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Jul 29

The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products

E(3)-equivariant neural networks have demonstrated success across a wide range of 3D modelling tasks. A fundamental operation in these networks is the tensor product, which interacts two geometric features in an equivariant manner to create new features. Due to the high computational complexity of the tensor product, significant effort has been invested to optimize the runtime of this operation. For example, Luo et al. (2024) recently proposed the Gaunt tensor product (GTP) which promises a significant speedup. In this work, we provide a careful, systematic analysis of a number of tensor product operations. In particular, we emphasize that different tensor products are not performing the same operation. The reported speedups typically come at the cost of expressivity. We introduce measures of expressivity and interactability to characterize these differences. In addition, we realized the original implementation of GTP can be greatly simplified by directly using a spherical grid at no cost in asymptotic runtime. This spherical grid approach is faster on our benchmarks and in actual training of the MACE interatomic potential by 30%. Finally, we provide the first systematic microbenchmarks of the various tensor product operations. We find that the theoretical runtime guarantees can differ wildly from empirical performance, demonstrating the need for careful application-specific benchmarking. Code is available at https://github.com/atomicarchitects/PriceofFreedom.

  • 4 authors
·
Jun 16, 2025

On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion models

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.

  • 2 authors
·
Feb 6, 2024

Emotional Expression in Low-Degrees-of-Freedom Robots: Assessing Perception with Reachy Mini

Emotion expression is central to human--robot interaction, yet little is known about how people interpret affect on robots with sparse, non-anthropomorphic expressive capabilities. This study examined how people perceive emotional expressions displayed by Reachy Mini (Pollen Robotics and Hugging Face), a low-degree-of-freedom (low-DoF) robot with a constrained and distinctly non-human expressive repertoire. In an online within-subjects study, 100 participants viewed 10 short video clips of Reachy Mini expressing different emotions and, for each clip, identified the perceived emotion, rated its valence and arousal, and evaluated the robot on social-perception traits. Exact emotion recognition was modest overall and varied considerably across expressions, with anger, sadness, and interest recognized more reliably than emotions such as love, pleasure, shame, and disgust. However, participants were generally more successful at recovering broader affective meaning than exact emotion labels, particularly along valence and arousal dimensions. Emotional expressions also shaped social evaluation, as positive expressions were perceived as warmer and more sociable than negative ones, and animacy varied less across conditions. These findings suggest that even constrained robotic expressions can communicate affective meaning and influence social impressions, positioning Reachy Mini as a useful benchmark for studying affective communication in low-DoF robots.

  • 3 authors
·
May 11