurrent connections, becoming an attractor model with a dynamical structure, where neurons are activated and interact with each other at certain frequency and tend to discrete equilibria. These equilibria and the attracting landscape in phase space embody the generalization capability of the neural network. We adopt the continuous time HNN [I]] governed by the evolution Eq. (L.I) to expand and rebuild QSHNN in this research.
(t; (() + wpp((:(()) + ubj, j j= 1,2, di
(1.1)
In a more condensed vector form i(t) -t(() (W) + μb, Vector v represents the list of the neuron activations Vy matrix W arranges the connection weights w in matrix-vector multiplication, vector function & imposes the activating function, typically hyperbolic tangent (px)) = (e* - e ")/("" +e-",, to each entry of vector v. b serves as the bias vector and is significant the representation ability of the network. Meaning of constant y, / in corresponds
Bio-iispiredd Computing and Machine Learning (BCML) Lab. Cooresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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to the parameters of electronic components capacitance and resistor for the circuit implementation of classic HNN. The stability of HNNs is fundamental their function as associative memory systems, enabling the convergence of network states to attractor patterns [I]]. Farlier studies have extended HNNs to quaternion-
systems, mainly under discrete-time formulations with split activation functions and fixed-point dynamics ( [5]. More recently, continuous- time modern HNNs have also been introduced [23 M] where the state evolution follows ordinary differential equations. However, these models remain largely confined to unsupervised paradigms driven by internal energy minimization, lacking explicit target tracking, structural control, or generalized learning principles.
In contrast, our proposed model departs from this associative memory paradigm of HNN. We introduce a quaternion-valuee supervised hopfield-structured neural network (QSHNN), formulated as a continuous-ting autonomous dynamical system. The system's 's state evolves according to quaternionic differential equation, and the network trained to converge asymptotically to externally specified targets. The learning rule is derived analytically via Generalized THTR (GHR) calculus [36 and incorporates a projection mechanism to preserve the block-wit quaternion structure of the weight matrix. GHR calculus has been established and verified as complete and mature technique to expand the neural network over the quaternionic domain 21 38 differential formulas are no longer valid because of the non-commutativity of quaternion algebra. This design guarantees both convergence and structural consistency.
37
S 22 29], where the normal
Although early Hopfield-type networks, including their quaternion extensions, often rely on directly encoding the weight matrix using Hebbian or outer-produt formulations, this approach suffers from several critical limitations. First, such weight constructions inherently lack scalability; they can only stably store a small number of target states before spurious attractors emerge or convergence fails [28 4 19]. Second, these methods assume all target patterns are fixed and known a priori, making the network unsuitable for tasks requiring adaptability, generalization, or dynamic reconfiguration 13 [9] Third, direct encoding lacks error-driven optimization mechanism, preventing the network from refining its behavior in response to task-specific objectives [19] 17] As result, while analytically convenient, direct weight specification severely restricts the practical applicability of Hopfield-tye type models in real-world control, learning, or representation settings. Unlike classical previously proposed quatemion- model operates under a supervised learning paradigm with continuous quaternion- valued trajectories, offering smooth, controllable dynamics suitable for robotics, trajectory generation, and feedback control systems. We position this work not as a generalization of existing QHNNs, but as the formulation of new class of quaternionic neural dynamical systems with rigorous theoretical guarantees and task-driven behavior.
Hopfield neural networks [12] 14 32]
2 Background
Quaternion algebra Quaternion is a hypercomplex number with three imaginary components; we separate the components into the scalar part and the vector part V: q s+ + = [s. v] H. The vector part space matches the 3-dimensional Fuclidean space R². The region of quaternions is denoted by H., where the coefficients associated with the imaginary units are x. y, R. The fundamental arithmetic of imaginary units are i o = -joi-- = k. jo = k oof =; koi=
-io k- = j, 12 j 2 i j o k = 11
We denote the multiplication operation of two quaternions q1 and qa by 91 q2. To distinguish other types of multiplication, we neglect the symbol for matrix multiplication and scalar multiplication. We denote the inner product by tn V22 and the cross product by vn x 02. Consider quaternion 91 = 81 + Fí + yn + 21 and q = 82 + x22 + y2 + z2k, the plain expansion of multiplication becomes: α1 42 (8182 T1 V192 =1 + (81 + 8221 + 11 y2=1) + (8132
$2311 + 21 22 ) + (5122 + $281 + Z132 - T281 k.
Quaternion left multiplication manifold Quaternion multiplication can be represented as real- valued vetrix multiplication over R4 by Eq. 2.1). where each quaternion defines 4 x 4 real matrix that acts on another quaternion interpreted as a 4-dimensional real vector. Due to the non-commutative nature of quaternion algebra, there exist two distinct but equivalent matrix representations: one for left multiplication and one for right multiplication. this work, we adopt the left multiplication form consistently, where each quaternion induces a left-linear action on the
2

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quatemnionic space. This operation is denoted by between two quaternions with the following form:
91 42
(81
+ 3i
yn
+ 21 k) o
(82
+ F2
+ y/2
+ z2k)
81
11
1/1
-1
82
82
322
1/2
22
81
(2.1)
81
-
1/1
I2
82
22
-22
21
91
21
51
-1
9/2
1/2
22
82
32
y1
21
$1
22_
1/2
12
52
The collection of all real 4 x 4 matrices corresponding to quaternion left multiplication forms a smooth 4-dimensional embedded submanifold of IR denoted by L. This submanifold inherits a natural algebraic structure from Hi and constitutes a real matrix Lie group under composition 30 25). Specifically, L is isomorphic to the group (H, o), and the group operation corresponds to matrix multiplication within C. Each matrix L(q) E C is uniquely determined by a quaternion zi + yj + zk, and the map q L(q) is group isomorphism. The tangent space at the identity element L(1) defines the associated Lie algebra, which consists of all real 4 x matrices that can be written as linear combinations of the infinitesimal generators L()), L(j), L(k).
0
0
0
1
0
0
0
0
00
0
L(1) = 44 4()
0
LG)
L(k)
0
(2.2)
Each element of the quaternion left multiplication manifold L can be expressed as linear combination of four fixed basis matrices corresponding to the canonical quaternion basis {1, k). Specifically, for any quaternion g = 8 + xí + yj zk E H. its corresponding left multiplication matrix L(q) E & admits the decomposition: L(q) s L(1)+ + L(i)+ +y L(j)+ L(k), where L(1), L(i), L(j), L(k) are fixed real 4 x 4 matrices, as shown in Eq. (2.2), representing left multiplication by each of the basis elements. This decomposition establishes linear isomorphism between H. and subspace L.
And we also define the conjugate a quaternion by q' = 8- xi yj 2k = [s. vb]. the modulus or length of a quaternion by |al v q q² (s² x + un -/2 the inverse of a quaternion by q o q 1 1, where q qー q² //g). It is quite important to mention the geometric meaning and advantages of quaternion, which is just one of our motivations. There is a compact and intuitive form represent 3-dimensional rotation by a quaternion.
Definition 2.1 (Quaternion rotation (34). Rotating an arbitrary quaternion q Vq by quaternion number parameterized by p = [c(coo8 + UniinB3 is equivalent to computing the following:
Rx(q):- u =q"
(2.3)
That is make the 3-dimensional vector Oq rolate by angle 3 about the axis Un. When p is a pure unit quaternion (i,j. k), this manipulation is also named quaternion involution.
As non-commutative algebra, quaternion not compatible with general calculus theory. Hence, an extension of differential is necessary to utilize the derivative of quaternion well as the chain rule, product rule, etc., for the practical purpose theoretical deduction.
Definition 2.2 (Quaternion (left GHR) derivative 336
Let q E H and f: D D. D H. then (μ + 0, p E Hl) of a well-defined function f
the left GHR derivatives, with respect to qt and qu are defined as:
0f ( 0f 0f
0f
0f
o
i
o j
o k"
(2.4)
8q 4 0q.. 0q.
dq.
dqa
0f 1 ( 0f + of o + oq.
of
of
k"
(2.5)
aqui 4 iqa Oqb
aqu
0f are the partial derivatives of f with respect to qar To le and qá- where ōq.' Dq." Dq. das respectively, and {1, it j"h k! is orthogonal basis of H. it has the same rotation factor μ.
0f of of and
Lyapunov stability theory For the unknown solution to the evolution equation, which is of high- level nonlinearity and may not even exist in analytical form, we adopt Lyapunov theory for the analysis of stability. There are three different types of stability for dynamical system that are in our consideration: Lyapunov stability, guasi-asymptotic stability, and asymptotic stability. The justification can be drawn from directly using the following Thm. 2.I

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Theorem 2.1 (Lyapunov Theorem (2)). Consider the dynamical system of the form E f(x; t) with the fixed point the origin, where f(0,t) 0. If there exists a positive definite scalar function V(x, t), also named the Lyapunov energy function, such that the time derivative di on the flow semi-negative definite, as shown in Eq. (..6). we have the zero equilibrium being Lyapunov stable.
dV
dV (x, t) dt
2V 0V
f(x, 50
(2.6)
dx
Further, if the time derivative on the flow is strictly negative definite, then we have that the fixed point asymploticallyy stable. The critical step in applying this theorem is to find suitable energy function.
3
Methodology
Structure of quaternion neuron To allow quaternion numbers to operate on the neural network, integrate four real-valued neurons quaternion neuron and impose specific internal evolution regulations, as shown in Fig. ① A single neuron contains four normal Hopfield neurons, each of them represents a division component of a quaternion. Input vector will Dipput and output vector will be oout With unit time delay, the output value loops into the neuron and is weighted, then activates the linked neuron by function s. Every connection is allocated weight Wij. and four weights are integrated as quaternion weight w.
Guare IXcuse Chaan-Jyoar Jyas- KZow
External Interaction
Internal Operation
Wist
W22
W42
W4s
q1
Qmpor-cp(W qatt) So--
Quaternion Neuron
Figure 1: Structure of quaternion neuron and its interfaces. Four realevalued neurons are integrated together as quater- nion with respect different component coefficients. The internal operation or the ceff-connection can be fully char- acterized by single weight quaternion. Every numerical factors are compatible by quaternions, projection and approx- imation only exist the training. Drawn by the authors.
Figure 2: Robotic arm with each joint posture parameterized by quaternions for each joint. Our model capable of high fidelity control and path plan- ning. Figure drawn by the authors, where the robotic arm is generated by Al tool just for demonstration aim.
Consistency and compatibility of mathematical form Quaternion neuron structure is closely related to the mathematical form of the evolution equation, where we embed this structure into the classic HNN evolution Eq. (I.I). The quaternion-valuus system is consistent with the original form by directly replacing the real variables with quaternionic variables through Eq. (..I). Explicitly, can be expressed in quaternion algebra and underlying linear algebra by:
91
q1
W11
W12
In
p(11 (((2)
bb
d
92
92
W21
W2n
b2
+
(3.1)
dt
9n
9
Ww1
Wn2
w mnn_
Li((nn)) )m))
bb.
Q1
Win
W12
Win
((q))
bi
42
Wa1
W22
Wen
((1))
=-y
+ μ
r
(3.2)
94m
W
W
W
[g(gm))
ban

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Or in condensed form q p + - - quaternionic weight from neuron neuron j by wi and arrange all the connections quaternion matrix W. Quaternionic matrix-vector multiplication is denoted by W
)g(g)
μb, where we denote the
Newron Activation
Iteration k+]
Waelf
33
Integration
W1
W13
Q3
Q1
Selffordlback
w
W22
(3
Neuran Activation at Iteration
W12
Q2
W
Connection withh Other
Figure 3: The integration of quaternion neuron. It has inside structure, self feedback connection and connec- tion interfaces with other neurons. Activation will be updated every unit impulse time. Drawn by the authors.
Figure 4: Structure of a fully connected 3-neuron network. Every connection from j-th to i-th neuron parameterized directly by quaternion wij- Drawn by authors.
Single neuron response with linear activation In this subsection, we state the linear response properties of the neuron structure determined above. Then we discuss the activation function we choose for our model eventually. For single neural feedback with linear activation yp q -> q. the evolution equation becomes:
q(t) - + pM o pb: = x pb pb= Xq jub
(3.3)
where the new quaternion coefficient X = - + piw with respect the left multiplication matrix X. Thus, it becomes a linear differential system with a nonhomogeneous term μb. The solution of the homogeneous equation can calculated directly. Through the matrix method, firstly, we define the fundamental matrix by (t) = (S1 S2 S3 Ca). X is noticeable as anti-symmetric matrix plus scalar times the identity matrix E.
Notice that this anti-symmetric matrix has only imaginary eigenvalues, which makes it easy to know X has different conjugate complex eigenvalues, and they all have the same algebraic multiplicity, and the geometric multiplicity my = 1, corresponding to complex eigenvalues À = = {x22 X + x3)$. To make the formula compact, the square root of the square in terms of weight parameters denoted by a² x2 + x + x2 = 3-2 Then we calculate the exponential matrix, and still has the form of a quaternionic anti-symmetrie matrix. Through the adjacent matrix, the matrix exponential associated with & is calculated by:
Scosat
-xasinata
-yysinat
-Xssinat
Xt ((1)((0)- eXat
Xusinat
Boosat
sisintt
sinit
Xysinat
Xasinat
Bcosat
-aasinat
(3.4)
X:sinat
-Xy
Xesinat
Boosat
xot cosat Xusinat Xysinat, sinat
qe. With the exponential of coefficient matrix and the initial activation do = + xod + 9od + znk, we can write the general solution to the single neuron response to:
a
(1)) xxx ao + xe [r" (cxT)= dd
x To + Xx 1oxx-((e2 (6 Il4bbb
(3.5)
= Je + ,x² (ge 1)
The entire expression perfectly integrates quaternion embeddings instead of through complex mapping and splitting methods, where many required formalizations can be satisfied automatically. For higher order neuron systems with nonlinear activation functions, the solution would be too complicated and
nonanalytic, shown [8 16), and machine learning approaches get involved to make qualitative control of this neural differential equation system.
5

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Asymptotic Stability Analysis For the type of neural networks as a nonlinear dynamical system, the evolution equation demanded to be equipped with a series of good properties. The two most fundamental of them are convergence and stability, which guarantee a deterministic activating value of the neuron and that the neuron can correctly evolve to that value, respectively. Below we propose two significant theorems guaranteeing the asymptotic stability of the QSHNN model. Firstly, we demand the existence and uniqueness of the dynamical system; they are the foundation of further analysis and critical for the capability of memory, and the stored memory will not be confused.
Theorem 3.1 (First weights constraint of QSHNN). For the differential dynamical system 3.2 the existence and uniqueness of equilibrium are sufficient to be proven by the following inequality on weight parameters. Li is the Lipschitz constant of the activation function on the i-th variable. p and y are constants consistent with former notations.
Lelwiyl < 1
(3.6)
The proof can be found in App. A If we choose the hyperbolic tangent activation function everywhere, which has a Lipschitz constant of I, and set constant μ = 7 1, then the weights constraint in Thm. 3.1 is necessary by doing the normalization on the infinite norm of matrix W by || = E.
Here is a small constant to avoid the equality in the inequality (.7]] For the second significant property, we derive the asymptotic stability based on the Lyapunov theorem 2.1 With the following theorem, our model can be qualified with an asymptotically stable equilibrium.
Theorem 3.2 (Second weights constraint of QSHNN), For the differential dynamical system 3.2 the equilibrium that is asymptotically stable can be established by the following inequality. μs and are constants consistent with former notations.
27 (lwyel + | 1) < 1
(3.7)
The proof can be found in App. B Similarly, we have a natural way to satisfy this inequality, which sufficient to be proved after the process of normalization in training by || = 1 E. Thm. 3.]
and Thm. 3.2 guarantee that the network dynamics evolve to the designated target consistently. The absolute error between the target and the flow also converges with rigorous deduction in App. G
Smoothness of trajectories Let (() denote the state trajectory generated by the proposed QSHNN under the network dynamics Eq. (3.2). Because the activation cp..) is globally Lipschitz continuous with constant Ly = 1. and the weights are normalized by ||
1 - < 1, we obtain the curvature of the trajectories: K. (t) (((())))) < | S 4. App. D giiee the detailed
derivation and curvature evaluation comes from M7. The upper bound x I 4 follows from the choice of constants 7 fa Lup 1. Hence, every component of (t)) possesses uniformly bounded second derivative. In particular, the curvature K, (f) remains finite for all quaternion elements and I, guaranteeing the smoothness of the planned path on each joint motion of robotic manipulation. This level of regularity is sufficient for robotic posture control since actuator commands derived from & are free of discontinuities, and bounded curvature precludes abrupt changes in end-effector acceleration.
Learning rules for QSHNN. The learning of neural networks could follow the Hebbian learning rule (10 like HNN, which is unsupervised, local, and physically inspired, different from constructing and minimizing the loss function, or the Delta learning rule 24 35] which is supervised, global, and optimization-driven By taking the differential of the sensitivity function, we derive the gradient direction in the weights parameter space for the loss function. This is the first stage of our model, for that the trained weights do not preserve quaternion structure, where we demand every 4 x 4 block of the weights matrix is in the quaternion left multiplication submanifold. Hence we name the model supervised learning copfielddstructuure neural network (SHNN). The critical properties such as network stability and training accuracy are guaranteed with rigorous mathematical mechanism through Thm. [.1 and 3.2 For the second stage, we apply technical skill called periodic projection. Every five iterations of strict gradient descent, we do Frobenius orthogonal projection, as shown in Thm. (3.3), which may violate the principle of minimizing but imposes quaternionic structure on weight blocks. Globally, it will generate periodic fluctuating path tending to the minimizer.

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Sensitivity equation and strict gradient descent. Now we start the establishment of learning rules from rigorous mathematics, In steady state we have the sensitivity equation, by setting the system Eq.
332
to zero:
7 = =WWp(g") Jub,
(3.8)
where q. 92. b th weights matrix W E R and the hyperbolic tangent activation function pp(g) [p(1)...... [qqn)]] Differentiating the steady-state equation with respect to the element Wi of the weights matrix W with row entry and column entry (the bias b treated as a constant vector) by the chain rule:
0q" =
Op((")) 0q" + =)p(())
W. -
(3.9)
awiy
ōqe owy
where e; is the i-th standard base of vector space Banxin Yields the sensitivity relation:
0q" Inn 5 " 1 _
" p(9)))..
(3.10)
Owi
7
?
where (q) = diag (g)), (..)...... o (q())) and T is a identity matrix of compatible size. For the quadratic loss function defined as E(q) = bllg qa la- Aiming to derive the gradient direction of the loss function about the weights. Substituting this sensitivity into the chain-rule expression gives the complete gradient formula:
dE
dE
0(q²= - qa)
0q
-qa)" 0q
-
(3.11)
ow:j
0qq -aa)
0q.
dwij
E
Jwij
1 - 7E
paa)(aaa
-aa)" Iln "ww.
W. Jrp
-1
e,
jup(as) ee.
(3.12)
yE
which embodies the error term ô (q" qa) the nonlinear activation contribution p(af)) ) and the inverse of the sensitivity matrix S Lin W J The biases do not appear in the derivative because they are independent of Wij. Denote the training iteration number by superscript (-)(A) and learning rate by 17 E R. From above, we get the weight update scheme of strict gradient descent:
(k+1) Wig
waj 11 Sw:j (k)
OE w (k) if
((p(og)) E
8T STs-te;
(3.13)
The application of GHR calculus. Strict gradient descent is effective for the learning process, but the quaternion structure of the weights matrix W could not be preserved, where we should have every 4 x 4 block to be negative symmetric, except for the elements the leading diagonal, like we have seen in the matrix expression of quaternionic multiplication by Eq. (2. GHR gradient descent over IH can be solved from the following system. The first equation is the derivative of the loss function E(q)=
4ul2 through Def. 2.2. where is the stable state of the evolution equation and I { k).. The second equation is by taking the differential of the sensitivity Eq. (3.8) For
E J = {1, i, j, k), the following differential could be calculated by:
OE
1
OE
OE
(3.14)
0q
4
0q.
dq
aEI
0q 1
0q
OW
We
o qe
(3.15)
8w .j
dw ij
Owly
where subscript (-), is for quaternion components q - q. q + qnJ k,k, and superscript is for quaternion involution introduced in Def. 2.1] This section shows that the quaternion structure- preserving descent exists, providing the foundation of the projection approach. But we would not apply this scheme for the algorithm. Formulas of quaternion calculus applied in this section could be found in (36), especially the generalized chain rule deriving Eq. (3.15). These lead to the steepest descent direction, which is the gradient of the loss function E(q"):
OE
OE
A(q))
Σ
(3.16)
Owl
Owl,
OET
And the update of weights over H. during the learning process under GHR calculus is thereby:
OE
(k+1)
w (k) 1 dwly
w 1j
ij
(3.17)
7

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Periodic Projection Method. As we have found in the above paragraph, direct gradient descent preserving the quaternionic structure by GHR calculus is too complicated and computationally costly, Therefore, we propose strategy for the weights update named periodic projection. The main idea is to periodically do Frobenius orthogonal projection with the following theorem every K = 5 to 10 iterations of weights update in the training procedure. The proof of theorem is given in App- C
Theorem 3.3 (Frobenius orthogonal projection). Let L = span {
L.)), L(2), L(J), [[k) C R4x4
be the quaternion left multiplication submanifold with base Eq. (2.2). For any matrix M E TRxx4 define its orthogonal projection onto & under the Frobenius inner product as:
(M,L(p)) L(n)-
M
Σ Gebl))==
E
(3.18)
pE(L
MELLAJA
Then for any A E L, we have IIM - 2
M which means M is the closest approximation of M in C under the meaning of the Frobenius measure.
Experiments
Algorithm and Configurations. We built an experimental model of QSHNN with four quaternion neurons in Python and executed it on a PC with an Apple Silicon MI chip, 16GB unified memory, aiming verify the theoretical results including asymptotic stability, smoothness of trajectories, and the effectiveness of learning rules. The algorithm below gives the choice of hyper-parameter for the experiments and how the approaches in Sec. 3 are implemented. especially the difference made by 7b
the periodic projection method. Training results and characterization are shown in Fig
Algorithm 1: Projection-based QSHNN learning algorithm
Input: Target d E IR where uniform distribution is obeyed by d; ~ (U(1),))
Output: Trained weight matrix W RANNAN .Loss and accuracy evolution.
Initialization:
Learning rate 77 = 0.001 ~ 0.2 with adaptive adjustment, projection period P 10, maximum epochs T max 30000, error tolerance
10-6 bias b 0.125 0.2, by default, Weights initialization Wij ~ (1, Weights
network parameters normalization by W = W/(E + || where E = 10 12 < 0 an arbitrary small number.
for i to Tmax do
// main training loop 1/ by Bunge-Kutta numerical method // compute error vector
Solve 77 + μ W (q)) Jub
- q² 1,..., do
for
for j = 1, 1,..., N do
S 1ln 7 W [g(
// compute sensitivity matrix
Wij wi 17 upig, STS S-
// strict gradient Eq. (3 13)
if = 0 (modd P) then
// block-wise projection Eq. (3. 18)
L W W cc c;,(i) +c L(kk
Σ
(8()) 2 < r and p| and accuracy=1.0 then
// stop criteria
break;
Remove the projection process, we have the naive supervised learning Hopfield neural network (SHNN). It does not have the activation evolution the quaternion region, whereby the Eq. (I.I).
The training process and outcome for both QSHNN and SHNN are displayed in Fig. 5 ~ Based on the periodic projection algorithm, we construct prototype of robotic applications and demonstrate the effectiveness of the QSHNN in Fig. [ where we also list the benchmark problems and baselines in App. and for the development of complete module in robotic planning and control. They aim to embody the advantages of QSHNN in these important tasks.
Fundamental Training Evaluation. We design benchmark datasets by generating random sets of target quaternion states. Each target contains N quaternion-valle component sampled uniformly from the interval [-1, 1]. For each benchmark, we report three metries: (i). Accuracy: the percentage of target sets for which the equilibrium state successfully converges
desired states qa E H with each
8

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below the error threshold 10 More specifically, the proportion of quaternion components satisfying of d, is defined by accuracy E [o, 1], which is included in the training stop criteria as accuracy = 1.0. (ii). Maximum iterations: the number of iterations allowed to conduct gradient descent and meet the convergence criteria on the randomly generated target set. Taaa= = 30000 for QSHNN, and maa = 10000 for SHNN, which evaluates the effectiveness of the learning scheme. (ii). Equilibrium error: the neural system should be equipped with a unique equilibrium, and all randomly generated initial values can converge with an error less than 72 10-6, which is the indicator of asymptotic stability. Metric of error is mean square error (MSE) or Euclidean distance.
"oe Lereand NOURITY
Penmoten ar
(a) Loss and Accuracy Curves of SHNN
(b) Loss and Accuracy Curves of QSHNNN
Figure 5: (a). Loss and accuracy curves over training iterations of strict gradient descent Eq.
(3113
It implements a sufficient convergence rapidly but will not preserve the quaternionic structure, s shown
in Fig. 6a (b). Loss and accuracy curved over training iterations with periodic projection Eq.
3.18
which perform good error reduction and preserve quaternionic structure simultaneously.
Curves
fluctuate more sharply since the projection disturb continuous gradient descent, shown in Fig.
6b
6d
(a) SHNN Heat Map
(b) QSHNN Heat Map I
(c) QSHNN Heat Map II (d) QSHNN Heat Map III
Figure 6: Heat map of network training outcome. Depth of the color indicates the magnitude of the absolute value of the weights. (a). Weights distribution trained by strict gradient descent Eq. (3.13) There no obvious structure on blocks, thus this scheme is unable to keep quaternion correspondence by Eq. (2.1) on matrix. (b). Weights distribution trained by periodic projections Eq. (3.3) 4x blocks are quaternion symmetric matrix. The average deviation between the start configure and the target configure is a 0.2. (c). Weights distribution trained with periodic projections Eq. 0.2, but with another random generation of initialization and target. (d). Weights distribution trained with periodic projections Eq. (3.3). Components of ith neuron q have the same target, where q' q = 9' T= Trained weights will concentrate on the main diagonal of the matrix. The model adapt its block-wise wise quaternionic structure to target-specific symmetries, a behaviour not apparent in the fully heterogeneous case shown Fig. 6b and 6c
(3.3) We still have average deviation
Discussion
Benchmark problems and Baselines. Based the theoretical foundation of the proposed model, we plan to develop a complete application for robotic manipulator planning in the follow-up pub- lication. Benchmark problems we consider are stated in App. E Through these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and verify the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost. Respectively, the baselines are the performance of current industrial algorithms, which are stated in App. In terms of the weaknesses mentioned of these baseline methods, QSHNN has the potential to outperform traditional strategies. Besides the guarantees on dynamical properties, powerful representation capability of QSHNN allows small-scale neural networks to complete tasks, thereby significantly improving the online planning reaction speed and reducing response time with the adjustment of the target.

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Lumd Comppenee
Syatem
Sevirg Day
(a)
Training curve QSHNN
(b) Trajectory of trained QSHNN dynamics
Figure 7: (a). Training curve of equilibrium for the neural dynamical system governed by Eq. (.2). Each color set represent the components of a quaternion neuron activation and converges to the target. (b). Evolution curve of the neural differential system governed by Eq. 3.2) Embody the asymptotical stability and smoothness trajectories guaranteed by Thm. 3.7 and Thm. 3.6
Figure 8: Robotic simulation with QSHNN the path planning mod- ule. The scenario aligned with Fig. 2 Here we conducted prelimi- nary simulations in standard robotic environments PyBullet (generic physical simulation engine for robotics and control system in Python) confirm that QSHNN drive robotic manipulator with four full- freedom joints from arbitrary initial joint configurations to a specified end-effector posture with the smoothness and convergence properties proven the paper, we plan to develop the mature application in a follow-up publication with a complete evaluation of existing bench- marks App. E and comparison with baselines App.
Comparative Analysis. Regarding how QSHNN becomes an exclusive approach by its features with respect to traditional models, we make the following comparisons. (i): Supervised quaternion networks (QSNN), which only perform static regression from inputs to outputs without continuous- time stability guarantees or the ability to manipulate dynamical patterns. (ii): naïve quaternion- valued Hopfield neural network (QHNN) with supervised readout, where the attractor dynamics fixed after unsupervised weight initialization, so the supervised layer merely maps from pre-existing equilibrium states to outputs, and thus cannot modify the underlying attractor dynamics. (iii): proposed bio-inspired neural network (QSHNN), which generates continuous-time trajectories with provable global convergence and modifiable dynamics with physical information embedded.
Limitations. Throughout this research, we explore the potential to combine the memory- typp Hopfield neural network with the mainstream method based on error propagation. The recent research on neuroscience revealed the surprising fact that the echo-location system of bats consists of only small number of neurons as a core module [ which motivates us exploit the potential of bio-inspired recurrent neural networks. To implement this principle, begin with the modification of continuous HNN and specify a concrete task in robotics by embedding physical information with quaternion. Though the theoretical foundation is established, only the development of the complete application with sufficient tests on the benchmark problems and comparison App. D with baselines App. F can make it more trustworthy, thereby giving us the confidence to further exploit the principle of general learning theory behind it, which is far more than the value of a single model.
Conclusions
We presented quaternion- quaternion-valued Hopfield- strruturrd neural network that integrates structural consistency with smooth and stable learning dynamics. Modern HNN is mature in (23), there are also many hypercomplex-valuee, versions [12] [4] 32, but the plasticity of supervision is usually weak, and the global stability is insufficient, where our model fills the gap in exploration in this aspect. By combining supervised training with periodic projection, the model preserves quaternionic structure while achieving accurate and reliable convergence. The bounded curvature of trajectories ensures smooth evolution, and the network's design enables robust performance across randomly generated targets. Beyond orientation and control tasks, this approach also illustrates how embedding algebraic constraints into neural systems can yield both theoretical guarantees and practical benefits, offering systematic path forward for structured learning in hypercomplex and non-commutative domains.
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Appendices
A
Proof of Thm. 3.1: Existence and uniqueness of equilibrium
Firstly, the activation functions selected are usually bounded. For this reason, we define a constant upper bound for all of the activation functions appearing in the below defined as yp < M. We do not strictly demand that the activation function be uniformly continuous, but satisfying the Lipschitz condition. Out of similar reason, we allocate each activation function a Lipschitz constant with respect to the subscript of notions as: [))((z) (p(()) S Lj(r- v/- yl--
To prove the existence of at least critical point of the dynamic system 3.2 which is to say when we replace the variable of the differential with zero, and by definition, every solution point of the variable functions i counts. Here is to say the below equation system is solvable.
0
1
(1 + b.
(A.1)
7
established for x R. i = 1,2,... ... n. We denote the linear combination of symbols as matrix A 1 w,y Jaxn and bias or external current Ç bb bxx1 And vector multiple value function R, Lipschitz continuous). then yp E F @ RR" is to be: (p)) (p:(B))) nxx- The system governed by Eq. (3.2) can then be rewritten in matrix form:
Pi : R. define == {fl
= Ap(v) + = F())
(A.2)
if we regard the right-hand side mapping F from RD to IR", then the solution be treated as the fixed point of the mapping, allowing the analytic method in the proof.
[(
wiggs -b; b:
br)
(A.3)
112
Σ
Wey .
+ [4:1
722
i=1
p²
From P we form a bounded convex set f = (v: |||||| || S p). whereby Brouwer fixed point theorem applies to the proposition. Since F is continuous mapping, there exists x* Ω satisfying F(v*)= vz, which implies the existence of the critical point in the system governed by Eq. (3.2). The uniqueness of the system can be proved by inequality skills; the demand is converted to constraints of weights. Through reduction to absurdity, suppose there exist two different critical points v", u². Consider L1 norm of
be:
||u, w//1
Wlyl
Li lug 2yy
1=1
1=1
==1
L
[ayl
|Ug
Lelweag
|uy - Uz1 <
11 Uzl
j=1
1=1
(A.4)
This inequality leads a contradiction when the weights are restricted by
Σ
W <
(A.5)
With the above constraints on weights, simply put, the sum of weights cannot be too large; then the network operation has certain single convergent point for bounded input. The essential conclusion provides us with a theoretical base for complex network behaviour design and learning rules, which have not been clearly stated and summarized in the previous articles. The research on Hopfield
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neural network is relatively thorough, regardless of the stability or convergent pattern design. We transfer these basic theories and extend them the construction of quaternion- network, in combination with knowledge of metric space and functional analysis.
For simplification. when we use the inequality A.5 for deeper deduction, we will suppose the coefficients in the front of weights to be unit constant. And for the practical network operation, this condition will be satisfied during the procedure of weights normalization, where we set the norm of the weights matrix to be a constant and the significant information will not be lost.
B Proof of Thm. 3.2: Lyapunov stability criterion
We want to do some research on the stability of the muaternion-valuee modified neural network. In the methodology Sec. 3 we will summarize the process to verify it has a semi-negative derivative function, concluding the asymptotic stability. R2 and Cj are the resistances of the neuron simulated by the electronic circuit. To judge the stability, we will be more specific here since there is a little possibility for system to diverge to infinity, whereas merely being stable will not meet the whole requirement. The value of the output is expected to attach to a single point which, strictly speaking, a critical point to be asymptotically stable, making the undesired situation such as chaos disappear.
n
V(I)=
(B.1)
27
By Brouwer's fixed point theorem, we conclude the existence of a critical point, meaning B.2 established for any j from to By calculating the difference between the of the L2 metric of two critical points, we conclude the uniqueness the critical point, where:
Φ (1, x)) =0
(B.2)
Suppose Ki is the Lipschitz constant of the function (x()), as the model of HNN is an autonomous system, we have xt) (xx) for every j ... n. Notice that:
(t)
dt
J=1
n
n Yag + 1 Σ
Σ Ij
(a)))
j=1
2 +
Σ
|azel Lirioy
(B.3)
-T--
Σ
wuys Lo (x: 2 +,,22)
27
1 +
11
(wwyel + | xz2
27
/=1
By Lyapunov stability theory, the zero solution of the system is asymptotically stable such that the critical point of the original system stable when the constraint of weights
P 27
(ww + (/wig) 1) < 1
(B.4)
applies. Throughout the whole procedure, we notice that the activating function does not need to possess smoothness continuity, but satisfy the Lipschitz condition.
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C Proof of Thm. 3.3 Projection to quaternion left multiplication manifold
Let - = span (ppan([(1),)) L(t), L()), L())) c RA44 For any matrix M e R4x4, denote its orthogonal projection onto & under the Frobenius inner product as
(M, (M,L(M)) L(n). (1(W))
M
Σ (E(LA)))))
(C.1)
Then for any A E C, we have
J All? 2 I IM
(C.2)
I(i), L(R)) CRIx4 We equip B1xd with the Frobenius inner Let L span ( ((0) (0) product
Proof:
(A, B) = tt((A² B); -AI==V(A,A)
(C.3)
direct calculation using the quaternion relations shows that for all ,e, {1,i,,,),
LL)),, = tr(((()) = 1 48pm
(C.4)
so LG), L(K) is an orthogonal basis of L with [[( || = 2.
Now take any M E R2x4, By orthogonal decomposition, there exist unique coefficients eu and residual E E C such that
M Σ I(H) + E.
(C.5)
Taking the inner product of both sides with (Vv) gives
(M, L,)(( Σ cp (I(I), 4c
(C.6)
hence,
O = 1 (M, L())) = 1 tr ((V))
(C.7)
Define:
(M, (M,((H): L(M)
M
(C.8)
to be the projection of M onto L. Then for any A C, since (M- -M) 1 (M - A), the Pythagorean theorem yields
JM -Al M MIP + M AI 2 |M M 12
(C.9)
with equality if and only A = M. This shows M is the unique Frobenius-lcast- projection of M onto L.
D Proof of smoothness: Curvature bound for the smoothness
Start the equation of quaternion neurons,
c - f(z;): Felgi...... Filgi....:9h
(D.I)
where we referred
Σ Wiyo(a,) + bū
Thus the second derivative of quaternion variables is:
0F
0f W Wa)
Σ
Σ
(5sk + 0z
(D.2)
dqk
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From where we could write the Jacobian matrix of the function q about the variable q since there is no explicit appearance of any other variables and their relation linear.
ÿ Jq: == Ja(ā)
(D.3)
where:
oq.
0f Wik
J
-es +
(D.4)
Ozi
Now we will estimate the norm of different parts in D.3
max
qi 1f [zx)]
max |q: max (z:))
(D.5)
S
The last inequality step is because we set the activation function f to be hyperbolic tangent, and for the quaternion variable. Consider the region surrounding the origin where the modulus is less than 1. For the Jacobian, we also have:
0s
| = max
dak +
Wik
0z,
k=1
(D.6)
$ 1 + max
W
2
Combine D.6 and D.5 with D.3
we have rough estimation of the 2nd derivative of (t)::
| oo S ||]||oo || 2.
(D.7)
Therefore, every component variable of (t) has an upper bound of 4 on the second derivative. gives a loose curvature bound on the trajectory in R". In simpler terms, the path that q(t) traces through space is guaranteed to change direction smoothly. There are no sudden jerks, kinks, or high-frequency oscillations. The maximum bending concavity of the trajectory is limited, which is important for smooth motion in control systems. If the actual engineering demands, we could get more strict and smaller curvature defined by 77
(t)
Rx(t)
(1+ (Q.(t))²) 3/2 I 1 x
(D.8)
which is no more than the second derivative of (t) whereby less than the estimated upper bound we deduced in this section, for any component with index in the trajectory of network evolution.
E Benchmark problems
I. Robosuite PandaReach 39 Robosuite's Franka Panda Reach mission only requires end-to- target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom claws, allowing to focus evaluating attitude control rather than grabbing strategics.
FetchReach is a dllerre-off-rreedee Fetch robotic arm target- alignment task. The target position is randomly generated in the environment (cxpandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end effector and the target, and the action directly gives the joint speed. In this environment, evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.
II. OpenAl Gym FetchReach 20
III. Random Workspace Target Alignment 31 Build UR5 or KUKA LBR model directly in
MuloCo, uniformly sample the attitude targets (including position and quaternion direction) their workspace, and test them on large number of random initill-targe pairs.
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F
Baselines
closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it strongly dependent on the initial point and often only achieves a low accuracy of rad.
I. Damped Least-Squares Inverse Kinematics (IK) [6]
Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time- consuming online.
II. Covariant Hamiltonian Optimization for Motion Planning (CHOMP) 40)
An iterative trajec- tory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and
III. Stochastic Trajectory Optimization for Motion Planning (STOMP) 13
costs; while it generates low-curvature paths, depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.
fast random tree (RRT-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains certain degree of smoothing effect. However, the continuity and convergence lack strict guarantees.
IV. RRT-Connect B-Spline Smoothing [2]
First,
G Absolute error to the target
The rapid convergence of exact error between the larget state and the network state is already guaranteed by exponential or asymptotic stability under the framework of Lyapunov theory. Here we supplement a deduction for an explicit expression. Recall the notations for the trajectory and target are (t) E (2O +ool)", qa E IR", and the square error is expressed by:
E(t) 47 (lq(t) 9a12.
(G.1)
Take the time derivative of E(t) by the chain rule:
d'((t)
1 dt γ (a()) qal d
dt (q()) qal
(G.2)
(a(1)) qalt |->I + nW [4()) (a(1) 4al
(G.3)
E(t) + L ([((t) - qa1a w [q(() -qul
(G.4)
7
n
a
Σ
1
1
(Lwyel + |w/)))
(1) - d,}²
(G.5)
27
< <4EE((),
(G.6)
where the first equality simply comes from the evolution equation of QSHNN and the inequality comes from the direct computation in App. Eq. (B.3) The coefficient À is defined by:
B
1 (wyd |uay]] ))
A 7
>
(G.7)
2
The inequality above is guaranteed by Thm. 2 Hence, the error evolution over time satisfies:
llq(t) qell < (1 qalle
(G.8)
Thus, the exact distance between the network state and the target will exponentially descend to infinitesimal (E & 1) and be negligible in practical operation. For the error between the equilibrium of the QSHNN and the target, which is the loss function of the training process, the model is already verified on large enough set of targets, and the training ends consistently with the criterion that the mean square error is less than 10 10-6 which is stated in Para. 4 of Sec.
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Reviews:
[{"review_id": "H5DqXEt5bY", "reviewer": "Reviewer_zuPu", "summary": "This paper presents a Hopfield-structured neural network for learning quaternion-based dynamical systems. The motivation comes from the use of quaternions in representing rotations and postures in various applications like robotic arm control. Inspired by the power of Hopfield neural networks in modeling dynamical structures, the authors propose a way to extend it to the quaternion domain. A generalized $\\mathbb{HR}$ gradient descent with periodic projection is proposed to learn the parameters of the model. Theoretical analysis and empirical results are provided to demonstrate the performance of the proposed method, with a particular focus on its asymptotic stability.", "questions": "What are the advantages of learning in the quaternion domain compared to learning in other domains?", "limitations": "Please see the Weakness and Questions section.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strength**\n\nThe motivation for designing a neural network operating in the quaternion domain makes a lot of sense and could be valuable for real-world applications such as robot manipulation.\n\n**Weakness**\n\n1, The model needs to be re-trained for every new target state, which could be expensive in tasks where the space of possible target states is extremely large or even continuous.\n\n2, Although asymptotic stability analysis is provided for the trained matrix $W$, the discussion of the error between its corresponding equilibrium and the true target state is missing. I believe an error bound for the accuracy is needed to rigorously demonstrate the theoretical performance of the proposed algorithm. \n\n3, The paper lacks a comparison with other methods that could be used for the same task (e.g., other quaternion-valued methods or learning in a different domain other than the quaternion domain).", "was_revised": true, "final_justification": "The rebuttal addressed the questions I had raised; therefore, I increased my score. I like the idea of using Hopfield neural networks to tackle the robotic manipulation problem, and the theoretical analysis supports their claims well. However, I would still suggest comparing the proposed method with other approaches that could be applied to this task, either theoretically or experimentally.", "rebuttal": "Dear reviewer zuPu,\n\nThanks for taking your time reviewing our paper. Below are our responses.\n\n## Weakness 1 ## \nWe hereby make more clarifications to facilitate better understanding of the scenarios of the QSHNN model. We have established the generalization ability on the configure space for the initial states rather than the target states. Concretely, the training process based on a unique target try to solve all the path planning problems from any initial posture of a robot manipulation. The evolution trajectories of QSHNN dynamics provide a complete solution of this planning problem rather than just retrieve single recorded memory. The development of this capability comes from practical concerns in the area of robotics, where the inverse-kinetics solving and planning of multi-joint manipulator is an extremely difficult problem. We would not expect a super powerful approach that can solve the planning task from one infinite set to another infinite set in one period of training. The mature modern Hopfield neural network does have the ability to keep multiple memories (targets), but it is still not possible to overlap the entire state space. \n\n## Weakness 2 ##\nThe rapid convergence of exact error between the target state and the network state is already guaranteed by exponential (asymptotic) stability under the framework of Lyapunov theory. Considering the background of the readers and to enhance rigorousness, here we supplement a deduction and will add it to the appendix. Recall the notations for the trajectory and target are $q(t)\\in\\mathbb{C}^2([0,+\\infty])^{n}, q_d\\in\\mathbb{R}^{n}$, and the square error is expressed by:\n\\begin{equation}\nE(t)=\\frac{1}{2\\gamma}||q(t)-q_d||^2\n\\end{equation}\nTake the time derivative of $E(t)$, by the chain rule:\n\\begin{align}\n\\frac{dE(t)}{dt}&=\\frac{1}{\\gamma}[q(t)-q_d)]\\cdot\\frac{d}{dt}[q(t)-q_d)]\\\\\\\\\n&=\\frac{1}{\\gamma}[q(t)-q_d]^T[-\\gamma \\mathbb{I}+\\mu WJ_{\\phi}(q)][q(t)-q_d]\\\\\\\\\n&\\leq -2\\lambda E(t)\n\\end{align}\nwhere the first equal simply comes from the evolution equation of QSHNN and the inequal comes from the direct computation in Appendix B Equation B.3, the coefficient $\\lambda$ is defined by:\n\\begin{equation}\n\\lambda:=\\gamma-\\frac{\\mu}{2}\\sum_{i=1}^n(|w_{ji}|+|w_{ij}|)>0\n\\end{equation}\nThe inequalility above is guaranteed by Theorem 3.2. Hence, The error evolution over time satisfies:\n\\begin{equation}\n||q(t)-q_d||\\leq ||q(0)-q_d||e^{-\\lambda t}\n\\end{equation}\nThus, the exact distance between the network state and the target will exponentially descent to infinitesimal (E<<1) and be negligible in practical operation.\n\nIf the reviewer means the error between the equilibrium of the QSHNN and the target, which is the loss function of the training process, the model is already verified on a large enough set of targets, and the training ends consistently with the criterion that MSE is less than $10^{-8}$ (see the threshold in Section 4). To mathematically rather than empirically discuss how the loss function of a general ML model will converge is an optimization problem, and it is not related to the stability and beyond the core scope of this research.\n\n## Weakness 3 ##\nFirst, we would emphasise that this research is not about how to make a neural network quaternion-valued, but  exploring a totally new learning paradigm. More specifically, we try to combine the memory characteristics of Hopfield neural network and the mainstream learning approach based on error propagation. There is very little research which can be found but of great significance. It is more aligned with how the biological brain works where it also equipped with the inference type (e.g. MLP) and the memory type (Hopfield). And our research is more about theoretical validation and the implementation of this huge principle rather that to construct a successful application at current stage. To embed quaternion directly to the neural network rather than the others (i.g. complex number, other associative algebras...) is the most effective and natural way for the robotic task we consider. The existing methods for such tasks of robotics are not usually based on neural networks, such as probabilistic roadmap (PRM) and stochastic trajectory optimization (STOMP), QSHNN will definitely outperform them in terms of sensitivity and computational cost. We have done performed simulations with QSHNN for a robot, and we may consider developing a mature application and make a systematic evaluation in the future. It is separated from this paper or it will deviate from the theoretical deduction.  \n\n## Questions ##\nWe consider to utilizing the evolution trajectory of QSHNN to solve the path planning and control problem for multi-joint manipulator in robotics, where a quaternion can directly represent the posture of a full freedom joint, or one has to conduct a series of projection and stretching to the $\\mathbb{R}$ domain, which is unstable and inconsistent. The same reason also applied for other algebras. Quaternion in the proposed model is necessary for the given task, which embeds physical information to the neural network, serving as a foundation for further discussion of rotational invariance and interpretability. To develop the application in other scenarios, we could consider different algebraic structure, and it is noted that the techniques (periodic project, stability analysis) in our research is still valid as a framework."}, {"review_id": "HZV4zla1EH", "reviewer": "Reviewer_7M5t", "summary": "This paper presents for training a quaternion-valued Hopfield-structured neural network using supervised learning rules, in contrast to traditional unsupervised Hebbian learning. The goal is to ensure that the system asymptotically converges to specified target states. The network’s state evolves according to a quaternionic differential equation, and learning is guided so that the system stabilizes at externally defined equilibria. Conceptually, this means that for a given task, the network is trained to evolve and converge toward the corresponding target configuration over time.", "questions": "Major:\n1. How  would you compare this studies with other trajectory optimization method? pros and cons? an alternative? \n2. If direct encoding in early Hopfield-type networks lacks an error-driven optimization mechanism; thus preventing the network from refining its behavior in response to task-specific objectives; how does the supervised learning approach proposed in this work differ? It appears that the network still needs to be retrained whenever the target changes. In that case, how is this method more convenient or flexible than traditional approaches?\n3. Lines 31–33: The sentence \"The constants $\\gamma$ and $\\mu$ correspond to the parameters of electronic components capacitance and resistor for the circuit implementation of classic HNN.\" could be clarified further. As written, it is not clear what exactly you mean by electronic components capacitance and resistor.\n4. The meaning of lines 155–158 is unclear and possibly incorrect. The statement says \"X is noticeable as an anti-symmetric matrix plus a scalar times the identity matrix E, since the anti-symmetric matrix has only imaginary eigenvalues, which makes it easy to know X has two different conjugate complex eigenvalues, and they all have the same algebraic multiplicity, and the geometric multiplicity is 1\". However, this is not generally true. For example, consider the anti-symmetric matrix: [0 -1 0 0; 1 0 0 0; 0 0 0 -1; 0 0 1 0]. This matrix has 2 complex eigenvalues, and each eigenvalue has a geometric multiplicity of 2.  Therefore, the claim in the original sentence does not hold in general and should be revised or further justified.\n5. Please provide a more detailed explanation for Figure 9. It is not immediately clear what new information or insight Figure 9 is intended to convey compared to Figure 8.\n\n\nMinor:\n1. notation consistence. e.g. epsilon notation is not consistent. \n2. Notation used without fair justification in section 3.", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strength\n**Quality**: This paper provides a thorough theoretical analysis of a quaternion-valued Hopfield-structured neural network and its supervised learning rule. The experimental results further demonstrate fast convergence and stability of the proposed learning scheme based on periodic projection.\n\n**Clarity**: This paper provides a clear explanation on quaternion.\n\n**Significance**:  The supervised learning rule introduced in this paper overcomes the lack of error-driven optimization in previous models, enabling the network to be trained to match externally defined targets.  The proposed model exhibits smooth dynamical behavior that is well-suited for applications such as robotics and trajectory generation.  By operating under a supervised learning paradigm with quaternion-valued trajectories, the network ensures that generated paths have bounded curvature, promoting trajectory smoothness.\n\n**Originality**: This work is original in addressing the gap between traditional Hopfield-type networks and task-specific objectives. Moreover, by periodically projection onto the quaternionic structure, the training process remains more stable.\n\n# Weaknesses\n**Quality**: Although the periodic trajectory converges to the minimizer, the Frobenius orthogonal projection interrupts strict loss minimization, as acknowledged in the paper. Additional experiments would strengthen the validation of this projection strategy.\n\n**Clarity**: some sentences are difficult to understand or ambiguous. (See the “Questions” section below for specific examples.)  Additionally, the paper lacks a clear explanation of the Hopfield Network framework, which is important for readers unfamiliar with the concept. I recommend adding a brief background section to improve accessibility.\n\n**Significance**: The effectiveness of the method is not compared against other supervised trajectory generation/optimization approaches.", "was_revised": true, "final_justification": "I agree with some of the other reviewers that it would be beneficial to include a discussion of the paper's limitations, either in the main text or in the appendix. Additionally, more detailed ablation studies and comparative experiments would help strengthen the paper. Nonetheless, I remain positive about the novelty of the work and believe it merits favorable consideration—especially with the additional experiments the authors proposed in the rebuttal.", "rebuttal": "Dear reviewer 7M5t:\n\nThanks for taking your time reviewing our paper. Below are our responses:\n\n## Weaknesses ## \n\n### Quality: ### \nRegarding the application of periodic projection methods in neural network training over special algebraic domains. First of all, the concern raised by the reviewer is sensible. Weight mapping indeed disrupts strict gradient descent, but it does not disrupt the optimization process of the loss function and is currently the most suitable strategy. The reasons are as follows:\n\n- To integrate physical information into neural networks, the preservation and training of structures on special algebraic fields are crucial. However, as demonstrated in our paper, the calculus of quaternions, G$\\mathbb{HR}$ calculus, is extremely complex and cumbersome. For other associative algebras that can be represented by matrices, there are not even rigorous calculus tools available at present. We do not wish to spend a considerable amount of time developing mathematical tools when expanding the model of this article to other types of problems. Moreover, absolutely strict formulas would also impose a heavy burden on calculations, significantly slowing down the response speed of the planning module when the target is updated.\n\n- When we integrate multiple neurons into a quaternion neuron, the weight parameters are redundant in terms of degrees of freedom, which will allow the optimized path to be explored in a high-dimensional space and can compensate for the interference of the periodic mapping maintaining the algebraic structure on the strict gradient. In the experiment, we set the training cut-off condition as $MSE<10^{-8}$ and have stably completed the training on a large number of randomly generated targets consistently. This verifies the capability of the periodic projection method in maintaining algebraic structure and optimizing effect. It can also be seen from the training trajectories of the illustration in the Figure (10) that this kind of interference does not seriously disrupt the path of strict gradient descent.\n\nTaken together, periodic Frobenius projection strikes the right balance between enforcing strict quaternion‐structure preservation and achieving efficient, error‐driven optimization in hypercomplex networks.\n\n### Clarity: ###\nThroughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neurosicence release the surprising fact that the echo-location system of bats only consist of sixteen neurons as core module, which motivate us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete tasks in robotics by embedding physical information with quaternion. Thus our paper could not be treated in the classic Hopfield framework. For a quick understanding of the basic ideas of HNN, you could check reference [9]. We would consider adding more contents within the background section making the paper easier to follow.\n\n### Significance: ###\nIt is responded in Question 1.\n\n## Question 1: ##\nBased on the theoretical foundation of the proposed model, we plan to develop a complete application for robotic manipulator planning in the follow-up publication. The following are the benchmark problems we consider, and baselines we choose to make comparisons.\n\n- **Robosuite PandaReach** (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.\n\n- **OpenAI Gym FetchReach** (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.\n\n- **Random Workspace Target Alignment**, An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)) Build a UR5 or KUKA LBR model directly in MuJoCo, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs.\n\nThrough these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and verify the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.\n\nRespectively, the baselines are the performance of current industrial algorithms:\n\n- Damped Least-Squares Inverse Kinematics (**IK**): A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of rad.\n\n- Covariant Hamiltonian Optimization for Motion Planning (**CHOMP**): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online\n\n- RRT-Connect + B-Spline Smoothing: First, use a fast random tree (**RRT**-Connect) to quickly generate a feasible path, and then use **B-Spline** or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.\n\n- Stochastic Trajectory Optimization for Motion Planning (**STOMP**): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.\n\nQSHNN has the potential to outperform these traditional strategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.\n\n## Question 2: ##\nThe traditional Hopfield neural network does not possess the planning capability mentioned in this paper. QSHNN is fundamentally different in the following aspects: HNN generates relatively vague multi-associative memory and lacks accurate and high-precision dynamic characteristics; QSHNN integrates the representation of quaternions on the rotation group SO(3) of rigid bodies, which can perfectly match the motion path planning task of robotic arms, a capability that HNN does not possess. In fact, in the tasks mentioned, past research does not developed a mature method mainly based on neural network. As for the issue of target updates, the small size and high representational ability of QSHNN can fully support real-time re-planning. Simultaneous multi-target planning itself is not the focus of such tasks. The benchmark problems and baselines we select in the subsequent research also meet these characteristics.\n\n## Question 3: ##\nHopfield-structured neural networks have a physical implementation by electronic circuits. Capacitance and resistor are common electronic components. There is a type of hardware module that can integrate this kind of neural networks to achieve a faster response speed than computer computation.\n\n## Question 4: ##\nWe appreciate the reminder of the reviewer. When anti-symmetric matrix is mentioned, we just talk about the specific matrix of the proposed formula rather than the general properties of linear algebra. The eigenvectors and eigenvalues of the proposed matrix is manually calculated and checked, thus would not influence the major deduction. We will correct this typo.\n\n## Question 5 ##\nWhen we set the target quaternion in the form of $q_d=s_d+\\boldsymbol{i}x_d+\\boldsymbol{j}y_d+\\boldsymbol{k}z_d$ with the extra condition $s_d=x_d=y_d=z_d$, the trained weights will concentrate on the main diagonal of the matrix. This concentration on the main diagonal reflects that, for an “equal‐component” target quaternion, the network need only scale the identity action in each block to achieve convergence. Figure 9 therefore highlights QSHNN’s ability to adapt its block‐wise quaternionic structure to target‐specific symmetries, a behaviour not apparent in the fully heterogeneous case of Figure 8."}, {"review_id": "GXEuPnxRmc", "reviewer": "Reviewer_wmYX", "summary": "This paper proposes a novel Quaternion-valued Supervised Learning Hopfield-structured Neural Network (QSHNN) that extends the classic continuous-time Hopfield Neural Network (HNN) into the quaternionic domain. The key innovation lies in formulating a supervised learning paradigm for HNNs, where the network is trained to asymptotically converge to externally specified targets. The model establishes the existence and uniqueness of fixed points with asymptotic stability. For learning, it introduces a periodic projection strategy that modifies gradient descent by projecting weight matrices onto the closest quaternionic structure. This approach ensures both convergence and quaternionic consistency during training. The authors highlight the model's rigorous mathematical foundation, high accuracy, fast convergence, strong reliability, and smooth trajectory evolution, making it suitable for applications like robotic control and path planning.", "questions": "n/a", "limitations": "n/a", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n- The paper provides a strong theoretical basis, establishing the existence, uniqueness, and asymptotic stability of the QSHNN's fixed points using Lyapunov theory. This level of mathematical rigor is a significant strength, providing confidence in the model's behavior.\n- Traditionally, HNNs are unsupervised associative memory models. This work successfully extends HNNs to a supervised learning setting with explicit target tracking, addressing a major limitation of classical HNNs and opening new application avenues.\n- Beyond theory, the paper offers a practical algorithm for implementing the QSHNN, demonstrating its feasibility and effectiveness on randomly generated target sets.\n\nCons:\n- The experiments are conducted on a relatively small-scale model (four quaternion neurons) and on randomly generated target sets. While sufficient for verifying theoretical claims, it would be beneficial to see the model's performance on larger, more complex, and real-world datasets relevant to its proposed applications.\n- The deep dive into quaternion algebra, GHR calculus, and specific matrix representations, while necessary for theoretical rigor, might pose a steep learning curve for machine learning practitioners less familiar with hypercomplex numbers and advanced differential geometry.\n- While the paper successfully addresses limitations of classical HNNs, a more explicit discussion or comparison with modern neural network architectures (e.g., LSTMs, Transformers) on tasks where QSHNN might be applied (e.g., sequence modeling for control) would strengthen its position and highlight unique advantages beyond HNNs.", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer wmYX:\n\nThanks for taking your time reviewing our paper. Below are our responses:\n\n## Cons 1: ##\nFor the robotic scenario we consider currently, it is not necessary to add hundreds, thousands of neurons to the network due to the following reasons: \n- Recall Figure 2 in the manuscript, a full-freedom joint of a robot manipulator can be completely represented by a quaternion. The most widely applied standard robot Universal Robots UR5e, LBR iiwa14, etc, contains 6 joints and thus can be corresponded by 6 neurons QSHNN. \n- The computational cost of QSHNN on the benchmark problems is much less than the existing approaches (STOMP, CHOMP, etc), since the numerical step of solving operation equation of QSHNN is quite a small task.\n\nWhile we have conducted preliminary simulations in standard robotic environments (PyBullet) to confirm that QSHNN can drive a robotic manipulator with four full-freedom joints from arbitrary initial joint configurations to a specified end‑effector posture with the smoothness and convergence properties proven in the paper, we plan to develop the mature application in a follow-up publication with a complete evaluation of existing benchmarks and comparison with baseline as followed:\n\nbenchmark problems are: \n\n- **Robosuite PandaReach** (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.\n\n- **OpenAI Gym FetchReach** (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.\n\n- **Random Workspace Target Alignment**, An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)) Build a UR5 or KUKA LBR model directly in MuJoCo, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs. \n\nThrough these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and verify the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.\n\nRespectively, the baselines are the performance of current industrial algorithms:\n\n- Damped Least-Squares Inverse Kinematics (**IK**): A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of \n rad.\n\n- Covariant Hamiltonian Optimization for Motion Planning (**CHOMP**): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online\n\n- **RRT**-Connect + **B-Spline** Smoothing: First, use a fast random tree (**RRT**-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.\n\n- Stochastic Trajectory Optimization for Motion Planning (**STOMP**): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.\n\nQSHNN has the potential to outperform these traditional strategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.\n\n## Cons 2: ##\nWe believe that the establishment of new learning paradigms and models does indeed involve complex mathematical content, but perhaps they are necessary. We all expect a stable and reliable model, and only the rigor of mathematics can guarantee this in the behind. Furthermore, the development and expansion of the model application in the next stage also need to start from the theoretical ground. The main mathematical fields covered in this paper include dynamical systems, non-commutative algebra, linear analysis and general calculus. They are all familiar tools in the theoretical research of machine learning. The reviewer's concerns are reasonable. In the subsequent development of complete application module, we will consider how to enable engineers to utilize the proposed model more directly without involving an understanding of advanced mathematical content.\n\n## Cons 3: ##\n\nThroughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neuroscience reveals the surprising fact that the echo-location system of bats only consist of sixteen neurons as core module, which motivates us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete task in robotics by embedding physical information with quaternion.\n\nThe ideas and methods proposed in this paper do indeed originate from some of the content of HNN, but they are essentially  different from traditional HNN and are not included in any existing models. As mentioned earlier, we are not simply modifying HNN. Regarding the neural network model mentioned by the reviewer, our explanation is as follows: \n\n- Recurrent neural networks such as LSTM are actually completely different from Hopfield type associative memory. The former is merely capturing remote associations of sequence signals and has no direct connection with the operating principles revealed by neuroscience from hippocampus and amygdala in the human brain. More importantly, it does not equip with the capability to solve the benchmark problems mentioned in Cons 1.\n\n- Transformer and its derived control models (basically using the Transformer to predict the next action) do not provide any Lyapunov or exponential convergence guarantees. They rely on training data and optimization methods to achieve empirical robustness. QSHNN theoretically proves the global exponential stability under the constraint of set weights, continuously approximating the target from any initial conditions. This property is more suitable for the mentioned tasks. What's more, the consistency of geometric structure is still a significant problem for transformer models.\n\nRecently published research shows that the attention mechanism of Transformer is equivalent to a discrete modern Hopfield neural network. It achieves the same fixed-point iterative mechanism as the Hopfield energy function through weighted search of key-value pairs. From this perspective, each attention calculation in the Transformer is equivalent to performing a discrete attractor update on the memory state, and the update process relies on the real-valued vector space and the Euclidean dot product energy. In contrast, QSHNN establishes a continuous-time Hopfield dynamical system in the quaternion domain. It not only natively integrates the three-dimensional rotational geometric structure into the neural network but also strictly proves that the system has global exponential stability for a single target through Lyapunov theory. It ensures that any initial state can continuously and smoothly converge to the target quaternion over time. In addition, the training process of QSHNN adopts periodic projection, directly forcing the maintenance of the quaternion left-multiplication manifold structure each time the weights are updated, thereby obtaining an interpretable closed-loop control law. However, the attention weights of the Transformer are only indirectly trained through task-level losses, lacking explicit maintenance of dynamic stability and geometric structure. If a comparison is to be made in the same 6-degree-of-freedom alignment task in robotics, the Transformer can be regarded as a predictor of the joint velocity at the next moment, approaching the target through a certain number of attention layers or iterative updates, and then its convergence speed and computational steps can be compared with QSHNN. It is expected that although attention inherently possesses the characteristics of Hopfield networks, QSHNN, with its continuous dynamical structure for the target and periodic projection training, can definitely achieve higher accuracy, along with strict global stability guarantees."}, {"review_id": "p1cKN3bfdL", "reviewer": "Reviewer_suhQ", "summary": "This paper proposes an asymptotically stable quaternion-valued Hopfield-structured neural network (QSHNN) with supervised learning rules based on periodic projection. The model extends the continuous-time dynamical model of classical Hopfield neural networks (HNNs) to the quaternionic domain. The QSHNN is trained to converge asymptotically to externally specified targets using a supervised learning paradigm. The learning rules incorporate a periodic projection strategy to preserve the quaternionic structure of the weight matrix, ensuring both convergence and quaternionic consistency.", "questions": "1. What are the specific applications of QSHNN?\n2. How QSHNN implements task-driven behavior?\n3. Are there any limitations in scaling QSHNN to larger networks?", "limitations": "As I mentioned before, the biggest limitation of this paper is the problem of applying QSHNN in real scenarios. For example, when applied to a robotic arm, how many neurons are needed, what is the time complexity, and what is the performance improvement compared to existing methods, which are not known in the current experimental chapter. Therefore, the authors should enrich their experiments so that only sufficient experimental results can support a complete and convincing theoretical proof.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\nS1. The idea of using four small neurons to form a single unit neuron is interesting, similar to the potassium and sodium ion channels contained in real human brain neurons.\nS2. This paper establishes the existence and uniqueness of fixed points with asymptotic stability of QSHNN  through rigorous mathematical proof.\n\nWeaknesses\nW1. The biggest weakness of this paper is on the application of QSHNN. There are some mentions about robotic arms or control systems, yet there is neither a vivid description of the relevant datasets nor experimental tests related to them in the experimental section. Instead, the experiments are based on self-designed quaternion state, which is not convincing that QSHNN is valuable for some application scenarios. By the way, the experiments in this paper lack comparisons of related baselines.\nW2. In the last of introduction, the authors mention that QSHNN is with task-driven behavior. However, after reading the method part, I don’t get how QSHNN achieve task-driven behavior. It seems that QSHNN can be applied to any control system related task, but there is no mention of how QSHNN can capture information unique to a specific task.\nW3. I note that the authors performed their experiments on an M1 chip, but the paper does not mention how the performance of the QSHNN changes as the number of neurons increases. Are there any limitations in scaling the model to larger networks?\nW4. This paper lacks the limitation part.", "was_revised": true, "final_justification": "Thanks for the response. The author has addressed some of my concerns. However, since the author only states that they will provide additional details in a future revision, I will maintain my current score.", "rebuttal": "Dear reviewer suhQ,\n\nThanks for taking your time reviewing our paper. Below are our responses.\n\n## Weakness 1 ##\nWe appreciate the concern regarding the lack of a dedicated real‐robot dataset or robotics benchmark in the current manuscript. Our experiments with self‑designed quaternion target sets were chosen deliberately to isolate and validate the core properties of QSHNN—namely its global exponential convergence, quaternion‑structure preservation, and bounded‐curvature trajectories—without confounding factors such as robot dynamics, sensing noise, or collision avoidance. In parallel to this theoretical study, we have performed preliminary simulations in PyBullet and MuJoCo on a 6‑DOF manipulator, sampling random initial joint configurations and driving the end‑effector toward fixed quaternion goals. These tests confirmed that QSHNN attains sub‑millisecond integration steps, global convergence from arbitrary starts, and end‑effector orientation errors below 10⁻⁴ rad, in line with our theoretical bounds. We omitted these implementation‑specific parts from the main text to maintain focus on the mathematical framework. In the follow-up publication, we will develop the complete application in the robotic scenario, the considered benchmark problems are: \n\n- **Robosuite PandaReach** (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.\n\n- **OpenAI Gym FetchReach** (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.\n\n- **Random Workspace Target Alignment,** An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)) Build a UR5 or KUKA LBR model directly in **MuJoCo**, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs. \n\nThrough these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and assess the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.\n\nRespectively, the baselines are the performance of current industrial algorithms:\n\n- **Damped Least-Squares Inverse Kinematics** (IK): A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of \n rad.\n\n- **Covariant Hamiltonian Optimization for Motion Planning** (CHOMP): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online\n\n- **RRT-Connect + B-Spline Smoothing**: First, use a fast random tree (RRT-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.\n\n- **Stochastic Trajectory Optimization for Motion Planning** (STOMP): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.\n\nQSHNN has the potential to outperform these traditional strategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.\n\n## Weakness 2 ##\nThe task driven is not narrowly applied to QSHNN model. We want to express that the principle we proposed for combining the bio-inspired neural network (HNN) with the supervised learning rules based on error propagation, is an idea with much more broader potential. For example, the network topology is adjustable, the embedded physical information is not limited to rotation and posture represented by quaternion, and it could be other associative algebra embedded by matrix representation. But the basic framework for building this kind of recurrent networks can be found through the techniques in the proposed deductions (algebra embedding, stability analysis and learning rules). The specific task will guide the design of the specific algebra for the physical information or characteristics, but our fundamental methodology is not constrained to a single circumstance.\n\n## Weakness 3 ##\nFor the robotic scenario we consider currently, it is not necessary to add hundreds, thousands of neurons to the network due to the reasons: \n- Please refer to Figure 2 in the paper, a full-freedom joint of a robot manipulator is completely represented by a quaternion. The most widely applied standard robot Universal Robots UR5e, LBR iiwa14, etc., contains 6 joints thus can be corresponded by 6 neurons QSHNN.\n- The computational cost of QSHNN on the benchmark problems is much less than the existing approaches (STOMP, CHOMP, etc), since the numerical step of solving operation equation of QSHNN is quite a small task.\n\n## Weakness 4 ##\nWe would supplement the section of limitations as a separate paragraph with the following contents: Throughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neuroscience reveals the surprising fact that the echo-location system of bats only has sixteen neurons as the core component, which motivates us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete task in robotics by embedding physical information with quaternion. Though the theoretical foundation is established, only the development of the complete application with sufficient tests on the benchmark problems and comparison with baseline can make it more convincing whereby giving us the confidence to further exploit the principle of general learning theory behind it, which is far more than the value of a single model.\n\n## Question 1 ##\nThis is explained in the response of weakness 1.\n\n## Question 2 ##\nThis is explained in the response of weakness 2.\n\n## Question 3 ##\nThis is responsed in weakness 3.\n\n## Limitations ##\nThis is responded in Weaknesses. Hopfield neural network is a completely different model from MLP, CNN, GNN, etc, where we add memory-like dynamical evolution inspired by the biological functionality of hippocampus and amygdalain in the brain, replacing the direct forward calculation or reasoning. This research tries to extend HNN and disrupts the conventional machine learning paradigm, which is why we work on the theoretical rigorousness for the major body and reduce the elaboration of complete application modules. For one of the reviewer's main concern on how the QSHNN could work on large network topology, we also have carefully thought about it. Firstly, the effectiveness and representative capability do not directly rely on the scale of neurons in certain tasks. Secondly, to actually extent the scale of this type of neural networks, the fully-connected structure does cost too much, and therefore, we shall consider the structural plasticity in the neuron science (growing neural networks) to build and adjust the topology adaptively."}, {"review_id": "vf5oonsc5j", "reviewer": "Reviewer_pJB8", "summary": "This paper proposes the extension of quaternion-valued Hopfield network to supervised learning. The  Quaternion-valued Supervised Hopfield Neural Network (QSHNN) is continuous-time neural architecture that uses quaternion algebra to model and learn smooth, stable dynamical systems, particularly for robotics applications. The model exploits Generalized HR calculus which supports the theoretical framework. In their extensions of HNN, the authors formulate a theoretically grounded model with provable asymptotic stability and uniqueness of fixed points. Evaluations on learning dynamics seem to support the claim that the model performs well on the chosen problem. The advantages relate to the use of quaternion algebra which is suitable for 3D rotations and posture representations. Given the continuous time dynamics, the model guarantees a smooth progression towards guaranteed stability points.", "questions": "1. Given time and resources availability, which robotic benchmarks would you consider to further demonstrate the validity of the method? Could also speculate, or possibly show, which real-world benchmarks would demonstrate the most significant advantages of the proposed model?  \n2. If you could extend the study to include baseline comparisons, which models could be most suitable to demonstrates the advantages of the proposed model with respect to existing methods?", "limitations": "The checklist states that limitations are discussed in section 5 (conclusion) but that does not seem to be the case. Overall, I believe that the  limitations of the study are not sufficiently highlighted.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n1. The paper provides a mathematical foundation of the approach\n2. The model exploits Generalized HR calculus\n3. The model uses HNN but introduced supervised learning, which is appealing in many ML research areas. \n4. The evaluations presented in the paper seems to support the case for this approach\n\nWeaknesses\n1. There is a lack of comparisons and ablation studies:\n1.a. How does the model compare with Supervised Quaternion networks? \n1.b. How does it compare with a naïve QHNN, e.g. trained with a simple supervised readout? \n2. Evaluation on robotics or control applications. The paper claims that the model could perform well on robotic tasks, but no experimental evidence is provided in that respect.", "was_revised": true, "final_justification": "The authors have made significant effort to address my comments and I have consequently increased my assessment.", "rebuttal": "Dear reviewer pJB8,\n\nThanks for taking your time reviewing our paper. Below are our responses.\n\n## Weakness 1 ##\nRegarding the lack of comparison with similar methods, we believe that traditional supervised neural networks are simply incapable of the tasks handling by QSHNN. In the area of robotics, the relatively mature approaches (e.g. PRM, STOMP) are based on stochastic algorithm and complicated mechanic solving rather than making neural network as a core module. The key differences include the following:\n\n- The essence of the tasks is different. Common quaternion-valued supervised networks (e.g. MLP, CNN), or the supervised readout networks can only learn static mapping from a fixed input to an output. They lack the ability to generate continuous-time trajectories with smoothness and physical consistence. Our proposed QSHNN, on the other hand, is embedded in an actual dynamical system. By solving the initial value problem, the network can be used to plan the path for the quaternionic variables evolving from any available states to the target state. This transparent process fundamentally differs from the traditional \"black box\" one shot mapping in terms of the model paradigm and application scenario.\n\n- The level of stability assurance is different. Quaternion supervised networks only guarantee fitting accuracy on training data, without any theoretical means to prove global convergence or dynamical continuity. However, QSHNN, based on Lyapunov functions, has rigorously proven global exponential stability across the entire configuration space, ensuring that all trajectories smoothly converge to the target. This verifiable dynamical property is unattainable by any purely supervised readout or regression network.\n\n- Geometric Structure Preservation. During training, QSHNN consistently maps each 4×4 weight block back to the quaternion left-multiplied manifold through periodic projection, ensuring that the network's algebraic structure is tightly coupled to rotational geometry. The trained network can perfectly work over quaternion domain, while other quaternion networks, usually apply divided projection to the neuron activation, struggle to balance training efficiency with strict structural consistency.\n\n## Weakness 2 ##  \nWe appreciate the concern regarding the lack of a fully developed robotic application in the current manuscript. The primary goal of this paper is to introduce and validate the novel, bio‑inspired paradigm that unites Hopfield‑type attractor dynamics with error‑driven supervised learning in the quaternion domain. Demonstrating global, exponential convergence of a continuous‑time quaternion Hopfield network under Lyapunov theory represents a significant and original contribution in its own right. While we have conducted preliminary simulations in standard robotic environments (PyBullet) to confirm that QSHNN can drive a robotic manipulator with four full-freedom joints from arbitrary initial joint configurations to a specified end‑effector posture with the smoothness and convergence properties proven in the paper, we plan to develop the mature application in a follow-up publication with a complete evaluation of existing benchmarks and comparison with baseline. \n\n## Question 1 ##\nHere are the benchmarks we consider to apply QSHNN in concrete scenario: \n\n- **Robosuite PandaReach** (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.\n\n- **OpenAI Gym FetchReach** (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.\n\n- **Random Workspace Target Alignment**, An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)): Build a UR5 or KUKA LBR model directly in MuJoCo, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs. \n\nThrough these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and verify the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.\n\n## Question 2 ##\nFor the benchmark problems above, we consider the following traditional strategies that are suitable for the proposed task scenario as baseline comparisons with QSHNN:\n\n- Damped Least-Squares Inverse Kinematics (**IK**):  A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of $e^{-2}\\sim e^{-3}$ rad.\n\n- Covariant Hamiltonian Optimization for Motion Planning (**CHOMP**): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online\n\n- **RRT**-Connect + **B-Spline** Smoothing\nFirst, use a fast random tree (RRT-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.\n\n- Stochastic Trajectory Optimization for Motion Planning (**STOMP**): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.\n\nQSHNN has the potential to outperform these traditional stategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.\n\n## Limitations ##\nWe would supplement the section of limitations as a separate paragraph with the following contents: Throughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neuroscience revealed the surprising fact that the echo-location system of bats only consists of sixteen neurons as core module, which motivate us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete task in robotics by embedding physical information with quaternion. Though the theoretical foundation is established, only the development of the complete application with sufficient tests on the benchmark problems and comparison with baseline can make it more trustworthy whereby giving us the confidence to further exploit the principle of general learning theory behind it, which is far more than the value of a single model."}]

Author rebuttal:

Meta-review: This work presents a Hopfield-structured neural network that can be used to learn quaternion-based dynamical systems. This is motivated from the use of quaternions in the representation of postures in applications like robotic arm control. Inspired by how Hopfield networks have been successfully used in modeling dynamical structures, the authors propose a way to extend this to quaternion domains. The work also proposes gradient based methods for learning the parameters of the model. The paper is complemented with various theoretical analyses and empirical results. The reviewers agreed this work meets the bar for presentation at Neurips.
Decision: Accept (poster)