To make sense of the world our brains must analyze high-dimensional datasets streamed by our sensory organs. Because such analysis begins with dimensionality reduction, modeling early sensory processing requires biologically plausible online dimensionality reduction algorithms. Recently, we derived such an algorithm, termed similarity matching, from a Multidimensional Scaling (MDS) objective function. However, in the existing algorithm, the number of output dimensions is set a priori by the number of output neurons and cannot be changed. Because the number of informative dimensions in sensory inputs is variable there is a need for adaptive dimensionality reduction.

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We propose Convexity-Driven Projection (CDP), a boundary-free linear method for dimensionality reduction of point clouds that targets preserving detour-induced local non-convexity. CDP builds a $k$-NN graph, identifies admissible pairs whose Euclidean-to-shortest-path ratios are below a threshold, and aggregates their normalized directions to form a positive semidefinite non-convexity structure matrix. The projection uses the top-$k$ eigenvectors of the structure matrix. We give two verifiable guarantees. A pairwise a-posteriori certificate that bounds the post-projection distortion for each admissible pair, and an average-case spectral bound that links expected captured direction energy to the spectrum of the structure matrix, yielding quantile statements for typical distortion. Our evaluation protocol reports fixed- and reselected-pairs detour errors and certificate quantiles, enabling practitioners to check guarantees on their data.

We introduce a unified framework for evaluating dimensionality reduction techniques in spatial transcriptomics beyond standard PCA approaches. We benchmark six methods PCA, NMF, autoencoder, VAE, and two hybrid embeddings on a cholangiocarcinoma Xenium dataset, systematically varying latent dimensions ($k$=5-40) and clustering resolutions ($ρ$=0.1-1.2). Each configuration is evaluated using complementary metrics including reconstruction error, explained variance, cluster cohesion, and two novel biologically-motivated measures: Cluster Marker Coherence (CMC) and Marker Exclusion Rate (MER). Our results demonstrate distinct performance profiles: PCA provides a fast baseline, NMF maximizes marker enrichment, VAE balances reconstruction and interpretability, while autoencoders occupy a middle ground. We provide systematic hyperparameter selection using Pareto optimal analysis and demonstrate how MER-guided reassignment improves biological fidelity across all methods, with CMC scores improving by up to 12\% on average. This framework enables principled selection of dimensionality reduction methods tailored to specific spatial transcriptomics analyses.

Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations between triples of points and other points. More significantly, based on this curvature profile, we introduce a quantitative measure to evaluate the effectiveness of data representations, such as those produced by dimensionality reduction techniques. Furthermore, Our experiments demonstrate that this curvature-based analysis can be employed to estimate the intrinsic dimensionality of datasets. We use this to explore the large-scale geometry of empirical networks and to evaluate the effectiveness of dimensionality reduction techniques.

The Engineers' Salary Prediction Challenge requires classifying salary categories into three classes based on tabular data. The job description is represented as a 300-dimensional word embedding incorporated into the tabular features, drastically increasing dimensionality. Additionally, the limited number of training samples makes classification challenging. Linear dimensionality reduction of word embeddings for tabular data classification remains underexplored. This paper studies Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA). We show that PCA, with an appropriate subspace dimension, can outperform raw embeddings. LDA without regularization performs poorly due to covariance estimation errors, but applying shrinkage improves performance significantly, even with only two dimensions. We propose Partitioned-LDA, which splits embeddings into equal-sized blocks and performs LDA separately on each, thereby reducing the size of the covariance matrices. Partitioned-LDA outperforms regular LDA and, combined with shrinkage, achieves top-10 accuracy on the competition public leaderboard. This method effectively enhances word embedding performance in tabular data classification with limited training samples.

Dimensionality reduction can distort vector space properties such as orthogonality and linear independence, which are critical for tasks including cross-modal retrieval, clustering, and classification. We propose a Relationship Preserving Loss (RPL), a loss function that preserves these properties by minimizing discrepancies between relationship matrices (e.g., Gram or cosine) of high-dimensional data and their low-dimensional embeddings. RPL trains neural networks for non-linear projections and is supported by error bounds derived from matrix perturbation theory. Initial experiments suggest that RPL reduces embedding dimensions while largely retaining performance on downstream tasks, likely due to its preservation of key vector space properties. While we describe here the use of RPL in dimensionality reduction, this loss can also be applied more broadly, for example to cross-domain alignment and transfer learning, knowledge distillation, fairness and invariance, dehubbing, graph and manifold learning, and federated learning, where distributed embeddings must remain geometrically consistent.

In order to cope with this "curse of dimensionality," we

To get a finer assessment of the individual impacts of each sub-term of ClassNeRV stress (Equation 3), we perform an ablation study.

DR may be supervised by taking advantage of class information.