SI-SDR: Scale-Invariant Signal-to-Distortion Ratio

• SI-SDR is a scale-invariant metric that objectively quantifies signal fidelity by removing time-invariant scaling effects, making it ideal for audio and biomedical applications.
• It improves the assessment of signal separation, enhancement, and denoising by measuring error orthogonal to the scaled reference, ensuring fairness compared to traditional SDR metrics.
• SI-SDR is employed as a differentiable loss in modern neural models, yielding significant gains in speech, music, and physiological signal processing.

The scale-invariant signal-to-distortion ratio (SI-SDR) is a widely-adopted objective metric for evaluating and optimizing the fidelity of signal separation, enhancement, and denoising algorithms, particularly in speech and music domains. Unlike earlier SDR metrics that are sensitive to global gain, SI-SDR computes the distortion after removing the influence of time-invariant scaling, ensuring that perceptually irrelevant amplitude mismatches do not affect the score. SI-SDR is also crucial as a differentiable loss function for training end-to-end neural architectures, driving advances in separation quality and robustness across multiple modalities. The following sections provide a detailed exposition of SI-SDR: definition and mathematical formulation, relationship to traditional metrics, algorithmic implementations in source separation, role as a training objective, empirical findings across audio and biomedical tasks, and known challenges or limitations.

1. Mathematical Definition and Core Properties

SI-SDR quantifies the distortion between an estimated signal $\hat{s}$ and a reference clean signal $s$ after optimally rescaling the reference:

$$\alpha = \frac{\langle \hat{s}, s \rangle}{\| s \|^2}$$

$$\text{SI-SDR}(s, \hat{s}) = 10 \log_{10} \left( \frac{\| \alpha s \|^2}{\| \hat{s} - \alpha s \|^2} \right)$$

Here, $\alpha s$ is a projection of $\hat{s}$ onto $s$, and the numerator quantifies the energy in the “target” component, while the denominator measures the residual error. This orthogonal projection ensures the error is strictly perpendicular to the scaled reference, achieving scale invariance: if $\hat{s}$ differs from $s$ only by a multiplicative gain, SI-SDR is maximized and insensitive to the gain value (Roux et al., 2018, Ganchev, 29 Mar 2024).

2. Comparison to Traditional SDR and Alternative Metrics

In classical BSS_eval SDR, a full time-invariant FIR filter or arbitrary reference rescaling is permitted, which can artificially mask significant distortions by warping the reference signal (Roux et al., 2018). This practice may inflate SDR scores even for signals with severe spectral deletions or gain manipulations. SI-SDR corrects this by limiting the adjustment to a single scalar, making the error term orthogonal to the reference and robust to gain gaming.

Other metrics include:

• SIR/SAR: SI-SDR can be decomposed into scale-invariant signal-to-interference (SI-SIR) and signal-to-artifacts (SI-SAR) ratios, where SI-SAR has shown better correlation with perception for certain stems in music separation (Jaffe et al., 9 Jul 2025).
• Perceptual metrics (PESQ, STOI): While PESQ and STOI better predict subjective intelligibility, SI-SDR’s computational efficiency and differentiability render it attractive as an optimization criterion (Ganchev, 29 Mar 2024).

3. SI-SDR as a Loss for End-to-End Neural Architectures

Recent neural models in speech separation, enhancement, and denoising employ SI-SDR directly as the training loss, either standalone or combined with auxiliary objectives:

• Speech Separation: Conv-TasNet, Chimera++, SpEx, TF-GridNet, Deep Attractor Networks, and other models maximize SI-SDR measured on time-domain outputs, yielding substantial gains in separation quality across the WSJ0-2mix and related benchmarks (Wang et al., 2018, Heitkaemper et al., 2019, Xu et al., 2020, Wang et al., 2022).
• Biomedical Signal Denoising: In photoplethysmography (PPG) denoising, SI-SDR loss complements MSE to enforce scale-invariant waveform fidelity, crucial for retaining physiological features (e.g., heart rate) (Chiu et al., 13 Oct 2025).
• Multi-Channel and Latent Space Optimizations: SI-SDR is adapted to multi-channel contexts (e.g., adaptive reference selection (Dai et al., 5 Jun 2024)) or latent-code loss formulations (e.g., two-step separation where maximizing SI-SDR in latent space lower-bounds time-domain performance (Tzinis et al., 2019)).

SI-SDR-based Loss Examples

Framework	Loss Formulation	Context
Conv-TasNet	$-\,\text{SI-SDR}(\text{ref},\text{output})$	Speech separation
DPNet	$L = L_{MSE} + \lambda_1 L_{SI-SDR}$	PPG denoising
SpEx, TF-GridNet	$$-\sum_c 10 \log_{10} (\|\alpha s^{(c)}\|^2 / \|\alpha s^{(c)} - \hat{s}^{(c)}\|^2)$$	Speaker separation

Such loss functions are differentiable, enable permutation-invariant training, and are compatible with adaptive algorithms (Nakajima et al., 2018, Kolbæk et al., 2019).

4. Empirical Impact and Benchmarking Results

SI-SDR is the primary metric for reporting separation and enhancement performance in major benchmarks:

• Speech Separation (WSJ0-2mix): Integrating iterative phase reconstruction inside training and using novel mask activations (e.g., convex softmax) yields SI-SDR improvements from ∼11 dB up to 12.6 dB (Wang et al., 2018).
• Music Source Separation (MUSDB18): While SI-SDR and SDR predict quality for vocals, SI-SAR and embedding-based metrics have better concordance for drums and bass stems (Jaffe et al., 9 Jul 2025).
• Speaker Extraction: Multi-scale and multi-task time-domain networks achieve SI-SDR values up to 14.6 dB, outperforming frequency-domain or i-vector-based approaches (Xu et al., 2020).
• Biomedical Signals: DPNet achieves lowest MSE (6.7×10⁻³), highest cosine similarity (0.961), and HR-MAE of ∼1 bpm, demonstrating that SI-SDR enforced denoising retains signal morphology for clinical endpoints (Chiu et al., 13 Oct 2025).

5. Adaptations and Extensions

SI-SDR has been extended or adapted for specific contexts:

• Masking-Based Multi-Channel Enhancement: Adaptive reference channel selection based on highest output SI-SDR is superior to fixed-channel schemes, improving training on array data with distributed sources (Dai et al., 5 Jun 2024).
• Modified SI-SDR (mSI-SDR): For binaural signals, mSI-SDR computes joint distortion across left and right channels via concatenation and omits explicit scaling, enabling better balance between enhancement and spatial ambience (Hsu et al., 2023).
• VAD-Masked SI-SDR (mSI-SDR): In multi-task architectures, mSI-SDR incorporates voice activity detection masks to focus emphasis on speech-active frames, yielding superior VAD performance and real-time suitability (Tan et al., 2020).
• Convolutive-Invariant SDR (CI-SDR): In multi-channel reverberant environments, a short FIR filter replaces the scalar scaling in SI-SDR to permit invariance to channel-dependent convolution, markedly improving ASR error rates over conventional SI-SDR loss (Boeddeker et al., 2020).

6. Challenges, Limitations, and Evaluation Pitfalls

Despite its compelling properties, SI-SDR has notable limitations:

• Noisy References: When clean references for training and evaluation contain additive noise, SI-SDR is provably upper-bounded by reference SNR, regardless of separation quality (Jepsen et al., 20 Aug 2025). Models can be incentivized to reproduce noise, leading to higher SI-SDR but degraded perceptual quality. The negative correlation between SI-SDR and perceived noisiness (as evaluated by NISQA.v2) underscores that higher SI-SDR does not always reflect cleaner outputs.
• Perceptual Alignment: SI-SDR may not always match human perception, particularly in musical separation for stems where spatial artifacts dominate (e.g., vocals), or masking and artifacts drive perceptual quality (drums, bass) (Jaffe et al., 9 Jul 2025).
• Reference Channel Ambiguity: In multi-channel frameworks, reference selection is non-trivial; dynamic SI-SDR-based channel selection mitigates suboptimal results but requires per-example decision logic (Dai et al., 5 Jun 2024).
• Real-Time and Low-Latency Constraints: Framewise adaptations (as in mSI-SDR) are needed for SI-SDR to be meaningful in streaming scenarios.

7. Conclusion and Future Outlook

The scale-invariant signal-to-distortion ratio is a cornerstone metric for objective assessment and training of signal separation, enhancement, and denoising algorithms. Its mathematical rigor, differentiability, and invariance to time-invariant scaling have directly enabled key advances in audio processing, biomedical signal analysis, and array-based enhancement. SI-SDR continues to evolve—through modifications for multi-channel, joint-task, and real-time contexts—and is subject to ongoing scrutiny regarding perceptual alignment, especially with noisy ground truths and stem-specific behaviors. Future research directions include development of SI-SDR-compatible metrics that better reflect subjective quality, mechanisms for reference cleaning or noise-robust evaluation, and hybrid metric strategies combining SI-SDR with perceptual embedding measures.