A toy example of this transformation is presented in Figure 5. Moreover, we can divide all of these individuals into two cases: 1. R (x We can divide all of these individuals into three cases: 1. π Figure 6: Jointly optimizing the decision policy and the counterfactual explanations can offer additional gains. Employment Length: How long the applicant has been employed. FICO Score: The applicant's FICO score, which is a credit score based on consumer credit Annual Income: The declared annual income of the applicant. Marital status: Whether the person is married or single.

At the same time, it is hard to translate operations into their non-Euclidean domain given the lack of closed-form expressions.

Learning has become a core component of the modern information systems we increasingly rely upon to select job candidates, analyze medical data, and control "smart" applications (home, grid, city).

Learning has become a core component of the modern information systems we increasingly rely upon to select job candidates, analyze medical data, and control "smart" applications (home, grid, city).

Estimating the data uncertainty in regression tasks is often done by learning a quantile function or a prediction interval of the true label conditioned on the input.

Estimating the data uncertainty in regression tasks is often done by learning a quantile function or a prediction interval of the true label conditioned on the input.

In the existing literature, a common approach to approximate the metric is to first find a map that preserves Euclidean lengths instead of distances.

This is the Appendix for "Self-Supervised Learning Disentangled Group Representation as Feature". Table .1 summarizes the abbreviations and the symbols used in the main paper.Abbreviation/Symbol Meaning Abbreviation SSL Self-supervised Learning SL Supervised Learning DCI Disentangle Metric for Informativeness IRS Interventional Robustness Score EXP Explicitness Score MOD Modularity Score LR Logistic Regression GBT Gradient Boosted Trees OOD Out-Of-Distributed Symbol in Theory U Semantic space X V ector space I Image space G Group G ( x) Group orbit w.r .t.G containing the sample x ϕ Image generation process U I φ Visual representation I X f Semantic representation U X m The number of decomposed subgroups Symbol in Algorithm P Partition of dataset P Learned partition through Eq. (3) P Set of partitions used in Eq. (2) N Number of training images θ "Dummy" parameter used by IRM I Image X List of abbreviations and symbols used in the paper. Section A provides the preliminary knowledge about the group theory. Section D presents the additional experimental results. 1 A Preliminaries Groups often arise as transformations of some space, such as a set, vector space, or topological space. The set of clockwise rotations w.r .t. its centroid to retain We say this group of rotations act on the triangle, which is formally defined below.