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EquiSELD: Efficient training of equivariant sound event localization and detection networks
Goksenin Yuksel, Marcel van Gerven, Kiki van der Heijden
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Summary
The paper introduces EquiSELD, an equivariant attention network for Sound Event Localization and Detection (SELD) that processes First-Order Ambisonics (FOA) signals. It leverages exact O(3) symmetry (rotations and reflections) by processing paired streams of invariant scalars and equivariant intensity vectors. EquiSELD outperforms prior equivariant networks (like CGNet) and non-equivariant baselines (like SELDNet) on both simulated (TAU2021) and real-world (STARSS23) datasets, achieving superior performance with significantly fewer parameters and training costs.
Entities (10)
Relation Signals (7)
EquiSELD → evaluatedon → TAU2021
confidence 95% · We evaluate our method on two established SELD tasks: TAU2021... EquiSELD achieves a summary SELD score of Es=0.40
EquiSELD → evaluatedon → STARSS23
confidence 95% · STARSS23 is a recorded dataset... EquiSELD achieves a summary SELD score of Es=0.56
First-Order Ambisonics → exhibits → O(3) equivariance
confidence 95% · First-order Ambisonics (FOA) signals exhibit exact O(3) symmetry: The rotation or reflection of the FOA signal modifies the direction of arrival
EquiSELD → uses → O(3) equivariance
confidence 95% · EquiSELD... is an equivariant attention network... processing first-order Ambisonics as paired streams of O(3)-invariant scalars and equivariant intensity vectors
EquiSELD → outperforms → SELDnet
confidence 90% · EquiSELD additionally surpasses the performance of non-equivariant SELD networks of a similar size
EquiSELD → outperforms → CGNet
confidence 90% · EquiSELD outperforms prior equivariant networks... EquiSELD has 2.9x fewer parameters and uses approximately 19x less GPU hours than CGNet-STS
CGNet → uses → SO(3) equivariance
confidence 90% · CGNet-STS is an existing SELD network that incorporates equivariance to SO(3) rotations
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Abstract
Abstract:First-order Ambisonics (FOA) signals exhibit exact O(3) symmetry: The rotation or reflection of the FOA signal modifies the direction of arrival of the sound sources, while preserving the sound sources themselves. Prior attempts to utilize this spatial symmetry of FOA to improve the efficiency and robustness of sound event detection and localization (SELD) systems either learned only an approximation of the symmetry through rotation-based augmentation or relied on computationally expensive methods to integrate equivariance. Furthermore, prior work focused exclusively on SO(3) equivariance , leaving the potential of incorporating O(3) equivariance for SELD tasks unclear. To address these limitations, we developed EquiSELD. This equivariant attention network processes first-order Ambisonics as paired streams of O(3)-invariant scalars and equivariant intensity vectors, producing an invariant activity magnitude and an equivariant DOA with a Multi-ACCDOA readout. To compare the impact of O(3) versus SO(3)-equivariance, we designed a matched SO(3)-only variant. EquiSELD outperforms prior equivariant networks on both simulated scenes with measured RIRs and recordings of real-world sound scenes at a fraction of the training cost. EquiSELD additionally surpasses the performance of non-equivariant SELD networks of a similar size on the simulated real-world sound scenes and achieves competitive performance on the real-world sound scenes.
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- Source: https://arxiv.org/abs/2609.23156v1
- Canonical: https://arxiv.org/abs/2609.23156v1
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EquiSELD: Efficient training of equivariant sound event localization and detection networks Goksenin Yuksel Marcel van Gerven Kiki van der Heijden Abstract First-order Ambisonics (FOA) signals exhibit exact O(3)O(3) symmetry: The rotation or reflection of the FOA signal modifies the direction of arrival of the sound sources, while preserving the sound sources themselves. Prior attempts to utilize this spatial symmetry of FOA to improve the efficiency and robustness of sound event detection and localization (SELD) systems either learned only an approximation of the symmetry through rotation-based augmentation or relied on computationally expensive methods to integrate equivariance. Furthermore, prior work focused exclusively on SO(3)SO(3) equivariance , leaving the potential of incorporating O(3)O(3) equivariance for SELD tasks unclear. To address these limitations, we developed EquiSELD. This equivariant attention network processes first-order Ambisonics as paired streams of O(3)O(3)-invariant scalars and equivariant intensity vectors, producing an invariant activity magnitude and an equivariant DOA with a Multi-ACCDOA readout. To compare the impact of O(3)O(3) versus SO(3)SO(3)-equivariance, we designed a matched SO(3)SO(3)-only variant. EquiSELD outperforms prior equivariant networks on both simulated scenes with measured RIRs and recordings of real-world sound scenes at a fraction of the training cost. EquiSELD additionally surpasses the performance of non-equivariant SELD networks of a similar size on the simulated real-world sound scenes and achieves competitive performance on the real-world sound scenes. Index Terms: Sound event localization and detection, Ambisonics, Group equivariant networks †address: 1Dept. of Machine Learning and Neural Computing, Donders Institute, Radboud University, Nijmegen, the Netherlands 2Mortimer B. Zuckerman Institute, Columbia University, New York, USA 1 Introduction Sound Event Localization and Detection (SELD) provides a joint spatio-temporal characterization of acoustic scenes, supporting applications ranging from computational auditory scene analysis to immersive audio and smart environments [14, 21]. First-order Ambisonics (FOA) has become the de facto standard input for SELD systems [6, 19, 22]. The mathematical definition of the FOA signal ensures that the sound scene’s directional components transform equivariantly under any rotation or reflection of the scene, while its omnidirectional component (W) remains invariant. This spatial equivariance is a natural candidate to improve the efficiency and robustness of SELD systems [17]. Yet, most SELD systems process FOA with standard CNNs or Transformers that do not encode spatial equivariance directly, instead relying on rotational data augmentation to inject the spatial symmetry statistically [13, 20, 6]. However, this approach consumes data and computational resources to learn a constraint that could be hard-coded and, even though such rotational transformations are physically exact in the Ambisonic domain, the trained network only approximates the desired symmetry. Group-equivariant networks address this limitation by incorporating equivariance into the network directly. Sato et al. [17] employed Clebsch–Gordan product networks [9] to integrate SO(3)SO(3) rotation equivariance into a unified rotation, scale, and time-translation framework. However, their architecture does not take into account reflection equivariance, is computationally expensive, and cannot represent overlapping instances of the same sound class (common in real-world acoustic scenes, [19, 15]). Outside the audio domain, equivariant attention networks have gained traction for processing sets of geometric features, combining the expressivity of transformers with exact symmetry guarantees [11, 5] and building on the permutation-equivariant set attention of [10]. A complementary line of work achieves equivariance through typed scalar–vector streams, in which invariant scalars gate equivariant vector channels via their norms [4, 7]. Inspired by this direction, we propose, to the best of our knowledge, the first SELD network that is exactly equivariant to the full orthogonal group O(3)O(3). EquiSELD11 1 Code&Models: https://github.com/labhamlet/equiSELD is an equivariant attention network, processing each time–mel token as a pair of typed streams. Each token comprises O(3)O(3)-invariant scalars and equivariant first-order intensity vectors. By computing attention weights and all nonlinearities solely from invariant scalars, and mixing vector channels via bias-free linear maps, we achieve exact equivariance. Our network accomodates overlapping sound sources of the same type, outputting Multi-ACCDOA [18] activity as an invariant magnitude vector and DOA as an equivariant unit vector. We furthermore isolate O(3)O(3)’s contribution with a matched SO(3)SO(3)-only variant. The results demonstrate that EquiSELD produces exactly equivariant predictions, surpasses the performance of an existing SO(3)SO(3)-equivariant network [17] with 2.9×2.9× fewer parameters and 19×19× less training time, and obtains a lower SELD score than SELDNets trained with and without exact O(3)O(3) data augmentation. 2 Method Figure 1: EquiSELD architecture. Five invariant scalar features (orange) and two equivariant intensity vectors (blue) are processed with frequency attention, attention pooling, and attention across time and tracks. The network produces multi-ACCDOA outputs for C sound classes for K tracks of which direction estimates are equivariant. ISAB = induced set attention block; PMA = pooling by multihead attention [10]. 2.1 Invariant and Equivariant FOA Features First-order Ambisonics (FOA) comprises an omnidirectional channel W and three directional channels =[X,Y,Z]⊤B=[X,Y,Z] . Under rotations or reflections Q∈O(3)Q∈ O(3), W is unchanged and ↦QB QB. For each time–frequency bin of an STFT (1024-sample window, 20 ms hop), we compute the complex intensity =W∗i=W^*B, its active and reactive components a=ℜi_a= \i\ and r=ℑi_r= \i\, and channel powers EW=|W|2E_W=|W|^2 and EB=∥2E_B= ^2. Features are pooled to 64 mel bands, with ⟨⋅⟩m · _m denoting the weighted frequency sum under the m-th mel filter. The normalized intensities are p,m=⟨p⟩mE¯m+ϵ,p∈a,r,I_p,m= _p _m E_m+ε, p∈\a,r\, (1) where E¯m=⟨12(EW+13EB)⟩m E_m= 12(E_W+ 13E_B) _m is the mean band energy and ϵ=10−8ε=10^-8 is used throughout for numerical stability. Each time–mel bin contains two equivariant vectors, stacked as rows, and five invariant scalars: m _m =[a,m⊤r,m⊤], = bmatrixI_a,m \\ I_r,m bmatrix, (2) m _m =[ℓW,ℓB,∥a,m∥,∥r,m∥,cm]⊤. = [ _W, _B, _a,m , _r,m ,c_m ] . Here ℓX=log(⟨EX⟩m+ϵ) _X= ( E_X _m+ε), X∈W,BX∈\W,B\, are log band energies, and cmc_m is the cosine similarity between a,mI_a,m and r,mI_r,m 2.2 Equivariant Token Blocks A token (,)(s,V) pairs dsd_s scalar channels with dvd_v vector channels, stored as rows j⊤v_j of V. The group action is Q⋅(,)=(,Q⊤)Q·(s,V)=(s,VQ ). A block F is equivariant if F(Q⋅)=Q⋅F()F(Q·T)=Q· F(T) for every Q∈O(3)Q∈ O(3) and every token collection T . Input tokens (ds=5d_s=5, dv=2d_v=2) are mapped to higher dimension by a scalar linear layer and a vector map (3). Throughout, (⋅)W_(·) and U are learnable real matrices; σ(⋅)σ(·), diag(⋅) (·), [⋅;⋅][\,·\,;\,·\,], and ∥⋅∥F · _F denote the logistic sigmoid, a diagonal matrix, vertical concatenation, and the Frobenius norm. Vector linear maps. Vector channels are mixed without bias: ℒv()=v.L_v(V)=W_vV. (3) The same weights act on all three spatial coordinates, giving ℒv(Q⊤)=ℒv()Q⊤L_v(VQ )=L_v(V)Q . Channel norms. The per-channel norms ()=[∥1∥2+ϵ,…,∥dv∥2+ϵ]⊤. ν(V)= [ _1 ^2+ε,…, _d_v ^2+ε ] . (4) satisfy (Q⊤)=() ν(VQ )= ν(V). Within the token blocks, vector information enters the scalar stream only through these invariant norms. Normalization. Scalars use LayerNorm [2], denoted LNLN. Vectors are normalized by their RMS channel norm [7] and rescaled by learnable channel gains γ: v()=diag()dv−1∥F2+ϵ.N_v(V)= ( γ)V d_v^-1 _F^2+ε. (5) Since ∥Q⊤∥F=∥F _F= _F, the denominator is invariant and v(Q⊤)=v()Q⊤N_v(VQ )=N_v(V)Q . Gated feed-forward block. An MLP produces scalar updates and one gate per vector channel: ~ V =ℒv,1(v()), =L_v,1 (N_v(V) ), (6) =GELU(1[LN();(~)]+1), =GELU (W_1[LN(s); ν( V)]+b_1 ), [Δ;] [ ;\,g] =2+2, =W_2h+b_2, ′ =+Δ, =s+ , ′ =+ℒv,2(diag(σ())~). =V+L_v,2 ( (σ(g)) V ). where ℒv,1,ℒv,2L_v,1,L_v,2 follow (3), and 1,2b_1,b_2 are learnable scalar biases. Both Δ and g are invariant. Applying each gate uniformly to a vector’s three coordinates therefore preserves equivariance, as do the vector maps and residual addition. Equivariant attention. Following equivariant attention networks [5], we compute attention weights from invariant token descriptors: ϕi=LN(i)+(v(i)). φ_i=LN(s_i)+U\, ν (N_v(V_i) ). (7) For one head with query/key width dhd_h, let i=qϕiq_i=W_q φ_i and j=kϕjk_j=W_k φ_j. The attention weights are Aij=softmaxj(i⊤jdh).A_ij=softmax_j\! ( q_i k_j d_h ). (8) Here i indexes query tokens and j indexes key/value tokens. The same weights update both streams: i′ _i =i+os∑jAijvsϕj, =s_i+W_o^s _jA_ijW_v^s φ_j, (9) i′ _i =i+ℒov(∑jAijℒv(v(j))). =V_i+L_o^v ( _jA_ijL_v^v (N_v(V_j) ) ). The matrices vs,osW_v^s,W_o^s are scalar value and output projections, and ℒv,ℒovL_v^v,L_o^v are their vector counterparts (3). Because ϕi φ_i is invariant, queries, keys, and attention weights are unchanged under Q. For vector values j=ℒv(v(j))Z_j=L_v^v (N_v(V_j) ), aggregation satisfies the equivariance constraint. Channel concatenation, residual addition, and composition preserve this property, so all blocks above are equivariant to rotations and reflections. 2.3 SELD Architecture EquiSELD adapts Set Transformer attention [10] to paired scalar–vector features, using ds=128d_s=128, dv=32d_v=32, and 2.25M parameters. Figure 1 specifies layer order, token counts, and block repetitions. Temporal vector convolutions are bias-free and share kernels across Cartesian components. We add learnable mel-band positional embeddings to the scalar stream, shared across time. Frequency attention and pooling follow the induced set attention block (ISAB) and pooling by multihead attention (PMA), respectively. ISAB/PMA seeds have learnable scalar components only, with seed vectors fixed at zero. Time–track attention operates across time within each track and across tracks within each frame. Output heads combine invariant activities with normalized equivariant directions into multi-ACCDOA outputs [18] for C classes and K tracks per class. 3 Experiments Table 1: SELD performance on STARSS23 and TAU2021, with training cost on TAU2021 (single A100). For STARSS23, “Det.” is the number of the 13 classes the model detects and the parenthesised LE is the class-averaged localization error restricted to those classes. We show the jackknife estimates of the ℰSE_S under the test-set recordings. STARSS23 TAU2021 Model Par. GPU h↓ ER↓ F↑ LE (det.)↓ LR↑ Det.↑ ℰS↓E_S ER↓ F↑ LE↓ LR↑ ℰSE_S↓ SELDNet 0.75M 0.16 0.74 13.7 55.7∘55.7 (33.1∘33.1 ) 26.5 11 0.66 [0.61, 0.69] 0.68 27.5 35.8∘35.8 42.8 0.55 [0.53, 0.56] SELDNet 2.25M 0.18 0.75 16.1 51.9∘51.9 (28.6∘28.6 ) 29.5 11 0.65 [0.62, 0.68] 0.68 29.8 22.3∘22.3 46.6 0.51 [0.50, 0.52] SELDNet+ 2.25M 0.83 0.57 25.4 68.2∘68.2 (18.5∘18.5 ) 30.0 09 0.60 [0.57, 0.63] 0.61 40.5 32.3∘32.3 49.4 0.47 [0.46, 0.49] CGNet-STS-MultiACCDOA 6.53M 14.5 0.80 11.3 78.3∘78.3 (33.1∘33.1 ) 24.5 09 0.72 [0.68, 0.74] 0.73 30.1 25.3∘25.3 52.0 0.51 [0.50, 0.53] CGNet-STS 6.28M 12.1 – – – – – – 0.79 26.2 31.9∘31.9 55.1 0.54 [0.52, 0.55] EquiSELD 2.25M 0.77 0.72 24.9 34.3∘34.3 (22.1∘22.1 ) 39.7 12 0.56 [0.51, 0.59] 0.64 49.8 16.4∘ 63.9 0.40 [0.38, 0.41] EquiSELD-SO(3) 2.25M 0.77 0.69 21.4 31.7∘31.7 (19.4∘19.4 ) 30.9 12 0.58 [0.50, 0.61] 0.60 51.3 15.9∘15.9 63.9 0.38 [0.37, 0.40] AttnSELD 2.25M 0.77 0.97 13.6 52.6∘52.6 (29.5∘29.5 ) 30.6 11 0.70 [0.68, 0.74] 0.71 42.6 17.9∘17.9 57.6 0.45 [0.44, 0.47] AttnSELD+ 2.25M 0.85 0.82 25.1 33.6∘ (21.5∘21.5 ) 44.9 12 0.58 [0.51, 0.61] 0.67 42.2 18.8∘18.8 56.7 0.45 [0.43, 0.46] 3.1 Setup Datasets. We evaluate our method on two established SELD tasks: TAU2021 [15] and STARSS23 [19]. TAU2021 is a synthetically generated dataset based on real recorded room impulse responses (RIRs) featuring moving sound sources and interfering noise (sampling rate = 24 kHz). It comprises six folds, each containing approximately 100 minutes of audio. We use folds 1–4 for training, fold 5 for validation, and fold 6 for testing. STARSS23 is a recorded dataset of real-world listening scenes that presents a more challenging acoustic environment with up to six overlapping sound events (sampling rate = 24 kHz). All STARSS23 systems are trained on the real dev-train split only. Training and evaluation. We adopted the Multi-ACCDOA output format with K=3K=3 tracks for each of C sound classes and trained models on 5 s segments using the ADPIT loss [18] and a detection threshold of 0.50.5. We trained models for 100 epochs on TAU2021 and for 200 epochs for STARSS23. We used a longer training regime for STARSS23 to avoid underfitting. Evaluation metrics. Following the conventional evaluation of SELD systems, we report the macro-averaged, location-dependent error rate (ER20∘ER_20 ) and F-score (F20∘F_20 ). Here, a prediction counts as a true positive only if the DOA falls within 20∘20 of the target. We furthermore report the class-dependent localization error (LECDLE_CD) in degrees and localization recall (LRCDLR_CD) [16]. We use the ℰSE_S to quantify overall SELD performance: ℰS=14[ER20∘+(1−F20∘)+LECD/180∘+(1−LRCD)].E_S= 14[ER_20 +(1-F_20 )+LE_CD/180 +(1-LR_CD)]. (10) 3.2 Model space and training parameters EquiSELD. We optimized EquiSELD with AdamW [12] (lr=3×10−4lr=3× 10^-4, β=(0.9,0.95)β=(0.9,0.95), and weight decay 0.050.05). We applied gradient clipping at 1.01.0 and used a linear warmup over 2000 steps followed by cosine decay down to 1%1\% of the peak learning rate. EquiSELD-SO(3). To quantify the potential benefit of O(3)O(3) equivariance of EquiSELD in comparison with SO(3)SO(3) equivariance, we introduce an SO(3)SO(3)-equivariant version of the model by mixing a third, axial-vector channel — the cross product a×rv_a×v_r — into the vector stream. As the product of two polar vectors, this feature transforms as ↦det(Q)Qv (Q)\,Qv. This is identical to a polar vector under proper rotations, but sign-flipped under reflection. It therefore reduces the guaranteed symmetry from O(3)O(3) to SO(3)SO(3). The variant is otherwise identical to EquiSELD in architecture, parameter count, and training recipe. AttnSELD. To compare equivariant and non-equivariant parameterizations at matched model size, we construct AttnSELD. This architecture is similar to EquiSELD but releases the constraints that guarantee equivariance through four modifications: (i) vector channel mixings gain learned 3×33× 3 spatial mixing matrices and per-channel biases; (i) attention logits and the feed-forward block read raw three-dimensional vector components rather than their norms; (i) vector normalization permits per-component affine transformations; and (iv) inducing points and the pooling head carry learnable, non-zero vector parts. Since exposing raw components increases the input dimensions of the scalar projections, we adjust the vector width to dv=12d_v=12 (dv=32d_v=32 in EquiSELD) resulting in nearly 2.25 M parameters as in EquiSELD. AttnSELD+. To assess to what extent the non-equivariant version of EquiSELD can learn equivariance through O(3)O(3) augmentation, we introduce AttnSELD+. Our augmentation implementation follows the channel-first [13]. We sample an element from O(3)O(3) for each instance in the batch and apply on-the-fly augmentations. It is continuous over the O(3) elements. SELDNet. We adopt the SELDNet variant that served as the official DCASE 2023 Task 3 baseline [19]. The CRNN of [1], extended with two multi-head self-attention layers and a Multi-ACCDOA output [18]. This model first extracts a concatenated mel-spectrogram and normalized active intensity vector (IV) using a CNN front-end, then processes the resulting frame-wise features with a GRU followed by multi-head self-attention. We train two variants: the conventional recipe with 0.75 M parameters, and a larger parameter-matched version with 2.25 M parameters (obtained by increasing the GRU hidden dimension from 128 to 240). All SELDNet variants follow the published training recipe using the Adam optimizer [8] and a learning rate of 10−310^-3. We also introduce SELDNet+, which follows the same data augmentation schema as the AttnSELD+. CGNet-STS and CGNet-STS-MultiACCDOA. CGNet-STS is an existing SELD network that incorporates equivariance to SO(3)SO(3) rotations, time translations, and scale changes through Clebsch–Gordan product networks [9]. We re-implemented the architecture from the original paper and verified functional equivalence against the authors’ reference code22 2 https://github.com/nttrd-mdlab/group-equiv-seld. However, the original implementation outputs separate DOA and SED predictions, which prevents the model from handling overlapping sound events of the same class. We therefore adapted CGNet-STS to the Multi-ACCDOA framework by widening its final layer to emit K×CK× C first-order direction vectors (where K is the number of tracks and C is the number of classes) and widening the GRU readout to produce matching logits zk,cz_k,c. The final Multi-ACCDOA vector is then formed as k,c=σ(zk,c)k,c/∥k,c∥a_k,c=σ(z_k,c)\,u_k,c/ _k,c . As we found that the Multi-ACCDOA version performs better on the synthetic scenes than the original implementation (Section 4), we continued with this version for real-world STARSS23 dataset. 4 Results 4.1 SELD Results On the simulated scenes with measured RIRs (TAU2021), EquiSELD achieves a summary SELD score of ℰS=0.40E_S=0.40, outperforming all versions of SELDNet (lowest ℰS=0.47E_S=0.47) and CGNet-STS (lowest ℰS=0.51E_S=0.51, Table 1). Crucially, EquiSELD has 2.9×2.9× fewer parameters and uses approximately 19×19× less GPU hours than CGNet-STS to achieve this performance, showcasing the effectiveness of equivariant parametrization via the equivariant transformers framework. Finally, on the more challenging and limited real-world scenes (STARSS23), EquiSELD achieves a summary SELD score of ℰS=0.56E_S=0.56, once more surpassing all versions of SELDNet (lowest ℰS=0.60E_S=0.60) and CGNet-STS (ℰS=0.72E_S=0.72). Rotation vs. rotation and reflection equivariance. EquiSELD-SO(3) achieves comparable summary SELD scores as EquiSELD under O(3)O(3) equivariance for both the synthetic and real-world scenes (Table 1, see overlapping jackknife confidence intervals). These results indicate that the gain in effectiveness of O(3)O(3) equivariance (rotation and reflection equivariance) is comparable to the gain of effectiveness of SO(3)SO(3) equivariance (rotation equivariance only). EquiSELD without equivariance constraints. AttnSELD performs substantially worse on the synthetic scenes (TAU2021, ℰS=0.45E_S=0.45) as well as the real-world scenes (STARSS23, ℰS=0.70E_S=0.70, Table 1). These findings highlight the impact of the equivariance constraints on model performance. Training AttnSELD+ with O(3)O(3) augmentation improved performance on the real-world scenes at a small increase of the training cost, achieving a similar summary score as EquiSELD (ℰS=0.58E_S=0.58, Table 1). However, performance of AttnSELD+ did not improve on the synthetic scenes (ℰS=0.45E_S=0.45, Table 1). Undetected classes. On STARSS23, LECDLE_CD assigns 180∘180 to undetected classes, coupling class coverage with localization precision. SELDNet+ and CGNet-STS each detect nine of 13 classes, yielding overall errors of 68.2∘68.2 and 78.3∘78.3 , respectively, versus 18.5∘18.5 and 33.1∘33.1 over detected classes. These restricted averages involve different class subsets and are therefore not directly comparable. 4.2 Orientation Robustness We evaluate the spatial equivariance of each model on a sweep of 74 elements of O(3)O(3): 12 Fibonacci-sphere directions with 3 in-plane rolls each, plus the identity, paired with their parity partners −Q-Q. On test-fold recordings comparing f(Qx)f(Qx) against Qf(x)Qf(x), EquiSELD achieves exact equivariance (maximum deviation for EquiSELD = 2.8×10−72.8× 10^-7). Table 2 shows that SELDNet’s scores range from 0.65 to 0.82 and AttnSELD’s range from 0.70 to 0.86. Variants with O(3)O(3) augmentations exhibit smaller residual variation as well, but EquiSELD attains this without augmentations and gives the best absolute performance. Table 2: SELD score over O(3) elements on STARSS23. Model Mean Std Min Max SELDNet 0.76 ± 0.03 0.65 0.82 SELDNet+ 0.60 ± 0.01 0.60 0.62 AttnSELD 0.75 ± 0.04 0.70 0.86 AttnSELD+ 0.58 ± 0.01 0.57 0.59 EquiSELD (ours) 0.56 ± 0.00 0.56 0.56 Figure 2: Data scaling. Shaded areas are 95% confidence intervals. 4.3 Data Scaling To compare how the effectiveness of inherent network equivariance scales with data quantity, we depict performance on TAU2021 as the training set increases from one to four folds (100100–400400 min, Figure 2). EquiSELD improves monotonically from ℰS=0.55E_S=0.55 to ℰS=0.40E_S=0.40, while SELDNet+, AttnSELD and AttnSELD+ show less improvement. This graph shows that for simulated scenes with measured RIRs, EquiSELD outperforms AttnSELD+, especially at larger training-set sizes. 5 Conclusion We proposed EquiSELD, an O(3)O(3)-equivariant attention network that encodes the rotation and reflection symmetry. The inherent equivariance of this low-parameter network requires no O(3)O(3) augmentation to build rotation and reflection robustness. Experimental results show that EquiSELD outperforms a prior equivariant SELD network and the SELD baseline system on both synthetic and real-world scenes. Moreover, EquiSELD has 2.9×2.9× fewer parameters and requires 19×19× less training time than the preceding equivariant network. Finally, we show that the benefit of inherent equivariance scales with data size, consistent with findings on scaling equivariant transformers more broadly [3]. Future work will explore softening this hard constraint — through soft-equivariance penalties — to exploit the elevation priors that strict O(3)O(3) and SO(3)SO(3) symmetry cannot, and extending the network to higher-order Ambisonics. 6 Acknowledgments This work used the Dutch national e-infrastructure with the support of the SURF Cooperative using grant no. EINF-14624. References [1] S. Adavanne, A. Politis, J. Nikunen, and T. Virtanen (2019) Sound event localization and detection of overlapping sources using convolutional recurrent neural networks. IEEE Journal of Selected Topics in Signal Processing 13 (1), p. 34–48. External Links: Document Cited by: §3.2. [2] J. L. Ba, J. R. Kiros, and G. E. Hinton (2016) Layer normalization. Note: arXiv preprint arXiv:1607.06450 External Links: 1607.06450, Link Cited by: §2.2. [3] J. Brehmer, S. Behrends, P. D. Haan, and T. Cohen (2025) Does equivariance matter at scale?. Transactions on Machine Learning Research. 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