Beyond Instance-Level Alignment: Dual-Level Optimal Transport for Audio-Text Retrieval
Authors: Wenqi Guo
Shikui Tu
Lei Xu
Venue: ICLR 2026 Poster (2026)
Keywords: Audio-Text Retrieval
Cross-Modal Matching
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: Cross-modal matching tasks have achieved significant progress, yet remain limited by mini-batch subsampling and scarce labelled data. Existing objectives, such as contrastive losses, focus solely on instance-level alignment and implicitly assume that all feature dimensions contribute equally. Under small batches, this assumption amplifies noise, making alignment signals unstable and biased. We propose DART (Dual-level Alignment via Robust Transport), a framework that augments instance-level alignment with feature-level regularization based on the Unbalanced Wasserstein Distance (UWD). DART constructs reliability-weighted marginals that adaptively reweight channels according to their cross-modal consistency and variance statistics, highlighting stable and informative dimensions while down-weighting noisy or modality-specific ones. From a theoretical perspective, we establish concentration bounds showing that instance-level objectives scale with the maximum distance across presumed aligned pairs, while feature-level objectives are governed by the Frobenius norm of the transport plan. By suppressing unmatched mass and sparsifying the transport plan, DART reduces the effective transport diameter and tightens the bound, yielding greater robustness under small batches. Empirically, DART achieves state-of-the-art retrieval performance on three audio-text benchmarks, with particularly strong gains under scarce labels and small batch sizes.

Published as a conference paper at ICLR 2026
BEYOND INSTANCE-LEVEL ALIGNMENT: DUAL- LEVEL OPTIMAL TRANSPORT FOR SUDIO-TEXT RE-
TRIEVAL
Wenqi Guo', Shikui Tu Lei Xu
1 Shanghai Jiao Tong University, Shanghai, China
2 Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ), Shenzhen, China (wonqiguo, tushikui, leixu]es jtu. edu.
ABSTRACT
Cross-modal matching tasks have achieved significant progress, yet remain lim- ited by mini-batch subsampling and scarce labelled data. Existing objectives, such as contrastive losses, focus solely on instance-level alignment and implicitly as- sume that all feature dimensions contribute equally. Under small batches, this assumption amplifies noise, making alignment signals unstable and biased, We propose DART (Dual-level Alignment via Robust Transport), a framework that augments instance-level alignment with feature-level regularization based on the Unbalanced Wasserstein Distance (UWD). DART constructs reliability-weighte marginals that adaptively reweight channels according to their cross-modal con- sistency and variance statistics, highlighting stable and informative dimensions while down-weighting noisy modality-specific ones. From theoretical per- spective, we establish concentration bounds showing that instance-level objectives scale with the maximum distance across presumed aligned pairs, while feature- level objectives are governed by the Frobenius norm the transport plan. By suppressing unmatched mass and sparsifying the transport plan, DART reduces the effective transport diameter and tightens the bound, yielding greater robust- ness under small batches. Empirically, DART achieves state-of-the-arr retrieval performance on three audio-text benchmarks, with particularly strong gains under scarce labels and small batch sizes.
INTRODUCTION
Audio-text retrieval is fundamental cross-modal matching task that supports applications in mul- timedia search (Elizalde et al.) (2019) and content understanding (Oncescu et al. 2024). The key challenge lies in learning aligned representations that capture semantic correspondences between heterogeneous modalities, enabling the retrieval of audio clips given text queries and vice versa. Existing approaches, including learn-to-mated to-match frameworks (Luong et al. 2024 Shi al. Li al. 2019), contrastive learning (Jia et al.) 2021 Radford et al. [2021] Mei et al. 2022 Wu et al. 2023) and triplet losses (Wei et | [202 [ /Zng et al.) 2022), can viewed under unified inverse optimal transport (IOT) perspective (Shi et al. 2023 where paired supervision is used to learn a shared metric between audio and text features.
In practice, this metric is optimized from mini-batches. As the batch size decreases, sampling vari- ance increases, amplifying feature fluctuations and making the learned metric more susceptible to noise and bias. One key reason is the reliance on instance-level similarity, typically measured by Euclidean distance or cosine similarity. Such measures collapse each audio-text pair into a single scalar and implicitly treat all embedding dimensions equally informative. However, audio and text embeddings are heterogeneous: stable semantic cues often coexist with modality-specife noise and transient patterns across dimensions. Such uniform collapse allows few noisy channels domi- nate the scalar score, yielding unstable alignment signals and biased gradients even for semantically matched pairs. Prior weignil- methods (for example, Luong et al. 2024) mitigate this
"Corresooniing authors

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Published as a conference paper at ICLR 2026
by rescaling dimensions, but they still reduce the weighted embeddings to single pairwise score, leaving volatile channels coupled to the learning signal, particularly under small- bbtch training.
This observation motivates moving beyond purely instance-level alignment. To mitigate instabil- ity and bias, we propose DART (Dual-level Alignment via Robust Transport), which augments instance-level alignment with feature-level regularization. At the instance level, DART adopts an 1OT objective to enforce tight alignment between paired audio and text samples. At the feature level, DART treats each embedding channel as matching unit and minimizes the Unbalanced Wasserstein Distance (UWD) between audio and text features. Noisy channels tend to incur larger transport costs, which leads UWD to assign them less mass and thus suppress spurious alignment, while stable semantic channels with smaller costs are preferentially matched. This intuition is sup- ported by our empirical analysis in Appendix (Fig. 2) where injecting synthetic noise into feature channels yields a monotonic increase in their standardized transport cost. Beyond this implicit fil- tering, DART introduces Reliability-Awar it computes a reliability score based variance, kurtosis, and cross-modal correlation. The scores are normalized into reliability-aware marginals in UWD, steering the transport plan toward high- reliability channels that consistently encode stable semantic cues, while downweighting volatile or modality-speciin ones.
Marginals (RAM) as priors in UWD. For each channel,
From theoretical perspective, establish that instance-level and feature-level alignments exhibit fundamentally different concentration behaviors. For standard instance-level alignment, the mini- batch objective aggregates pairwise similarities, and we prove that its concentration bound scales with the maximum distance (Dmax) among positive pairs in the batch. This reliance on an extremal, worst-case quantity explains the sensitivity of vanilla IOT outlier samples and noisy labels. In contrast, our proposed feature-level formulation yields bounds governed by the Frobenius norm of the optimal transport plan (||P*1])) .. By shifting the controlling quantity from a volatile maximum distance to an aggregate norm, DART provides tighter guarantees and greater robustness, particu- larly small-batch regimes where Dmax prone to high variance. Overall, our contributions are threefold:
We introduce DART, a dual-level alignment framework that augments instance-level IOT with structural feature-level regularization, enabling robust cross-modal retrieval by simul- taneously optimizing sample and channel relationships.
We design reliability-aware marginals that incorporate statistical cues (variance, kurtosis, correlation) to reweight feature channels, effectively suppressing modality-speciff noise within the transport process.
We provide theoretical analysis connecting feature-level alignment to tighter concentra- tion bounds. Extensive experiments demonstrate that DART achieves state-of-the-ar per- formance on three benchmarks, showing significant gains under challenging small-batch and limited-label conditions.
PRELIMINARIES
2.1
ENTROPIC OPTIMAL TRANSPORT AND INVERSE OPTIMAL TRANSPORT
Entropic optimal transport (EOT) extends classical OT by adding an entropy regularization term, which improves computational efficiency and yields smooth couplings (Cuturi 2013). Given empir- ical measures μ and v, EOT solves
min NEU(a.))
(1)
where E mx is the ground cost matrix (C d(z,, U,)), -III= H. (In Il. - 1) is the negative entropy, and > 0 is the regularization parameter. The feasible set U U(, v) = (II E
RM mmXn
II1, = IITHIT 1m = y) enforces the marginal constraints.
In cross modsl retrieval, the ground cost is often unknown and must be learned. Inverse optimal transport (IOT) (Dupuy et al. 2016 [i et al. 2019 /Sturrt & Wolfram 2020) parameterizes the cost C (e.g., via neural network) and optimizes 0 such that the induced coupling IT aligns with
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the observed ground truth matching II:
min L (ii II",, where II = arg min NEU(..))
(
(2)
0
where (-Il-) denotes the Kullback-Leibler divergence, and f, E.g are the dual potentials obtained from the Sinkhorn iterations.
2.2 AUDIO-TEXT RETRIEVAL AS 1OT.
Audio-text retrieval aims to align audio clips with their corresponding text captions. Given audio and caption data pairs D = {z2,, 3-) where represents the i-th audio sample and yi the associated text caption, the goal between audio queries and text captions. Specifically, audio samples are encoded via an audio encoder fo(-), while text captions are encoded via a text encoder ge(-). A distance function d(:;)) such Euclidean or cosine distance, is employed to measure the similarity between the audio and text embeddings, denoted as Ci = d(ffa (zi), (WW)))
learn cross- mapping that enables bidirectional retrieval
In the IOT framework, the network is optimized to minimize these distances for semantically matched pairs while maximizing them for unmatched ones, effectively treating the pairwise dis- tance matrix as a parameterized ground cost. During training, the EOT solver (Eq. [ generates coupling matrix (0,6) that encodes the inferred matching relationships. The model is optimized by minimizing the KL divergence between the induced coupling (9.6) and an observed ground-truth matching II. where II 1/n if and 0 otherwise. This objective encourages the model to assign low transport costs to semantically matched pairs while penalizing misalignments.
At inference, given a set of audio samples Xust = { and caption samples Yest {uy }=1"
the retrieval of a caption for query audio x; is performed by selecting the sample that yields the minimum distance (or maximum similarity):
exp (d (fo (xt) 90 (W;)))
arg min
(3)
Ene k= exp (d (fo (xt) 96 (yk)
2.3 LIMITATIONS OF INSTANCE-LEVES 1OT.
While the IOT perspective provides unifying view, it reveals critical limitations in practice. In mini-bated training. the cost matrix is estimated from partial data and aggregates all embedding di- mensions uniformly. This uniform pooling ignores the inherent heterogeneity of audio and text embeddings. For example, for the caption "A drone is whirring", the embedding typically en- codes meaning in distributed manner: some channels (or directions) contribute more to object- identity cues such as drone, while others are more informative about sound-attribute cues such as the whirring acoustic pattern. n practice, while many latent channels effectively represent these sta- ble semantic components, others inevitably encode modality-specific noise or unstable variations.
Concretely, the instance-level distance d(x,, uy) collapses these diverse sub-components through a uniform summation (c.g., d(z,, y₁) = (F - ))a)²). Because of this mandatory aggregation, even a small subset of noisy or high-variance channels can disproportionately inflate the distance for semantically matched pairs. As result, the learned metric becomes biased toward spurious fluctuations rather than true semantic alignment. This intuition is formalized in Section 4 where Theorem D shows that the concentration bound for the instance-level IOT loss is governed by the maximum alignment distance Dmax max =0 d(x;, .₁ Consequently, the learning signal dominated by these inflated outlier pairs, especially in small-batch regimes.
3 DART: DUAL-LEVEL ALIGNMENT VIA ROBUST TRANSPORT
3.1 MINI-BATCH INSTANCE-LEVEL IOT
Given a mini-batch of i audio-text pairs (X', y²), the encoders fo and gs produce embeddings U R kxd, and va E Rxd, The cost matrix is defined as
C ( 1 d( d
4,21 V))))
(4)
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Yoa dimi
Fest timn
Vest dona
Brank
Reliability-Awe Marginals
Semantically
Noise: Prrivaee
lligh Kumeess
Relevuat Clannel
Channel
Channel
Audio
Audio Encoder
Fiom
Auudoo
Audio
High-Reiibbilit
Low-Reiabilit
Low-Relibbllty
Aadio Represeraations
Audio
Foss, dlm)
Fost, Sind
Feat dien
Foar slmnd
large volume of wateer
T
Uniulmeed
boat motor burnes while..
Text Encoder
Text
OT
Test
a creckon
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Text Represcniationns
Festur--Love Cost Maxiis
Tranport Pias P
Figure 1: An overview of the proposed DART framework. DART aligns audio and text modalities through both instance-level optimization using the Inverse Optimal Transport (IOT) objective and feature-level optimization via channel-wise distribution alignment. The latter minimizes the Un- balanced Wasserstein Distance (UWD) with reliability-awan marginals to guide the transport plan toward stable semantic channels while suppressing noisy modality-speciie ones.
where .) :) is distance metric (Euclidean in implementation). An entropy-regularrized OT solver (Sinkhorn) produces coupling (9,6),) and the IOT objective minimizes the divergence to the ground-truth matching:
(o Φ) KLIII" (1.)))
(5)
which reduces to log (T(0)))) under one-to-one alignment.
This mini-batch IOT formulation is widely used and provides baseline for retrieval tasks. How- ever, it estimates costs from partial data and aggregates all embedding dimensions uniformly, which makes it sensitive to batch variance and noisy channels. We therefore extend IOT with feature-level optimization, as detailed next.
3.2 FEATURE-LEVEL DISTRIBUTION ALIGNMENT
Audio and Text Feature-Level Representations. In DART, each feature dimension is treated as an independent distribution across the mini-batch and aligned across the two modalities. Given the audio feature matrix Uh and text feature matrix Vh, the j-th column of these matrices interpreted distribution of the j-th feature across the mini-batch samples. Specifically, for the audio and text modality:
fo (29),
96 (vi),
(+(-))=
v*(r)))
(6)
So (ah))
go (vi)
Here, fo (a:)) denotes the value of the j-th feature dimension for the i-th audio sample. Similarly, on the text side, each feature dimension corresponds to a k-dimensional vector representing the distribution of this feature across the mini-batch samples.
Ground Metric (Feature-Level). The Wasserstein distance has become widely adopted metric for measuring the discrepancy between probability distributions, as it considers both distributional shifts and underlying geometric structures Panaretos & Zemel (2019). Leveraging this property, DART promotes aligmment at the feature level between audio and text modalities via optimal trans- port. Specifically, let U R and V E R&xd, denote the audio and text feature matrices for

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the b-th mini-batch of size k. with du and de as their respective feature dimensions. DART constructs feature cost matrix C/ E Rdd xd,, whose jith entry measures the Euclidean distance be- tween the distributions of the i-th audio feature dimension and the j-th text feature dimension within that mini-batch:
C. Cpeatree (.. - (..)))ll.
(7)
Here, (i, i) and (:. the k-dimensional vectors corresponding to the i-th and the j-th features of audio and text, respectively, across the samples in the b-th mini-batch.
(Unbalanced) Wasserstein Distance between Feature Distributions in Mini-Batch. In many real-world scenarios, feature distributions across modalities (e.g., audio and text) are inherently misaligned due noise, missing data, and variations feature quality or scale. Such discrepan- cies become more pronounced in randomly sampled mini-batches, where the total "mass" and sup- port of the distributions may differ across modalities. Consequently, as required by the traditional Wasserstein distance, may result in suboptimal alignment. To alleviate this, DART utilizes the un- balanced Wasserstein distance (UWD), which relaxes the mass-conservation constraint by allowing mass "leakage".
Formally, given the cost matrix Cd Feature the UWD is the optimal value of the following optimization problem, with p6 as the transport plan:
Pd arg min p p Xd.
CCrauuuee po) + (KL(P"1a, I La(("")
where (, denotes the Frobenius inner product, and 1a., (1d,) is vector of ones of length du (d.). (Ceaniie (0.0)) pb). represents the overall transport cost, capturing how dissimilar the audio and text feature distributions within the mini-batch. The second term adds KL-based regularization that penalizes discrepancies between the marginals of the transport plan Pa and the uniform distributions u², and v The parameter controls the trade-off between minimizing the cost and maintaining mass consistency.
The first term, (
Mini-Batee Feature-Level Loss. Once pd is obtained, the feature-level UWD loss within the 6-th mini-batch is defined by the total transport cost:
C0wD(O, o) (Featurre Pd)
3.3 RELIABILITY-AWARE MARGINALS (RAM)
To further guide feature-level transport toward stable semantic channels and away from volatile or modality-specific ones, DART builds reliability-awart marginals that act as priors in the unbalanced OT objective, steering mass allocation toward informative features and reducing the influence of noisy or unstable dimensions.
Channel Reliability Estimation. Given audio and text embeddings U Rxxd and V e gxd for a mini-batch size k, the reliability of the j-th channel is estimated from three complementary statistics:
(10) where corr denotes normalized cross-modal correlation, var captures variance instability, and kurt measures heavy-tailednee
ry o (com(UU v²(.))) var (Ur(U))))) V (+,)) - kurt (t.)), ('()))
(--) is the sigmoid function. A higher score ry E (0. 1) indicates that channel j is more likely to capture stable cross-modal semantics. Detailed definitions of these statistics and their computation are provided Appendix B
Normalization into Marginals. tions:
The reliability scores are normalized into probability distribu-
n E, rj 6 Σ,
(11)
where r = (ri....,TT) Ta) is the vector channel reliabilities. These marginals replace the uniform in the UWD formulation, biasing the transport plan toward reliable channels.
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-elia UWD Loss. reliability-ann aware feature-level loss:
Substituting the marginals into the UWD formulation yields the
RUWDR Φ) min (0) P820 Feature! po) + + u/) KL((P)) l1 (12)
Here the KL terms penalize deviations of the transport marginals from the reliability priors (u², v²). As a result, channels with higher reliability scores receive larger marginal mass, encouraging Pe to allocate more transport to them. This reduces the overall cost term (Creatuee Po), effectively lowering the feature-level loss and constraining the solution toward semantically stable dimensions while suppressing noisy ones.
Stabilization via EMA. To prevent fluctuations in reliability estimation from small batches, DART aggregates per-channel scores across distributed workers and updates them using exponential moving average (EMA). Specifically, for each channel j. the global reliability score r at step is updated as
(t)
5 = -r + (1 - 3))
(13)
where ((() is the score from the current mini-batch and 3 E (0, 1) is the smoothing coefficient, which we 0.9 in all experiments. This EMA update ensures that transient spikes drops in small batches do not immediately affect the marginals.
DART then integrates this reliability-wware UWD loss into the overall training objective encour- age cross-modal alignment. The total loss is given by:
B Σ
Lixal min 0.6
(A,) (l
(14)
where (0, Φ) the loss defined 5 and À is hyperparameter that balances the two losses.
4 CONCENTRATION BOUNDS FOR LIOT AND LUWD
Theorem (Concentration of Instance-Level IOT Loss). Let ö (0,1) and m be the fixed mini- batch size. Suppose the log function is T--Lipschitz on [c, 1] and the optimal transport plan II satisfies
Hg 1]. Define the maximum alignment distance over the ground-trutt support II
D Dmax
max d(Ii, U;),
(15)
(i-i): TT,>00
namely the largest distance among audio-text pairs labeled as matches. Then, with probability at least 1 8:
eL,²
1og(2/8)
C LioT| < (Dmax + el2 (0g2 (mm)+10 (m) V
(16)
2
2B
where the number of training batches.
Theorem (Concentration of Feature-Leva UWD Loss). Let 5 E (0, 1) and consider the feature-
level UWD loss LUwD in equation Suppose the mini-batch cost matrix (0),0) estimated from i.i.d. paired samples, with variance bounded by o².
Feature E Rdd xd,, is
Then, with probability at least
L - LüwDl s |P
6m +
2T f
(17)
20 log(2/8) is the regularization parameter in equation and P is the optimal feature-level transport plan.
where C.. =
=
m

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Table 1: Retrieval performance on AudioCaps (AuC) and Clotho (Clo) datasets. All methods trained with batch size of 256, consistent with the settings reported in the original papers, except for the models in the second block (rows with CNN/BPE encoders), where the batch size is reduced to 6 due to GPU memory constraints. For DART variants, w/ RAM denotes using reliability- aware marginals, while w/o RAM reduces uniform marginals.
Method
Encoder
T + A (AuC)
A
(AuC)
+ A (Clo)
A + T (Clo)
R@1
R@10
R@1 1
R@ 10
R@1
R@10
R@ 1
R@ 10
Oncescu et al.
2021
28.1
79.0
33.7
83.7
9.6
40.1
10.7
40.8
(Mc1 cl al.
2022
Audio:
33.9
82.6
39.4
83.9
14.4
49.9
16.2
50.2
Deshmukk ei al.
2022
ResNer38
33.1
80.3
39.8
84.6
15.8
49.9
17.4
54.3
[Wu ei al.
2023
Text:
36.7
83.2
45.3
87.7
12.0
43,9
15.7
51.3
1 Juong
2124
BERT
39.10
85.78
49.94
90.49
16.65
52.84
22.10
56.74
DAKI w/o
RAM
40.20
85,45
$4.44
90.59
17.30
S3.35
22.48
57.03
DART w/ RAM
41.67
85.97
55.27
90.38
17.18
54.52
23.54
58,85
WWng et al. 2023 A: CNN
33.72
83.59
39.14
82.24
16.63
51.98
20.47
55.50
DART w7o RAM
T: BPE
33.12
81.93
43.30
84.11
19.67
$7.18
26.50
63.25
DART w/ RAM
33.42
82.53
43.30
84.11
20.07
59.08
26.79
62.00
Chen et al. 2023
Beats
91.2
66.9
96.7
36.7
74.4
25.9
64.7
DART w/o RAM
T: BERT
56.2
93.2
71.1
97.3
37.0
75.9
27.5
68.9
DART w/ RAM
56.9
93.2
72.1
97.0
37.5
75.9
27.9
69.5
The two bounds highlight key distinction between instance-level and feature-level formulations. For the instance-level loss Theorem the deviation is controlled by the largest alignment distance Daax among audio-text pairs labeled matches. Because mini-batches only contain restricted subset of samples, their feasible matching set is limited. When the correct partner of a sample is absent from the batch (e.g., due to label noise), the transport plan may forced assign mass to a higher-cost alternative. This inflates the effective Daax in mini-batch training compared to the global dataset, leading a looser concentration bound and larger variance in gradient estimates.
In contrast, the feature-level bound in Theorem 2 depends on the Frobenius norm of the transport plan || This term measures the squared sum of all transport assignments across channels, so the deviation is controlled by the overall mass distribution rather than dominated by single worst-case pair. As a result, occasional noisy or high-cost channels contribute only marginally to the bound, while the majority of stable semantic channels reduce the effective variance. This aggregation effect makes the bound inherently tighter and less sensitive to outliers, thereby providing greater robustness under small batches or noisy labels.
EXPERIMENTS
We evaluate the effectiveness and generalization of DART audio-text retrieval benchmarks and beyond. We compare DART against standard baselines including contrastive learning (Radford) al. 2021 Jia et al. 2021). triplet losses (Wei al.) 2021), and OT-based methods (Shi et al. 2023). All methods are trained under the same conditions unless otherwise noted. Detailed implementation settings are provided in Appendix
DART consistently enhances overall audio-text retrieval performance. We present DART on the AudioCaps and Clotho datasets, comparing them with of-ie-o-tteerttttt methods using the R@ and R@ 10 metrics. To ensure fair comparison, we categorize the baselines based on their au- dio/text encoder architectures and adopt identical model settings, including batch sizes, for each group. As shown in Tab. D DART consistently superior or comparable performance across all en- coder settings. For instance, with BeSRe138+BERT encoders on AudioCaps, DART outperforms the strongest baseline Luong et al. (2024) by 4.5% (A-T) and 1.1% (T-A) in R@ Similar gains observed Clotho, where DART leads in both R@ and R R@ 10. Despite matching the ONE-PEACE' |Wann et al. (2023) constrained batch size of 2 (required due to model scale), DART achieves superior performance in of key metrics while maintaining comparable results in others.
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Table 2: Retrieval performance on the AudioCaps dataset under varying semi-supervised and noisy conditions. The top rows show semi-supervised settings with 20% and 40% unlabeled data, while the bottom rows represent noisy correspondence settings with 20% and 40% of captions replaced by unrelated ones. All methods use batch size of 32.
Condition
Method
Text - Audio
Audio + Text
R @
R@5
@@ 10
R@!
R@5
R @ 10
Triplet loss
15.34
48.34
66,88
24.29
52.83
69.84
-uperviuee (20% Unlabeled)
Contrastive loss
28.58
65.55
81.50
35.63
68.42
80.36
(Luong al. 2024
32.93
67.43
80.89
39.81
70.53
82.44
DART
34.85
70.44
83.34
45.03
76.28
86.62
Triplet loss
0.1
0.52
1.06
0.1
0.52
1,46
Semi-Supervised (40% Unlabeled)
Contrastive loss
20.58
53.96
70.72
27.37
58.72
75.21
(uong et al. 2024
28.58
62.69
77.19
35.00
69.27
79.72
DART
33.24
69.55
82.74
43.67
74.39
87.46
Triplet loss
16.82
46.39
62.71
19.64
46.39
59.77
Noisy Labels (20% Noisy)
Contrastive loss
25.80
61.56
78.16
33.33
66.66
78.78
(Luong et al. 2024
31.32
67.11
80.48
38.35
73.77
84.85
DART
32.87
67.77
81.06
43.57
73.98
86.72
Triplet loss Contrastive loss
0.58
1.58
2.13
1.14
4.91
8.98
Noisy Labels (40% Noisy)
22.23
55.90
72.76
26.95
59.03
73.24
(Long al. 2024
26.20
61.31
76.17
34.37
65.30
77.84
DART
29.67
65.30
80.20
37.09
67.18
80.45
DART remains robust under small batches and noisy or semi-supervised labels. We first eval- uate DART's performance under noisy and semi-supervised conditions the AudioCaps dataset. Noise is introduced by randomly replacing text captions with unrelated ones at ratios 20% and 40%, while semi-supervised settings simulate scenarios where portion of the data lacks labels entirely by randomly masking parts of the label information (in II). In this experiment, small batch size of 32 to test DART's performance under limited negative samples and noisy data conditions. The results in 2 show that DART maintains stable retrieval performance even with re- duced negative samples and noisy inputs, demonstrating its resilience to input perturbations. This robustness in challenging settings highlights DART's capacity to generalize well even when faced with noisy data and limited label availability. These findings underscore DART's suitability for large-scale, real-world applications where data quality and label availability may be limited, and computational resources constrained.
set
Zero-Shot Sound Event Detection. We evaluate DART's generalization ability by conducting zero-shot sound event detection on the ESC-50 dataset. Models are pretrained on the AudioCaps dataset for the audio-caption matching task and applied directly to ESC-50 without additional fine- tuning. Following Luong et al. (2024), all classes the test set are converted to template captions, such "This is sound of class." As shown in 4 we report the @ 1, R @ 5, R @ 10, and mAP scores for models trained with three types of IOT loss under different constraints: triplet loss, contrastive loss. matching loss (as used in Luong et al. (2024)), and proposed DART, with consistent batch size 128 for models. DART achieves the highest R@ score 80.75%, outperforming triplet loss (71.25%) contrastive loss (72.25%), and matching loss (79.25%). It also shows competitive performance in @@5 and R @@@ 10, closely matching the results of offLoong Luong et al. (2024). This demon- strates DART's superior generalization to unseen sound events in a zero-shot setting, Notably, the matching loss in Luong et al. (2024) is similar to our LIoT in 5 and the improvements highlight how the feature-level ZUwD DART enhances alignment between audio and distributions, boosting performance.
DART introduces negligible GPU memory overhead compared to instance-level baselines. A potential concern is whether feature-level transport increases GPU memory consumption. For
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batch size of k and feature dimension d=-52,, the feature-level cost matrix CFeature (9.066 involves computing df = 512² pairwise distances, each costing O(k) operations. Storing all intermediate results in float32 requires only a few megabytes: the embedding matrices U V R occupy about 64KB each when k=32, and both the cost matrix CFeattree and the transport plan Pd require roughly 1MB each. The unbalanced Wasserstein loss LUWD in Eq. (16) is computed as a point- wise product (CFeature, P"), requiring no additional buffers. Importantly, P in Eq. (15) can be computed on CPU via offloaded OT solvers and is detached from the gradient graph, so during backpropagation only Creatre contributes gradients. In practice, this means DART introduces only ~2MB of extra GPU memory with no additional GPU cost for optimizing P'. Moreover, reliability estimation (variance, kurtosis, and cross-modal correlation) can be precomputed or updated offline, further ensuring that DART fits within the same GPU memory budget as instance-level IOT methods. For extremely high-dimensional encoders (e.g., d > 2048), one can further ensure scalability by first projecting features to a lower dimension d' S 1024 with a lightweight linear layer before computing feature-level OT, or by applying low-rank approximations of the cost matrix (such Nyström-type methods) to reduce the effective quadratic dependence on
DART generalizes effectively to other cross- modal tasks such as image-text retrieval. Beyond audio-text retrieval, we evaluate DART on the MSCOCO dataset for image-text match- ing. As shown in Tab. 3] DART consistently improves both image-ttxt and text- retrieval compared to strong baselines. This demonstrates that the proposed dual-level alignment and reliability-aware marginals are not tied to the audio-text domain, but transfer naturally to other heterogeneous modalities. The results highlight DART's potential as a general solution for cross-modal matching tasks.
Table 3: Image-text retrieval performance on MSCOCO dataset. Results are reported in R@ 1.
Method
Image Texx
Text
Shi et al.) DART (ours)
19.15
20.90
21.27
23.34
Ablation Study. We conduct ablation studies to understand the contribution of each component in DART.
First, the dual-level objective is necessary. As shown in Appendix TO using only the feature-level loss LUwD fails to recover cross-modal correspondences (R@1 = 0), whereas the instance-level IOT loss Lior provides a standard baseline. Jointly optimizing both yields the best performance, confirming their complementary roles: Lior anchors global sample alignment, while LuwD acts as a structural regularizer to filter noisy feature directions.
Second, ReliabilityAwware Marginals (RAM) are both effective and interpretable. Replac- ing RAM with uniform marginals ("DART w/o Reliability" in Tab. I consistently degrades retrieval accuracy, indicating that treating all channels equally is suboptimal. Further decom- position of RAM into its core variants (Table(5) reveals three key insights: (i) Correlation alone unstable: While simple cross-modal correla- tion (corr) may slightly improve one retrieval direction, it often hurts the other (e.g., A-T R@I drops from 51.52% to 50.05%), leading to a lower mean R@1 1 than the uniform baseline. This suggests that correlation is susceptible to spurious signals in mini-batches. (ii) Stability statistics are crucial: Both EMA variance (emavar) and kurtosis (kurt) provide consistent gains over the uniform baseline. They serve as effective sta- bilizers by down-weighting high-varian verianereeeee channels and penalizing heavy-tailed, outlier-domintted dimensions. (iii The full RAM design is robust: it achieves the best overall performance (mean R@1 1: 45.55) with the highest A-TT @@1 (52.56) while keeping T- competitive, demonstrating that all three statistics are necessary to safeguard against different types of noise while maintaining peak accuracy.
Table 4: The zero-shot sound event detection on the ESC50 test set, the R@1 score is equivalent to accuracy.
Loss
Audio + Sound
R@1 1
R@5
R@ 10
mAP
Triplet
71.25
91.75
95.75
80.09
Contrastive
72.25
93.00
96.75
80.84
1OT
79.25
97.5
99.25
87.09
DART
80.75
97.25
99.75
87.78
Finally, sensitivity analyses confirm that DART is robust to variations in key hyperparameters. Fi- nally, study the sensitivity to key hyperparameters. Tab. 12 shows that DART yields consistent

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Table 5: Core RAM variants AudioCaps (ResNet38-BERT) batch size 64).
Marginal Design
A-T R@
A-+ R@ 10
T-A R@1
T-*A R 10
Mean R
uniform (w/o RAM)
51.52
90.80
38.31
85.77
44.92
(correlation)
50.05
90.60
38.64
85.22
44.35
emavar (EMA variance)
51.83
90.49
38.52
85.56
45.18
kurt (kurtosis)
51.93
90.60
38.64
85.74
45,29
RAM (full)
52.56
90.60
38.54
85.56
45.55
gains across mini-batch sizes and is particularly beneficial in the small- regime (e.g., clear. R @ improvements at k=8 and k=32), while remaining competitive at larger batches. Tab. Tab.[ffure further in- dicates that performance is stable under moderate changes of the loss weight À (0.1-0.7), with only
minor variation retrieval metrics. Alternative marginal choices for UWD (Tab. g exhibit similar trends, confirming that DART is robust reasonable design variations.
6 RELATED WORKS
Cross- matching a fundamental challenge in multi-modal learning, aiming to establish meaningful correspondences between two modalities, such text-image (Jia et al. 2021 Rad- ford et al. 2021 Wei et al. 2020). text-audio (Wu et al., 2023 Deshmukh et al.) 2022). by align- ing their underlying distributions. Recent advancements have focused leveraging metric learn- ing techniques to learn joint embedding spaces where semantically similar instances from different modalities are mapped close to each other, using methods such triplet loss (Mei al. 2022 Wei
et al. 2020), contrastive learning (Radford et al., 2021] Yang et al. 2022). and matching loss (Shi
al. Despite their success, these approaches treat all embedding dimensions equally, implicitly assuming that each channel contributes similarly to semantic alignment. In practice, however, many dimensions may be noisy, redundant, or modality-specifte making uniform treatment suboptimal. Although Luong et al. (2024) introduces per-channel coefficients that reweight feature dimensions, their formulation only assigns weights to corresponding dimensions across modalities (c.g., audio j-th dimension with text j-th dimension), which effectively assumes one-to-onc channel aligmment. Such constraint overlooks potential cross-channel correspondences (e.g., an audio rhythm dimen- sion aligning better with textual verb-related dimension), thereby limiting flexibility. Moreover, their method still relies heavily on sample-level supervision, leaving it vulnerable to small-batch batch variance and noisy labels.
7 CONCLUSION
We presented DART, a dual-level alignment framework for audio-text retrieval that couples instance- level IOT with feature-level regularizer based on unbalanced optimal transport, targeting the sen- sitivity instance-level objectives to feature-level noise and mini-batch sampling variance. DART moves beyond treating embeddings holistic units: it treats embedding channels matchable units and, through reliability marginals, concentrates transport mass on channels that consis- tently encode stable semantic while downweighting volatile modality-specific ones. From a theoretical standpoint, we analyzed the mini-batch instability of instance-level alignment and de- rived concentration bounds showing worst-positiv dependence on the maximum paired distance, whereas the feature-level formulation is controlled by the Frobenius norm of the transport plan, yielding tighter guarantees under small batches and noisy supervision. Empirically, DART im- proves retrieval accuracy on three benchmarks across encoder choices, with pronounced gains in small-batch settings, and it also transfers to image-text retrieval, suggesting that the feature-level view is not tied to specific modality. Beyond performance, DART provides an interpretable mech- anism for robustness: the transport plan reveals which channels are emphasized, and RAM offers lightweight way to inject stability priors without introducing heavy additional networks. This in- dicates that feature-level regularization is not merely a heuristic, but a principled complement to lnstance-level matching when channel-wise noise dominate scalar similarities.
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ACKNOWLEDGEMENTS
This work was supported by the Science and Technology Commission of Shanghai Municipality (24510714300), and National Natural Science Foundation of China (62172273).
REFERENCES
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LLM USAGE STATEMENT
We used large language models (e.g., ChatGPT) general-purpose writing assistants improve the readability and clarity of the manuscript. The research ideas, methodology, experimental design, and analysis were conceived and carried out entirely by the authors.
APPENDIX OVERVIEW
The appendix is organized into the following sections with additional analysis, proofs, and imple- mentation details:
In Section
we provide full training pseudocode of DART.
In Section B we detailed implementation of the reliability-aware marginal estimation mod- ule.
In Section we present theoretical analysis, including notation, mini-batch sampling paradigms, and concentration bounds for LioT and LUWD.
In Section we describe implementation details such as hardware, software stack, and training setup.
In Section
we summarize dataset statistics for AudioCaps, Clotho, and ESC-50.
In Section detail evaluation metrics and baselines.
In Section
we list the complete hyperparameter settings for all encoder configurations.
In Section analyze the effect of the weighting parameter in Eq. equation 14
Section we study alternative marginal distributions for the feature-level loss.
In Table 9
evaluate the effect of temperature values retrieval performance.
In Table 10
we conduct ablation studies on the two loss components Cior and LUWD.
In Table II
we show the effectiveness of LuwD as complementary constraint under dif- ferent sample-level objectives..
In Table we investigate the impact of varying mini-batch sizes on DART's performance.
A ALGORITHM
Algorithm DART
Input: Initialize audio encoder fe. text encoder Io. training data pairs D, number of mini-batches B
repeat
for b 1 to B do
Sample (X6, yb) from
Embeddings U fx(),, V = bY*b;
Compute cost matrices Co Sample. C Peeaure by and 7
Solve EOT that I ( (amppe))) by
Lorr KI IT with label IT eye
Solve UOT that P UUT(UC Cb
Feature) by
LUWD (Ch (eetuure Pb)
Lioaa LIOT + XLUWD
0.0 -- 0, Φ 1) V 0.6 Liotal'
end for until converged
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B RELIABILITY SCORE COMPUTATION
In this section, we provide the detailed implementation of the reliability-aware marginal estimation module described in Section 3.3 The procedure consists of three steps: computing channel-wise statistics, aggregating them into reliability scores, and stabilizing the estimates across training.
Channel-wise Statisties, For mini-batch size k, the audio and text embeddings are denoted U V Rxxd For the j-th feature channel, compute:
1. Cross-modal correlation:
(U), Rjj (i.j) - -T}
(t))( = (t)] u,ll2 (:,j) 8yl12
(18)
where Uj and Vj are the sample means of the j-th channel. This normalized correlation measures semantic consistency between modalities.
2. Variance instability:
var U (:. (+.)) Var(U + Var(V
(19)
Larger values indicate unstable or noisy activations across samples.
3. Kurtosis (heavy-tailedness):
kurt (U,))), (*())) = Kurt(U*(..))) + Kurt(V
(20)
where Kurt(=) = -|(=-E) denotes standardized fourth-order moment. High kurtosis
indicates outliers or bursty patterns.
Score Aggregation. The channel reliability score ry is defined as:
T = (corr(U () - var(U" (:.J), v² (:) (:.)( ().)))
(21)
where o--) is the sigmoid function. Higher values indicate more stable and semantically reliable channels. The scores normalized into probability marginals:
(22)
where r = (r1...., ra).
Stabilization Across Training. Since rj is computed per mini-batch, it can fluctuate due to ran- domness in sampling. To stabilize the estimates, adopt:
EMA smoothing: Reliability scores are updated across training iterations using exponen- tial moving average:
(t-1) (4) 3 ( + (1 (3 batch
where 8 [0, 1) is a smoothing coefficient.
Hysteresis rule: Thresholds This To decide whether activate or suppress channel, avoiding oscillations.
Warm-up: All channels are considered active during the first Twarn iterations.
Freeze: After fixed epoch, the selection of reliable channels can be frozen for stability.
Top-K filtering: Optionally, only the K most reliable channels are retained to reduce noise further.
Implementation Note. All reliability statisties (correlation, variance, kurtosis) computed in- dependently from the forward- forward-backward graph and can be executed off-GPU. This avoids additional GPU memory usage and ensures minimal runtime overhead.
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C THEORETICAL ANALYSIS
C.I SYMBOL DEFINITIONS
Let the full-batch optimal coupling matrix be II E _nxx for n data pairs. For mini-batch stochastic optimization:
Let So. To C [r]] denote the sample index sets of size m for batch b.
Let og [m] + S₁ and Bg [m]] + T. be index mapping functions that link local mini- batch indices to global indices.
Let II Rmxm denote the local coupling matrix for batch where [I q) correspond- ing to the pair (on(p), (g(g) in the global coupling matrix.
Then we formalize two distinct mini-bate sampling paradigms:
Definition 3 (Global coupling matrix under Non-overlapping Mini-Batch Sampling). Let
II R= be the global coupling matrix constructed from non-overlapping mini-batches ((S,,T))
(1),-(1),,,0000000) if Absiie So and j E To: 0,
II (i,j)
(23)
otherwise.
Proof Roadmap.
The theoretical analysis proceeds in three steps:
Inner Optimization (Coupling Matrix). We first establish the e-strong convexity of the entropy-regularizzed OT objective (Lemma 4), which allows bounding deviations between the mini- batch coupling II and the full-batch optimum II via the functional gap.
Mini-Batch Concentration. Using Hoeffding's ' inequality, we derive concentration bounds on the mini-batch OT objective (Lemma (5). showing how the deviation shrinks with the number of batches B and depends on the maximum alignment distance M.
Loss-Level Bounds. Finally, we transfer these results to the loss functions: Theorem for the instance-level L1or and Theorem 2 for the feature-level LUwD. highlighting their different dependence on D versus P
Together, these steps show why feature-level regularization yields tighter bounds and better robust- ness under small batches noisy data.
C.2 CONCENTRATION BOUNDS OF LIOT
In this subsection, we analyze the deviation introduced by mini-batch optimization in solving the original Inverse Optimal Transport (IOT) problem (5). The IOT problem comprises two sequential stages: (1) an outer minimization stage learning representations, and (2) an inner optimization stage for computing the coupling (i.e., the probability matching matrix) between point sets. The inner stage corresponds to a standard Entropy-Reed Regulaaiied Optimal Transport (EOT) problem. Our analysis focuses quantifying the discrepancy between the full-batch solution II" and the mini- batch solution =
Lemma (e-Strongly Convexity of EOT Objective). Consider the objective function f()) : u(,,v) + for the
Optimal Transport (EOT) problem, defined as:
f(I) - (1,,
where U(ps, v) denotes the set of coupling matrices with marginals μ and v, is the cost func- tion, and HIII) is the entropy term. Then is e-strongly convex over the relative interior of U(, v). Specifically, for any feasible coupling matrix II, E U(p, v) and the EOT optimizer II" arg mintaa f (II), it holds that:
2 11
II P
S(LL) (II").
(24)
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Proof. For any coupling matrix II E relint((((141z1) (where Hy > 0), the Hessian operator of is given by:
f(II = e d-di (1/I1i))a))
where the diagonal operator acts on the vectorized matrix. For any tangent direction D E Tu(uy) we have:
D I
(D, ('
D
where the inequality follows from Hy & 1 in the probability simplex. This establishes the e-strong convexity.
According to the Frobenius norm deviation between II and II is upper-bounded by the func- tional value difference, which allows us analyze the convergence through the functional gap that: III I ()((I))) f(I
Lemma 5 (Concentration Bound of Mini-Batch EOT Objective). Let E (0, and B 2 1, we have bound between f (III)) and f(II") depending the number of batches B that
log (2/8)
IIII f(II') S M
(25)
2B
with probability at least
Here, M D + (2 + 1).
Proof. The proof can be found in Fatras et al. (2019). Lemma 3], and we restate it here for better understanding.
To simplify the problem, we first derive an upper bound for the function value fIIT") of any fea- sible coupling matrix II mm within a mini-batch. For any X; f and Y; V, have |X Yill S M, implying C²2y I M for all (i,j). The Shannon entropy E(IIT) -isiysm H log H satisfies 0 < E(TI") < log2(m²) where the maximum entropy occurs when II is uniform. Applying the triangle inequality, we have
(
| = ( IT Co) -
S
Σ 154.5m
Ci
+
Σ IsiJsm
nt log IT +1)
D Σ
I + (logg(mm² +
Isissm
< D ((2 1) M.
(26)
The mini-batch sampling process be modeled sequence of independent trials over all pos- sible mini-batch configurations. For each trial E [B), let 1, E {0,1], be a Bernoulli random variable indicating whether a specific mini-batch configuration (among the total (m) possible con- figurations) is selected. Therefore, we then have
8(II8) f(II')
Σ
wb.
(77)
B
b=1
represents the deviation between the actual selection status the mini-batch pair (S,, in trial and its expected selection probability. f(II') denotes the objective function value corresponding to the selected mini-batch configuration, and w/ reflects the contribution of trial to the overall deviation from the expected value. Since the variables [wb independent, centered (E[wr) 0) and bounded by M, can apply Hoeffding's 's inequality, which gives:
where wa - (1,- S(TT). Here, 1, 1,-1/((m)
1/((m))
2t
P([((I))) s(IT)) > >)) +) =P
Σ
Wb > < exp(
(28)
BM2
1=1
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To derive high-pun high-probability bound, set the right-hand side equal to confidence parameter 8: 21²
2 exxp(- BM=
6, solving for yields:
log (2/6)
M
(29)
2B
Theorem (Maximal Deviation Bound in the Inner Optimization Stage of LIOT Let & (0, 1), B 1 and the mini-batch size be fixed, have a maximal deviation bound between II and II" that
II II
| f(IT*)
cM log(2/6)
1 2 2B
(30)
with probability at least Here, D + <(2 (mg + 1).
Proof. Here, the first inequality is obtained by the strong convexity of EOT (4). The second in- equality follows from the triangle inequality by introducing the intermediate solution IT Apply- ing S to bound the mini-batch estimation error, we derive the final result.
Before deriving the deviation bound in the outer optimization stage of Liors recall that the mini- batch objective function is defined
B
k
T log (II (1)0))
(31)
where II is a permutation matrix representing the ground-truth matching, with II; correctly paired audio and text instances.
indicating
The full-batch loss function is then given by:
LiT = - /1 #(I, y) log (I,9) du(x)dv(y))
(32)
Here, μ and discrete measures defined the finite sets X = (11, ....... }. respectively: μ = 1 Em= 8zy, V -j=1= ōyz. The function म ( y) represents the probabilistic coupling between the elements of and Y.
{21, F2 22. Tm and
Theorem (Concentration Bound of C 55). Let 10, B 2 and the mini-batch size m be fixed. Suppose the log function is L-piscchitz over [e, 1) for some 0, where II E [e, 1] for all batches. Then, with probability at least - 8, the maximal deviation between the mini-batch loss
L and the full-batch loss fior satisfies:
c
Lion
12 II
IIo
(33)
cML?
1og(2/8)
- 2
(34)
2B
Here, M = D D ((2log((()) + 1).
Proof. Step 1: Lipschitz continuity of loss difference. Since II is permutation matrix, it satisfies Th for correctly paired instances and otherwise. Thus, the loss function simplifies to:
B log :
B L0r
log II Cior log II
(35)
B
b=1
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Both II and II correspond to the predicted probabilities of correct matches, their values tend to be close to 1. Therefore, assuming that the logarithm function is L- Lipschitz over the domain of II is reasonable. This implies the following bound:
| fiorl
|log
I
log II
(36)
(Lipschitz condition)
(37)
- L.
I 1
(Element-wise difference to vector 2-norm)
(38)
1=1
- IT'
|P
(Vector 2-norm to matrix Frobenius norm).
(39)
Step Concentration via pre-established bound. By applying the concentration result in 6 wc
directly obtain the 33
C.3 CONCENTRATION BOUND OF LUwD p
Assumption (Statistical Model of Feature Extractors). The feature encoders So and gs satisfy:
fa(a); Mfs + (1,( [x=x(0r,(x]]] 0, Vur=~x(0,,x(z)) = j
(40)
9o(y); Poss 6g.((1), Ey->yoo, 0, Verg~>(0g, ogg
(41)
where Pf Pg.j are expected feature values, and (2,)(2), (V) are zero-mean noise terms.
Lemma (Concentration of Feature-Level Cost Matrix). Let Zu == (xx); - go(y)) denote the difference in the i-th and j-th feature dimensions of paired embeddings, where {(Ikk ys) ] are sampled i.i.d. and suppose each Zy is uu--aaussian with zero mean and bounded variance proxy S o²)). Given the empirical cost entry as:
o² (i.e., Z[Zy] = 0, Z
offeani := =11((0) -960
ZE
and let the full-batch cost be: Cf == mm- (B(((B)()))) (lgo()))ll Then, with probability at least 1 8, the deviation between empirical and expected feature- level cost satisfies:
20 log(2/8)
C
Beat
Cf
Fear)
m
Proof. We begin by computing the expectation of B
C,
m
m
E
B ij
(Fo(x)); (1k)))
Σ
E [[(Hss + 88,6) (19.3 + 8)
m
E [((ura -Mg.s) + - 3)
E [(Di + 0z)
Σ +2 1
k=1
1=1
m (o² D2,))
(42)
To simply, we C Gaussian, and the squared terms Z2 are sub-exponential. According to the concentration incquality for sub-exponential variables, have:
BBeat)
リボ= Zzz where Zμ = fo(xx); (gx)) Since Zμ are sub-
-
21)
t - アユ )
P
< 2eexp min
(43)
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According 42
substitute E[Z]]
Di o² and S =Ek=1 Zg for t e/m and obtain:
P P(SS m( + em) S exp (-o$m)
ot
(44)
Using the inequality
E[S] 2 /B[]
VS - S--VESS
<
with B[S] m(Di + +²²)
we derive:
( Beaay - mm(D + +
m(D + + 2c) SP <P((Sm(DD)
22 m + 02)). (45)
Combining with the result, we obtain:
2 +
P
c
B
mD + o2) 2c) s exxp
(46)
Recall
VmDrg. sjo and that v m((2 + (via Taylor expansion), we can write:
that
Cj
VmD +
2Dg
0²
Cf
C(feeat
VS VmDis sct
(47)
2Dij
Theorem 10 (Feature-level Loss LUWD (9)
Concentration). Assume the feature-level cost matrix (Feat) E R is computed from i...d. paired samples, and each entry satisfies the deviation
bound:
FeattB
Feett kj
S Cm
with probability at least 8,
sj
where cm
V
20 log(2/8)
Then the unbalanced OT loss satisfies the deviation bound:
[uor( (C Bffea)BB Lvor
Cuor(Ctffaain Cffeeatie
[P'll cm +
1
Cm
2X
with probability at least - 8, where A = e + regm the strong convexity constant of the UOT objective.
Proof. Leveraging the strong convexity of unbalanced optimal transport (with coefficient À + ): (Cffeaa/)) Luor Cffeaaty
1
|
(FFea))
(Feat) ||F + 21
(Feet)))
P-
(48)
Based on Lemma the theorem follows.
D IMPLEMATION DETAILSS
The experiments conducted Linux workstation equipped with Intel() Xeon(R) Gold 6226R CPU (2.90GHz) and an NVIDIA A100-PCIE-40GB GPU. The detailed implementation code provided in the supplementary materials.
19

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Published as a conference paper at ICLR 2026
E DATASETS DETAILS
We evaluate DART on three widely used datasets: AudioCaps Kim et al. (0019). Clotho Drossos al. (2020), and ESC-50 Piczak (2015). covering audio-text retrieval and sound event detection lasks.
AudioCaps is the largest audio captioning dataset, containing approximately 50K audio-caption pairs. All audio clips are sourced from AudioSet Gemmeke et al. (2017). a large-scale dataset for audio tagging. The training set consists of 40,582 audio clips, each 10 seconds long and paired with single
caption. In contrast, the validation and test sets contain 494 and 957 audio clips, respectively, with each clip accompanied by five ground-truth captions.
Clotho is an audio captioning dataset collected from the Freesound platform, featuring audio clips 30 seconds. We use the second version of the dataset for our
of varying durations between experiments. The training set includes 3,839 audio clips, while the validation and test sets contain over clips each. Every audio clip paired with five human-annotated captions.
ESC-50 is an environmental sound classification dataset designed for sound event detection, con- sisting of 2K labelled recordings across 50 sound classes. Since our goal is to evaluate DART's transferability, we use only the test set, which contains 400 audio clips.
Baselines. We compare DART against state-of-the-art audio-text retrieval models, including On- cescu al. (202T), Mei al. (2022), Deshmukh al., (2022), Wu et al. (2023). [uong al. (2024). Wang et al. (2023) and Chen et al. (2023), ensuring consistency in evaluation settings. All baseline results are directly sourced from their respective papers for fair comparison. Furthermore, we vestigate the impact of different training objectives, including contrastive and triplet loss, to analyze
DART's adaptability under various learning paradigms.
EXPERIMENTAL SETUP
Evaluation metrics. We evaluate DART using Recall at Rank (R@k), standard metric for modal retrieval. @@k k measures the proportion of queries where at least one ground-truth match appears in the top-k retrieved results. Formally, for a query of size N, R@k computed as:
R@k 1 Σ (rank(y)) k),
(49)
where I(.) is the indicator function that returns the correct match y is ranked within the top-k, and O otherwise. A higher @@k indicates better retrieval performance. We report R@ R@5, and R @ to compare DART with baseline methods.
G HYPERPARAMETERS
Tab. 6 provides a comprehensive overview of the hyperparameters employed across all baseline models. The three sections correspond to different encoder configurations evaluated in the main comparison table: (1) ResNer38 (audio) + BERT (text), (2) CNN (audio) + BPE (text), and (3) Beats (audio) BERT (text). All settings align with those used in the respective baselines to ensure fair comparison and match the configurations reported in their original papers.
H EFFECT OF THE WEIGHTING PARAMETER
We analyze the effect of the weighting parameter À the overall loss function 14 presents the retrieval performance under different values of X. The results indicate that DART is robust to variations in X, with consistent performance across the tested range. The best performance is 0.7, achieving an R@ I score of 40.41% for text-to-audio retrieval and 53.70% for audio-to-text retrieval. Notably, even at = 0.1, the model performs well, with R@ scores of 40.31% and 53.29% for text-to-audio and audio-to-text retrieval, respectively. This suggests
observed
20

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Published as a conference paper at ICLR 2026
Table Detailed hyper- pappemtrraettrrs used in training for the retrieval experiments reported in Table !
Hyperparameters
AudioCaps
Clotho
Batch size
256
256
Optimizer
Adam
Adam
Learning rate
5 x 10
10
Weight decay
0.0
0.0
Total epoch
10
0.5
0.5
0.03
0.03
0.05
0.05
Batch size
6 Adam W (0.9,0.999) 1 x 10
Optimizer
AdamW
Adam B
(0.9.0.999)
Learning rate
1 x 10-6
Weight decay
0.01
0.01
Total epoch
10
10
0.5
0.5
0.03
0.03
0.05
0.05
Batch size
256
256
Optimizer
AdamW
Adam W
Adam Ø
(0.9, 0.98)
(0.9, 0.98)
Learning rate
x 10-7
10-7
Weight decay
0.01
0.01
Total epoch
10
10
0.5
0.5
0.03
0.03
0.05
0.05
21

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Published as a conference paper at ICLR 2026
that while the feature-level alignment provided by LUwD contributes to optimal performance, the underlying IOT framework also plays critical role in ensuring DART's robustness. This flexibility underscores DART's ability to effectively balance the contributions of different loss components, enabling robust cross-modal retrieval across diverse settings.
Table 7: Retrieval performance (%) under varying values (Eq. 14 with small fixed mini-batch size of 32.
À
Text + Audio
Audio + Text
R@1
R@5
R@ 10
R@1
R@5
@@ 10
0.1
40.31
75.23
86.22
53.29
83.07
90.17
0.3
39.97
75.04
85.66
51.51
82.23
90.28
0.5
39.94
75.07
85.64
53.08
83.49
90.59
0.7
40.41
75.06
86.22
53.70
81.50
90.28
I EFFECT OF THE MARGINALS IN LUWD
We analyze the effect of the marginal distributions used in the feature-level loss defined in Eq. Specifically, we consider the following initialization strategies for the source and target marginals:
Uniform Distribution: This baseline assumes prior knowledge of feature importance and sets both marginals to uniform weights, assigning equal mass to all feature dimensions.
Feature Norm- Based Initialization: this setting, the marginal distributions are derived from the & norm of the corresponding feature dimensions. The underlying motivation is that dimensions with higher magnitude may carry more semantic or discriminative infor- mation, and thus should be assigned greater mass in the transport plan.
Feature Sitaance-Based Initialization: Here, we use the empirical variance of each fea- dimension across the batch to form the marginals. The rationale that dimensions exhibiting higher variance across samples are likely to be more informative and discrimi- native for downstream alignment.
Table Retrieval performance (%) under different marginal distribution in
Marginal
Text
Audio
Audio
Text
R@D
R@5
@@ 10
R@1
R@5
@@ 10
Uniform (l)
32.87
67.77
81.06
43.57
73.98
86.72
Le Norm-based
33.24
68.54
81.40
42.00
73.87
85.37
Variance-based
33.10
68.25
80.64
43.88
73.24
85.89
EFFECT OF THE TEMPERATURE VALUES IN LUWD
Table 9: Retrieval performance (%) under different temperature values with a small fixed mini-batch batch
size of 32.
Temperature
Text
Audio
Audio -> Text
R@I
R@5
R@ 10
R@I
R@5
R @ 10
0.05
36.46
71.95
83.94
46.39
78.05
88.29
1.0
35.59
71.43
83.97
48.69
77.33
86.42
10.0
37.47
71.60
83.72
46.81
75.76
86.00
100.0
35.34
71.37
83.85
46.71
78.89
89.13
inf (cot)
36.13
72.20
83.66
46.81
77.74
88.09
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Published as a conference paper at ICLR 2026
K ABLATION STUDY ON THE TWo Loss
We analyze the contribution of individual loss components to the overall performance. 12 presents the results of the ablation study on the loss components, comparing the effects of LioT and LuwD individually and in combination. Using only LUWD results in poor performance, with R@) scores close zero, highlighting the necessity of correspondence labels for cross-modal alignment tasks. When both Lior and LuwD are combined, the model achieves the best performance, which demon- strates the complementary nature of the two loss components.
Table Ablation study on the training loss components using the AudioCaps dataset with a small fixed mini-batch size of 128.
LOSS
Text - Audio
Audio
Text
R@1 1
R@5
R@ 10
R@1
R@5
R@ 10
LOwD
0.12
0.52
0.98
0.10
0.31
0.41
Lior
39.28
74.50
85.85
52.45
80.25
90.17
Cior LUWD
39.94
75.07
85.64
53.08
83.49
90.59
L EFFECTIVENESS OF FEATURE-LEVEL Loss AS A COMPLEMENTARY CONSTRAINT
Our feature-level loss LUwD is designed to capture fine- tieeggained alignment between modalitt-specific dimensions, and intended to be used as a complementary constraint rather than a standalone ob- jective. Specifically, while many existing retrieval systems are trained with sample-level objectives such as contrastive loss. triplet loss, more advanced models like m-LTM, our method is compati- ble with all of them.
In this section, we conduct controlled ablation study to demonstrate that LUwD can be seamlessly integrated with different sample-level losses and consistently improves retrieval performance across the board. Table I summarizes the results on the AudioCaps dataset (same setup as in the main paper). We observe that for each baseline loss, adding our feature-level loss leads noticeable gains in both A-tT and T-+A retrieval tasks.
This confirms that our approach serves as a modular and universally beneficial component that strengthens the representation alignment across modalities without conflicting with the core retrieval objective.
Table 11: Retrieval performance on the AudioCaps dataset with different sample-level objectives, evaluated with and without the feature-level loss LUwD under a small fixed mini-batch size of 32.
Method
A R@
T-A R@
Triplet loss
37.72
32.85
+ LUWD
38.24
32.10
Contrastive loss
38.24
31.07
+ LuWD
39.18
32.14
1OT loss
Luong al. 2024
41.69
32.39
+ LUWD
41.79
32.87
L.1 EFFECT OF BATCH SIZES (NUMBER OF SAMPLES PER BATCH)
Second, we examine the impact of batch size on DART's performance, particularly focusing on the role of LUwD- As shown in 12 the benefits of LuwD This finding is particularly relevant for real-world applications where computational resources are often constrained. By providing feature-level alignment, LUwD enables DART to maintain strong
pronounced with smaller batch sizes.
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Published as a conference paper at ICLR 2026
Noise Strength vs OT Transport Cost
0.900
Noisy cost
0.850
OT Transport Cost
0.825
0.775
0.750
0.725
0.700
0.25
0.30
Noise Std
Figure 2: Effect of injected noise OT cost. We fix clean pair of audio-ter text feature distributions and inject i.i.d. Gaussian noise with increasing standard deviation a into one modality. For each a, we plot the OT cost between clean features cleann>>clean) and between noisy and clean features (ncesy-)clean). The noisy OT cost grows monotonically with and is consistently larger than the clean baseline, empirically supporting the claim that stronger channel noise leads larger transport costs.
performance despite having fewer negative samples, making well-suited for large-scale deploy- ments with limited resources.
Table 12: Batch).
Retrieval Performance (%) with Varying Mini-Batch Sizes (Number of Samples per
LOSS
Text - Audio
Audio Text
R@1
R@5
@@ 10
R@1
R@5
R@ 10
SOTA
Luong al.
2024
20.44
49.95
65.54
32.91
63.74
77.11
8
LIoT LUWD
24.24
57.57
72.49
35.21
65.93
78.78
SOTA
Luong al.
(2024)
33.77
69.94
82.44
43.36
74.19
85.78
32
LoT + LUWD
36.46
71.95
83.94
46.39
78.05
88.29
SOTA
Luong et al.
(2024)
39.28
74.50
85.85
52.45
80.25
90.17
128
Lior LUWD
39.94
75.07
85.64
53.08
83.49
90.59
M ABLATION STUDY FOR RAM
Table 13: Corr-based RAM variants AudioCaps (ResNet38-BERR) batch size 64).
Marginal Design A-T R@1
T-+A R@ 1
Mean R@
corr-gap corr-burt 50.99
51.83
38.12
44.97
38.75
44.87
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Published as a conference paper at ICLR 2026
Channel Reliability VS. standardized OT cost
6.9 10
Pearson Correlation log(C)): -0.379
x.P
standardized OT cost
6.6x 100
6.5
6.4 109
6.3 109
10 (Noisy)
Highest 10 (Reliable)
0.2
0.3
0.4
0.5
0.6
0.7 0.8 0.9
Relisbility
Figure 3: Relationship between reliability scores and standardized OT cost in trained DART model. For each feature channel j. we compute its reliability score ry and standardized OT cost Čj. Each point corresponds to one channel, and we report the Pearson correlation p(r, log C)) in the legend. Channels with low reliability (highlighted red) concentrate in the high-cost region, while high- reliability channels (highlighted in green) lie in the low-cost region, indicating that RAM success- fully down-weights noisy, high-cost channels in the transport.
N VISUALIZATION OF THE PROPOSED RAM
To verify that RAM truly identifies and suppresses noisy channels, we relate our reliability scores {ry} to the transport signal produced by DART itself. In trained model, the feature-level UWD module induces a channel-wist transport cost reflecting the inherent difficulty aligning each chan- nel across modalities. We denote this empirical, solver-induced quantity the actual OT cost. Since costs vary with feature scale and batch statisties, standardize them obtain Cj. removing scale and batch effects ensure comparability across channels. larger C2 means channel j is less consistent mooasl (more noise-like or podality-specifien and thus should receive less transport mass.
As shown in Fig. Yj is negatively correlated with the standardized OT cost, with Pearson p(r, log C) = -0339.. Channels with the lowest reliability scores (red) concentrate in the high-cost region, while the highest-reliabilit channels (green) lie predominantly in the low-cost region. This result supports two key points. First, the transport objective itself assigns higher costs to channels that behave noisily or inconsistently across modalities, corroborating our intuition that such chan- nels are poor carriers of shared semantics. Second, RAM is well aligned with this transport cost structure: by using Fj to form reliability-aware marginals, RAM assigns lower marginal weights to high-cost channels and biases the transport plan toward low-cost channels that more consistently encode stable semantic cues. Consequently, spurious alignment signals from small set of volatile channels are less likely dominate the matching process, and transport mass concentrated on channels that support robust cross-modal alignment.
25


Reviews:
[{"review_id": "8lz0ZINB7d", "reviewer": "Reviewer_P5Cb", "summary": "The paper proposes DART, a framework that enhances cross-modal retrieval by combining instance-level alignment with feature-level regularization based on the Unbalanced Wasserstein Distance. By reweighting embedding channels according to cross-modal consistency, DART suppresses noisy dimensions and stabilizes learning under small batches and scarce labels.", "questions": "The ablation study needs a major revision.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "* The paper goes beyond conventional instance-level contrastive alignment by introducing a feature-level regularization mechanism based on UOT and IOT. This dual-level design elegantly captures both instance-wise and dimension-wise consistency, addressing the long-standing assumption that all embedding dimensions are equally informative.\n\n* The authors provide clear theoretical analysis showing how instance-level alignment objectives scale with the maximum distance among aligned pairs, while feature-level regularization scales with the Frobenius norm of the transport plan. This theoretical distinction explains DART’s robustness to noise and its improved generalization under small batch regimes.", "weaknesses": "* The ablation study is poorly written so that it is not clear how each critical design choice is justified.\n\n* In the Introduction, the authors claim that ‘noisy channels tend to incur large transport costs.’ However, no empirical evidence is provided to support this claim, and it remains unclear whether the proposed feature alignment method effectively addresses this issue.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IuE2d14PoA", "reviewer": "Reviewer_LD5Q", "summary": "The authors propose a dual-level optimal transport framework that integrates instance-level alignment with feature-level regularization via Unbalanced Wasserstein Distance and Reliability-Aware Marginals, achieving more robust and stable cross-modal (audio–text) retrieval under small-batch and noisy-label conditions.", "questions": "1. As mentioned in Weakness 1, the reliability score formula corr - var - kurt seems heuristic. Can the authors provide more theoretical or empirical evidence for choosing this specific combination?\n2. As noted in Weakness 2, the O(d^2) complexity is a potential threat. Have the authors considered methods to mitigate this issue in high-dimensional settings (e.g., d > 2048)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The authors propose a novel and reasonable solution that incorporates instance-level inverse optimal transport and feature-level unbalanced Wasserstein regularization within a dual-level optimal transport framework.\n2. Introducing reliability-aware marginals (RAM) to reweight feature channels based on cross-modal statistics is intuitive and effective.", "weaknesses": "1. Lack of Deeper Ablation for RAM Components: The existing ablation experiment table (1) only compares \"DART w/ RAM\" with \"DART w/o RAM\". I think more fine-grained ablation experiments should be provided. For example: (1) Use only corr, (2) Use only corr-var, (3) use other combinations (such as weighted sum). Without these experiments, we cannot determine if components like kurtosis are necessary or if the current combination is optimal.\n2. Computational scalability concerns: The feature-level OT module introduces a cost matrix of size *d×d*, leading to quadratic complexity in feature dimensionality.   The paper does not provide sufficient discussion on scalability to high-dimensional encoders such as CLIP or BEATs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FPDFyV6Lzw", "reviewer": "Reviewer_kiaK", "summary": "This paper presents a dual-level alignment via robust transport (DART) method for audio-text retrieval. The proposed method has been explained in detail and experiments have been condudcted for evaluation.", "questions": "According to Table 1, using Beats as audio encoder achieves better performance. Please explain why this configuration was not adopted in following experiments.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "This proposed method combines both instance-level alignment and feature-level regularization for cross-modal retrieval. Plenty of experiments have been condcuted to answer four key questions.", "weaknesses": "The novelty of the proposed method needs more clarification. I don't think the introduction section gives a clear explanation on the connection between the proposed method and existing ones. Accroding to the Related Work section, channel-level considerations have been made in previous methods. Thus, the contribution of this paper would be not so significant if it only combines both instance and feature-level alignments. Besides, the paper writing still needs improvement. For example, there are ?? at line137/142.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lt6ZCPjIrn", "reviewer": "Reviewer_BWQA", "summary": "The paper proposes DART, a framework for cross-modal retrieval that addresses the limitations of purely instance-level alignment under small batch sizes and scarce labels. DART combines the conventional instance-level IOT objective with a novel feature-level regularization based on the UWD. It employs RAM to reweight feature channels based on statistical cues (correlation, variance, kurtosis), guiding the transport plan towards stable semantic dimensions and suppressing noisy ones. The framework demonstrates state-of-the-art results on audio-text benchmarks, and generalizing to image-text retrieval.", "questions": "1. See weakness\n2. In eq. 10, how to calculate kurt?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. The introduced feature-level alignment using UWD is new. It tries to address the known weakness of traditional contrastive losses (and instance-level IOT) that treat all feature dimensions equally.\n2. DART achieves better performance compared to previous works.\n3. Experiments on image-text retrieval shows the generalization ability of the proposed DART.", "weaknesses": "1. The writing of this paper is below ICLR's standard. For example, in Line 137 and 142, ``??'' apprear in the main text instead of the reference to equations, figures, or tables. In Eq. (6), x and y lack explanations.\n2. Sec. 2.3, limitations of instance-level IOT lacks of theoretical or experimental evidence to support the claims.\n3. Novelty is limited: The idea of learning feature importance (reweighting dimensions) is not new in cross-modal retrieval, as noted by the authors themselves when discussing Luong et al. (2024) . While DART's method uses cross-channel transport and richer statistics, the incremental novelty over existing per-channel weighting schemes is largely due to the expensive UWD machinery, which leads to a major concern about complexity.\n4. RAM is based on simple, first-order statistics (cross-modal correlation, variance, and kurt). These static, hand-designed proxies for ``semantic stability'' may be insufficient for highly complex, evolving feature spaces. The reliance on simple statistics makes the mechanism feel more like a heuristic than a deep, learned principle. The ablation also shows that RAM improves performance (Tab 1, DART w/ RAM vs w/o RAM) with limited improvements.\n5. Also, it would be interesting to see which part of RAM is most important.\n6. Lack of running time, memory comparison.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)