This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

Conformal prediction is a distribution-free uncertainty quantification method that has gained popularity in the machine learning community due to its finite-sample guarantees and ease of use. Its most common variant, dubbed split conformal prediction, is also computationally efficient as it boils down to collecting statistics of the model predictions on some calibration data not yet seen by the model. Nonetheless, these guarantees only hold if the calibration and test data are exchangeable, a condition that is difficult to verify and often violated in practice due to so-called distribution shifts. The literature is rife with methods to mitigate the loss in coverage in this non-exchangeable setting, but these methods require some prior information on the type of distribution shift to be expected at test time. In this work, we study this problem via a new perspective, through the lens of optimal transport, and show that it is possible to estimate the loss in coverage and mitigate arbitrary distribution shifts, offering a principled and broadly applicable solution.

We consider the empirical distribution of the embeddings of a k-layer polynomial GNN on a semi-supervised node classification task and prove a central limit theorem for them.

Conditional sampling is a fundamental task in Bayesian statistics and generative modeling.

Many real-world problems require reasoning across multiple scales, demanding models which operate not on single data points, but on entire distributions. We introduce generative distribution embeddings (GDE), a framework that lifts autoencoders to the space of distributions. In GDEs, an encoder acts on sets of samples, and the decoder is replaced by a generator which aims to match the input distribution. This framework enables learning representations of distributions by coupling conditional generative models with encoder networks which satisfy a criterion we call distributional invariance. We show that GDEs learn predictive sufficient statistics embedded in the Wasserstein space, such that latent GDE distances approximately recover the W2 distance, and latent interpolation approximately recovers optimal transport trajectories for Gaussian and Gaussian mixture distributions.

We consider the empirical distribution of the embeddings of a $k$-layer polynomial GNN on a semi-supervised node classification task and prove a central limit theorem for them. Assuming a community based model for the underlying graph, with growing average degree $\nu_n\to\infty$, we show that the empirical distribution of the centered features, when scaled by $\nu_{n}^{k-1/2}$ converge in 1-Wasserstein distance to a centered stable mixture of multivariate normal distributions. In addition, the joint empirical distribution of uncentered features and labels when normalized by $\nu_n^k$ approach that of mixture of multivariate normal distributions, with stable means and covariance matrices vanishing as $\nu_n^{-1}$. We explicitly identify the asymptotic means and covariances, showing that the mixture collapses towards a 1-D version as $k$ is increased. Our results provides a precise and nuanced lens on how oversmoothing presents itself in the large graph limit, in the sparse regime. In particular, we show that training with cross-entropy on these embeddings is asymptotically equivalent to training on these nearly collapsed Gaussian mixtures.

A.1 Training dataset pre-processing We used 40000publicly available videos from YouTube which were available in a spatial resolution of at least 1920 1080 pixels. In an attempt not to skew the distribution of content too far from what may inform biological representation learning, we excluded most artificial content such as screenshots and videos of computer games. To reduce video compression artifacts and prevent systematic downsampling artifacts, each segment was then spatially downsampled to a randomized height between 128 and 160. Each segment was then separated into 15 pairs of neighboring frames, and a randomly placed, but spatially colocated patch of 64 64 pixels was cropped out of each frame pair. The order of the frame pairs was then randomized in a running buffer, and all RGB pixel values were normalized to the range between 0 and 1 before being fed into the model.

We consider the fixed-budget best arm identification problem where the goal is to find the arm of the largest mean with a fixed number of samples. It is known that the probability of misidentifying the best arm is exponentially small to the number of rounds. However, limited characterizations have been discussed on the rate (exponent) of this value. In this paper, we characterize the minimax optimal rate as a result of an optimization over all possible parameters. We introduce two rates, Rgo and Rgo, corresponding to lower bounds on the probability of misidentification, each of which is associated with a proposed algorithm. The rate Rgo is associated with Rgo-tracking, which can be efficiently implemented by a neural network and is shown to outperform existing algorithms. However, this rate requires a nontrivial condition to be achievable. To address this issue, we introduce the second rate Rgo . We show that this rate is indeed achievable by introducing a conceptual algorithm called delayed optimal tracking (DOT).

In recent years, the widespreadavailability of large-scale, high-dimensionaldatasets has driven significant interest in clustering algorithms that are both computationally efficient and robust to distributional shifts and outliers. The classical clustering method, K-means, can be seen as an application of the Lloyd-Max quantization algorithm, in which the distribution being quantized is the empirical distribution of the points to be clustered. This empirical distribution generally differs from the true underlying distribution, especially when the number of points to be clustered is small. This induces a distributional shift, which can also arise in many real-world settings, such as image segmentation, biological data analysis, and sensor networks, due to noise variations, sensor inaccuracies, or environmental changes. Distributional shifts can severely impact the performance of clustering algorithms, leading to degraded cluster assignments and unreliable downstream analysis. The field of clustering has a rich history. One of the most popular algorithms in this field is theK-means (KM) algorithm, introduced by [1], which computes centroids by iteratively updating the conditional mean of the data in the Voronoi regions induced by the centroids. However, standardK-means is sensitive to initialization and, in general, converges only to a local minimum.