Despite its widespread adoption, Adam's advantage over Stochastic Gradient Descent (SGD) lacks a comprehensive theoretical explanation. This paper investigates Adam's sensitivity to rotations of the parameter space. We observe that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis in practice. This reveals that conventional rotation-invariant assumptions are insufficient to capture Adam's advantages theoretically. To better understand the rotation-dependent properties that benefit Adam, we also identify structured rotations that preserve or even enhance its empirical performance. We then examine the rotation-dependent assumptions in the literature and find that they fall short in explaining Adam's behaviour across various rotation types. In contrast, we verify the orthogonality of the update as a promising indicator of Adam's basis sensitivity, suggesting it may be the key quantity for developing rotation-dependent theoretical frameworks that better explain its empirical success.

Recent advances in Multimodal Large Language Models (MLLMs) have significantly improved 2D visual understanding, prompting interest in their application to complex 3D reasoning tasks. However, it remains unclear whether these models can effectively capture the detailed spatial information required for robust real-world performance, especially cross-view consistency, a key requirement for accurate 3D reasoning. Considering this issue, we introduce Viewpoint Learning, a task designed to evaluate and improve the spatial reasoning capabilities of MLLMs. We present the Viewpoint-100K dataset, consisting of 100K object-centric image pairs with diverse viewpoints and corresponding question-answer pairs. Our approach employs a two-stage fine-tuning strategy: first, foundational knowledge is injected to the baseline MLLM via Supervised Fine-Tuning (SFT) on Viewpoint-100K, resulting in significant improvements across multiple tasks; second, generalization is enhanced through Reinforcement Learning using the Group Relative Policy Optimization (GRPO) algorithm on a broader set of questions. Additionally, we introduce a hybrid cold-start initialization method designed to simultaneously learn viewpoint representations and maintain coherent reasoning thinking. Experimental results show that our approach significantly activates the spatial reasoning ability of MLLM, improving performance on both in-domain and out-of-domain reasoning tasks. Our findings highlight the value of developing foundational spatial skills in MLLMs, supporting future progress in robotics, autonomous systems, and 3D scene understanding.

A.1 Dataset Examples450 In this section of the appendix, we present a detailed overview of several representative tasks from451 each category included in REASONINGGYM. For each task, we describe its structure, complexity452 parameters, and provide examples.453 A.1.1 complex_arithmetic(Algebra)454 Find the solution of an arithmetic operation involving complex numbers.455 The spiral order is clockwise, starting from the top-left corner. Predict the corresponding output grid by applying the rule you found.

Joint-embedding self-supervised learning (SSL) commonly relies on transformations such as data augmentation and masking to learn visual representations, a task achieved by enforcing invariance or equivariance with respect to these transformations applied to two views of an image. This dominant two-view paradigm in SSL often limits the flexibility of learned representations for downstream adaptation by creating performance trade-offs between high-level invariance-demanding tasks such as image classification and more fine-grained equivariance-related tasks. In this work, we propose seq-JEPA, a world modeling framework that introduces architectural inductive biases into joint-embedding predictive architectures to resolve this trade-off. Without relying on dual equivariance predictors or loss terms, seq-JEPA simultaneously learns two architecturally separate representations for equivariance-and invariance-demanding tasks. To do so, our model processes short sequences of different views (observations) of inputs.

The growing importance of mRNA therapeutics and synthetic biology highlights the need for models that capture the latent structure of synonymous codon (different triplets encoding the same amino acid) usage, which subtly modulates translation efficiency and gene expression. While recent efforts incorporate codon-level inductive biases through auxiliary objectives, they often fall short of explicitly modeling the structured relationships that arise from the genetic code's inherent symmetries. We introduce Equi-mRNA, the first codon-level equivariant mRNA language model that explicitly encodes synonymous codon symmetries as cyclic subgroups of 2D Special Orthogonal matrix (SO(2)). By combining group-theoretic priors with an auxiliary equivariance loss and symmetry-aware pooling, Equi-mRNA learns biologically grounded representations that outperform vanilla baselines across multiple axes. On downstream property-prediction tasks including expression, stability, and riboswitch switching Equi-mRNA delivers up to 10% improvements in accuracy. In sequence generation, it produces mRNA constructs that are up to 4 more realistic under Fréchet BioDistance metrics and 28% better preserve functional properties compared to vanilla baseline. Interpretability analyses further reveal that learned codon-rotation distributions recapitulate known GC-content biases and tRNA abundance patterns, offering novel insights into codon usage. Equi-mRNA establishes a new biologically principled paradigm for mRNA modeling.

Rotary Positional Encodings (RoPE) have emerged as a highly effective technique for one-dimensional sequences in Natural Language Processing spurring recent progress towards generalizing RoPE to higher-dimensional data such as images and videos. The success of RoPE has been thought to be due to its positional equivariance, i.e. its status as a relative positional encoding. In this paper, we mathematically show RoPE to be one of the most general solutions for equivariant positional embedding in one-dimensional data. Moreover, we show Mixed RoPE to be the analogously general solution for M-dimensional data, if we require commutative generators - a property necessary for RoPE's equivariance. However, we question whether strict equivariance plays a large role in RoPE's performance. We propose Spherical RoPE, a method analogous to Mixed RoPE, but assumes non-commutative generators. Empirically, we find Spherical RoPE to have the equivalent or better learning behavior compared to its equivariant analogues. This suggests that relative positional embeddings are not as important as is commonly believed, at least within computer vision. We expect this discovery to facilitate future work in positional encodings for vision that can be faster and generalize better by removing the preconception that they must be relative.

Designing metal-organic frameworks (MOFs) with novel chemistries is a longstanding challenge due to their large combinatorial space and complex 3D arrangements of the building blocks. While recent deep generative models have enabled scalable MOF generation, they assume (1) a fixed set of building blocks and (2) known local 3D coordinates of building blocks. However, this limits their ability to (1) design novel MOFs and (2) generate the structure using novel building blocks. We propose a two-stage MOF generation framework that overcomes these limitations by modeling both chemical and geometric degrees of freedom. First, we train an SMILES-based autoregressive model to generate metal and organic building blocks, paired with a cheminformatics toolkit for 3D structure initialization. Second, we introduce a flow matching model that predicts translations, rotations, and torsional angles to assemble the blocks into valid 3D frameworks. Our experiments demonstrate improved reconstruction accuracy, the generation of valid, novel, and unique MOFs, and the ability to create novel building blocks.

Equivariant models have recently been shown to improve the data efficiency of diffusion policy by a significant margin. However, prior work that explored this direction focused primarily on point cloud inputs generated by multiple cameras fixed in the workspace. This type of point cloud input is not compatible with the now-common setting where the primary input modality is an eye-in-hand RGB camera like a GoPro. This paper closes this gap by incorporating into the diffusion policy model a process that projects features from the 2DRGB camera image onto a sphere. This enables us to reason about symmetries in SO(3)without explicitly reconstructing a point cloud. We perform extensive experiments in both simulation and the real world that demonstrate that our method consistently outperforms strong baselines in terms of both performance and sample efficiency. Our work, Image-toSphere Policy (ISP), is the first SO(3)-equivariant policy learning framework for robotic manipulation that works using only monocular RGB inputs.

Large language models (LLMs) suffer from forgetting of upstream knowledge when fine-tuned. Despite efforts on mitigating forgetting, few have investigated how forgotten upstream examples are dependent on newly learned tasks. Insights on such dependencies enable efficient and targeted mitigation of forgetting. In this paper, we empirically analyze forgetting that occurs in N upstream examples of language modeling or instruction-tuning after fine-tuning LLMs on one of M new tasks, visualized in M N matrices. We show that the matrices are often well-approximated with low-rank matrices, indicating the dominance of simple associations between the learned tasks and forgotten upstream examples. Leveraging the analysis, we predict forgetting of upstream examples when fine-tuning LLMs on unseen tasks with matrix completion over the empirical associations. This enables fast identification of most forgotten examples without expensive inference on the entire upstream data. Despite simplicity, the approach outperforms prior approaches that learn semantic relationships of learned tasks and upstream examples with LMs. We demonstrate the practical utility of our analysis by showing statistically significantly reduced forgetting as we upweight predicted examples for replay during fine-tuning.

Despite its widespread adoption, Adam's advantage over Stochastic Gradient Descent (SGD) lacks a comprehensive theoretical explanation. This paper investigates Adam's sensitivity to rotations of the parameter space. We observe that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis in practice. This reveals that conventional rotation-invariant assumptions are insufficient to capture Adam's advantages theoretically. To better understand the rotation-dependent properties that benefit Adam, we also identify structured rotations that preserve or even enhance its empirical performance. We then examine the rotation-dependent assumptions in the literature and find that they fall short in explaining Adam's behavior across various rotation types. In contrast, we verify the orthogonality of the update as a promising indicator of Adam's basis sensitivity, suggesting it may be the key quantity for developing rotation-dependent theoretical frameworks that better explain its empirical success.