Continuous Orthogonal Mode Decomposition: Haptic Signal Prediction in Tactile Internet Mohammad Ali Vahedifar, Mojtaba Nazari, Qi Zhang
arXiv:2604.09446v1 [eess.SP] 10 Apr 2026
DIGIT and Department of Electrical and Computer Engineering, Aarhus University, Denmark
Abstract—The Tactile Internet demands sub-millisecond latency and ultra-high reliability, as high latency or packet loss could lead to haptic control instability. To address this, we propose the Mode-Domain Architecture (MDA), a bilateral predictive neural network architecture designed to restore missing signals on both the human and robot sides. Unlike conventional models that extract features implicitly from raw data, MDA utilizes a novel Continuous-Orthogonal Mode Decomposition framework. By integrating an orthogonality constraint, we overcome the pervasive issue of “mode overlapping" found in state-of-theart decomposition methods. Experimental results demonstrate that this structured feature extraction achieves high prediction accuracies of 98.6% (human) and 97.3% (robot). Furthermore, the model achieves ultra-low inference latency of 0.065 ms, significantly outperforming existing benchmarks and meeting the stringent real-time requirements of haptic teleoperation. Index Terms—Mode Decomposition, Haptic Signal Prediction, Tactile Internet, Ultra-low latency and High Reliability.
I. I NTRODUCTION The Tactile Internet (TI) is envisioned to enable a human operator to interact with a remote robot by receiving realtime force feedback, allowing intuitive execution of complex tasks [1]. However, TI faces two critical challenges. First, communication delay between the human operator’s command and the robot’s (teleoperator’s) response results in a perceived loss of transparency and system instability. Second, packet loss or missing signals can cause the signals from both sides to become uncorrelated, thereby failing to accurately preserve the subtle statistical properties of the motion on the robot side. The TI research community has shown growing interest in integrating general Machine Learning (ML) methods for signal prediction; such models offer promising avenues toward more stable and practical remote control [2]–[5]. The main limitation of these approaches is that they operate directly on raw time-series data. While such models can minimize statistical prediction error, they often fail to distinguish the underlying physical mechanisms driving the signal. From the perspective of semantic communication, a haptic signal is not a monolithic stream; but it is a superposition of distinct physical events, such as the operator’s low-frequency This research was supported by the TOAST project, funded by the European Union’s Horizon Europe research and innovation program under the Marie Skłodowska-Curie Actions Doctoral Network (Grant Agreement No. 101073465), the Danish Council for Independent Research project eTouch (Grant No. 1127- 00339B), and NordForsk Nordic University Cooperation on Edge Intelligence (Grant No. 168043). Authors’ e-mails: {av, qz}@ece.au.dk.
intent, high-frequency environmental textures, and sudden contact transients. Predicting force (F ), velocity (V ), or position (P ) on raw data forces the model to learn these conflicting dynamics simultaneously, which can lead to poor generalization on complex tasks. This motivates us to primarily focus on prediction in the “mode" domain. This approach posits that haptic signals should be treated not as single, patched objects, but as composites of distinguishable components. This enables the predictive model to learn the specific behavior of each component infrastructure independently. The main contributions of this paper are as follows: i. Continuous-Orthogonal Mode Decomposition (C-OMD): We enforce orthogonality in the continuous mode space. This formulation enables zero mode interference. ii. Mode-Domain Architecture (MDA): We introduce a tailored architecture that operates in the decomposed mode domain for bilateral signal prediction in TI. MDA achieves significantly higher accuracy and 8-30× inference speedup. II. R ELATED W ORKS Signal decomposition techniques are adaptive tools for breaking down complex waveforms into their constituent components. Among these approaches, Variational Mode Decomposition (VMD) [6] defines modes as amplitude-modulated frequency-modulated (AM–FM). The VMD requires the number of modes to be specified a priori, which is often a challenge in practice. Building upon VMD, extensions such as Variational Mode Extraction (VME) [7] and Successive Variational Mode Decomposition (SVMD) [8] aim to alleviate parameter sensitivity through iterative mode extraction and residual analysis. However, VME only extracts a desired mode rather than performing a full decomposition, and SVMD replaces the parallel structure of VMD with successive computations that depend on previously extracted modes. This sequential dependency reduces compatibility with GPU-based implementations, where parallelism plays a critical role in overcoming computational bottlenecks. These approaches rely on solving approximated augmented Lagrangian formulations. Due to these approximations, complete mode separation is not theoretically guaranteed, and mode interference may still occur. Moreover, existing methods either require the number of modes to be specified in advance or sacrifice parallelizability. These limitations motivate the development of a new framework that preserves the parallel
structure of VMD. We recast the problem of selecting the number of modes as an empirical grid search and provide theoretical guarantees for complete mode separation. In this work, we propose C-OMD. The key idea is to enforce orthogonality directly in the mode space, enabling principled and theoretically grounded separation. III. P RELIMINARIES Definition 1 (Intrinsic Mode). An intrinsic mode is modeled as an amplitude-modulated and frequency-modulated (AM-FM) signal: mk (t) = Ak (t) cos(ϕk (t)), (1) where the instantaneous amplitude Ak (t) and the instantaneous frequency ωk (t) = ϕ′k (t) vary much more slowly than the phase ϕk (t) itself. While (1) provides the physical representation of a mode, this highly non-linear formulation is not directly amenable to variational optimization or the enforcement of global orthogonality constraints. However, as noted in VMD [6], a smoothly varying AM-FM signal naturally exhibits a compact frequency spectrum centered around a mean frequency ωk . Therefore, to formulate our C-OMD, we rely on the following bandwidthbased characterization to mathematically bound the modes. Definition 2 (Mode). A function mk ∈ H 2 (R) is a mode with center frequency ωk ∈ R+ if its spectral energy is concentrated near ωk , i.e., the normalized spectral variance satisfies: Z ∞ 2 (ω − ωk )2 |m̂k,+ (ω)| dω ≤ ϵk (2) Bk2 := 0 Z ∞ 2 |m̂k,+ (ω)| dω
Definition 4 (Analytic Signal). For a real-valued signal m(t) ∈ L2 (R), the analytic signal is: m+ (t) = m(t) + j H[m(t)],
where, H[.] is the Hilbert Transform [9]. The Fourier transform is supported only on non-negative frequencies: m̂+ (ω) = 2m̂(ω) for ω ≥ 0 and m̂+ (ω) = 0 for ω < 0. IV. C ONTINUOUS -O RTHOGONAL M ODE D ECOMPOSITION The central contribution of C-OMD is to augment the orthogonality constraint between modes. The C-OMD problem 2 seeks modes {mk }K k=1 ⊂ H (R) and center frequencies K {ωk }k=1 ⊂ [0, ∞) that solve:
Definition 3 (Mode Decomposition). Let Gij ≜ ⟨mi , mj ⟩ denote the inner product between mode i and mode j. f ∈ L2 (R) is a collection {mk }K k=1 of finite and predefined K number of intrinsic modes mk (t). K X
mk (t) = f (t),
Gij = 0,
∀ i ̸= j.
(3)
k=1
We work in the Hilbert space L2 (R) of square-integrable functions equipped with the inner product: Z ∞ ⟨m, n⟩ = m(t) n(t) dt (4) −∞
where n(t) denotes the complex conjugate of n(t), and the induced norm ∥m∥L2 = ⟨m, m⟩1/2 . We additionally require modes to be in the Sobolev space H 2 (R) = {m ∈ L2 (R) : m′′ ∈ L2 (R)} to ensure that Fourier-domain integrals involving ω 2 are finite, and that Parseval’s isometry applies.
K X
min
{mk }, {ωk }
s.t
(i)
k=1 K X
2 ∂t mk,+ (t) e−jωk t L2 (6) mk (t) = f (t)
(reconstruction)
k=1
(ii)
Gij = 0, ∀ i ̸= j
(orthogonality)
where mk,+ (t) = mk (t)+j H[mk (t)] denotes the analytic signal. The reconstruction constraint (6.i) ensures that the modes collectively account for the entire signal. The orthogonality constraint (6.ii) ensures that each mode carries no duplicated information. To handle the two constraints in (6), we introduce a Lagrange multiplier λ(t) ∈ L2 (R) and a quadratic penalty parameter α > 0 for the reconstruction constraint. A symmetric matrix of Lagrange multipliers Γ = [γij ] ∈ RK×K (with γij for i ̸= j) and a penalty parameter β > 0 for the orthogonality constraints. The augmented Lagrangian reads:
0
for some tolerance ϵk > 0, where m̂k,+ (ω) is the Fourier transform of the analytic signal of mk .
(5)
L Γ, λ) =
K X
X 2 2 ∂t mk,+ (t) e−jωk t L2 + α f − mk L2
k=1
+⟨λ, f −
X
mk ⟩ + β
k
X
2
|Gij | +
i̸=j
k X
γij Gij
i̸=j
(7) P 2 The quadratic penalty α ∥f − mk ∥ provides good convergence at finite weight (corresponding to a Bayesian prior with noise variance ∝ 1/α), while the Lagrangian multiplier λ(t) enforces P the constraint exactly in the limit. The quadratic penalty β i̸=j |Gij |2 penalizes inter-mode correlation. A. Fourier-Domain Reformulation By Parseval’s isometry, the entire Lagrangian (7) can be ˆ evaluated in the Fourier domains: 2 Z ∞ K Z ∞ X X 2 (ω − ωk )2 |m̂k,+ (ω)| dω + α fˆ(ω)− m̂k (ω) dω −∞
k=1 0
k
Z ∞ X X 2 ˆ + λ̂(ω) f (ω)− m̂k (ω) dω + β Ĝij + γij Ĝij . −∞
k
i̸=j
(8)
Mode Update: Taking the first variation of (7) with respect to m̂k (ω) and setting it to zero yields the mode update. For non-negative frequencies ω ≥ 0: m̂n+1 (ω) = k
A 1 + 2(ω − ωk )2 /α
(9)
X λ̂(ω) 1 X γkj A:=fˆ(ω)− m̂i (ω)+ β ⟨mnk , mnj ⟩+ m̂j (ω) − 2α α 2 i̸=k j̸=k (10) Center Frequency Update: The center frequency update is unchanged from VMD [6], being independent of the orthogonality constraint: Z ∞ 2 ω m̂n+1 (ω) dω k (11) ωkn+1 = Z0 ∞ 2 m̂n+1 (ω) dω k 0
which places ωk at the center of gravity of the mode’s power spectrum. Dual Ascent: The Lagrange multipliers are updated by dual (gradient) ascent: X λn+1 (t) = λn (t) + τλ f (t) − mn+1 (t) , (12) k k n+1 n γij = γij + τγ ⟨mn+1 , mn+1 ⟩, i j
enforces bandwidth and reconstruction, while a subsequent Newton–Schulz step enforces orthogonality by projecting the current mode estimates onto the nearest set of orthogonal functions. The two steps alternate, each correcting for the constraint violations introduced by the other, until both constraints are jointly satisfied. The projection is realized via the Gram matrix of the mode system, which we now introduce. C. Newton–Schulz Iteration for Orthogonality Enforcement The Newton–Schulz iteration orthogonalizes a system of functions by operating on the K × K Gram matrix while linearly combining the functions themselves. Given K candidate modes {m1 (t), . . . , mK (t)} ⊂ L2 (R), the Gram matrix G ∈ CK×K is defined by: Z ∞ Z ∞ m̂i (ω) m̂j (ω) dω, (14) mi (t) mj (t) dt = Gij = −∞
−∞
where the second equality follows from Parseval’s theorem. The modes are orthogonal if and only if G is diagonal. (0) (0) Let {mk (t)}K k=1 be the initial set of modes. Compute G via (14) and normalize: (0)
(0)
mk (t) ← P
mk (t) (0) mj
j
i ̸= j,
(13)
(15)
L2
Then iterate: for m = 0, 1, 2, . . .,
where τλ , τγ > 0 are the dual step sizes.
C(m) = 23 IK − 21 G(m) ,
B. Limitations of Penalty-Only Orthogonality Enforcement Eq. (9)Pincorporates orthogonality through the correction term − α1 j̸=k (β⟨mnk , mnj ⟩ + γkj /2)m̂j (ω) in the numerator. While this steers the modes toward mutual orthogonality, the enforcement is indirect: the correction is computed from the inner products of the previous iterate, and the Wiener filter then modifies the spectral coefficients to simultaneously satisfy bandwidth, reconstruction, and orthogonality–competing objectives weighted by α and β. This has two practical consequences: i. P Slow convergence to orthogonality. The quadratic penalty β i̸=j |Gij |2 penalizes non-orthogonality but does not project onto the orthogonal manifold. For finite β, the modes are approximately orthogonal, and achieving high precision requires β → ∞, which destabilizes the bandwidth objective. ii. Conflict with the Wiener filter structure. The denominator 1 + 2(ω − ωk )2 /α enforces spectral compactness around ωk . The orthogonality correction in the numerator subtracts components correlated with other modes, but this subtraction is then re-weighted by the bandwidth filter. Consequently, a single Wiener-type update yields a trade-off rather than an exact simultaneous enforcement of compactness and orthogonality. To address this, we augment the ADMM (Alternating Direction Method of Multipliers) iteration with an explicit orthogonal projection step after the Wiener filter update. This follows the principle of alternating projection: the Wiener filter
1/2 .
2
(m+1)
mk
(t) =
K X
(m)
(m)
Ckj mj
(16) (t),
(17)
j=1 (m+1)
Gij
(m+1)
= ⟨mi
(m+1)
, mj
⟩.
(18)
D. Per-Frequency Orthogonalization To achieve exact orthogonality while preserving spectral structure, we orthogonalize the modes at each frequency independently. For each frequency ω, define the vector of spectral amplitudes: v(ω)⊤ = [m̂1 (ω), m̂2 (ω), . . . , m̂K (ω)] ∈ C1×K .
(19)
The per-frequency orthogonalization operator P : (L2 (R))K → (L2 (R))K acts on the spectral amplitudes at each frequency ω by applying a K × K transformation to v(ω): v⊥ (ω) = P v(ω), (20) where P ∈ CK×K is obtained by running Newton–Schulz on the frequency-integrated Gram matrix: Z ∞ G= v(ω) v(ω)H dω, (21) −∞
and applying the resulting transformation globally across all frequencies. This is equivalent to the functional Newton– Schulz of Section IV-C is expressed in the Fourier domain.
V. E XPERIMENTS & P ERFORMANCE E VALUATION
(a) Mode-Domain Architecture Human Side
Robot Side
m1h (t−W : t), ... , mKh
m1r (t−W : t), ... , mKr
Per-Mode TCN Encoder
Per-Mode TCN Encoder
separate per mode k
shared weights w/ human
Cross-Mode Self-Attn
Cross-Mode Self-Attn
[z1h , ... , zKh ]
Linear Coupling
Cross-Side Cross-Attn Qh ← z̃ h , Kr, Vr ← z̃ r
Wc z̃ h
[z1r , ... , zKr ]
K/V Exchange Kh , Vh ↔ Kr , Vr
Linear Coupling
Cross-Side Cross-Attn Qr ← z̃ r , Kh, Vh ← z̃ h
Wc z̃ r
Add & LayerNorm
Add & LayerNorm
Per-Mode TCN Decoder
Per-Mode TCN Decoder
Dilated deconv → H
Dilated deconv → H
m̂1h (t+1 : t+H), ... , m̂Kh
m̂1r (t+1 : t+H), ... , m̂Kr
h ẑ h =LN(zattn +zlch )
Reconstruct f̂ h
P
L = Lpred + λ1 Lrecon + λ2 Lorth + λ3 Lrel , 1 X 2 m̂k (t+h) − mk (t+h) 2 Lpred = KH
r ẑ r =LN(zattn +zlcr )
Reconstruct f̂ r
h k m̂k
Dataset. We use real-world haptic data traces captured with a Novint Falcon haptic device inside a Chai3D virtual environment [10]. The dataset records 3-DoF position, velocity, and force measurements at fs =1000 Hz from both the human operator and the teleoperated robot across five manipulation tasks. All reported results are averaged over 10 independent runs, with training using 70% of the data and testing using 20%, and 10% of the data reserved as a validation set. Training. MDA is trained by a four-component loss: (22) (23)
k,h
2 Lrecon = fˆ − f 2
Lorth =
P
r k m̂k
2 G − IK F
1 X |m̂k (t+h) − mk (t+h)| Lrel = KH max(|mk (t+h)|, τ )
(24) (25) (26)
k,h
Fig. 1. Mode-Domain Architecture. C-OMD decomposes each haptic signal into K orthogonal modes processed by per-mode Temporal Convolutional Networks (TCN) encoders and Cross-Mode Self-Attention within each side. Cross-Side Cross-Attention (main path) lets the human latent attend to robot keys/values and vice versa; a parallel Linear Coupling branch (dotted skip) provides a stable residual Wc z̃ added before LayerNorm, ensuring gradient flow even when attention weights are noisy. Fused latents ẑ h , ẑ r are decoded and summed to reconstruct the predicted signals. (a) Per-Mode TCN Encoder
(b) Per-Mode TCN Decoder
Mode mk (t−W : t)
Fused latent ẑk ∈ Rd
Dilated Conv d=1 ReLU + LayerNorm
Dilated Conv d=4 ReLU + LayerNorm
Dilated Conv d=2 ReLU + LayerNorm
(c) Cross-Side Cross-Attention with Residual Coupling
Qh =z̃ h WQ
zlch =Wc z̃ h Linear Coupling
Kr =z̃ r WK Vr =z̃ r WV
h
(d) Cross-Mode Self-Attention
Tokens: [z1 , z2 , ... , zK ]
r ⊤
(K ) h =softmax Q √ zattn d
Dilated Conv d=2 ReLU + LayerNorm
r V
Q, K, V projections
⊤
√ softmax QK
d
h +z h ) Add & LayerNorm ẑ h =LN(zattn lc
Dilated Conv d=4 ReLU + LayerNorm
Dilated Conv d=1 ReLU + LayerNorm
Latent zk ∈ Rd
Linear Proj. Rd → RH
m̂k (t+1 : t+H)
V
Add & LayerNorm Feed-Forward + Norm
ẑkh : human latent (with robot context)
Feed-Forward + Norm
[z̃1 , ... , z̃K ]
Fig. 2. (a) Per-mode TCN encoder: dilations d=1, 2, 4 with residual skip. (b) Per-mode TCN decoder: inverted dilations d=4, 2, 1 with linear projection Rd → RH . (c) Cross-Side Cross-Attention with residual coupling detail: zattn and zlc =Wc z̃ are summed and layer-normalized. (d) Cross-Mode SelfAttention over K mode latents.
Proposition 1 (Band-Limitedness Preservation). Perfrequency orthogonalization (20) preserves the spectral support of each mode: if m̂k (ω) = 0 for |ω − ωk | > Bk , then m̂⊥ k (ω) = 0 for |ω − ωk | > Bk , provided P is blockdiagonal with respect to non-overlapping frequency bands. In overlapping bands, the orthogonalization redistributes energy between modes but does not introduce new spectral content.
Lpred is the primary per-mode MSE prediction loss over window H. Lrecon penalizes reconstruction error after summing predicted modes back to the full signal, ensuring the modes remain jointly consistent. In Lorth the Gij =⟨mi , mj ⟩, enforces explicit orthogonality of the decomposed modes. Lrel is a relative error term with threshold τ =0.01 to avoid division by near-zero values during low-force regions near contact transitions. This term prevents the model from ignoring perceptually important small-force events that pure MSE training tends to underweight. Loss weights are set to λ1 =0.1, λ2 =0.01, and λ3 =0.05. In our experiment, we evaluate prediction accuracy and inference time across different prediction window sizes (W ∈ {1, 5, 10, 25, 50, 100}). At training time, the MDA decoder observes a mixture of ground-truth and model-predicted mode values as its input sequence [11]. Let ϵe = 1 − e/E denote the teacher-forcing ratio at epoch e out of E total epochs. With probability ϵe the decoder receives the true data sequence; otherwise, it receives the model’s own prediction from the previous step. Inference. At inference time, when a haptic packet is lost or delayed beyond the predefined deadline, the model enters an autoregressive restoration mode: the predicted signal output from the previous step is concatenated with the available history and fed back as the encoder input for the next step, without any ground-truth observations. Architecture. We used Mamba [12], Transformer [13], and MDA (An overview of the proposed architecture is presented in Fig. 1, while detailed views of each section are provided in Fig. 2.) as our Neural Networks (NNs) architectures. All models are trained with Adam (β1 =0.9, β2 =0.999), weight decay 10−4 , and gradient clipping at norm 1.0. Learning rates are 3×10−4 for Mamba and Transformer+Fourier and 5×10−4 for MDA. All NNs were trained for 300 epochs on a single NVIDIA RTX Ada 6000.
SVMD
C-OMD
Human
Robot
(b)
Mamba
100
Accuracy
100
90
100
10−1
80
101
90
100
10−1
80
50
1 5 10
25
50
100
10−1
80
10−2
10−3
VMD Human: 91.2%→65.0% | 0.627→4152.13 ms VMD Robot: 89.5%→63.0% | 0.624→4923.19 ms SVMD Human: 93.8%→69.6% | 2.461→101579.60 ms SVMD Robot: 92.2%→67.7% | 2.574→128309.87 ms C-OMD Human: 97.5%→80.4% | 0.070→1.11 ms C-OMD Robot: 95.9%→78.0% | 0.074→1.18 ms
60 10−4
10−5
90
70
10−3
VMD Human: 90.0%→62.8% | 0.561→3121.65 ms VMD Robot: 88.3%→60.5% | 0.617→3918.56 ms SVMD Human: 92.3%→67.3% | 2.287→71335.37 ms SVMD Robot: 90.7%→65.3% | 2.504→89482.50 ms C-OMD Human: 96.2%→77.8% | 0.071→1.00 ms C-OMD Robot: 94.7%→75.3% | 0.075→1.07 ms
101
10−2 70
60
MDA (Ours)
100
10−2 70
Latency Constraint
(c)
101
Accuracy (%)
Inf. Time
Transformer
50
100
1 5 10
25
Prediction horizon (samples)
50
10−3
VMD Human: 92.7%→66.8% | 0.521→2181.18 ms VMD Robot: 91.0%→64.8% | 0.558→2666.24 ms SVMD Human: 94.9%→71.9% | 2.041→47725.18 ms SVMD Robot: 93.4%→69.6% | 2.173→59251.30 ms C-OMD Human: 98.6%→83.4% | 0.066→0.86 ms C-OMD Robot: 97.3%→81.2% | 0.065→0.96 ms
60 10−4
10−5
50
100
1 5 10
25
Prediction horizon (samples)
50
Inference Time (seconds)
VMD
(a)
10−4
10−5
100
Prediction horizon (samples)
Fig. 3. (a,b,c) Prediction accuracy (left axis, solid lines) and inference time right log-axis, dashed lines) vs. prediction window W ∈ {1, 5, 10, 25, 50, 100} samples for VMD (□), SVMD (△), and C-OMD (⋆) across three architectures. The green line marks the TI latency constraint ∆t = W ms. (b) C-OMD: Accuracy & Inference Time vs. K 100
(c) SNR Robustness 100
Accuracy
0
W=1 W=5
95
10−1
W=10
−5 90
90
80 75
80
10−2
W=100
70
70
10−3
Inference Time
C-OMD—H C-OMD—R
VMD—H VMD—R
−10 80
−15
−20
70
−25 60
60
65
Inference time (s)
W=50
85
Accuracy (%) — Force
W=25
Accuracy (%)
Accuracy (%)
90
SVMD—H SVMD—R
C-OMD—H C-OMD—R
10−4
60
Degradation (%)
(a) Method Comparison (W = 5) 100
VMD—H VMD—R
−30
SVMD—H SVMD—R
−35
50 50 2.5
5.0
7.5
10.0
12.5
Sliding Window ID
15.0
17.5
20.0
2
3
4
5
6
Number of modes K
7
8
30
25
20
15
10
5
0
SNR (dB) ← degrading
Fig. 4. (a) Prediction accuracy over sliding windows across three architectures at window sizes W =5. (b) Prediction accuracy (left axis, solid) and inference time (right log-axis, dashed) vs. number of modes K ∈ {2, . . . , 8} for C-OMD with MDA architecture, evaluated on W ∈ {1, 5, 10, 25, 50, 100} samples. (c) SNR robustness on the Force signal: accuracy (%, left axis) and relative degradation (%, right axis, faint) as channel SNR degrades from 30 dB to 0 dB.
VI. R ESULT AND D ISCUSSION Accuracy and inference time evaluation across various window size are depicted in Fig. 3. The strongest configuration, COMD+MDA, achieves 98.6% (Human) and 97.3% (Robot) in W =1, gracefully degrading to 82.4% and 80.1% in W =100, while VMD with the same backbone as MDA collapses to 65.3% and 63.1%, a degradation rate more than twice as steep. The inference time comparison is equally decisive for realtime deployment. The parallel orthogonalization of C-OMD is ≈8× faster than VMD and ≈30× faster than SVMD, with C-OMD+MDA completing a full forward pass in under 0.06 ms at W =1, well within the 1 ms TI budget even at W =100. SVMD, despite offering a modest accuracy gain over VMD (+2–+5%), pays a prohibitive latency penalty from its sequential mode extraction, consistently exceeding 1.7 ms at W =1 and trending toward or beyond the latency constraint at longer windows (Fig. 3, green line). This makes SVMD architecturally incompatible with strict haptic deadlines.
Each point in Fig. 3.(a,b,c) represents the average accuracy across of all window slides for a specific W . For example, when W = 5, the point reflects the average accuracy over 20 window slides for the next 100-sample prediction. To examine this in greater detail, Fig. 4.(a) illustrates the accuracy changes across each sliding window for the 100-sample prediction with W = 5. The architecture is fixed to MDA while the decomposition method varies: C-OMD achieves a mean accuracy of 98.6% (Human) and 97.3% (Robot), outperforming VME by +3.7 and VMD by +6.0, with a consistently faster coldstart convergence (within 3–4 windows). This confirms that explicit spectral orthogonality is the dominant performance driver: eliminating cross-mode leakage stabilizes the latent representation from the first window onward. The most striking finding from Fig. 4.(b) is that more modes do not mean better performance, a result that challenges the conventional assumption that richer signal decompositions always benefit from prediction. For VMD and SVMD (Result
TABLE I T OTAL GFLOP S : DECOMPOSITION + MODEL FLOATING - POINT OPERATIONS PER FORWARD PASS (W =100, K=3, H=100). I NF. ( MS ): INFERENCE TIME AT W =1 PER SIGNAL TYPE AND SIDE . ACC . (%): MEAN ± STD OVER 5- FOLD AT W ∈ {1, 10, 100}. G REEN = BEST; UNDERLINE = SECOND - BEST.
Inf. (ms) — Human Inf. (ms) — Robot Arch.
Decomp. GFLOPs
F
V
P
F
V
P
Accuracy (%) — Human
Accuracy (%) — Robot
W =1
W =1
W =10
W =100
W =10
W =100
VMD Mamba SVMD† C-OMD
0.276 1.847 0.132
0.487 0.491 0.480 0.506 0.511 0.499 89.8±0.8 81.4±1.1 61.2±1.9 88.1±0.9 79.7±1.2 59.0±2.1 1.946 1.962 1.929 2.024 2.040 2.007 92.3±0.7 84.6±1.0 65.8±1.7 90.7±0.8 83.0±1.1 63.5±1.9 0.061 0.063 0.061 0.064 0.065 0.064 96.2±0.5 90.1±0.8 76.4±1.4 94.7±0.6 88.5±0.9 74.1±1.5
VMD Trans. † +Fourier SVMD C-OMD
0.318 1.983 0.174
0.508 0.514 0.502 0.529 0.535 0.522 91.2±0.7 83.5±1.0 63.5±1.8 89.6±0.8 81.8±1.1 61.2±2.0 2.037 2.054 2.019 2.118 2.136 2.100 93.8±0.6 86.4±0.9 68.1±1.6 92.2±0.7 84.8±1.0 65.9±1.8 0.063 0.064 0.063 0.066 0.067 0.066 97.4±0.4 91.8±0.7 79.2±1.3 95.9±0.5 90.1±0.8 77.0±1.4
VMD SVMD† C-OMD
0.233 1.728 0.089
0.443 0.448 0.438 0.461 0.466 0.456 92.6±0.7 84.9±1.0 65.3±1.7 91.0±0.8 83.2±1.1 63.1±1.9 1.764 1.778 1.749 1.835 1.849 1.819 94.9±0.6 87.8±0.9 70.2±1.5 93.4±0.7 86.1±1.0 68.0±1.7 0.059 0.060 0.059 0.062 0.062 0.061 98.6±0.3 93.4±0.6 82.4±1.2 97.3±0.4 91.8±0.7 80.1±1.3
MDA
provided in GitHub repository due to page limit), accuracy peaks at K=4–6 depending on the window and then degrades, because additional quasi-orthogonal modes introduce redundant spectral content that the MDA cannot distinguish from noise. C-OMD breaks this pattern entirely: its explicit orthogonality constraint forces each mode to carry unique, non-overlapping information, so the optimal K collapses to just K=3 across all windows W ∈ {1, . . . , 100}, achieving 98.6% at W =1 and 82.4% at W =100, gains of up to +17% over VMD at the same K. The inference time axis reinforces this advantage where C-OMD remains the fastest method. In Fig. 4(c) under channel impairment, C-OMD loses only ≈16 as SNR falls from 30 dB to 0 dB, while VMD sheds more than 34 over the same range; the reason is structural, orthogonality confines noise injected into one frequency band to that mode alone, preventing it from cascading across the decomposition in a way that neither VMD nor SVMD can prevent. Taken together, the two panels establish a decompositionfirst, attention-second design principle: C-OMD provides the representational precondition for high accuracy, and cross-side attention yields an orthogonal, additive gain on top. Each point in Fig. 4(a) represents the average accuracy across 9 outputs corresponding to the three spatial dimensions (X, Y, Z) of the three features: F, V, and P. To provide a more detailed view of Fig. 4(a), the average of each of the three features in three dimensions is provided in Table I. Table I delivers an unambiguous message: C-OMD+MDA is the Pareto-optimal choice across all measured axes. It achieves 0.089 GFLOPs, representing a 2.5–3.6× reduction compared to VMD (0.23–0.32) and a massive 19.1–22.4× reduction compared to SVMD (1.7–2.0). Furthermore, with an inference time of approximately 0.06 ms, it provides an ≈ 8× speedup over VMD and an ≈ 30× speedup over SVMD. Most importantly, it maintains the highest accuracy at every window size, reaching 98.6% at W = 1 and 82.4% at W = 100. In short, C-OMD’s explicit orthogonality simultaneously reduces computation and improves prediction; there is no accuracy– efficiency trade-off to navigate.
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