While artificial neural networks are known as universal approximators for continuous functions, many modern approaches rely on overparameterized architectures with high computational cost. In this work, we introduce the Barycentric Neural Network (BNN): a compact shallow architecture that encodes both structure and parameters through a fixed set of base points and their associated barycentric coordinates. We show that the BNN enables the exact representation of continuous piecewise linear functions (CPLFs), ensuring strict continuity across segments. Given that any continuous function on a compact domain can be uniformly approximated by CPLFs, the BNN emerges as a flexible and interpretable tool for function approximation. To enhance geometric fidelity in low-resource scenarios, such as those with few base points to create BNNs or limited training epochs, we propose length-weighted persistent entropy (LWPE): a stable variant of persistent entropy. Our approach integrates the BNN with a loss function based on LWPE to optimize the base points that define the BNN, rather than its internal parameters. Experimental results show that our approach achieves superior and faster approximation performance compared to standard losses (MSE, RMSE, MAE and LogCosh), offering a computationally sustainable alternative for function approximation.

Data with underlying nonlinear structure are collected across numerous application domains, necessitating new data processing and analysis methods adapted to nonlinear domain structure. Riemannanian manifolds present a rich environment in which to develop such tools, as manifold-valued data arise in a variety of scientific settings, and Riemannian geometry provides a solid theoretical grounding for geometric data analysis. Low-rank approximations, such as nonnegative matrix factorization (NMF), are the foundation of many Euclidean data analysis methods, so adaptations of these factorizations for manifold-valued data are important building blocks for further development of manifold data analysis. In this work, we propose curvature corrected nonnegative manifold data factorization (CC-NMDF) as a geometry-aware method for extracting interpretable factors from manifold-valued data, analogous to nonnegative matrix factorization. We develop an efficient iterative algorithm for computing CC-NMDF and demonstrate our method on real-world diffusion tensor magnetic resonance imaging data.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank $k$ tangent subbundle on $\mathbb{R}^d$, $k

Principal component analysis (PCA) is a workhorse of modern data science. Practitioners typically perform PCA assuming the data conforms to Euclidean geometry. However, for specific data types, such as hierarchical data, other geometrical spaces may be more appropriate. We study PCA in space forms; that is, those with constant positive (spherical) and negative (hyperbolic) curvatures, in addition to zero-curvature (Euclidean) spaces. At any point on a Riemannian manifold, one can define a Riemannian affine subspace based on a set of tangent vectors and use invertible maps to project tangent vectors to the manifold and vice versa. Finding a low-dimensional Riemannian affine subspace for a set of points in a space form amounts to dimensionality reduction because, as we show, any such affine subspace is isometric to a space form of the same dimension and curvature. To find principal components, we seek a (Riemannian) affine subspace that best represents a set of manifold-valued data points with the minimum average cost of projecting data points onto the affine subspace. We propose specific cost functions that bring about two major benefits: (1) the affine subspace can be estimated by solving an eigenequation -- similar to that of Euclidean PCA, and (2) optimal affine subspaces of different dimensions form a nested set. These properties provide advances over existing methods which are mostly iterative algorithms with slow convergence and weaker theoretical guarantees. Specifically for hyperbolic PCA, the associated eigenequation operates in the Lorentzian space, endowed with an indefinite inner product; we thus establish a connection between Lorentzian and Euclidean eigenequations. We evaluate the proposed space form PCA on data sets simulated in spherical and hyperbolic spaces and show that it outperforms alternative methods in convergence speed or accuracy, often both.

Since solving State-of-the-art algorithms for Approximate Nearest Neighbor Search the problem exactly requires an expensive exhaustive scan of the (ANNS) such as DiskANN, FAISS-IVF, and HNSW build data dependent database - which would be impractical for real-world indices that indices that offer substantially better accuracy and search span billions of objects - practical interactive search systems use efficiency over data-agnostic indices by overfitting to the index Approximate Nearest Neighbor Search (ANNS) algorithms with data distribution. When the query data is drawn from a different highly sub-linear query complexity [10, 18, 24, 30] to answer such distribution - e.g., when index represents image embeddings and queries. The quality of such ANN indices is often measured by query represents textual embeddings - such algorithms lose much k-recall@k which is the overlap between the top-results of the of this performance advantage. On a variety of datasets, for a fixed index search with the ground truth -nearest neighbors (-NNs) in recall target, latency is worse by an order of magnitude or more for the corpus for the query, averaged over a representative query set. Out-Of-Distribution (OOD) queries as compared to In-Distribution State-of-the-art algorithms for ANNS, such as graph-based indices (ID) queries. The question we address in this work is whether ANNS [16, 24, 30] which use data-dependent index construction, algorithms can be made efficient for OOD queries if the index construction achieve better query efficiency over prior data-agnostic methods is given access to a small sample set of these queries. We like LSH [6, 18] (see Section A.1 for more details). Such efficiency answer positively by presenting OOD-DiskANN, which uses a sparing enables these indices to serve queries with > 90% recall with a sample (1% of index set size) of OOD queries, and provides up to latency of a few milliseconds, required in interactive web scenarios.

Many monogenic disorders cause a characteristic facial morphology. Artificial intelligence can support physicians in recognizing these patterns by associating facial phenotypes with the underlying syndrome through training on thousands of patient photographs. However, this ‘supervised’ approach means that diagnoses are only possible if the disorder was part of the training set. To improve recognition of ultra-rare disorders, we developed GestaltMatcher, an encoder for portraits that is based on a deep convolutional neural network. Photographs of 17,560 patients with 1,115 rare disorders were used to define a Clinical Face Phenotype Space, in which distances between cases define syndromic similarity. Here we show that patients can be matched to others with the same molecular diagnosis even when the disorder was not included in the training set. Together with mutation data, GestaltMatcher could not only accelerate the clinical diagnosis of patients with ultra-rare disorders and facial dysmorphism but also enable the delineation of new phenotypes. GestaltMatcher uses a deep convolutional neural network to improve recognition of rare disorders based on facial morphology. The framework detects similarities among patients with previously unseen syndromes, aiding discovery of new disease genes.

Six significant new methodological developments of the previously-presented "metastimuli architecture" for human learning through machine learning of spatially correlated structural position within a user's personal information management system (PIMS), providing the basis for haptic metastimuli, are presented. These include architectural innovation, recurrent (RNN) artificial neural network (ANN) application, a variety of atom embedding techniques (including a novel technique we call "nabla" embedding inspired by linguistics), ANN hyper-parameter (one that affects the network but is not trained, e.g. the learning rate) optimization, and meta-parameter (one that determines the system performance but is not trained and not a hyper-parameter, e.g. the atom embedding technique) optimization for exploring the large design space. A technique for using the system for automatic atom categorization in a user's PIMS is outlined. ANN training and hyper- and meta-parameter optimization results are presented and discussed in service of methodological recommendations.

Random embedding has been applied with empirical success to large-scale black-box optimization problems with low effective dimensions. This paper proposes the EmbeddedHunter algorithm, which incorporates the technique in a hierarchical stochastic bandit setting, following the optimism in the face of uncertainty principle and breaking away from the multiple-run framework in which random embedding has been conventionally applied similar to stochastic black-box optimization solvers. Our proposition is motivated by the bounded mean variation in the objective value for a low-dimensional point projected randomly into the decision space of Lipschitz-continuous problems. In essence, the EmbeddedHunter algorithm expands optimistically a partitioning tree over a low-dimensional — equal to the effective dimension of the problem —search space based on a bounded number of random embeddings of sampled points from the low-dimensional space. In contrast to the probabilistic theoretical guarantees of multiple-run random-embedding algorithms, the finite-time analysis of the proposed algorithm presents a theoretical upper bound on the regret as a function of the algorithm's number of iterations. Furthermore, numerical experiments were conducted to validate its performance. The results show a clear performance gain over recently proposed random embedding methods for large-scale problems, provided the intrinsic dimensionality is low.