For labels taking values in a finite metric space, we introduce techniques new to weak supervision based on pseudo-Euclidean embeddings andtensor decompositions, providing anearly-consistent noise rate estimator.

For labels taking values in a finite metric space, we introduce techniques new to weak supervision based on pseudo-Euclidean embeddings andtensor decompositions, providing anearly-consistent noise rate estimator.

These noisy outputs are modeled and aggregated into a single higher-quality pseudolabel. Any conventional supervised end model can be trained on these pseudolabels.

Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data analysis, this paper investigates the stability of magnitude: its continuity properties with respect to the Gromov-Hausdorff topology. We show that magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces. Yet, we find evidence to suggest that it may be 'generically continuous', in the sense that generic Gromov-Hausdorff limits are preserved by magnitude. We make the case that, in fact, 'generic stability' is what matters for applicability.

Estimating optimal phylogenetic trees or hierarchical clustering trees from metric data is an important problem in evolutionary biology and data analysis. Intuitively, the goodness-of-fit of a metric space to a tree depends on its inherent treeness, as well as other metric properties such as intrinsic dimension. Existing algorithms for embedding metric spaces into tree metrics provide distortion bounds depending on cardinality. Because cardinality is a simple property of any set, we argue that such bounds do not fully capture the rich structure endowed by the metric. We consider an embedding of a metric space into a tree proposed by Gromov. By proving a stability result, we obtain an improved additive distortion bound depending only on the hyperbolicity and doubling dimension of the metric. We observe that Gromov's method is dual to the well-known single linkage hierarchical clustering (SLHC) method. By means of this duality, we are able to transport our results to the setting of SLHC, where such additive distortion bounds were previously unknown.

Many of the foundations of machine learning rely on the idealized premise that all input and output spaces are infinite, e.g.~$\mathbb{R}^d$. This core assumption is systematically violated in practice due to digital computing limitations from finite machine precision, rounding, and limited RAM. In short, digital computers operate on finite grids in $\mathbb{R}^d$. By exploiting these discrete structures, we show the curse of dimensionality in statistical learning is systematically broken when models are implemented on real computers. Consequentially, we obtain new generalization bounds with dimension-free rates for kernel and deep ReLU MLP regressors, which are implemented on real-world machines. Our results are derived using a new non-asymptotic concentration of measure result between a probability measure over any finite metric space and its empirical version associated with $N$ i.i.d. samples when measured in the $1$-Wasserstein distance. Unlike standard concentration of measure results, the concentration rates in our bounds do not hold uniformly for all sample sizes $N$; instead, our rates can adapt to any given $N$. This yields significantly tighter bounds for realistic sample sizes while achieving the optimal worst-case rate of $\mathcal{O}(1/N^{1/2})$ for massive. Our results are built on new techniques combining metric embedding theory with optimal transport

Theoretical results from discrete geometry suggest that normed spaces can abstractly embed finite metric spaces with surprisingly low theoretical bounds on distortion in low dimensions. In this paper, inspired by this theoretical insight, we highlight normed spaces as a more flexible and computationally efficient alternative to several popular Riemannian manifolds for learning graph embeddings. Normed space embeddings significantly outperform several popular manifolds on a large range of synthetic and real-world graph reconstruction benchmark datasets while requiring significantly fewer computational resources. We also empirically verify the superiority of normed space embeddings on growing families of graphs associated with negative, zero, and positive curvature, further reinforcing the flexibility of normed spaces in capturing diverse graph structures as graph sizes increase. Lastly, we demonstrate the utility of normed space embeddings on two applied graph embedding tasks, namely, link prediction and recommender systems. Our work highlights the potential of normed spaces for geometric graph representation learning, raises new research questions, and offers a valuable tool for experimental mathematics in the field of finite metric space embeddings. We make our code and data publically available.

Weak supervision (WS) is a rich set of techniques that produce pseudolabels by aggregating easily obtained but potentially noisy label estimates from a variety of sources. WS is theoretically well understood for binary classification, where simple approaches enable consistent estimation of pseudolabel noise rates. Using this result, it has been shown that downstream models trained on the pseudolabels have generalization guarantees nearly identical to those trained on clean labels. While this is exciting, users often wish to use WS for structured prediction, where the output space consists of more than a binary or multi-class label set: e.g. rankings, graphs, manifolds, and more. Do the favorable theoretical properties of WS for binary classification lift to this setting? We answer this question in the affirmative for a wide range of scenarios. For labels taking values in a finite metric space, we introduce techniques new to weak supervision based on pseudo-Euclidean embeddings and tensor decompositions, providing a nearly-consistent noise rate estimator. For labels in constant-curvature Riemannian manifolds, we introduce new invariants that also yield consistent noise rate estimation. In both cases, when using the resulting pseudolabels in concert with a flexible downstream model, we obtain generalization guarantees nearly identical to those for models trained on clean data. Several of our results, which can be viewed as robustness guarantees in structured prediction with noisy labels, may be of independent interest. Empirical evaluation validates our claims and shows the merits of the proposed method.