Our code is available at: https://github.com/LINs-lab/ReLA .

TIDA intrinsically: (a) builds hierarchical vMF distributions to cluster samples and discovery taxonomic context by optimizing Eq.

The first subset (in red) is utilized to evaluate a traditional accuracy-basedlossfunction `a,suchasthecrossentropy. This benchmark is based on a loss function designed to incentivize the trained model to produce the smallest possible conformal prediction sets with the desired coverage (e.g., 90% ifα = 0.1). The hybrid training procedure is similar to Algorithm 1, in the sense that it relies on analogous soft-sorting, soft-ranking, and soft-indexing algorithms toevaluate adifferentiable approximation Wi oftheconformity scoreWi in(8). Above, the second equality follows directly from the fact thatS(x,U;π,t), defined in (A2), is by construction increasing in t, and therefore Y / S(x,U;π,1 α) if and only if min{t [0,1]:Y S(x,U;π,t)}>1 α. The proof consists of showing that`a and`u are separately minimized by ˆπ = π,although only approximately inthelatter case.

Topological Data Analysis (TDA) provides a pipeline to extract quantitative topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to what extent a given object exhibits some topological properties. These losses can then be used to perform topological optimization via gradient descent routines. While theoretically sounded, topological optimization faces an important challenge: gradients tend to be extremely sparse, in the sense that the loss function typically depends on only very few coordinates of the input object, yielding dramatically slow optimization schemes in practice. Focusing on the central case of topological optimization for point clouds, we propose in this work to overcome this limitation using diffeomorphic interpolation, turning sparse gradients into smooth vector fields defined on the whole space, with quantifiable Lipschitz constants. In particular, we show that our approach combines efficiently with subsampling techniques routinely used in TDA, as the diffeomorphism derived from the gradient computed on a subsample can be used to update the coordinates of the full input object, allowing us to perform topological optimization on point clouds at an unprecedented scale. Finally, we also showcase the relevance of our approach for black-box autoencoder (AE) regularization, where we aim at enforcing topological priors on the latent spaces associated to fixed, pre-trained, black-box AE models, and where we show that learning a diffeomorphic flow can be done once and then re-applied to new data in linear time (while vanilla topological optimization has to be re-run from scratch). Moreover, reverting the flow allows us to generate data by sampling the topologically-optimized latent space directly, yielding better interpretability of the model.

Let Cost(π) be the value of weak OT functional for a plan π, i.e., Cost( π) We are going to use our Theorem 3.1. As a result, every plan is optimal.Proof of Proposition 3.3. According to our Theorem 3.2, one only has to ensure that Anyway, this is indifferent for us. It remains to upper bound the first term in (23). Formula (12) for the optimal drift follows from [38, Proposition 4.1] From our Proposition 3.3 it follows that For other ϵ > 0, the analogous equivalence holds true.