Source-linked AI summary
Signed Wasserstein distance for room impulse response comparison
Linda Fabiani, Sebastian J. Schlecht, Filip Elvander
TL;DR
Room impulse responses are sparse, signed signals whose reflections can shift with small parameter changes, challenging point-wise similarity measures. The paper connects EDC comparison to optimal transport and proposes a signed Wasserstein distance for raw RIRs. Its perturbation studies report useful geometric loss landscapes and improved room-dimension accuracy over considered baselines, with promising room-parameter optimization performance.
Problem
RIR reflection shifts can make point-wise losses poorly informative, complicating similarity assessment for these structured signals.
Source: §1 · p. 1Method
The paper relates equal-total-energy-normalized EDC distances to Wasserstein-1 and introduces a signed Wasserstein distance that compares raw RIR waveforms while retaining sign structure.
Source: §1 · pp. 1–2Results
Takeaways & Limitations
The metric offers a transport-based way to compare raw RIR temporal structure while retaining signed reflections rather than treating interference as additive energy.
Source: §1 · p. 2, §4 · p. 2, §4.1 · p. 3The method focuses on RIR shape rather than scale, so normalization discards absolute gain information.
Source: §4.2 · p. 3
Abstract
from arXiv · showhide
The nature of room impulse response signals is often adverse to a clear evaluation of similarity: they are sparse, rapidly varying, signed signals resulting from the complex interaction of room reflections and phase relationships. Therefore, conventional point-wise metrics may be sensitive to small temporal shifts and provide poor loss surfaces for optimization. In this paper, we propose an optimal transport-based distance for RIR similarity that explicitly accounts for the structure of impulse responses, inspired by the signed Wasserstein distance formulation. Optimal transport provides a natural framework for this problem, by comparing signals based on how their underlying distributions are displaced rather than requiring a one-to-one correspondence. We evaluate the proposed distance against commonly used audio baseline metrics through a series of room-parameter perturbation studies. The results demonstrate the advantage of using the signed Wasserstein metric for geometric evaluation tasks and show a promising performance for room parameter optimization within a differentiable Image-Source model.
1. INTRODUCTION
Room impulse responses combine sparse reflections and reverberant tails, making similarity difficult to assess when small parameter changes shift reflection timing. The paper relates EDC distances to optimal transport and proposes a signed Wasserstein metric for raw RIRs, which performs well in geometric perturbations and room-parameter optimization.
1. INTRODUCTION
- Point-wise losses can become poorly informative when reflections are temporally misaligned, while spectral losses depend on the chosen time-frequency resolution.RIRs contain sparse early peaks and a reverberant tail, and small room-parameter changes can produce large deviations.Source: §1 · p. 1
- Under equal-total-energy normalization, the L1 distance between EDCs equals the Wasserstein-1 distance between their temporal energy distributions.This gives a transport interpretation to a commonly used room-acoustics distance.Source: §1 · p. 2
- The proposed signed Wasserstein-1 distance compares raw signed RIRs while preserving their non-local structure and sign information, unlike an energy-only treatment of interference.The classical Wasserstein-1 formulation does not directly apply to signed signals.Source: §1 · pp. 1–2
- The signed Wasserstein metric provides smooth, informative loss landscapes for geometric perturbations and improves room-dimension accuracy over the considered baselines.The evaluation includes source-position shifts, room-dimension scaling, and room-parameter optimization.Source: §1 · p. 2
2. OPTIMAL TRANSPORT
The section frames optimal transport as comparing distributions by the geometric cost of moving mass. It introduces Wasserstein distances and a one-dimensional expression for Wasserstein-1 based on cumulative distributions.
2. OPTIMAL TRANSPORT
- Optimal transport compares distributions by finding the least-cost way to move one into another, with cost tied to their geometric domain.The Wasserstein-p distance uses powers of Euclidean distance as its transport cost.Source: §2 · p. 2
- In one dimension, Wasserstein-1 has a particularly simple expression involving the cumulative distribution functions of the two measures.The paper identifies Wasserstein-1 and its signed-measure extension as central to the subsequent discussion.Source: §2 · p. 2
3. ROOM ACOUSTIC MODEL
The paper models room impulse responses as signed measures composed of discrete reflections. Their amplitudes and arrival times depend on room and source-receiver conditions, and parameter perturbations affect both.
3. ROOM ACOUSTIC MODEL
- Room impulse responses are modeled as atomic signed measures, with each reflection represented by an amplitude at an arrival time.The amplitudes may be positive or negative.Source: §3 · p. 2
- Reflection amplitudes and arrival times depend on room geometry, source and receiver positions, and physical constraints such as sound speed.The squared RIR is represented as a non-negative measure of instantaneous energy.Source: §3 · p. 2
- Small perturbations to underlying parameters affect both reflection amplitudes and arrival times, but point-wise metrics do not quantify the magnitude of temporal shifts.This motivates a comparison framework explicitly sensitive to temporal displacement.Source: §3 · p. 2
4. PROPOSED METRIC
The paper connects EDC comparison to Wasserstein-1 transport of temporal energy distributions, then introduces a signed Wasserstein-1 metric for comparing raw RIR waveforms. The signed formulation retains non-local comparison while respecting the sign structure of RIRs.
4. PROPOSED METRIC
- The proposed signed Wasserstein-1 metric compares raw RIR waveforms while retaining their sign structure and non-local comparison.It avoids treating interference effects as purely additive energy.Source: §4 · p. 2, §4.1 · p. 3
4.1. Connection Between EDC and Wasserstein-1
- Under equal-total-energy normalization, the L1 distance between EDCs equals the Wasserstein-1 distance between their temporal energy distributions.This gives EDC comparison an optimal-transport interpretation of how acoustic energy is distributed over time.Source: §4.1 · p. 3
4.2. Signed Wasserstein-1 distance
- The paper defines a signed Wasserstein-1 distance using positive and negative parts of signed measures, enabling transport-based comparison of signed RIR signals.The measures are normalized by their total integral, or sum in discrete time, to compare temporal structure while discarding absolute gain information.Source: §4.2 · p. 3
5. NUMERICAL EXPERIMENTS
Controlled perturbation studies compare the proposed distances on synthetic RIRs, and a differentiable Image-Source Model experiment tests parameter recovery. Signed W1 provides the clearest source-position and room-scaling loss profiles, the lowest room-dimension MAE, while ESR gives the lowest absorption MAE.
5. NUMERICAL EXPERIMENTS
- The perturbation studies use synthetic RIRs and compare the proposed metric with linear EDC, ESR, and multi-scale spectral loss.The experiments vary source position and room dimensions; the spectral baseline combines magnitude and log-magnitude distances across STFT resolutions.Source: §5 · p. 3, §5.1 · p. 3
5.1. Source-position shift
- Only signed W1 provides a clear, unique source-position minimum with a smooth increase toward maximum displacement; EDC identifies the minimum but varies little across much of the range.ESR and MSS landscapes are largely uniform, with some circular structures; the passage suggests equidistant source positions and coincident reflection times may explain these patterns.Source: §5.1 · p. 4
5.2. Room scaling variation
- Across uniform room-dimension scaling, signed W1 has the smoothest, most gradual distance profile, whereas MSS and ESR remain roughly constant except near the original dimensions.The passage attributes this profile to transport capturing the approximately common scaling of reflection delays.Source: §5.2 · p. 4, Figure 3 · p. 4
5.3. Room parameters inference
- Signed W1 achieves the lowest room-dimension MAE, while ESR achieves the lowest absorption-coefficient MAE in separate parameter-recovery experiments.The experiments use gradient-based optimization in a differentiable Image-Source Model; joint geometry and absorption optimization is left for future work.Source: §5.3 · p. 4, Table 1 · p. 4
6. CONCLUSION
The paper introduces a signed Wasserstein-based metric for comparing raw RIRs and evaluates it against commonly used point-wise distances. Perturbation studies show an advantageous loss landscape for room-dimension and source-localization tasks, with promising room-dimension optimization results in a differentiable Image-Source Model.
- The signed Wasserstein metric compares raw RIRs using signed reflections, which conventional Wasserstein-1 cannot directly handle.The paper reports robustness against commonly used point-wise distances and promising optimization results for room dimensions in a differentiable Image-Source Model.Source: §6 · p. 4