RESEARCH PAPER TECHNICAL LATEX

Abstract

We present a mathematically rigorous framework for multi-modal sensor fusion in real-time embodied interaction systems. Our approach, Recursive Polymodal Synthesis (RPS), fuses heterogeneous modalities with disparate sampling rates and noise statistics into a coherent latent representation suitable for generative control. The core mechanism is a proximal fixed-point iteration using spectral-norm-constrained relational operators, yielding contraction guarantees and a unique fixed point. We prove geometric convergence in $\mathcal{O}(\log(1/\epsilon))$ iterations to $\epsilon$-accuracy when $\alpha\lVert T\rVert_2<1$. The system uses modality encoders $\{E_m\}_{m=1}^M$, linear translators $\{T_m\}_{m=1}^M$ with $\lVert T_m\rVert_2\le \sigma_{\max}<1$, and an update $\Pprox_\alpha(\vz)=(1-\alpha)\Emap(\vx)+\alpha\,T(\vz)$. On synthetic data, the approach attains $99.94\%$ cross-modal coherence, validation loss $1.93\times 10^{-4}$, and $15$--$40$ ms CPU latency. Ablations confirm spectral constraints are necessary and reveal low effective rank structure in learned translators. RPS provides a principled, low-latency alternative to attention-based fusion with explicit robustness to missing modalities and tight stability bounds, enabling live performance, human--robot interaction, and adaptive interfaces.

Introduction

Problem formulation

- P1: Cross-modal coherence. Latents respect empirical dependencies so that $p(\vz_m\mid \vz_{-m})$ aligns with learned relationships.

- P2: Robustness to missing data. For any subset $S\subseteq[M]$, the partial map $\Phi(\cdot\mid S)$ is well-defined.

- P3: Real-time compute. End-to-end latency $L \le 50$ ms for embodied agency.

Naive concatenation fails P1, many attention-based schemes lack convergence guarantees (P2), and some iterative refinements blow the latency budget (P3). RPS addresses all three.

Contributions

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- Contraction-theoretic fusion. A proximal fixed-point update with spectral constraints ensures existence, uniqueness, and geometric convergence (thm:contraction).

- Architectural modularity. Encoders + linear relational translators + proximal blending enable learned cross-modal structure, graceful hallucination under dropout, and bounded iteration counts.

- Stable training pipeline. Alternating optimization with spectral projection; a multi-objective control loss balances accuracy, smoothness, range, velocity, and inter-channel diversity.

- Empirical validation and analysis. High coherence at low latency; ablations, sensitivity, spectral analysis, and information-theoretic diagnostics clarify what the model learns and why it works.

Preliminaries and notation

Bold lowercase denote vectors $\vx\in\R^{d}$, bold uppercase matrices $\W\in\R^{m\times n}$. Spectral norm $\lVert \W\rVert_2$; Frobenius norm $\lVert \W\rVert_F$; operator Lipschitz constant $\Lip(f)$. Concatenation $[\vz_1;\dots;\vz_M]$ and $[M]\defeq\{1,\dots,M\}$. For any linear map $A$, $\spec(A)$ denotes eigenvalues; for non-square $A$, $\spec(A^\top A)$ refers to squared singular values.

Recursive Polymodal Synthesis (RPS)

Modality encoders

Implementation sizes..
We use $M=4$ modalities with $(D_1,D_2,D_3,D_4)=(64,16,16,8)$ so $D=104$; hidden width $H_m=128$ per encoder.

Linear relational translators

proof.
For $\lVert \vz\rVert_2=1$, $\lVert T(\vz)\rVert_2^2=\sum_m \lVert \W_m\vz\rVert_2^2 \le \sum_m \lVert \W_m\rVert_2^2$.

Proximal fixed point

proof.
For any $\vz,\vz'$,
$\Pprox_\alpha(\vz;\vx)-\Pprox_\alpha(\vz';\vx)_2
= \alphaT(\vz)-T(\vz')_2\le \alpha\lVert T\rVert_2\vz-\vz'_2$.
Banach's theorem on $(\R^{D},\norm{\cdot}_2)$ gives existence, uniqueness, and geometric convergence.

corollary: Iteration budget.
With $\alpha=0.2$, $\sigma_{\max}=0.9$, $M=4$, lem:composite implies $\lambda\le 0.36$; $t\ge 7$ iterations suffices for $\epsilon=10^{-3}$ relative error. Empirically $t=3$--$5$.

Missing modalities and hallucination

proofsketch
Subtract the two fixed-point equations, unroll the contraction, and sum the geometric series bounded by $(1-\lambda)^{-1}$.
proofsketch

Latent normalization and mapping to controls

Training Objectives and Procedures

Alternating optimization

Control mapper loss

Optimization hyperparameters

Encoders/Translators: Adam, $\eta=10^{-3}$, weight decay $10^{-4}$, clip norm $1.0$, warmup $5$ epochs, cosine decay, early stop patience $15$, $\alpha=0.2$, $\sigma_{\max}=0.9$, $T_{\text{iter}}=5$. Mapper: AdamW, $\eta=5\times 10^{-4}$, weight decay $10^{-5}$, sequence length $T=50$, batch $32$.

Algorithms

Algorithm: Proximal Fixed-Point Inference with Modality Mask

\Require Input \vx=\vx_m_m=1^M, mask \mu∈0,1^M, encoders E_m, translator T, steps T_iter, blend \alpha
\State \vz^(0) ← [E_1(\mu_1\vx_1);\dots;E_M(\mu_M\vx_M)]
\Fort=0 to T_iter-1
\State \vz← T(\vz^(t))
\State \vz^(t+1)← (1-\alpha)\vz^(0)+\alpha \vz
\If\norm\vz^(t+1)-\vz^(t)_2 \le \epsilon break \EndIf
\EndFor
\State \Return \vz^(t+1)

Algorithm: Alternating Spectral-Constrained Training

\Require Dataset D, steps T_iter, \alpha, \sigma_\max
not converged
\State // Encoder step
\State Freeze T_m; for minibatches (\vx)\simD:
\State \vz^(0)← \Emap(\vx); iterate alg:prox to get \vz^*
\State Descend \nabla_\theta\sum_m\vz_m^* - T_m(\vz^*)_2^2
\State // Translator step
\State Freeze m; for minibatches (\vx)\simD:
\State Compute \Emap(\vx); descend \nabla_\W\sum_m\normm(\vx_m)-T_m(\Emap(\vx))_2^2
\State Project \W_m← (\sigma_\max/\max\sigma_\max,\lVert \W_m\rVert_2)\W_m
\EndWhile

Theoretical Analysis

Stability, sensitivity, and Lipschitz bounds

assumption
Each encoder satisfies $\Lip(\Enc{m})\le L_m$, so $\Lip(\Emap)\le (\sum_m L_m^2)^{1/2}\defeq L_E$.
assumption

proof.
Write the two fixed points for $\vx,\vx'$; subtract and bound using triangle inequality and contraction factor, then rearrange.

proposition: BIBO stability under additive noise.
If inputs are perturbed by $\delta\vx$ with $\norm{\delta\vx}\le \eta$, then the fixed point perturbation satisfies $\norm{\delta\vz^*}\le \frac{(1-\alpha)L_E}{1-\alpha\norm{T}_2}\,\eta$.

Complexity and memory

Per step: encoders $O(\sum_m H_m d_m + D_m H_m)$; translator $O(\text{nnz}(T))$ (dense: $O(D^2)$). With $T_{\text{iter}}$ steps, cost is $O(\text{Enc} + T_{\text{iter}}\cdot \text{Trans})$. GRU dominates latency at larger $H$; see tab:latency.

Relation to contractive residual maps

$\Pprox_\alpha(\vz)=\vz - \underbrace{\big(\vz-(1-\alpha)\Emap(\vx)-\alpha T(\vz)\big)}_{\text{contractive residual}}$.
For $\alpha\norm{T}_2<1$, the residual is a contraction, akin to contractive ResNets; the Neumann expansion in rem:closedform parallels resolvent expansions in monotone operator theory.

Experimental Results

Setup

Synthetic sessions: $f_s=\SI{100}{Hz}$, $T_{\text{session}}=6000$ frames; $N_{\text{train}}=10^5$, $N_{\text{val}}=2\times 10^4$. Encoders: $H_m=128$, outputs $(64,16,16,8)$. Translators: $\sigma_{\max}=0.9$. GRU: $H=256$, 2 layers, dropout $0.1$.

Encoder--translator behavior

Training loss decreases from $8.47\times 10^{-2}$ to $4.67\times 10^{-2}$; validation loss reaches $1.93\times 10^{-4}$. Coherence $\rho=0.9994$. Translators saturate the bound: $\lVert \W_m\rVert_2=0.900$.

Caption: Fixed-point convergence probability with threshold on validation.

Control mapper

Validation MSE $\approx 6.0\times 10^{-2}$; temporal autocorrelation decays at $\approx\SI{150}{ms}$; coverage of control range $\ge 82\%$ per channel.

End-to-end latency

Caption: Latency on Intel i7-1165G7 (single thread).

Robustness and ablations

Graceful degradation under modality dropout, additive noise, and temporal dropout. Removing spectral constraints leads to exploding $\lVert \W_m\rVert_2$ and eventual divergence; increasing $\alpha$ reduces iterations but raises validation error past $0.4$.

Caption: Effect of on convergence and validation loss.

Representation structure

SVD of $\W_{\text{mot}}$ shows $\epsilon$-rank $\approx 24$ at $\epsilon=0.01$; PCA of $\{\vz^*\}$: top $40$ components explain $95\%$ variance; axes correlate with energy, heart-rate, and beat-phase proxies.

Discussion

Comparison to baselines

Caption: Comparison summary. Higher coherence and lower latency are better.

Synthetic-to-real

Architectural priors and spectral constraints temper distribution shift; EMA normalization adapts scale drift. We anticipate $70$--$85\%$ zero-shot retention moving to $85$--$95\%$ with short adaptation by encoder fine-tuning while freezing $T$.

Limitations and next steps

Pending real-sensor evaluation; fixed modality and control cardinality; unidirectional control; lack of explicit multi-timescale hierarchy; CPU-only kernels. Future work: variable modality graphs, bidirectional couplings, multi-scale fixed points, GPU/quantized kernels, tighter sample complexity bounds.

Conclusion

RPS treats fusion as a contraction-mapped fixed-point problem with explicit spectral control, delivering unique coherent latents at low latency with robustness to missing data. The analysis explains stability and sensitivity; experiments support the design and reveal structured cross-modal dependencies. The framework generalizes beyond choreography to any real-time fusion task with strict latency budgets.

\appendix

Glossary of Symbols

Expanded Proofs

lemma: Invertibility of .
If $\alpha\norm{T}_2<1$, then $\I-\alpha T$ is nonsingular and $\norm{(\I-\alpha T)^{-1}}_2 \le 1/(1-\alpha\norm{T}_2)$.

proof.
Neumann series: $(\I-\alpha T)^{-1}=\sum_{k=0}^\infty \alpha^k T^k$ with geometric bound on partial sums.

proof.
First inequality from thm:contraction. The second uses rem:closedform and triangle inequality.

Synthetic Data Generation Details

Motion energy combines low-frequency Perlin-like noise with sinusoidal drive; HR uses an exponential lag filter of energy with Gaussian innovations; rhythm uses BPM drift and phase accumulation. Bounds ensure physical plausibility: hip angles within $\pm 45^\circ$, HR within $[60,190]$ BPM after clipping.

Reproducibility Checklist

- Full hyperparameters for all components (sec:hparams).

- Random seeds fixed per run; per-epoch EMA statistics checkpointed.

- Spectral normalization via $K=5$ power iterations; projection every step.

- Hardware: Intel i7-1165G7, single thread; Python 3.11, PyTorch 2.x.

Safety and Ethics

Though designed for consensual performance settings, any embodied sensing introduces privacy risks. We recommend on-device processing, opt-in consent logging, and auditable data retention policies. Spectral constraints improve stability, reducing erratic actuator outputs that could harm equipment or users.

References

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- Banach, S. Théorie des opérations linéaires. (1932).

- Miyato, T., et al. Spectral Normalization for GANs. ICLR (2018).

- Baltrušaitis, T., et al. Multimodal ML: A Survey and Taxonomy. TPAMI (2019).

- Parikh, N., Boyd, S. Proximal Algorithms. FnT in Optimization (2014).

- Cho, K., et al. Learning Phrase Representations using RNN Encoder–Decoder. EMNLP (2014).