The type of symmetry is typically fixed and has to be chosen in advance.

We also show that MatrixNet respects group relations allowing generalization to group elements of greater word length than in the training set.

Do the main claims made in the abstract and introduction accurately reflect the paper's Did you discuss any potential negative societal impacts of your work? Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Y es] Code and Did you specify all the training details (e.g., data splits, hyperparameters, how they Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? Did you include the total amount of compute and the type of resources used (e.g., type Did you mention the license of the assets? Did you include any new assets either in the supplemental material or as a URL? [Y es] We will provide our code. Did you discuss whether and how consent was obtained from people whose data you're If you used crowdsourcing or conducted research with human subjects... (a) The centered dot can sometimes be omitted if there is no ambiguity.

A good visual representation is an inference map from observations (images) to features (vectors) that faithfully reflects the hidden modularized generative factors (semantics). In this paper, we formulate the notion of good representation from a group-theoretic view using Higgins' definition of disentangled representation, and show that existing Self-Supervised Learning (SSL) only disentangles simple augmentation features such as rotation and colorization, thus unable to modularize the remaining semantics. To break the limitation, we propose an iterative SSL algorithm: Iterative Partition-based Invariant Risk Minimization (IP-IRM), which successfully grounds the abstract semantics and the group acting on them into concrete contrastive learning. At each iteration, IP-IRM first partitions the training samples into two subsets that correspond to an entangled group element.

We also show that MatrixNet respects group relations allowing generalization to group elements of greater word length than in the training set.

We extend Algorithm 1 to stochastic gradient descent (SGD). Algorithm 3 here modifies Algorithm 1 to allow transformations on both parameters and data. In this section, we derive the group actions for the test functions and multi-layer neural networks. More details about group theory can be found in textbooks such as Lang (2002). B.1 Continuous symmetry in test functions B.1.1 Ellipse Consider the following loss function with a 2 R However, we will only use the 2 variable version in the experiments.

We study the problem of conformal prediction (CP) under geometric data shifts, where data samples are susceptible to transformations such as rotations or flips. While CP endows prediction models with post-hoc uncertainty quantification and formal coverage guarantees, their practicality breaks under distribution shifts that deteriorate model performance. To address this issue, we propose integrating geometric information--such as geometric pose--into the conformal procedure to reinstate its guarantees and ensure robustness under geometric shifts. In particular, we explore recent advancements on pose canonicalization as a suitable information extractor for this purpose. Evaluating the combined approach across discrete and continuous shifts and against equivariant and augmentation-based baselines, we find that integrating geometric information with CP yields a principled way to address geometric shifts while maintaining broad applicability to black-box predictors.