Signal Embedding Energy (SEE)

• Signal Embedding Energy (SEE) is a metric that quantifies the rate at which semantic information is embedded into a signal's shape using physical and geometric principles.
• SEE leverages discrete shape encoding and semantic entropy to classify local signal configurations and assess noise interference in audio language models.
• Extensions to LALMs enable embedding-space noise mitigation via subspace projection techniques, resulting in improved model robustness and performance.

Signal Embedding Energy (SEE) is a metric and analytic construct designed to quantify the instantaneous rate at which a signal encodes semantic information through its shape, both at the waveform and model-embedding levels. Initially developed for time-domain signal analysis (Majumdar et al., 2016), SEE has recently been extended to quantify the internal perception of noise in Large Audio LLMs (LALMs), enabling direct evaluation of robustness and the design of effective mitigations (Zhang et al., 12 Jan 2026). SEE is grounded in physical and geometric principles of signal representation, formalized through energy-like quantities and structured subspace projections, and finds practical application in waveform analysis, semantic entropy calculation, activation-space noise measurement, and embedding-level denoising.

1. Physical Basis and Continuous-Time Formulation

In the continuous-time regime, a one-dimensional analog signal $s(t)$ is modeled as the trajectory of a unit-mass particle in a force field with one degree of freedom. By Newton's second law, the instantaneous force is $F(t) = s''(t)$. The power expended by the particle at time $t$ to trace this trajectory—interpreted as the rate of semantic information embedding in the signal's shape—is given by: $P(s(t)) = s''(t) s'(t)$ where $s'(t)$ and $s''(t)$ are the first and second derivatives, respectively. $P(s(t))$ represents the instantaneous energy used to encode semantic features locally. This formulation directly ties signal morphology to the physical principle of energy dissipation and underpins the notion that signal shape is the carrier of semantic information (Majumdar et al., 2016).

2. Discrete-Time Analogues and Shape Encoding Taxonomy

Digitized signals $s[n]$ inherit the SEE concept through finite differences:

• Backward difference $s'[n] = s[n] - s[n-1]$
• Forward difference $s'[n+1] = s[n+1] - s[n]$
• Second difference $s''[n] = s[n+1] - 2s[n] + s[n-1]$

Discrete SEE can be computed as left and right products: $$P(s[n-]) = s''[n] s'[n] ,\quad P(s[n+]) = s''[n] s'[n+1]$$ or as an ordered pair $(P(s[n-]), P(s[n+]))$. In local 3-point neighborhoods, the sign configuration $$(\text{sign}(s'[n]), \text{sign}(s''[n]), \text{sign}(s'[n+1]))$$ yields exactly 13 admissible patterns, each corresponding to a unique local geometric configuration; in analog settings, a similar classification yields 17 distinct shape encodings in the infinitesimal neighborhood (see Table below).

Discrete Config ID	Sign Pattern	Geometric Shape
1	$(−,−,−)$	Down peak
4	$(+, +, +)$	Up peak
10	$(+, +, −)$	Inflection
...	...	...

This exhaustive enumeration enables deterministic classification using finite automata and is foundational for assessing local semantic complexity (Majumdar et al., 2016).

3. Semantic Entropy and Regular Language Structure

To quantify the diversity of locally embedded shapes, semantic entropy is defined for a segment containing $N$ samples: $SE(s) = -\sum_{i=1}^{13} p(i)\, \log_2 p(i)$ where $p(i)$ is the empirical probability of configuration $i$ in the segment. Higher $SE(s)$ indicates more varied local geometries and thus richer semantic information content. The collection of all finite-length digital signals forms a regular language over the 13-letter shape alphabet, accepted by a deterministic finite automaton (DFA) and further extended to a weighted finite-state transducer (WFST) by associating continuous SEE or slope-based weights to transitions. This enables joint classification and quantitative assessment of signal events such as action potentials and speech phonemes (Majumdar et al., 2016).

4. SEE in Large Audio LLM Embeddings

Extending SEE to LALMs, Zhang et al. (2025) introduce a model-centric SEE metric that operates directly on the internal activations of audio encoders. For an input $x$, let $A^{(\ell)}(x)\in\mathbb{R}^{T(x)\times d_\ell}$ denote tokenwise activations at layer $\ell$. Structured calibration is performed as follows:

• Temporal mean-pooling: $$a^{(\ell)}(x) = \frac{1}{T(x)} \sum_t A_t^{(\ell)}(x)$$
• Collect semantic and pure-noise mean activations into $S^{(\ell)}$ and $N^{(\ell)}$
• Perform singular value decomposition (SVD) of both matrices
• Identify and select dominant noise directions $V_n$ orthogonal to all dominant semantic directions $V_s$, constructing a noise-only basis $Q^{(\ell)}$

At test time, activations are projected onto this basis and the signal embedding energy at layer $\ell$ is computed as: $$\mathrm{SEE}^{(\ell)}(x) = \frac{1}{T(x)} \sum_{t=1}^{T(x)} \| Z_t^{(\ell)}(x) \|_2^2 + \varepsilon$$ with $Z^{(\ell)}(x) = A^{(\ell)}(x) Q^{(\ell)}$, $\varepsilon$ a numerical stabilizer. The aggregate SEE score is summarized over critical layers: $$\mathrm{SEE}(x) = \frac{1}{|\mathcal{L}|} \sum_{\ell \in \mathcal{L}} \mathrm{SEE}^{(\ell)}(x)$$ SEE thus quantifies the energy of activation components attributed purely to noise, directly correlating with semantic interference and performance degradation (Zhang et al., 12 Jan 2026).

5. Experimental Assessment and Comparative Analysis

On benchmarks spanning speech-to-text, open-domain speech QA, music reasoning, and environmental sound classification, SEE exhibits a near-perfect negative correlation ($\rho \approx -0.98$) with generation success rate (GSR)—the likelihood that noisy-generated responses match clean ones. Empirical evaluation across Qwen-Audio, MiniCPM-o, and StepAudio models, and 10 diverse noise categories, demonstrates SEE’s superiority as a predictor of model breakdown compared to classical metrics such as waveform SNR or PESQ. Notably, waveform-level denoising methods (STFT Wiener filtering, wavelet thresholding, SEGAN, DFL) are only marginally effective for LALMs, often increasing SEE or failing to improve accuracy, due to the creation of new embedding artifacts in noise subspaces (Zhang et al., 12 Jan 2026). This suggests a fundamental mismatch between traditional denoising objectives and LALM noise sensitivity.

6. SEE-Driven Mitigation Strategies: SEEN and Subspace Intervention

To neutralize noise-induced semantic drift, Signal Embedding Energy Neutralization (SEEN) is introduced—a training-free, test-time projection that subtracts noise-subspace components from each activation: $$\widetilde{A}^{(\ell)}(x) = A^{(\ell)}(x) - \lambda C^{(\ell)}(x)$$ where $$C^{(\ell)}(x) = A^{(\ell)}(x) Q^{(\ell)} Q^{(\ell)\top}$$ and $\lambda \in [0, 1]$ tunes the intervention strength. SEEN requires no retraining or parameter updates and incurs zero additional audio latency. Empirically, SEEN achieves a 6.7% absolute improvement in GSR and >90% reduction in SEE compared to the best waveform denoiser, confirming the efficacy of embedding-space interventions (Zhang et al., 12 Jan 2026).

7. Implications and Future Directions

SEE establishes a paradigm wherein robustness against noise in multimodal LLMs is grounded in model-specific internal representations rather than acoustic fidelity. The framework shows that embedding-level detection and subtraction of structured noise components can outperform conventional denoising, indicating a fundamental gap between acoustic and semantic objectives. A plausible implication is that robustness research should prioritize embedding-aware diagnostics and adversarial subspace defenses, and integrate SEE-linked metrics into future training or architectural innovations. It is expected that further theoretical and empirical work will refine activation-space modeling and extend SEE to other input modalities and LLM architectures (Zhang et al., 12 Jan 2026).

• Semantic Information Encoding in One Dimensional Time Domain Signals (Majumdar et al., 2016).
• SEE: Signal Embedding Energy for Quantifying Noise Interference in Large Audio LLMs (Zhang et al., 12 Jan 2026).

Signal Embedding Energy (SEE) provides a rigorous, model-aware approach for quantifying both semantic information in classical signals and noise vulnerability in high-capacity audio LLMs, supplanting traditional fidelity metrics and informing next-generation robustness strategies.