Scoreq_nr: No-Reference Quality and Quaternionic Analysis

• Scoreq_nr is a family of no-reference quality assessment models that predict perceptual scores in speech, audio, and algebraic invariants using only processed input.
• In speech and audio quality estimation, Scoreq_nr models leverage contrastive and distributional losses on features from Mel-spectrograms and raw waveforms to achieve superior correlation with human scores.
• In algebra, Scoreq_nr computes adelic spinor norm images for quaternionic skew-Hermitian forms to determine spinor genera and class numbers through local-to-global invariant analysis.

Scoreq_nr refers to a family of “no-reference” or “non-intrusive” quality assessment functions and models. This term is notable in two principal domains: speech and audio quality estimation and the arithmetic of quaternionic skew-Hermitian forms. In the context of modern machine learning for perceptual quality assessment, Scoreq_nr designates neural models that predict subjective or algorithmic quality scores using only the distorted/processed input, without requiring access to a clean reference. In quaternionic arithmetic, Scoreq_nr computes adelic spinor norm images for lattices in quaternion algebras, determining spinor genera and class numbers via explicit local-to-global invariants.

1. Definitions and Theoretical Foundations

In speech and audio quality, Scoreq_nr denotes metrics and neural network predictors—such as the SCOREQ model family (Ragano et al., 2024, Mack et al., 29 Sep 2025, Faridee et al., 2022)—that estimate perceptual scores (e.g., MOS, MUSHRA, PESQ) solely from the processed signal. The core task is to directly map input features (usually derived from time-frequency representations) to quality estimates without relying on paired ground-truth references. Typical output targets include mean opinion scores (MOS), predicted PESQ, or estimated distributions of human scores.

In algebraic theory, Scoreq_nr(L) is a function that, for a skew-Hermitian lattice L over a quaternion algebra D/Q, returns the structure of local and global spinor norm images. These determine the spinor genus and, for indefinite forms, the class number (Arenas-Carmona et al., 2013).

2. Contrastive and Distributional Losses in Scoreq_nr Models

The speech Scoreq_nr models diverge from classical regression pipelines by integrating contrastive and/or distributional loss objectives. Standard L₂ regression does not enforce the ordinal structure of quality labels in the model's latent space: small MOS differences may be represented as distant vectors, and large MOS gaps may collapse in the embedding (Ragano et al., 2024). To address this, SCOREQ introduces a batch-all triplet loss for contrastive regression: $$L_{\text{triplet}} = \sum_{(a,p,n)\in\text{batch}} \max\{0,\, D(f(g(x_a)),f(g(x_p))) - D(f(g(x_a)),f(g(x_n))) + m\}$$ where $D(·,·)$ is Euclidean distance in embedding space, $m$ is a fixed or adaptive margin, and $(a,p,n)$ are anchor, positive, and negative samples ordered by MOS.

Distributional Scoreq_nr variants further predict the variance, histogram, or sample-specific ratings from the set of human opinions associated with each utterance. Auxiliary losses—such as cross-entropy or Wasserstein distance for histograms—can be combined with weighted MSE losses, leading to higher calibration and improved correlation with ground-truth MOS distributions (Faridee et al., 2022).

3. Model Architectures and Features

The no-reference Scoreq_nr architectural paradigm employs data-driven feature extraction, usually from Mel-spectrograms or raw waveform input. Typical designs include:

• Deep convolutional backbones, often combined with LSTM layers for temporal aggregation (Faridee et al., 2022, Mack et al., 29 Sep 2025).
• Projection heads for contrastive embedding, e.g., a fully connected layer with ReLU activation and optional ℓ₂ normalization (Ragano et al., 2024).
• Regression heads, e.g., a two-layer MLP with sigmoid or linear output for MOS or MUSHRA-score predictions (Mack et al., 29 Sep 2025).
• Optional auxiliary outputs for variance or histogram prediction.

Input preprocessing typically involves framing (e.g., 25 ms window with 10 ms shift), Mel filterbanks (e.g., 26 or 64 bins), dB compression, and optional normalization (Faridee et al., 2022, Mack et al., 29 Sep 2025, Ragano et al., 2024).

4. Empirical Performance and Best Practices

Quantitative studies consistently show that Scoreq_nr models outperform conventional L₂ regression baselines and earlier MOS-prediction approaches. For example, on diverse test sets, contrastive Scoreq_nr delivers Pearson correlation improvements of up to 0.06–0.10 points compared to L₂-trained wav2vec2-based regressors, especially out-of-domain (Ragano et al., 2024). For neural codec evaluation, SCOREQ_nr achieves Pearson ρ ≈ 0.81 and Spearman ρ ≈ 0.74, competitive with or surpassing other non-intrusive metrics (e.g., UTMOS, SHEET_SSQA) (Mack et al., 29 Sep 2025). Scoreq_nr models that predict full score distributions yield stack-ranked RMSE reductions of up to 0.016 and SRCC gains up to 1.0 percentage point (Faridee et al., 2022).

Saturation phenomena in the high-speech-quality regime are observed—Scoreq_nr metrics tend to flatten near the top of the MOS/MUSHRA scale, so reference-based or intrusive metrics should be used for fine discrimination among top-tier codecs (Mack et al., 29 Sep 2025). Fine-tuning with in-domain data can mitigate loss of robustness outside the originally trained distortion domains.

5. No-Reference PESQ and Alternative Domains

The concept of Scoreq_nr also includes PSQA-based estimators for non-intrusive PESQ prediction (Basterrech et al., 2012). Here, a compact feedforward neural network maps real-time network statistics (packet loss rate, mean loss burst size, packet-loss concealment flag) directly to PESQ scores, without reference audio. This model delivers RMSE ≈ 0.3 MOS under standard VoIP conditions, with negligible computational load.

In other modalities, referenceless quality estimation (Scoreq_nr analogues) has been extended to domains such as natural language generation using recurrent neural architectures to predict human judgment scores, with significant performance improvements when synthetic noisy data is introduced for training (Dušek et al., 2017).

6. Algebraic Scoreq_nr: Spinor Norms of Quaternionic Skew-Hermitian Forms

In arithmetic contexts, Scoreq_nr(L) computes the adelic spinor-norm subgroup for a lattice L in a quaternion algebra D/Q (Arenas-Carmona et al., 2013). The process analyzes local invariants at each p-adic or real place:

• At almost all primes, the spinor-norm image is either the full local multiplicative group or an index-2 subgroup, explicitly determined by Jordan decomposition and norm calculations.
• At p = 2, a detailed 6-case classification applies, depending on Jordan factors and parity of norm valuations. The global spinor-norm image is the idele product of local images. The index $[J_Q : Q^\times H_A(L)]$ determines the number of spinor genera, which equals the class number when L is indefinite at infinity.

7. Summary Table: Key Scoreq_nr Model Classes

Domain	Input/Output	Loss/Objective	Noted Performance
Speech NR	Mel/Raw→MOS/Hist	Contrastive triplet, (weighted) MSE, dist. loss	ρ=0.81–0.88, RMSE<0.5
Audio NR	LogMel→MOS/MUSHRA	Pairwise difference, ranking loss	ρ=0.81, τ=0.54
VoIP PSQA	Network stats→PESQ	Feedforward NN, MSE	RMSE≈0.3
NLG QE	MR+output→score	RNN, MSE, data augmentation	Pearson r: 0.27–0.35
Algebra	Lattice→idealic image	Local norm computations, global index	Determines class number

Scoreq_nr, across diverse modalities, enables efficient, robust prediction of quality or algebraic invariants in scenarios with missing or non-matching references by exploiting ordinal, distributional, or structural properties intrinsic to the data or objects under analysis.