Geometry of Uncertainty: Learning Metric Spaces for Multimodal State Estimation in RL
Authors: Alfredo Reichlin
Adriano Pacciarelli
Danica Kragic
Miguel Vasco
Venue: ICLR 2026 Poster (2026)
Keywords: Multimodal
RL
Primary area: reinforcement learning
TL;DR: Learns a dynamics-aligned latent metric where distance reflects minimal action steps, fuses multimodal observations via inverse-distance weighting (no noise model), and achieves robust state estimation with better RL performance.

Abstract: Estimating the state of an environment from high-dimensional, multimodal, and noisy observations is a fundamental challenge in reinforcement learning (RL). Traditional approaches rely on probabilistic models to account for the uncertainty, but often require explicit noise assumptions, in turn limiting generalization. In this work, we contribute a novel method to learn a structured latent representation, in which distances between states directly correlate with the minimum number of actions required to transition between them. The proposed metric space formulation provides a geometric interpretation of uncertainty without the need for explicit probabilistic modeling. To achieve this, we introduce a multimodal latent transition model and a sensor fusion mechanism based on inverse distance weighting, allowing for the adaptive integration of multiple sensor modalities without prior knowledge of noise distributions. We empirically validate the approach on a range of multimodal RL tasks, demonstrating improved robustness to sensor noise and superior state estimation compared to baseline methods. Our experiments show enhanced performance of an RL agent via the learned representation, eliminating the need of explicit noise augmentation. The presented results suggest that leveraging transition-aware metric spaces provides a principled and scalable solution for robust state estimation in sequential decision-making.

Published as a conference paper at ICLR 2026
GEOMETRY OF UNCERTAINTY: LEARNING METRIC SPACES FOR MULTIMODAL STATE ESTIMATION IN RL
Alfredo Reichlin, Adriano Pacciarelli, Danica Kragic & Miguel Vasco
Division of Robotics, Perception, and Learning KTH Royal Institute of Technology (al frei, anop,, dani, mi migullls @kttkv..sthss
ABSTRACT
Estimating the state of an environment from high-dimensional, multimodal, and noisy observations is a fundamental challenge in reinforcement learning (RI.). Traditional approaches rely probabilistic models to account for the uncertainty, but often require explicit noise assumptions, in turn limiting generalization. In this work, contribute novel method to learn structured latent representation, in which distances between states directly correlate with the minimum number of ac- tions required to transition between them. The proposed metric space formulation provides a geometric interpretation of uncertainty without the need for explicit probabilistic modeling. To achieve this, introduce multimodal latent transi- tion model and sensor fusion mechanism based on inverse distance weighting. allowing for the adaptive integration of multiple sensor modalities without prior knowledge of noise distributions. We empirically validate the approach on range of multimodal RL tasks, demonstrating improved robustness to sensor noise and superior state estimation compared to baseline methods. Our experiments show enhanced performance an RL agent via the learned representation, eliminating the need of explicit noise augmentation. The presented results suggest that lever- aging transition aware metric spaces provides principled and scalable solution for robust state estimation in sequential decision-making.
INTRODUCTION
Estimating the state of the environment from high-dimensional observations is a key challenge in Reinforcement Learning (RL). Compact low-dimensional representations of the sensory informa- tion have been shown to dramatically improve the performances of data-driven agents in synthetic (Anand et al,, 2019) and real-world (Finn et al. 2016) Florensa et al.) 2019) settings. In realistic scenarios, sensory information estimating the state of the system along with a measure of uncertainty allows for robust control, (Ackermann et al. 1993). When having access multiple sensors, state estimation can become more precise as these can compensate for one another. This, however, requires a more complex design of a multimodal robust state estimation model.
be unreliable due to external noises failures. In such cases,
Optimal solutions can be found by applying a Bayesian formulation to the problem. Bayesian filter- ing techniques allow for the maximum a-posteriori estimate of the state space even in case of uncer- tainties. These, however, require restrictions in terms of modeling the state distributions (Kalman 1960) using expensive Monte Carlo approaches (Gordon et al. 1993). Moreover, they gen- erally require an observation model and prior knowledge of the functional form of the uncertainty (Thrun 2002). this regard, deep learning solutions have shown promising results on the determin- istic representation of htihh-dimensional observations (Bengio et al. 2013) and complex dynamics (Hochreiter] 1997). Regarding uncertainty estimation, approaches included either training under noisy conditions by using variational methods (Krishnan al. 2015) using reformulation of Gaussian processes (Garnelo et al. 2018). Reliable state estimation under uncertainty remains, however, open challenge.
We address the problem of state representation from multi multi-sensory observations for RL. We propose to learn representation that aligns the sensory modalities and correlates temporal distances with

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Z:
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T
2t
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We propose METRICMM, novel state estimation model from noisy multimodal obser- vations. Each observation (01, on the left) is mapped into joint metric space (Z on the right) where distances from the latent dynamics prediction (2, .pr (pr(2t- at-1)) are correlated with their uncertainty.
Figure 1:
Euclidean distances. The key idea of this work is to recast the problem of uncertainty estimation geometrically in metric space. This, in turn, allows to significantly simplify the problem of state estimation, Figure [ The contributed model, Metric Learning for Multimodal State Estimation (METRICMM)) consists of an encoder for each modality and latent transition model trained with simple contrastive learning and prediction losses. We empirically show that the proposed represen- tation can be used train RL. agent variety of tasks. Moreover, demonstrate how the proposed state estimate robust to sensor noise of arbitrary nature without heing trained on any of these. Our work makes the following contributions:
1. Metric-consistent multimodal state space. We introduce METRICMM, which aligns sory modalities a shared latent space where temporal proximity corresponds to Euclidean distance, enabling simple geometric reasoning for control.
2. Robustness under unseen corruptions. We empirically demonstrate how METRICMM maintains high returns under seven corruption families (unseen during training). including settings where of three modalities are corrupted, consistently outperforming fusion and representation baselines.
RELATED WORK
Representation learning describes the problem of extracting a mcaningful representation from high- dimensional observations (LeCun al. 2015). This is of particular interest in reinforcement learn- ing, where the dimensionality of the observations scales exponentially with the data needed for learning (Kober al.) 2013). State representation learning. (Lesort et al. 2018), refers those representations that describe the underlying state of a sequential decision-makiin work in this direction has focused on estimating latent representation that low dimensional and invariant to distractors by using a reconstruction loss Munk et al.) (2016); Mnih (2016) or contrastive methods [askin al. (2020). These approaches, however, lose the original structure of the state space and can't 't be generalized to other kinds of disturbances in the observation space.
problem. Early
In the case of reinforcement learning, data has temporal structure due to the sequential nature of the problem. Model- based approaches make use of this structure to guide the learning of the rep- resentation and the policy. This can be done by learning transition model concurrently with the representation to either facilitate the learning of policy (Lillicrap) 2015
Haarnoja et al.) 2018) or planning (Ha Schmidhuber, 22188) Hafner et al.) 2019b; Janner et al. 2021). Moreover, this temporal bias has been used explicitly to guide the structure of the latent space of the learned repre- sentation to simplify the control problem (Watter et al.
2015 Zhang et al. 2019 Eysenbach et al.
"oode he found https:tttt//ttt httpa///////hh...coeceecc www..coooreeihhhi/nnnnooc
Modda
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While these methods use additional biases to simplify the learning problem or recover additional properties, they do not offer a viable solution to the problem of noise robustness.
2022
Bayesian filtering offers solution to the problem of noise robustness in state estimation (Haykin 2004). Classic methods of filtering offer provably optimal solutions in this direction but tend to be quite restrictive
the assumptions needed and the class of problems they can be applied to. Some of these assumptions have been successfully addressed by merging these models with modern versions of deep representation learning (Krishnan al., 2015 Karl et al.) 2016). These models, however, assume a noisy dataset to learn the uncertainty of the transition model and the observation model and restrict the model of the state estimate to be in a known form like a Gaussian distribution. Moreover, two major limitations are the need to learn a generative model for the update step and the inability to handle multimodal observations. In HHarnoja et al. (201)), they propose the use of a discriminative encoder for the observations to avoid learning the generative model. In Liu al. (2023), they overcome the multimodality problem by substituting the Kalman Gain with a Transformer's s attention module on the learned encoding of the different modalities. Contrary to these methods, our proposed model doesn't need to estimate the uncertainty explicitly allowing us to be agnostic to the kind of noise. Moreover, multimodality is easily addressed by aligning the representation with an invariant loss.
A generalization of these methods is described with the term State Space Models (SSM) where transitions and observations as well as noise are relaxed to arbitrary functions (Billings 2013).
Recurrent State Space Models (RSSM) learn explicitly the temporal relation of data using autore- gressive models (Hafner et al. 2019a), or sequenceeto-sequence models (Becker et al. 2024).
Uncertainty can be taken into account using a probabilistic version of this (Doerr et al. 2018). This has been done using variational methods and imposing a known form of the probability estimate in
the latent space using Variational AutoEncoders (VAE) (Kingma 2013). Alternative solutions to
avoid the reconstruction loss include the use of Prototypes (Deng et 2022)) or contrastive meth- ods (Becker et al. 2023). Similar to Bayesian filtering methods, uncertainty estimation remains a key challenge for these algorithms. Other forms of uncertainty estimation include Neural Processes
(Garnelo et al.] 2018 Jung et al. 2024), and Energy functions (Zhang et al. 2023). These have,
however, been applied only to supervised learning settings.
Metric learning in the context of reinforcement learning has recently begun to attract growing inter- est. Steccanella & Jonsson (2022) introduce the concept of temporal distance in a POMDP setting and propose a novel objective to approximate these distances in a normed space. This, in turn, can be used for planning either via model predictive control or reward shaping. Similar normed spaces have been proposed to derive an exploration policy (Park et al. 2023), a foundation for general data-driven policies (Park et al., 2024), or learning offline policy from sub-optimal demonstra- tions (Reichlin et al.] 2026). Eysenbach et al. (2022) use a learned metric to estimate the discounted state occupancy measure of policy within a policy-improvement loop. To address the problem of asymmetry in general POMDPs, Wang et al.
(2023) extends this formulation to quasimetric repre- sentations and shows how these can be used to learn an optimal value function for goal-conditioned offline RL. Overall, these methods leverage metric (or quasimetric) structure primarily to facili- tate policy learning or planning, whereas we instead exploit such spaces to address the problem of uncertainty estimation under sensory noise in multimodal state estimation.
BACKGROUND
Throughout the rest of the paper, we assume a Partially Observable Markov Decision Process (POMDP) defined by the tuple O A,). Here, S denotes the true underlying Marko- vian state space of the environment, OnN denotes the N available observation modalities, and A the action space. The transition function T S A + S assumed to be deterministic. The objective is to maximize the cumulative reward given by the function S x A - R. scaled by the discount factor y. Each observation provides a potentially noisy measurement of the true state s. We assume that each observation is sampled from unknown stochastic process, with independent noise affecting each modality, i.e. of ~ (o st).
The overall goal of RL is to estimate a policy that maximizes the expected cumulative reward. When s is not directly accessible, the policy is generally conditioned on the available obser- vations, i.e. OL:N Representation learning for RL can be defined as finding a suitable latent
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representation for the sensory observations, such that the policy, Z - A, achieves the maximum possible expected cumulative reward. That is, preserving the information necessary for optimal decision-making while discarding task-irrelevant noise. Stochasticity in the observa- tions requires modeling the representation as an estimation process which can be formulated as a Bayesian filtering problem. Via the Markov assumption,
write the estimation process cursively through Bayes, i.c. p(z; 2t-1, 0z-1, ) p(o,:N z))p(z; 2e-1, F2-1,at-1)/p(o0 (t1))///pp(oo/)) This is generally intractable when assumptions on the functional form of these probabilities can- not be made. Here, we simplify the estimation process and consider a deterministic representa- tion of the state, as such we consider the maximum a-posteriori (MAP) estimate of the state, i.e. 2t arg m p(zz 0,,00 ) arg max;, p(o," zx)p(== ž at-1). In literature, the second term (p(z) 24-1, generally referred to as the prediction step while the first one (p(o)"N z))) as the update.
4 METHOD
We propose a novel approach to learn a latent representation of multi-sensory observations, along with latent transition model, that enables robust state estimation independently of the nature of ob- servation noise. Our method learns a metric space in which distances between latent states correlate with the minimum number of actions needed transition between them in the environment. This provides a geometric interpretation of uncertainty, simplifying state estimation.
4.1 LATENT STATE REPRESENTATION
In an ideal scenario, having access to a perfect representation of the environment's dynamics would be sufficient to track the current state. However, in practice, approximation errors in the learned transition model and potential stochasticity in the POMDP dynamics must be accounted for. In sequential decision-making problems, these errors can compound over time, leading to divergence in the state estimate (Ross et al., 201T). To mitigate this issue, we incorporate information from multiple sensor modalities refine the state estimate each step, akin to the Bayesian filtering formulation. However, traditional filtering approaches require explicit uncertainty modeling, which may not generalize well across varying noise conditions.
Instead, we propose to structure the latent space such that state transitions and sensory observations be integrated without explicit probabilistic modeling. The key idea behind this work is that the uncertainty induced by transition model errors is generally local to its predictions, not in the raw observation space but in space where distances are induced by the system's 's dynamics. Specifically, we argue that the transition model's uncertainty should be interpreted in a space where distances correspond to the minimum number of actions required to transition between states. This motivates the construction of a metric space that aligns with the dynamics of the enviromment.
To formalize this, we define a metric space M = (Z, I1-112) induced by an idcal injective map bp: S + Z, where Z C Ih' is vector space, and distances are given by the Euclidean norm. Following previous work (Steccanella & Jonsson| [202]] Park al., 2024 [Eysebbach et al., [022] Wang et al. 2023 Park et al.) (023), we define Z such that distances in this space are correlated with the minimum number of actions needed to transition between their corresponding environment states, i.c., temporal distances. Inducing norm the measure of distance does not allow for asym- metry, i.c., a quasimetric. As already noted by Steccaanella & Jonsson (0022), such a formulation can only capture symmetrized approximation of these distances, e.g., min d(s1. .2), d(s $1 Injectivity between the two spaces allows us to use the formalism of MDP Homomorphisms (Ravin- dran & Barto 2001] Van der Pol et al., 2020). In the case of POMDPs, we do not have direct access to the state of the system and thus have to rely on sensory observations and dynamic predictions. As such, we can define an approximation of the dynamics of the environment in this new latent space, i.e. fT: Z x A + Z. Given this structure, we make the following assumption:
The transition error at time t constrained such that the predicted latent state (r at-1) lies within a small ball of states that are close in terms of action- based distance. That is, there exists an c-radius region in the latent space such that:
Transition Error Assumption
24 B( (zt-1, e)),
(1)
where B(2,c) = (2' E Z [d((z s..
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Intuitively, while the transition model may introduce small prediction errors, these enrors generally remain within neighborhood of states that require similar sequences of actions to transition be- tween. Nevertheless, the learned transition model accumulates emrors time. This requires the integration of sensory information to correct the latent state estimate. Each modality Oc provides an independent estimate of the state. In this regard, we define deterministic mapping from each observation space to the above-defined metric space Z:
(i :::00 O' Z, vi E [1,
where N represents the number of sensor modalities. We seek learn these mappings such that the latent representation is aligned between them and respects the notion of metric space previously defined.
4.2 SENSOR FUSION VIA INVERSE DISTANCE WEIGHTING
The information from the sensors can be used to correct the prediction of the latent transition model on the state of the environment. However, due to noise, different modalities might be more less reliable at any given time. We can make use of the geometry induced by the metric space to approx- imately model this uncertainty. Since do assume prior knowledge of noise distributions, we propose an adaptive weighting scheme based on the notion of distance induced by the metric space. The key intuitions are:
If an observation's latent encoding (o(( is close to the latent transition prediction ( -1-1). 1) then it is more likely be an accurate state estimate.
Conversely, if an observation encoding is far from the prediction, it is likely corrupted by noise or uninformative.
Thus, we weigh each modality's contribution using the inverse of its latent space distance to the transition model estimate. The final estimated state is:
21 Σ 12 - 22 + )
Σ 12
2
(3)
8112 +8
where 2 = (r(2 ()-1),,,,....... 2 (o()) and 8 = 10-5, This formulation follows MAP principle,
where more confident estimates (i.e., those that agree with the transition model) receive higher weights. This enables robustness without requiring explicit noise modeling.
4.3 LEARNING THE LATENT REPRESENTATION
We introduce three loss functions to enforce the desired metric structure of the latent space. We do not require noisy observations during training uncertainty. Whenever we refer to & or ă441 in the loss functions, we mean the average of the sensor encodings:
do not need an explicit estimate of the
2
Σ p,oo))
(4)
Assuming noise in the sensory observations, the mean is equivalent to the state estimate in Equa- tion
Contrastive Temporal Distance Loss We enforce temporal consistency by structuring the latent space such that successive states remain close, while randomly sampled states are further apart:
C [[] - 22 -
(5)
__ E log([=, ==]12]]
(6)
where 2, is randomly sampled state. A similar formulation for different applications was proposed in |Wang et al. (2023): Park et al.] (2023) 2024). The term L. ensures that the latent representa- tion does not collapse into a trivial solution where all states are mapped to the same point. By encouraging larger distances between random state pairs, we preserve meaningful geometry in the representation space. This term, however, is bounded by the triangular inequality given the positive term of the loss, i.e. L+. As pointed out in Wang al. (2023), maximizing negative distances, with C,+ as a local constraint, effectively spreads out every state as much as possible. turn, this results in representation where temporal distances recovered also for non-adjacent pairs.
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Return
Epochs
Epochs
Epocts
Epochs
notse-C
notce-D4
notso-0.0
nottenD.
Figure Mean and standard deviation over 5 seeds of the training return for SAC agent on the Hopper-v5 environment. The policies are trained with different state estimation modules (Lin- earComb, ConCat, CURL, GMC) and different amounts of Gaussian noise on the observations. With an increase in noise, the expected average return sensibly decreases for all the estimators. Epochs are thousands.
Latent Transition Loss transitions, we minimize:
To ensure consistency between the learned transition model and observed
CT = [ a.) - Et+1)²]
This enforces that the transition model accurately captures environment dynamics.
Multimodal Invariance Loss invariance loss:
To align representations across modalities, introduce
Line [o)( ()()))) vije e [1
This ensures that each modality is mapped the same learned latent metric space.
4.4 INTEGRATION WITH REINFORCEMENT LEARNING
The representation objectives described above combined lincarly to form the overall loss:
C - LT A + AnC + XaCinv:
where Å1, A2, Å3 are scalar weighting coefficients. The learned latent representation serves as the input to any standard reinforcement learning algorithm and conceptually independent of the specific policy optimization method used. In practice, we jointly optimize the representation loss L and the reinforcement learning objective in an end-to-end fashion. Unlike traditional approaches that rely on explicit noise augmentation uncertainty modeling, our formulation inherently accounts for observation uncertainty through geometry the latent space, simplifying training while improving robustness to corrupted missing sensory inputs.
EXPERIMENTS
Scope and setup. We study whether multimodal representations sustain control perfor- mance under severe observation perturbations and cross-modal mismatch. Our evaluation spans two suites. (i) MuJoCo: Hopper-v5, HalfCheetah-v5, Ant-v5, Walker2d-v5, Humanoid-v5, and InvertedPendulum-55 with synchronized RGB and depth streams. (ii) Fetch: 7-DoF manipula- tion (FetchPickAndPPacce-. stated, we train Soft Actor-Co Critic (SAC) end-to-end on top of the representation module and re- port mean return standard deviation over 5 seeds. All architectural and optimization details are deferred to the Appendix.
FetchSlide-v4) with RGB, depth, and point clouds. Unless otherwise
Corruptions and evaluation protocol. To probe robustness, we inject seven families of pertur- bations at test time (unseen during training): Gaussian (Hendrycks & Dietterich] 2019), Salt-and- Dietterich] 2019), Patches (Becker et al. 2023 Grigsby Qi 2020 Hansen & Wang 2021). Puzzle (Bucci et al. 2021 Noroozi & Favaro 2016). Texture (Becker et al. 2023
Pepper (Hendrycks
Liu et al. 2023: Hansen & Wang 2021). Failure (Poklukar et al. 2022; Skand et al. 2024). and
Hallucination (Zhang al. 2220)). These applied with different probabilities study how fast performances deteriorate. For MuJoCo, we corrupt one modality at a time isolate each sensor modality contribution to the overall state estimate; for Fetch, we additionally consider settings

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Failure Hogger-v5
Hallucination Ant-vā
Patches EaliChectal:v5
Retuurt
Puzzle Wa.ker2dv5
Texture Humannod-vv
Salt: and pepper IeveriodFooooolomoe
Rlotuon
Noise Frequency
Noise Erequency
Noise Prequency
LinerCoomb
Cur
GMC
CORAL
ConCat
AMD
Metrich1M
Figure 3: Mean and standard deviation 5 seeds and 50 trajectories of the testing return for a SAC agent on the Mujoco suite. The policies are tested with different state estimation modules and different amounts of noise (perturbations of modality time). METRICMMIs the only estimator that allows for a consistent return with high-frequency perturbations.
Table 1: Return of multimodal fusion methods under Patch corruptions applied simultaneously to two modalities on Fettch-PickAnPPlaan for increasing corruption probabilities. METRICMM pre- serves strong control performance across all corruption levels, while alternative fusion strategies degrade much more rapidly.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
-0.89 0.02
-1.99 1.14
-1.76
1.35
-2.57 + 2.11
-2.67
1.56
-2.17
1,28
Concat
-0.04 + 1.63
0.9
-2.84
0.76
-2.66 0.43
-2.33 0.63
-2.43
- 0.24
CURL
+
-1.55 -
-3.76
-.59
-3.4
-2.58
0.38
GMC
-0.01 1.16
-0.91
0.27
0.38
0.69
-1.88 0.89
-2.41 1.24
AMDF
2.2 1.32
0.93 1 0.62
0.54
-1.75
-2.16 1 0.35
-2.34
1 0.31
CORAL
-0.36 +: 0.43
0.23
-1.51
0.99
-1.23 0.64
-1.38
0.86
-1.47
0.94
MetricMM
1.91 + 1.09
1.87 +: 0.93
1.43 1.14
0.92
0.79
-0.91
0.26
-1.47 +: 0.1
where two of the three modalities simultancously corrupted. For each (task, corruption) tuple, we evaluate 50 episodes per seed and aggregate across seeds to obtain performance--everity curves (Fig. 3)
Baselines. We compare against six representative fusion/represennaatin methods, matching encoder capacity and tuning budget:
Linear Combination (LincarComb): a learned linear map that combines per-modality
latent features.
Concatenation (ConCat): feature concatenation followed by a shared projection.
CURL (Laskin et al.
2020): contrastive learning on augmented views each timestep.
without explicit cross- modal fusion.
GMC (Poklukar et al.
2022):
cross-modal contrastive alignment to learn a shared embed-
ding.
a-MDF (Liu et al. 2023): an attention attention-weighted differentiable Bayesian filter that fuses modality-specific encoders into a latent state.
CORAL (Becker et al. 2023); joint latent space learned via per-modality reconstruction and temporal contrastive alignment.
All models are trained jointly with SAC under identical data budgets, replay settings, and evaluation schedules. Additional implementation details, architectures and hyperparameters are deferred to the Appendix.
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Table 2: Return of multimodal fusion methods under Failure corruptions applied simultaneously two modalities on ch-Sidd, for increasing corruption probabilities. METRICMM preserves strong control performance across all corruption levels, while alternative fusion strategies degrade much more rapidly.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
5.64 1 0.92
3.31 + 0.98
1.41 +: 2.14
-1.21 +: 1.74
-3.58 + 1.17
-3.36
0.35
ConCat
7.01 +: 1.64
5.5 1.4
4.01 1 0.48
0.06
1.54
-1.61
1.73 1.73
-1.49
1.2
CURL
7,43 1: 0.43
5.34
1.89 2.35
-1.36
3.33
1
-3.69
1: 3.21
GMC
5.17 = 0.4
1.24
1.78
-1.77
1.77
-2.72 1,02
-3.11
= 1.59
AMDF
3.05 1.62
2.76 1.61
0,48:10
-0,67
-1.98
-2,41 2.25
CORAL
4.7 2.92
2.28 + 0.63
0.71
0.839
-1.87 1.8
-1.79 0.97
MetricMM
9.26 0.21
7.77 + 2.02 =
4.96
1.33 + 2.49
-0.23 3.58
robustness to single-modality corruption. Figure 3 summarizes perfor- mance as the perturbation probability increases for six tasks and diverse corruptions. We observe three consistent trends. (i) Robustness slope. Our method (METRICMM) exhibits the flattest degra- dation, preserving large fraction of the clean performance up to mid/high severities across tasks; in contrast, simple fusion baselines (LinearComb/CConaa) degrade steeply even for low frequency of corruptions, indicating that naive aggregation does not resolve cross-modal disagreements. (ii) Method ordering is stable across corruptions. Across failure, hallucination, texture, puzzle, patches, and salt-and-pepper, METRICMM maintains the top curve while CURL and GMC are competitive at low frequencies but drop sharply once the corrupted modality dominates the fused signal. (iii) High-DoF tasks are particularly sensitive. On Humanoid-v5 and HalfCheetah-v5, the gap between METRICMM and the best baseline widens with an increased perturbation's frequency, suggesting that principled cross-modal consistency becomes increasingly important control com- plexity grows.
Results on MuJoCo:
Results on Fetch: robustness under multi-modality corruption. We next corrupt two of the three modalities on manipulation tasks. Tables and 2 returns for patches and failure corrup- tions, respectively, as the probability of corruption per time step increases. Two effects stand out. (i) Majority corruptions. Even when the majority of sensory channels are degraded, METRICMM retains substantially higher returns, e.g., on Fetch Slide with failure noise, it remains above zero reward until very high severities, while all baselines collapse much earlier (Table 2). (ii) Graceful decay vs. collapse. On Fetch Pick-and- Place with patches, baseline returns turn negative quickly as the probability increases, while METRICMM degrades gradually and remains competitive inter- mediate frequencies (Table D). These results indicate that METRICMM can prioritize and re-weight the remaining reliable modality when others fail.
Training with noisy observations is not required (and can be harmful). Figure shows learn- ing curves on H5pper-v5 when injecting noise during training for four of the baselines. While moderate noise can induce invariances, it consistently slows exploration and lowers asymptotic re- turns across preprocessors; severe noise significantly delays the onset of learning. Practically, this requires knowing the corruption family priori and increases computation, whereas METRICMM attains robustness without any noisy training.
Every modality provides useful information. Not all observations are equally easy to exploit, and without explicit cross-modal alignment, policy often latches onto the cheapest signal. In Fetch, the point-cloud stream is the most informative for precise geometry but also the most demanding encode, making it especially vulnerable perturbations. To quantify reliance, Table reports performance when we corrupt only one modality at a high probability (0.99). When point clouds are the sole corrupted stream, most baselines exhibit little to no degradation, revealing a systematic over- reliance on RGB/depth and under-utilization of 3D structure. In contrast, our aligned representation distributes credit across modalíties and shows a more uniform sensitivity profile, indicating that each sensor contributes meaningfully to the leamnd state.
Temporally dependent noise. In realistic scenarios, noise is not necessarily time independent. For example, faulty sensor might produce persistent noisy observations for multiple consecutive time 4 we present results on the Fet chsuite for persistent sensor failure perturbations
steps. In Figure under 3 and 10 consecutive frames. METRICMM remains the most reliable method.
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Table 3: Return of multimodal fusion methods under Failure corruptions applied single modality with probability on ch- METRICMM remains robust independently of the dropped modality.
Model
all modalities
image only
depth only
point cloud only
Linear Comb
-1.14 1 2.41
-2.94 1,26
-1.96 +: 2.14
9.27 0.17
ConCat
1.53 1: 2.91
0.28 1 6.19
-2.38 1 0.67
7.37 + 2.11
CURL
0.16
-2.17 = 1.51
-0.06 + 8.26
8.15 =15:00 0.94
GMC
-1.76 1.76
-1.13
1.10
-4.52 4.01
7.67 0.53
AMDF
0.1 1.53
-2,62
2.87
-1.63 2.91
3.80 = 2.28
CORAL
-1.32 4.67
-3.87
4.63
-2.53 +: 1.15
7.56 2.03
MetricMM
8,46 + 0.92
7.29 + 1.39
8.90 = 0.96
7.87 = 0.65
Fetch PickAndPiacc, K
Fessh PickAndPlace, K
Fetci- Slide, K
Fetch Slide, X 10
Relumm
Proquency
Prequenry
Nosse Prequency
Uroquency
UnearComb
Curl
GMC
CORAL
ConCat
AMDF
MetricMM
Figure 4: Mean and standard deviation over 5 seeds and 50 trajectories the testing return for SAC agent the Fetch suite under time-penn umee-persistet sensor failure. The policies are tested with different state estimation modules and different amounts noise. Fach time the noise applied, persists for either 3 or 10 consecutive frames (K) that specific sensor modality. METRICMMis the only estimator that allows for consistent return with high-frequency perturbations.
5.1 ABLATIONS
We study the effects of the different components of the representation loss. For this, we consider an additional one-dimensional pendulum environ- ment where the goal to swing the pendulum up- side down, following (Silva et al., (0200. As sen-
-10
Roturn
sor modalities, we rely on images of the scene as well as sounds generated by the pendulum swinging and captured by three evenly distributed receivers. The amplitude and frequency of the received signal give us information on both the position and veloc- ity of the pendulum, thanks the Doppler effect. For this experiment, we consider the performances of METRICMM against two versions of itself, one trained without the invariance loss term (Ci..) and one without the metric loss pair (I+ and T..). With no invariance loss, the representation has no incen- tive to align the two modalities. Without the metric terms, the other hand, the representation loses the temporal structure and gets arranged to minimize the transition loss which compresses the space consider- ably. both cases, SAC is still able to learn a meaningful policy on top of these representations. However, the noise robustness ability is lost as depicted in Figure
-200
Noise Frequency
MetricMM
MetricMM no Cat MetricMM no 2, C.
Figure 5: Average return for a SAC agent on the one-dimensional pendulum environment and increasing level of Gaussian noise. A METRICMM exhibits robust performances, both of the ablation variants degrade with an increase noise.

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5.2 LIMITATIONS
Across locomotion and manipulation, METRICMM maintains strong control performance under severe and diverse corruptions, including cases where two of three modalities are degraded. The method remains robust without any noise-aware training, offering a practical route to resilient mul- timodal RL. Nevertheless, it has few important limitations.
Symmetric metric assumption. In its current form, METRICMM defines distance in latent space via a norm, and is therefore restricted to symmetric spaces. This pre- vents it from exactly representing quasimetric structures that frequently arise in RL, where the distance from to can differ from the distance from to s due to irre-
Return
versibility or constraints. As result, the learned metric can only provide a symmetric approximation of the true minimal-actio distance. While this limits the theoreti- cal fidelity of the representation, it is not catastrophic in practice. Distances, in this case, can still provide ap- proximation of uncertainty, e.g., pushing objects in the Fetch environments directionally constrained. The pol- icy, on the other hand, is not affected by this. The network does not explicitly rely on the measure of distance to es- timate the optimal action. Nevertheless, generalizing the representation to a quasimetric would allow for a tighter uncertainty estimation.
Noise Frequency
MetricMM
Concat 0
MetricMM a -
Concat
MetricMM n-
Concat 10
Figure Average return for a SAC
agent on the stochastic pendulum with Hallucination noise. With increased stochasticity (a) the performances duce, METRICMM remains robust while ConCat baseline quickly de- grades.
Deterministic dynamics. As discussed in the back- ground section, our formulation assumes deterministic dynamics the environment. tie transitions, the representation considers every possible next state to be adjacent to the current one and thus embeds it at a unit distance in Z. Moreover, the latent transition model estimates the barycenter of this next-state distribution. In moderately stochastic conditions, this is possible in practice. Figure/6fillustatie the degradation in performance under Hallucination noise for stochastic variation of the pendulum experiment for METRICMM compared to a ConCat baseline. While overall performance degrades as stochasticity increases, METRICMM retains a clear robustness advantage. Stochasticity the dynamics does require an increase in the dimensionality of the learned space to accommodate for an increase in the number of adjacent next-state and might be unfeasible practice. Moreover, modeling the latent transitions with deterministic module results in decrease in expressivity. Extending the framework to gen- eral stochastic setting might require a different notion of distance (e.g., replacing the norm-based distance with a divergence between predictive distributions) and different transition model.
the case of stochas-
CONCLUSION
We introduced novel approach for robust state estimation in reinforcement learning by learning structured latent space where distances reflect the minimum number of actions required to transi- tion between states. This metric space formulation enables geometric interpretation of uncertainty, eliminating the need for explicit probabilistic noise modeling. Our method integrates multi-sensory observations via inverse distance weighting, ensuring adaptive sensor fusion without prior knowl- edge of noise distributions. Additionally, contrastive and transition transition-consistenyy consistency losses enforce tem- poral structure, while an invariance loss aligns representations across modalities. We empirically demonstrated the effectiveness of our approach a diverse set of tasks, showing that it improves state estimation robustness in the presence of sensor noise and significantly enhances RL agent per- formance. Results confirm that the learned representation generalizes across different environments without requiring explicit noise augmentation. By leveraging the environment' 's dynamics within the latent space, our approach provides scalable and principled solution for robust state estimation. Fu- ture directions include adapting the method to non-stationary environments, evaluating adversarial robustness, and extending real-world robotics.
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7 ACKNOWLEDGEMENT
The work was enabled by the Berzelius resource provided by the Knut and Alice Wallenberg Foun- dation the National Supercomputer Centre. We further thanks the Swedish Research Council and the Knut and Alice Wallenberg Foundation. We thank Jens Lundell for the simulation of the point clouds code.
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A APPENDIX
A.1 EXPERIMENTAL DETAILS
Here, we provide the necessary details on the conducted experiments. For the MuJoCosuite, we augmented the observations to include both RGB images and depth images. For both, we used resolution of 84x84, 84. To ensure Markovianity in the observations, we stack 3 consecutive frames together for both the RGB images and the depth images. We do not use proprioceptive information. We keep the reward function to be the default one of the different environments. For the eescuui we augmented the observations to include RGB images, depth images and point clouds. Both RGB and depth images have resolution of 128x 128. We generate the point cloud observations by projecting rays from the camera position, given the depth of the environment. This results a variable number of points, each consisting of 3D position and RGB value. Again, for all the modalities, we stack 3 consecutive frames together. For these environments, we changed the reward function be linear combination of the negative Euclidean distance between the and-effector of the arm and the object and the negative Euclidean distance between the object and the goal (the latter scaled by the initial distance and multiplied by factor of 10). For the Pendulum experiment, the dynamics are defined as:
0, 3g
sin 02 +
mf2 at.
(10)
2
&, N (0, 0,
(11)
0.+1 (: + (0, + Ex) At, )
(12)
0.11 0, + 0c+1 At,
(13)
where g 10, = 1, = At 0.05. The sound observations generated by computing the frequency and amplitude a sound originating from the tip of the swinging pendulum at a constant frequency. The sound is perceived by 3 receivers each time step in the form of frequency (modeled via the Doppler effect) and an amplitude (modeled via the inverse-square law).
For each model and each environment, learn Soft Actor-Critic agent to maximize the reward of the environment. The list of relevant hyperparameters is described in Table 4 these are the same for every experiment.
Table 4: SAC hyperparameters.
Hyperparameter
Value
0.99
0.005
Learning rate actor
0.0003
Learning rate critic
0.001
Replay buffer size
100000
Batch size
256
Initial
Number critics
Number of simulations hetween updates
Number of workers
2
Target entropy
dim of actions
Number of hidden layers critic
Number neurons per layer critic
256
Non-linear activation critic
ReLU
Number of hidden layers actor
Number of neurons per layer actor Non-linear activation actor
256
ReLU
For every experiment, train the SAC agent together with its representation module end-to-end by minimizing the linear combination of all its losses. We train the agent until convergence, around 200 thousand steps for the McJJeCoenvirommenss and 400 thousand for the Fet.chenvironments. After
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convergence, we test the best-performing models for 50 trajectories. Noise is applied at random to any modality at every step (except the initial observation) with a different probability.
For every state estimation method the RGB and the depth images and a SetTransformer for the point cloud. The CNN consists convolutional layers with 32 filters each and kernel size and stride respectively of [4, 2]. [4, 2), [3, I] with a fully connected output layer. Each layer is followed by a ReLU non-linear activation func- tion. We encode each point cloud frame with a compact Se(Transformer. 6-D points (XYZ+RGB) are linearly projected to width dmodet-644 processed by two self-attention blocks (4 heads, Layer- Norm+residuals, forward width 2x). and pooled via multihead attention with a single learned seed (k=1). Over a sequence of T- frames, the pooled 64-D summaries are concatenated and mapped with linear layer to latent 2 of size d_ 64, then refined by a small MLP (64 -> dz,) to produce the final embedding.
Convolutional Neural Network (CNN) architecture for
Below the list of hyperparameters used for METRICMM and the baselines:
latent size d, 64. Per-modality encoders (CNN for images, Set Trans- former for point clouds). The loss hyperparameters are all 1, i.e., AT \1 X2 Xine
METRICMM:
per-modality encoders (CNN/Set Transformer) projecting to dz. learned lin- ear fusion over modality latents, capacity matched to ours (same dy, same encoder widths).
LinearComb:
ConCat: same per-modality encoders dy; fusion by feature concatenation (final latent dimension scales with number of modalities).
CURL: per- per-modality encoders to d, with contrastive head, InfoNCE temperature 0.2, momentum target encoder with EMA m=0.99; standard cropping as image augmentations and gaussian noise point cloud augmentations.
GMC: per-modality encoders to a shared embedding with cross-modal contrastive align- ment; InfoNCE temperature -00.3; same d, and encoder widths ours for fairness.
AMDF: attention-weighted differentiable filter over modality encoders with recurrent latent state; latent size d, and filter MLP widths matched ours; attention temperature/weights from the implementation defaults.
CORAL: per-modality encoders with joint latent space trained via reconstruction trastive objectives; InfoNCE temperature 7-0.1; same dy and encoder capacity as ours.
A.2 NOISES
For all the environments we experiment with the following families of corruptions as noise. Figure shows an example of the different noises for the MuJoCoAnt-v5 environment.
Gaussian: linear combination with samples from a Gaussian distribution. For RGB and depth images: (N, 25), for point clouds: (0 0.03). Can be applied all the modalities.
halt-and-pepper with probability 0.3 for MuJoCoand 0.8, each dimension the observa- tion is transformed to either the minimum maximum possible value. Can be applied to all the modalities.
portion at random of the image is masked, 30% for MuJoCooobeervations and 50% for Fetch. Can be applied to RGB and depth images only.
Patches:
Puzzle: the image is divided into a 3 by 3 grid and reshuffled. Can be applied to RGB and depth images only.
Texture: the background of the image is segmented and replaced with another image. Can be applied to RGB images only.
Failure: the entire observation is set to 0. Can be applied to all modalities.
Hallucination: the entire observation is substituted with another in-distribution observation from another trajectory. Can be applied to all modalities.
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Hallucination
Gaussian
Salt and pepper
79
Patches
Puzzle
43
Failure
Texture
13
10
Figure Examples of noises on the RGB images for the MoJoCoAnt-v5 environment.
A.3 ADDITIONAL MUJoCO RESULTS
In Figures and we provide additional results on the Mu.JoCoenviromment They illustrate the average reward and the standard deviation for every combination of noise for all the environments in the suite. Noise is applied to modality at time. Results are obtained from 50 evaluation trajectories after the models have fully trained.
A.4 ADDITIONAL FETCH RESULTS
Below, we provide additional results on the Fet Fetcehnnvronmeet The tables describe the average reward and the standard deviation for every combination of noise for the Pick and Place and the Slide environments. Noise is applied to either or modalities at time. Results are obtained from 50 evaluation trajectories after the models have fully trained.
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Table 5: Return of multimodal fusion methods under Failure corruptions applied to one modality on cetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
Linear Comb
-0.82 = 0.16
-1.2 = 1.38
-1,65 + 2.14
-1.53 - 1.68
-1.68 + 0.62
-1.37 + 0.76
ConCat
0.8 1.14
0.95 = 1.59
0.44 = 0.84
10499
-0.17 +
-0.92 0.47
CURL
1,58 1.0
1.08 1.11
-1.56
-2.08 0.2
-2.29
-2.56 + 1.36
GMC
1.15
-1.34
1.4 0.87
-1.51 0.77
-1.62
-1.63 163+0.00 0.54
-1.6 0.57
AMDF
2.73 + 1.11
2.09 = 1.08
1.02 + 0.93
-0.3 1
-0.83 + 1.98
-1.52 + 1.55
CORAL
-0.81 + 0.14
-0.84
0.12
-0.85 + 0.1
0.87
-1.04 + 0.38
-1.11 + 0.56
MetricMM
2.31 + 1.34
2,35 + 1.08
2.03
0.68
2.35 + 1.18
2.21 + 1.2
1.89 1.2
Table 6: Return of multimodal fusion methods under Gaussian corruptions applied to one modality on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
-0,84 + 0.54
-1.02 = 0.32
-0.78 = 0.64
-1.37 + 0.8
-1.42
0.85
-1.36 + 1.07
ConCat
1.36 + 1.06
0.28 = 0.36
-0.39 = 0.92
-1.89 0.44
-1.68
1.11
-1.51 + 0.66
CURL
2.97 + 1.32
1.33 + 1.23
0.13 0.99
-0.75
-1.11 0.41
-1.43 +
GMC
-0.24 1.86
-0.15 0.78
-0.55 0.71
0.942
-1.94 1.77
-0.85 + 0.4
AMDF
2.57 + 1.69
1.97 0.94
1.07 + 1.41
+0.04 + 1.08
-0.24 + 0.35
-0.56 + 0.28
CORAL
-0.36 + 0.59
-0.58 1:8:00 0.23
-0.57 + 0.24
-0.78 0.13
-0.61 + 0.19
-0.5 1 0.38
MetricMM
2.29 + 1.3
2.52 = 1.67
2.28 + 0.87
1.97 = 1.33
1.95 + 1.17
2.19 + 1.2
Table 7: Return of multimodal fusion methods under Hallucination corruptions applied to one modality on Fet ch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
-0,95 + 0.21
-1.44 + 0.58
-1.28 + 0.33
-1.57
+ 0.53
-1.82 + 0.7
-1.93
0.81
ConCat
-0.08 + 0.93
-0.77 0.47
-1.84 0.99
-2.38 + 0.46
-2.67 0.6
-2.44 + 1.27
CURL
2.26 0.81
2.44 = 0.65
0.69 0.3
0.32 0.26
-0.48 + 0.84
-0.83 0.48
GMC
0.28 1.29
0.12 1 0.81
-1.72
1.39
-1.42 1 1.17
-1.37
0.64
AMDF
2.43 1 1.19
1.03 1 0.78
0.25 1 0.34
0.7
-1.36 0.55
0.55
-1.21 0.16
CORAL
-0.43 +: 0.52
-0.58 1: 0.3
-0.59
- 0.33
-1.02 1 0.21
-1.24 1 0.35
-0.96 1 0.13
MetricMM
2.17 +: 0.73
1.99 + 1.23
1.85 + 1.09
1.85
0.96
2.11 + 0.84
2.08 + 1.05
Table 8: Return of multimodal fusion methods under Patches corruptions applied to one modality on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
-0,75 + 0.57
-1.07 = 0.87
-1,42 = 1.13
-1.36 0.94
-2.13 + 0.97
-2.59 + 1.68
ConCat
0.14 0.18
-0.55
0.29
-1.73 + 1.0
1.62
-1.47 1.11
-1.97 + 0.58
CURL
1.52 -: 0.83
-0.56
1.15
-2.43 1.3::000 0.93
-
1.14411.16
-2.67 + 0.32
GMC
-0.1
-1.78
0.78
0.94
-1.37
-1.92
-1.28
AMDF
2.76 + 1.44
1.28 + 1.03
0.56 - 0.89
-1.
-1.09 1:09::00 0.37
-1.98
1.28
CORAL
-0.59 + 0.19
-1.22 = 0.73
1.26
-1.23
-1.5 = 1.18
-1.13 - 0.62
MetricMM
2.33 + 1.16
2.13 + 1.32
1.89 + 1.21
2.22 1.51
2.13 + 1.39
2.23 + 1.56
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Fullure Ant-v5
Caussion Ast-45
Potches Ani-55
200
Purzle Ant-v5
Selt and pepper Ant-vS
Texture Ans-45
300
Patlrre HalfChhettbbed
Gaussian HellChhetabb-S
Hallucination HallCheetah-vS
21200
2000
200
0.5
Puzzle HalfCheetab-ed
Salt and pepper HallCheetabbv
Testure HalfCheetsh-ss
11080
200
Gauesian Hoppersu5
Hellucination Hopper-e5
Patches Horpersu5
200
Puzzle Hopper-u5
Salt and pepper Hopper-u5
Testure Hoppersu5
100
700
Notse Prequency
Noloe Prequency
Noise Frequency
UncarComb
Curl AMDF
GMC
CORAL
ConCat
MetricMM
Figure& 8: Mean and standard deviation over 5 seeds and 50 trajectories of the testing return for SAC agent on the Mujoco suite. The policies are tested with different state estimation modules and different amounts of noise (perturbations of one modality at time).
19

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Published as a conference paper at ICLR 2026
Pullure Humanood--s
Gaussian Humanold-rh
Holluctnation Humannodeh
300
200
300
1700
Patches Humancid-v5
Pazzle Humanold-v5
Solt and pepper Humancid-V5
200
1000
180
Fullure InvertedPendulumm5
Caussian InvertedPendulumm
Hallurination InvertedPeadulum-9
102
Ratches tavertedPendulums
Puzzle InvertedPenollumtt
Testure IevertedPendullmme
130
65
Fallure Wallier2dvS
Caussian Walker2d45
Hallucination Walker2d.w5
400
200
Patches Waller2dads
and pepper Walker2ddvv
Tenture Walker2d-s5
300
Noise Frequency
Noise Erequescy
Noise Frequency
UneerComb
Curl AMDF
GMC
CORAL
ConCat
MotricMM
Figure 9: Mean and standard deviation 5 seeds and 50 trajectories of the testing return for SAC agent on the Mujoco suite. The policies are tested with different state estimation modules and different amounts of noise (perturbations of one modality at a time).
20

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Published as a conference paper at ICLR 2026
Table 9: Return of multimodal fusion methods under Puzzle corruptions applied to one modality on ch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
-0.29 + 0.4
-0.5 = 0.58
-1.39 = 0.47
-1.23 0.74
-1.03
0.22
-1.66 + 0.78
ConCat
0.64 1.05
-0.25
1.48
-0.97 0.21
-1.18
-2.62 1.45
-1.71 0.31
CURL
2.79 1.14
2.79
1.02 0.19
+.44
-1.69
0.63
-2.66 0.32
GMC
-0.87 =:07:100
-0.97
0.55
-1.54
-
-1.31 0.42
-1.4 -1.4
AMDF
2.4
1.69 # 1.14
-0.67 = 1.79
-2.02 + 0.47
-2.61 + 0.6
-3.89 + 1.4
CORAL
-0.55 0.32
-0.64 0.11
-1.38 + 0.65
-1.53 + 0.99
-1.39 + 0.62
MetricMM
2.24 + 0.89
2,26 + 0.7
1.71 + 1.27
2.37
1.33
1.85 + 1.4
1.98 + 1.39
Table 10: Return of multimodal fusion methods under Salt and Pepper corruptions applied to one modality on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
-0.33 + 0.56
-0.68 = 0.19
-0.68 + 0.1
-0.83 = 0.02
-1.08 + 0.22
-0.72 + 0.1
ConCat
0.56 = 1.5
1.08 = 0.95
0.62 0.67
-0.76 = 1.26
-0.79 = 0.56
-0.89 + 0.43
CURL
3.56 0.93
2.33 + 0.88
1.64 0.85
0.49
0.15 + 1.29
-0.66 0.85
GMC
-0.15 0.96
0.96
-0.92 + 1.3
-1.64 1.06
-1.24
-1.9
-1.43 +
AMDF
2.54 1.07
1.83 + 0.7
1.68 + 0.55
-0.12 0.23
0.01 + 0.09
-0.83 + 0.43
CORAL
-0.39 + 0.48
-0.53 1 0.3
-0.62 - 0.16
-0.83 + 0.1
-0.88 + 0.12
-0.83 1 0.1
MetricMM
2.43 + 1.61
1.83 + 1.22
1.67 + 0.51
2.21 = 1.24
1.89 - 0.58
2.26 + 1.04
Table 11: Return of multimodal fusion methods under Texture corruptions applied to one modality on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
-0,22 + 0.97
-0,63 + 0.76
-0.42 + 1.38
-0.55 + 0.5
-0.36 + 0.41
-0.02
0.69
ConCat
1.56 + 0.78
0.89 = 0.96
0.71 = 1.69
-0.14 1.36
-0.62
2.25
0.86 + 1.91
1.91
CURL
2.39 + 1.46
0.09 - 1.02
-1.97 0.38
-2.9 0.35
-2.89 0.2
-3.2 0.31
GMC
-0.1 1 0.71
0.86
-1.41
1.45
-2.02 1 1.33
-1.6 1 1.01
0.97
AMDF
3,21 1: 2.03
2.36 1 1.66
1.26 + 1.34
0.1
-0.79 17991000 0.19
0.09 1 0.22
CORAL
-0.35 1: 0.54
-0.67 0.14
-0.99
1 0.3
-1.8 = 1.38
-1.54 1 1.08
-1.09 1 0.41
MetricMM
1.84 + 0.94
2.1 + 1.09
2.52 + 1.36
2.19
1.01
2.27 + 1.22
1.95 + 1.31
Table 12: Return of multimodal fusion methods under Failure corruptions applied simultaneously to two modalities on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
-1.32 + 1.36
-1.54 + 0.82
-1.36 = 0.54
-2.28
+ 1.32
-2.04 + 1.22
-2.37 + 1.16
ConCat
0.17 + 1.29
-0.25 0.1
-1.17 = 0.79
-2.37 = 1.99
-2.1 = 0.19
0.19
-2.78 0.99
CURL
1.32 + 0.43
-0.15
-2.39 :: 0.83
-.77
-2.55 0.65
-2.64
GMC
0.05 1.06
-1.64 0.63
-1.68 -
-2.8 0.38
-2.15 0.47
-2.28 0.45
AMDF
2.76 + 1.39
1.93 + 1.72
-0.18 - 1.06
-1.27 0.87
-2.39
1.21
-1.77 + 0,59
CORAL
-0.6 + 0.14
-1.09 = 0.43
-1.1 - 0.53
-1.34 0.82
0.82
-1.35 0.1
-1.83 - 1.51
MetricMM
2.22 + 1.74
2.44 + 1.48
1.65 + 0.64
-0.47 + 1.2
-1.51 + 1.25
-3.0 + 2.27
21

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Published as a conference paper at ICLR 2026
Table 13: Return of multimodal fusion methods under Gaussian corruptions applied simultaneously to two modalities on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
-0,54 + 0,67
-0,99 = 0.54
-1.64 0.8
-1.21
0.31
0.31
-1.91 + 0.78
-1.38 + 0.36
ConCat
-0.05 = 0.58
-1.05 0.66
-1.96 1 1.91
2.63:...7
-3.09 1.09:00 0.98
33.77 0.23
CURL
1.89
0.29 0.6
-1.5
-1.94
-2.11
-2.04 0.54
GMC
-0.1 1.09
-0.88
0.91
-1.59 +
-1.57 1.08
-1.22
0.44
0.44
-1.4 -1.4 0.52
AMDF
2.21 + 1.56
1.56 # 1.24
-0.27 # 0.45
-2.15 + 0.69
-2.65 0.96
-2.64 + 0.5
CORAL
-0,48 + 0.38
-0.59 + 0.25
-0.22
+ 0.25
-0.81 + 0.03
-1.1 0.44
-0.88 0.12
MetricMM
2.1 + 1.49
2,46 + 1.23
1.76 + 1.5
0.46 0.63
-0.83 + 0.39
-2.4 + 0.5
Table 14: Return of multimodal fusion methods under Hallucination corruptions applied simultane-
ously to two modalities on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
-0.24 0.61
-1.4 = 0.88
-1.73 = 0.65
-1.48 + 0.48
-1.92 + 1.1
-1.81 + 0.74
ConCat
-0.13
-0.92 1.23
-2.55 = 0.79
-2,69 = 0.62
-3552 0.62
-3.95
0.83
CURL
2.19
0.43 + 0.66
0.36
-2.22
2.24
-283 +
GMC
-0.12
-1.31 1:1::0666
-1.36 -
-2.11 + 1.02
-2.26 1.15
-2888 + 1.63
AMDF
0.91
-0.2 0.53
-0.77 = 0.69
-1.79 +
-2.85 + 0.23
-2.75 + 0.23
CORAL
-0,6 0.23
-0.92 - 0.38
-1.32 1.32:0.44 0.46
-1.46 0.54
-1.72 1: 0.8
-2.25 1: 1.28
MetricMM
2.36 + 1.18
2.13 = 1.33
1,86 + 1.07
0.71 0.46
-0.87 + 0.54
-3.46 + 1.15
Table 15: Return of multimodal fusion methods under Puzzle corruptions applied simultaneously to two modalities on Fe PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
-0.77 + 0.81
-1.09 + 0.35
-1.07 + 0.32
-1.97 + 0.9
-1.74 + 0.72
-1.63 0.61
ConCat
0.09 2.02
-1,83 + 1.13
-2.94 = 1.32
-266
-1.92 0.51
-2.01 0.67
CURL
2.61 + 1.13
1.34 = 0.86
0.86
-1.41
0.47
-.78
-1.77 0.03
-1.35 0.0
GMC
-1.15 1: 1.29
-1.27 1
-2.43
1.63
-1.37
-1.53
-2.22 1 1.2
AMDF
1.94 1 1.22
-0.04 1004:100 0.64
-1.9 1 1.32
22.4
-2.38 1 0.78
-2.31 0.74
CORAL
-0.34 1 0.56
-1.12 1 0.51
-1.22
1 0.52
-1.49 1 0.67
-1.26 0.3
-1.36 1 0.39
MetricMM
1.94 + 0.51
2.18 + 1.01
1.79 + 1.64
0.38
0.69
-1.05 + 0.49
-1.86 + 0.71
Table 16: Return of multimodal fusion methods under Salt and Pepper corruptions applied simulta- neously two modalities on Fetch PickAndPlace, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
-0,55 + 0.37
-0.57 + 0.2
-1.03 = 0,36
-1.21 + 0.46
-0.99 + 0.2
-0.96 + 0.16
ConCat
-0.02 = 1.22
0.25 = 1.43
-0,91 = 0.99
-2.06 - 1.24
-1.94 1.94=00 0.51
-1.38 + 1.37
CURL
2.53 0.66
2.08 + 0.79
0.12 1.15
-0.7
-1.17 0.58
-1.26 0.7
GMC
-0.45 -: 0.34
-1.34 0.76
-1.34 +
-1.27
-1.42 0.48
-1.72
AMDF
2,22 + 1.1
0.46 + 0.21
0.17 0.26
-0.73 + 0.34
-1.31 0.36
-1.38
0.26
CORAL
-0.56 + 0.26
-0.56 1 0.19
-0.66 = 0.08
-0.33 + 0.1
-1.25 1 0.61
-0.83 - 0.12
MetricMM
2.24 + 0.62
1.78 + 1.07
1.59 + 0.88
0.27 + 0.6
-0.8 + 0.83
-1.54 0.26
22

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Published as a conference paper at ICLR 2026
Table 17: Return of multimodal fusion methods under Failure corruptions applied to one modality on Petch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
6,36 + 0.71
4.79 + 2.13
4.58 + 0.47
1.15 + 1.23
0.5 + 1.76
-1.14 + 2.41
ConCat
6,5 6.5+0.11
5.49 0.12
3.49
1.86
2.8 0.96
2.48 2.0
1.53 = 2.91
CURL
6,95 0.95
0.82
5.68
0.92
2.72 1.97
1.27 1.93
0.16 = 3.09
GMC
5.65 + 1.48
4.39 +.99:1999
1.6
-0.42 + 1.81
-1.53
-1.76 1.31
AMDF
4.03 + 1.96
2.67
1.64
2.1
-0.21 0.58
0.17:0.55
0.1 1.53
CORAL
4.97 + 3.43
4.63
2.64
0.79 + 0.66
-0.35 1.21
-1,47 0.48
-1.32 + 0.67
MetricMM
8.98 + 1.49
9.31
0.84
9.12 + 0.51
8.25 4: 1.1
7.92 + 1.29
8.46 + 0.92
Table 18: Return of multimodal fusion methods under Gaussian corruptions applied one modality on Fetch Slide, increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
6.96 + 1.16
6.55 + 1.46
4.02
+ 0.4
3.64 + 1.64
3,6 1.24
2.17 + 2.3
ConCat
6.33 = 0.36
6.74 + 1.48
4.79
= 1.24
4.76 + 1.49
4.19
4.35 + 2.38
CURL
9.11 + 0.72
6.46 0.86
4.92
1.53
3.29
0.45
+
1.77 0.86
GMC
6.88 + 0.92
4.33 0.25
2.41 + 0.78
0.97
-1.24 - 1.5
-0.81 + 0.99
AMDF
4,56 + 2.47
2.7 1.05
1.65 + 0.82
0.32 0.43
0,26 1.18
-1.02 + 0.69
CORAL
5.25 1: 3.29
4.04 2.01
2.88 +: 2.27
1.59 1: 0.32
-0.16 1: 1.25
-0.24 1 0.74
MetricMM
9.34 + 0.61
9.54 +: 1.26
9.7 + 0.38
8.14 + 1.14
8,99 + 1.35
8.28 1.01
Table 19: Return of multimodal fusion methods under Hallucination corruptions applied to one modality on Fetch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
6.76 = 0.67
6.27 + 0.91
5.78 + 0.72
5.62 + 0.71
4.21 + 0.44
4.16 + 1.37
ConCat
7.14 = 0.05
6.61 + 1.72
1.233
5.21 = 1.12
4.04 0.99
2.57 + 1.11
CURL
8.96 0.8
7,99 1.5
7.58
5.96
0.64
4.87
0.87
4.72 - 0.75
GMC
7.07 =
+ 0.72
1.0
2.95 0.82
3.53
1.33...88
2.71
AMDF
3.62 = 1.73
3.75 1 1.69
4.06
1.92
2.28
1 1.17
2.25
4 0.99
1.62 1 1.18
CORAL
5.72 = 3.19
5.13
1: 2.82
4.02
3.01
3.54
1: 2.7
2.62 1: 1.68
2.28 1: 2.19
MetricMM
9.2 = 1.15
8.31 + 0.56
8.58 + 0.57
9.27
0.88
9.18 + 0.45
8.48 + 0.34
Table 20: Return of multimodal fusion methods under Patches corruptions applied to one modality on Fetch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
5.56 + 0.6
4.16 + 0.98
2.81
+ 1.12
1.96 + 2.34
0.35 + 1.86
-0.36 + 1.16
ConCat
7.5 0.94
4.7 0.56
3.54 + 0.9
2.75 + 1.69
1.3
1.3 + 2.44
CURL
8.42 + 1.34
5,95 1.43
4.53 + 1.76
1.41 + 4.01
-0.68
4.36
GMC
5.39 + 0.65
3.49 1.84
-0.25 2.09
-2.62
+ 2.24
-1.57
-3.09
1.44
AMDF
3.82 + 2.01
3.29 + 1.49
0.87 0.41
-0.06
+ 1.41
-0.62 1.0
-0.4 4 1.7
CORAL
3,94 + 2.84
2.08 + 1.11
-0.88 0.42
-0.92 1.4
-2.33 1.46
-1.85 + 0.41
MetricMM
8.75 0.7
9.19 + 1.06
9.37 + 0.95
8.13
0.09
7.26 + 1.01
8.51 + 0.79
23

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Published as a conference paper at ICLR 2026
Table 21: Return of multimodal fusion methods under Puzzle corruptions applied to one modality on Petch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
6,83 + 1.01
5,47 + 0.87
4.65 + 1.06
0.7 + 0.69
-0.18 + 0.6
-1.47 = 1.56
ConCat
6.35 + 1.87
5,95 1.73
4.38 + 1.99
2.26 2.15
1.56
2.42
-0.05 2.35
CURL
8.44 0.25
0.53
5,42
1.47
3.55 +
1.13
0.59 = 3.19
GMC
5.55 + 1.65
4.7 1.86
1.29
1.77
-1.75 1.02
-3.12 1.66
-2.94 1.66
AMDF
3.88 + 1.79
3.11 1.55
2.32 + 1.47
0.51 + 0.97
-0.85
1.75
1.75
-0.89 + 1.61
CORAL
4.95 + 2.95
3.87 2.33
0.56 + 1.75
-1.41 + 1.09
-2.0 0.96
-1.81 0.57
MetricMM
8.69 + 0.62
9.44 + 0.44
9.19 + 1.29
7.97 + 0.3
8.55 + 0.82
7.9 + 1.44
Table 22: Return of multimodal fusion methods under Salt and Pepper corruptions applied to one modality on Fetch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
6,48 0.14
5.76 = 0.24
5,64
+ 1.01
5.34 = 0.81
4.82 + 1.31
3.37 + 1.21
ConCat
6.41 1.52
6.05 = 1.48
6.48
1.03
= 1.03
5.18 = 1.59
3,49
2.32
2.32
3,6 - 2.02
CURL.
8.55 1.54
7.14 0.76
6,51 311+0.2
5.73
1.55
3.95
3.15 0.85
GMC
7.34 = 1.55
6.36 + 0.85
4.87 1.99
4.21
+ 2.5
2.46 1.51
1.16 2.65
AMDF
4.13 = 2.24
3.01 + 1.32
2,85 +.8
1.65 + 1.31
0.91
0.47
0,2 + 0.44
CORAL
5.04 = 3.53
4.3 2.26
5.29 +: 2.91
4.61 1: 2.6
2.46 1: 1.97
2.33 1 1.59
MetricMM
9.57 + 0.9
8.69 + 0.79
9.64 + 0.65
8.54 = 1.41
8.03 + 0.86
8.71 + 0.7
Table 23: Return of multimodal fusion methods under Texture corruptions applied to one modality on Fetch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
6,54 + 0.73
7.09 + 0.08
4.63 + 1.19
2.07 + 1.11
2.69 + 2.55
1.02 + 1.74
ConCat
7.32 + 1.39
5.64 4.641..3
5.32 + 2.07
4.93 +.98:28.6
4.59
3.52 - 3.85
CURL
8.01 + 0.92
5.71 0.92
1.27 2.85
-.22
-2.48 0.37
GMC
6.36 0.8
4.43 0.41
1.9
-2.15
-2.65 0.57
AMDF
3.28 1.83
2.69 1 1.71
1.03 0.35
-0.65
0.52
-1.5
-1.83 0.68
CORAL
4.49 1: 2.76
3.16 1: 2.03
1.71 4: 1.34
-1.02 0.28
-1.75 0.79
-2.43 1 0.86
MetricMM
9.45 +: 0.49
9.01 0.97
9.4 +: 0.65
7.54 0.18
9.02 4 1.12
7.6 + 0.47
Table 24: Return of multimodal fusion methods under Gaussian corruptions applied simultaneously to two modalities on Fetch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
6,65 + 1.03
4.15 + 0.62
3,28 + 0.66
-0.25 + 1.3
-0.15 + 1.23
-1.22 + 1.08
ConCat
6.05 + 0.65
5.69 + 0.73
4.63 # 1.14
2.64 + 1.41
0.87 + 1.13
0.31 + 1.46
CURL
7.06 + 0.71
6.68
0.11
3.93 0.39
-0.08 + 1.13
-1.12 1.51
-1.46
+1.4611..77 0.77
GMC
5.28 + 0.58
3.67 1.57
+
-2.04 4: 1.34
-2455 1.56
-3.75
2.28
AMDF
4.02 + 1.71
2.14 0.89
-0.11 0.61
-1.1 1.56
-2.81 = 1.7
CORAL
4.45 + 2.76
3.0 + 1.74
1.14 + 0.57
-2.03 + 1.51
-1.61 (661:00 0.74
-2.77 + 0.57
MetricMM
10.1 + 0.61
8.71 + 0.56
7.15 + 0.76
6.13 1.63
4.38 + 2.15
2.47 + 1.89
24

---

0.1 Published as a conference paper at ICLR 2026
Table 25: Return of multimodal fusion methods under Hallucination corruptions applied simultane-
ously to two modalities on Fetch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
6,64 + 1.36
5,88 = 0.37
2.73 + 0,68
1.51 + 0.36
-0.01 = 0.3
-0.71 = 0,4
ConCat
6.68 = 1.49
5.88 = 0.4
3.37 = 1.53
0.77 0.64
-1.2 1.42
-0.46 0.6
CURL
8,23 0.94
7.85
0.65
4.08 0.57
1.48 1.29
-0.15
0.21 0.68
GMC
6.62 +
4.99
+
1.82 1.34
0.03 0.39
-1.02
-1.86 - 0.47
AMDF
3.74 + 1.24
3.19 1.4 1.4
1.71 1.71:00 0.48
0.07 + 0.8
-0.29 +: 0.2
-0.51 + 1.08
CORAL
5.5 3.2
4.39
2.65
1.96 1.96 2.1
-0.03 + 0.8
-1.66 0.57
-1.36 + 0.56
MetricMM
8.67 + 1.48
8.88 + 0.55
8.15 + 1.19
6.94 0.87
5.94 0.97
3.32 1.65
Table 26: Return of multimodal fusion methods under Patches corruptions applied simultaneously to two modalities on Fetch Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LinearComb
5.31 + 0.75
5.32 + 1.59
1.85 + 2.11
-0.65
+ 1.31
-2.1 : 0.32
-2,05 = 0.92
ConCat
6.85 + 1.73
5.96 + 1.41
2.14 + 0.89
-0.66 + 1.37
-1.77 0.87
-1.93 = 0.61
CURL
8.31 1.27
1.27
5.58
1.18
+
-1.27
0.75
-2.74
-2.16 1.2
GMC
4.4
2.45 1.09
-0.65 1.59
-1.02 0.49
-2.23 1.04
-1.97
AMDF
3.24 1.94
2.33 + 0.72
0.74
-1.37 + 1.86
-2.6 2.96
-2.54 = 2.7
CORAL
3.69 1 1.61
1.53 + 1.32
-0.91 0.7.1
-2.46 1 0.85
-2.31 + 0.81
-2.1 1 1.27
MetricMM
8.62 + 0.73
7.81 + 0.74
4.03 + 0.74
1.74 1.77
-0.84 + 1.57
-0.12 - 1.46
Table 27: Return of multimodal fusion methods under Puzzle corruptions applied simultaneously to two modalities on Fe Slide, for increasing corruption probabilities.
Model
0.1
0.25
0.5
0.75
0.9
0.99
LincarComb
5.82 + 0.77
5.18 + 0.76
1.76 + 0.79
-1.67 + 0.55
- 0.62
-1.96
0.58
ConCat
7.14 0.98
5.31 + 1.1
2.02 + 1.54
2.03
-1.9
-2.73 + 1.48
CURL
7.42 + 0.74
5
2.67 1.25
0.96 0.69
-0.27 0.66
-0.76 +.7
GMC
6.25 0.32
1.22
0.08 0.94
-1.44
-2.37
1.9
AMDF
4.09 1 2.54
3.36
1.661220 2.0
0.55 1. 2.42
-1.5
-2.03 1.51
-1.76
1.74
CORAL
5.19 3.33
3.62 1 1.92
0.71 1: 0.33
-1.01 0.66
-2.71 0.14
-2.28
1: 1.06
MetricMM
9.12 1: 0.87
7.05 + 1.52
6.17 + 0.06
3.79 0.77
1.43 4 0.58
1.21 + 1.53
Table 28: Return of multimodal fusion methods under Salt and Pepper corruptions applied simulta- neously two modalities on Fetch Slide, for increasing corruption probabilities.
Model
0.25
0.5
0.75
0.9
0.99
LincarComb
6.3 4 0.8
6,5 + 1.13
5.02 + 1.15
1.6 + 1.72
-0.03 + 1.06
0.79 + 1.85
ConCat
5.56 2.56+10 1.4
0.62
3.58 + 1.86
1.11 + 2.76
0.29 + 1.74
-0.23 + 1.32
CURL
4.33:1.66
6.01 1011::00 0.28
4.61
1.34
1.83
2.3
-0.12 0.29
-1.18 0.43
GMC
0.45
5.36 1.72
1.97
1.76
0.75
2.56
-1.74
-1.08 3.09
AMDF
2.6 1.88
2.74 1.04
1.98 + 1.55
-1.35
+ 1.55
+ 1.97
-2.3 4 2.12
CORAL
5,22 +:
4.64 + 2.74
3.3 + 2.01
0.77 + 1.75
-0.14 + 0.24
0.11 = 0.52
MetricMM
9.2 + 0.94
7.18 2.01
6.79 1.56
5.79
1.27
2.97 + 2.65
3.23 + 2.2
25


Reviews:
[{"review_id": "wcnfuXXUdv", "reviewer": "Reviewer_PMUL", "summary": "This paper proposes a novel method for robust multimodal state estimation in POMDPs by learning a structured latent representation. The core idea is to create a metric space where the Euclidean distance between latent states directly correlates with the minimum number of actions required to transition between them.\n\nAll sensor modalities are encoded into this shared space, providing a geometric interpretation of uncertainty that avoids the need for explicit probabilistic noise models. The system fuses information by weighting each modality's contribution based on its inverse distance to the predicted state, thereby down-weighting corrupted or unreliable sensor data. Empirical results demonstrate that this approach significantly enhances an RL agent's performance and robustness against unseen sensor corruptions.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper introduces a novel geometrical view for handling state representation and uncertainty in POMDPs. By refraining uncertainty in geometric terms it bypasses the complexities and assumptions of traditional probabilistic models. This idea could inspire a new direction of research for handling partial observability and uncertainty in RL.\n\n- The paper is well written and the methodology is sound. Including implementation details that could make future reproduction of the algorithm and empirical results easy. \n\n- The experimental results are extensive and provide convincing evidence of the method's effectiveness.", "weaknesses": "- The paper fail in citing previous work from Steccanella et al. (2022), \"State Representation Learning for Goal-Conditioned Reinforcement Learning\". That work appears to be the first to propose the idea of minimum number of actions distance and motivates very similar objectives for learning an embedding space where distance between states in this embedding space approximates the minimum action distance, by means of leveraging local constraints and the useful upper-bound of the trajectory distance instead of simply trying to maximize the log distance. Please discuss and include this citation in your work. \n\n- The paper should make more clear to the reader that this approach assume deterministic dynamics. Is not clear to me how this approach will behave in the case of stochastic dynamics where the distance between states becomes an expectation and the minimum action distance will provide just a lower bound on that. And a broader discussion of this limitation is needed. The authors should explicitly state this assumption and dedicate a discussion to this limitation, outlining how the framework will behave in this scenario.\n\n- The paper uses a symmetric metric (Euclidean distance) to approximate the Minimum Action Distance, which is inherently asymmetric in most realistic environments. For instance, in environments with irreversible or asymmetric dynamics. By enforcing a symmetric metric, the model is forced to learn an inaccurate representation that cannot capture such crucial, directional aspects of the environment's dynamics. As noted in prior work (Steccanella et al., 2022), this forces only to learn a symmetric approximation of the true Minimum Action Distance. A more thorough discussion is needed to acknowledge this trade-off between simplicity and representational fidelity.", "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": "HFEz45oeiS", "reviewer": "Reviewer_Hf6h", "summary": "The paper addresses the important problem of robust state estimation from multimodal, noisy observations in RL. It proposes METRICMM, a method that learns a shared latent space and uses a simple geometric fusion rule, Inverse Distance Weighting (IDW), to combine sensor estimates. The core contribution is to reframe the problem of sensor uncertainty from a probabilistic one to a geometric one. The experimental results demonstrate a clear improvement in robustness over a wide range of baselines.", "questions": "See weaknesses", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The primary strength of this work is its core conceptual shift. Instead of relying on Bayesian filtering, which requires restrictive priors or generative models , METRICMM recasts uncertainty in geometric terms.\n\nThe inverse distance weighting (IDW) fusion mechanism is a direct and elegant consequence of the geometric formulation. The dynamics prediction z^t​ acts as a reliable \"anchor.\" Any sensor encoding that is geometrically distant from this anchor is naturally identified as noise and its contribution is automatically suppressed. This is far simpler and, as the evidence suggests, more robust than learned fusion mechanisms like attention ( \\alpha-MDF baseline ) or simple concatenation/linear combinations.\n\nMETRICMM consistently shows a much \"flatter\" degradation slope, maintaining high performance even as corruption frequency increases, while all baselines collapse quickly. Empirical results, albeit of simple benchmarks, are quite strong.", "weaknesses": "1) Naive Temporal Distance Loss: The paper's stated goal is to learn a space where distances correlate with the minimum number of actions required to transition between them. However, the actual loss function used is a massive oversimplification. This loss function does not model the minimum number of actions it models all single-step transitions are equidistant. It forces the distance between any two consecutive states to be 1, regardless of the optimality action taken. This is a naive implementation that contradicts the paper's core motivation.\n\n2) I think the above formulation will be optimistic when transitions are stochastic.\n\n3) The paper's core claim is that its learned metric representation is superior. However, the experiments confound this with a hard-coded, non-learned fusion rule (IDW). The high performance could also be due to the simple outlier-rejection property of IDW and not the learned representation. What will happen if the authors take the learned encoders from the strongest baselines (\\alpha-MDF or CORAL) and, at test time, replace their learned fusion module with the paper's exact IDW fusion rule?", "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": "FKl8sGYarz", "reviewer": "Reviewer_utqk", "summary": "The paper proposes a novel method for learning distance metrics between states in reinforcement learning. The approach is to learn mappings from observation modalities to latent states, and measure distance using a well-define metric in the latent state space. The learned state representation can then be used as the basis for reinforcement learning algorithms. The method is tested in a series of benchmarks.", "questions": "In the definition of POMDPs, do you assume that the underlying state space S is known to the learner? (even if it is not observable)\n\nThe proposed approach is to map *each* observation modality to the same latent space, in the hope that all observation modalities derived from the same state will map to approximately the same latent state. However, the mapping from states to observation modalities is stochastic, so a deterministic map is not likely to map all modalities to the same latent state. Why do you map all observation modalities to the same latent space? An alternative would be to have a different latent space for each observation modality and aggregate the distance measures in another way.\n\nFrom Equation (3) it appears as if you assume that the function \\varphi_T is known, is this indeed the case? I would have expected you to *learn* an approximate transition function in the latent space. If you do not assume that \\varphi_T is known, then I do not see how you can make an assumption regarding the transition error of \\varphi_T, since this depends on the quality of learning.\n\nWhat is the motivation for the exact form of Equation (3)?\n\nThe description of the experimental setup leaves a lot to be desired. What is the training pipeline? Do you train a distance estimate first, and then a deep RL policy, or is learning simultaneous? Is distance estimation done online or from offline data? What is the input to the deep RL algorithm (SAC), i.e. exactly how are states and state features defined? \n\nFrom where did you take the seven corruption families? Can you provide a reference?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "State representation learning is an important problem in reinforcement learning, and in this sense the paper makes a timely contribution. It also looks as if the proposed approach performs well in practice compared to other algorithms.", "weaknesses": "I believe that several concepts are not clearly explained, which makes it difficult to accurately evaluate the contribution. Mainly for this reason my opinion regarding acceptance at ICLR is on the negative side.\n\nApart from the cited paper by Wang et al., there are other works that explicitly learn a distance estimate between pairs of states. Concretely, the first work also measures distance as the minimum number of actions required to transition from one state to another.\n\nState Representation Learning for Goal-Conditioned Reinforcement Learning\nSteccanella & Jonsson, ECML 2022\n\nPark, Kreiman & Levine, ICML 2024\nFoundation Policies with Hilbert Representations", "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": "CstqllKHq4", "reviewer": "Reviewer_si1i", "summary": "Authors propose a new latent metric (MetricMM) for sensor fusion and state estimation. The key idea is that uncertainty can be inferred from distance thereby ridding the need for explicit uncertainty. Empirically this idea proved robust to different classes of noise in a dozen of RL control domains that other methods are brittle to.", "questions": "1. If I understand correctly, the noise is applied per frame by chance. What if the noise is temporally dependent, e.g., in the case of the \"failure\" noise, it is likely that a camera artifact persist for a few consecutive frames, \"sticky artifacts\" per se. How does that impact your methods and baselines?\n2. How are the seven families of perturbation designed? Puzzle and texture seems too artificial to be application-relevant.\n3. Each loss in eq (9) makes sense intuitively, though it would be more informative to show their role through a small ablation study even just in one domain and a few noise levels.\n\nDecision irrelevant suggestions:\n1. Table titles are cryptic. I understand you have lots of results to show (which is good signs), but it would be really helpful to expand in the title on the takeaway in natural language, at least those tables in the main body.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality and significance: The idea of utilizing the distance in latent metric space as a measure of sensor consistency thus reliability is novel as far as I am aware of. I like the motivation: it is new and simple (in a positive sense); it bypasses the noise modeling in sensor fusion which is indeed a pain unless one assumes Gaussian.\n\nQuality and clarity: Experimental setup are thorough and results are well organized and presented.", "weaknesses": "I lean towards accept because no detrimental weaknesses stand out to me, though I have a few questions below that will help me become more confident.", "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)